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EXTREME VALUE ANALYSIS OF TEMPERATURE AND RAINFALL: 
CASE STUDY OF SOME SELECTED REGIONS IN GHANA 
 
 
 
 
 
 
BY 
NKRUMAH STEPHEN 
(10552265) 
 
THIS THESIS IS SUBMITTED TO THE UNIVERSITY OF GHANA, LEGON IN 
PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE AWARD OF MPHIL 
STATISTICS 
 
 
JULY, 2017
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DECLARATION 
This is the result of research work undertaken by Nkrumah Stephen under the supervision of 
Dr. Kwabena Doku-Amponsah and Dr. Richard Minkah. I declare that the entity of the work 
contained therein is my own work, that I am the authorship owner thereof (unless to the 
extent explicitly otherwise stated) and I have not previously in its entirety or in part submitted 
it for obtaining any qualification. 
 
Nkrumah Stephen (10552265) Signed: ……………… Date: ……….… 
(Student) 
 
Dr. Kwabena Doku-Amponsah Signed: ……………… Date: ……….… 
(Principal Supervisor)                                                   
 
Dr. Richard Minkah    Signed: ………………. Date…………… 
(Co Supervisor) 
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ABSTRACT 
Rainfall and temperature are important climatic factors for crop production and affects 
humans’ basic needs such as health, shelter, food and water in Ghana. However extreme 
rainfall causes flooding which can lead to loss of lives, properties among others. Extreme 
temperature also causes drought, hot and cold spells among others which have adverse 
implications on human beings and agriculture. It is therefore important to know about 
occurrences of such extreme events and their chances of occurring. Extreme value theory 
gives us a statistical approach for the occurrences and the magnitude of these extreme cases 
or rare event that are beyond the scope of available data. In this thesis, we used “extreme 
value theory” to model the tail distributions of temperature and rainfall in Ghana. Daily data 
on rainfall and temperature was obtained and it spanned the period January 1960 to 
December 2012. “Extreme value theory” was used to estimate the extreme occurrence to be 
expected once every 5, 10, 20, 50 and 100 years respectively. It was observed that the 
extreme occurrences of temperature and rainfall can be modelled using Weibull and Frechet 
family of distributions in Ghana respectively. A maximum temperature of 34.7cC , 34.66oC , 
and 39.6oC  was predicted to occur in Accra, Ashanti and Northern regions respectively once 
every five years. Furthermore, it was estimated that an amount of 88.38mm, 85.61mm, and 
81.98mm will be recorded in Accra, Ashanti, and Northern regions once every 5 years 
respectively. It was recommended that Government should employ all necessary legislative 
policies to help manage the issue of Global warming considering the substantial rise in 
temperature in the years ahead based on our forecasted predictions. 
 
                                                             
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DEDICATION 
This work is dedicated to the Most High God for His loving kindness and mercies towards 
my life. 
Also to my parents, Mr. John Nkrumah and Mrs. Florence Nsiah for supporting my 
education. I am forever grateful for everything you have done for me. God richly bless you. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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ACKNOWLEDGEMENT 
This work could not have been completed without a certain amount of support, guidance and 
directions from some few individuals. 
My first and foremost appreciation goes to the Most High God for His all grace and mercies 
towards my life.  
My heartfelt love and appreciation also goes to both Dr. Kwabena Doku-Amponsah and Dr. 
Richard Minkah for all their sacrifice, support and guidance towards the completion of this 
work. 
I would also want to acknowledge Dr. Ezekiel Nii Noye Nortey and Mr. Issah Seidu for 
assisting me in various ways to complete this work. 
Many more thanks to my parents Mr. and Mrs. John Nkrumah and my siblings, Mrs. Martha 
Danso, Dr. Ali Abdulai Nkrumah, Mrs. Faustina Ampaabeng Kyeremeh, Eliah Mensah Hadjo 
and Yvette Nkrumah for supporting me. 
I also owe thanks to my colleagues Isaac Appiah, Obu-Amoah Ampomah, Owusu Yaw 
Kesse, Fred Fosu Agyarko, Emmanuel Kojo Aidoo, Enoch Sakyi-Yeboah and Ama 
Koranteng Fremah for their various contributions to this research work. 
May God Bless you all. 
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TABLE OF CONTENT 
DECLARATION ........................................................................................................................ i 
ABSTRACT ............................................................................................................................... ii 
DEDICATION ......................................................................................................................... iii 
ACKNOWLEDGEMENT ........................................................................................................ iv 
TABLE OF CONTENT ............................................................................................................. v 
LIST OF TABLES .................................................................................................................... xi 
LIST OF ABBREVIATIONS ................................................................................................ xiii 
CHAPTER ONE ........................................................................................................................ 1 
INTRODUCTION ..................................................................................................................... 1 
1.1 Background of the Study Area ............................................................................................. 2 
1.2 Research Problem and Justification ..................................................................................... 3 
1.3 Research Objectives ............................................................................................................. 4 
1.4 Significance of the Study ..................................................................................................... 4 
1.5 Scope and Contributions of the Study ................................................................................. 5 
1.6 Outline of the Study ............................................................................................................. 6 
CHAPTER TWO ....................................................................................................................... 7 
LITERATURE REVIEW .......................................................................................................... 7 
2.1 The Concept of Extreme Value Theory ............................................................................... 7 
2.2 Extreme Temperature......................................................................................................... 10 
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2.3 Extreme Rainfall ................................................................................................................ 11 
CHAPTER THREE ................................................................................................................. 13 
EXREME VALUE THEORY ................................................................................................. 13 
3.1 Extreme Value Theorem .................................................................................................... 14 
3.2 Generalised Extreme Value Distribution ........................................................................... 16 
3.3 Asymptotic Models for Minima......................................................................................... 17 
3.3.1 Parameter Estimation under the GEV ......................................................................... 19 
3.3.1.1 Maximum Likelihood Estimation ............................................................................ 20 
3.3.1.2 Probability Weighted Moments (PWM) .................................................................. 21 
3.4 Peaks-Over Threshold (POT) Method ............................................................................... 26 
3.4.1 Parameter Estimation under the POT Method (Maximum Likelihood Estimation) ... 29 
3.4.2 Estimation of other Parameters of Extreme Events .................................................... 30 
3.4.3 Threshold Selection ..................................................................................................... 31 
3.4.4 Statistical Choice of GPD Models .............................................................................. 33 
3.5 Comparison of ML and PWM Estimation Methods for GEV Distribution ....................... 34 
CHAPTER FOUR .................................................................................................................... 37 
RESULT AND DISCUSSION ................................................................................................ 37 
4.1 Basic Statistics ................................................................................................................... 37 
4.2 Greater Accra ..................................................................................................................... 40 
4.2.1 Fitting the Extreme Value Model ................................................................................ 40 
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4.2.1.1 Temperature .......................................................................................................... 40 
4.2.1.2 Rainfall ................................................................................................................. 47 
4.2.2 Fitting the Generalised Pareto (GP) Model ................................................................. 51 
4.2.2.1 Temperature .......................................................................................................... 51 
4.2.2.2 Rainfall ................................................................................................................. 58 
4.2.3 General Comments ...................................................................................................... 61 
4.3 Ashanti Region................................................................................................................... 63 
4.3.1 Fitting the Extreme Value Model ................................................................................ 63 
4.3.1.1 Temperature .......................................................................................................... 63 
4.3.1.2 Rainfall ................................................................................................................. 69 
4.3.2 Fitting the Generalized Pareto (GP) Model................................................................. 72 
4.3.2.1 Temperature .......................................................................................................... 72 
4.3.2.2 Rainfall ................................................................................................................. 80 
4.3.3 General Comments ...................................................................................................... 83 
4.4 Northern Region................................................................................................................. 83 
4.4.1 Fitting the Extreme Value Model ................................................................................ 83 
4.4.1.1 Temperature .......................................................................................................... 84 
4.4.1.2 Rainfall ................................................................................................................. 89 
4.3.2 Fitting the Generalized Pareto (GP) Model................................................................. 93 
4.3.2.1 Temperature .......................................................................................................... 93 
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4.3.2.2 Rainfall ............................................................................................................... 100 
4.4.3 General Comments .................................................................................................... 103 
CHAPTER FIVE ................................................................................................................... 104 
CONCLUSIONS AND RECOMMENDATION .................................................................. 104 
REFERENCES ...................................................................................................................... 106 
APPENDIX A ........................................................................................................................ 113 
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LIST OF FIGURES 
Figure 4.1a: Exponential Q-Q plot for minimum (Left panel) and maximum (right panel) 
Temperature ............................................................................................................................. 41 
Figure 4.2a: Diagnostic plot for the fitted GEV model for minimum Temperature ................ 44 
Figure 4.3a: Diagnostic plot for the fitted GEV model for maximum Temperature ............... 45 
Figure 4.4a: Exponential Q-Q plot of maximum Rainfall ....................................................... 48 
Figure 4.5a: Diagnostic plot for the fitted GEV model for maximum rainfall ........................ 49 
Figure 4.6a: Mean excess plot for minimum (left panel) and maximum (right panel) 
Temperature ............................................................................................................................. 52 
Figure 4.7a: Parameter estimates against threshold for maximum Temperature ..................... 53 
Figure 4.8a: Parameter estimates against threshold for minimum Temperature ..................... 54 
Figure 4.9a: Diagnostic plot for fitted GP for maximum Temperature ................................... 56 
Figure 4.10a: Diagnostic plot for fitted GP for minimum Temperature .................................. 57 
Figure 4.11a: Mean excess plot for maximum Rainfall ........................................................... 58 
Figure 4.12a: Parameter estimates against threshold for maximum Rainfall .......................... 59 
Figure 4.13a: Diagnostic plot for fitted GP for maximum Rainfall ......................................... 60 
Figure 4.1b: Exponential Q-Q plot for minimum (left panel) and maximum (right panel 
Temperature ............................................................................................................................. 64 
Figure 4.2b: Diagnostic plot for the fitted GEV model for minimum Temperature ................ 66 
Figure 4.3b: Diagnostic plot for the fitted GEV model for maximum Temperature ............... 67 
Figure 4.4b: Exponential Q-Q plot for maximum Rainfall ...................................................... 69 
Figure 4.5b: Diagnostic plot for the fitted GEV model for maximum Rainfall ...................... 71 
Figure 4.6b: Mean excess plot for minimum (left panel) and maximum (right panel) 
Temperature ............................................................................................................................. 73 
Figure 4.7b: Parameter estimates against threshold for maximum Temperature .................... 74 
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Figure 4.8b: Parameter estimates against threshold for minimum Temperature ..................... 75 
Figure 4.9b: Diagnostic plot for fitted GP for maximum Temperature ................................... 78 
Figure 4.10b: Diagnostic plot for fitted GP for minimum Temperature.................................. 79 
Figure 4.11b: Mean excess plot for maximum Rainfall .......................................................... 80 
Figure 4.12b: Parameter estimates against threshold for maximum Rainfall .......................... 81 
Figure 4.13b: Diagnostic plot for fitted GP for maximum Rainfall ........................................ 82 
Figure 4.1c: Exponential Q-Q plot for minimum (left panel) and maximum (right panel) 
Temperature ............................................................................................................................. 84 
Figure 4.2c: Diagnostic plot for the fitted GEV model for minimum Temperature ................ 86 
Figure 4.4c: Exponential Q-Q plot for minimum (left panel) and maximum (right panel) 
Rainfall ..................................................................................................................................... 90 
Figure 4.5c: Diagnostic plot for the fitted GEV model for maximum Temperature ............... 91 
Figure 4.6c: Mean excess plot for minimum (left panel) and maximum (right panel) 
Temperature ............................................................................................................................. 94 
Figure 4.7c: Parameter estimates against threshold for maximum Temperature ..................... 95 
Figure 4.8c: Parameter estimates against threshold for minimum Temperature ..................... 96 
Figure 4.9c: Diagnostic plot for fitted GP for maximum Temperature ................................... 98 
Figure 4.10c: Diagnostic plot for fitted GP for minimum Temperature .................................. 99 
Figure 4.11c: Mean excess plot for maximum Rainfall ......................................................... 100 
Figure 4.12c: Parameter estimates against threshold for maximum Rainfall ........................ 101 
Figure 4.13c: Diagnostic plot for fitted GP for maximum Rainfall ....................................... 102 
  
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LIST OF TABLES 
Table 4.1: Summary statistics of Data ..................................................................................... 38 
Table 4.2: ADF Stationarity Test on Temperature and Rainfall Data ..................................... 39 
Table 4.3a: GEV Parameter Estimate for Minimum Temperature .......................................... 42 
Table 4.4a: GEV Parameter estimate for Maximum Temperature .......................................... 43 
Table 4.5a: Exceedance Probabilities estimates for Temperature ........................................... 46 
Table 4.6a: Return Periods estimates for Temperature ............................................................ 47 
Table 4.7a: GEV Parameter estimate for Maximum Rainfall .................................................. 49 
Table 4.8a: Exceedance Probabilities estimates for Rainfall ................................................... 50 
Table 4.9a: Return Periods estimates for Rainfall ................................................................... 50 
Table 4.10a: GP parameter estimate for maximum Temperature ............................................ 55 
Table 4.11a: GP Parameter Estimate for Minimum Temperature ........................................... 55 
Table 4.12a: GP parameter estimate for Maximum Rainfall ................................................... 60 
Table 4.3b: GEV Parameter estimates for Minimum Temperature ......................................... 65 
Table 4.4b: GEV Parameter estimates for Maximum Temperature ........................................ 65 
Table 4.5b: Exceedance Probabilities estimates for GEV fit Temperature ............................. 68 
Table 4.6b: Return Periods estimates of GEV fit for Temperature ......................................... 68 
Table 4.7b: GEV parameter estimate for Maximum Rainfall .................................................. 70 
Table 4.8b: Exceedance Probabilities estimates for GEV fit for Maximum Rainfall ............. 71 
Table 4.9b: Return Periods estimates of GEV fit for Maximum Rainfall ............................... 72 
Table 4.10b: GP parameter estimate for maximum Temperature ............................................ 76 
Table 4.11b: GP parameter estimate for Minimum Temperature ............................................ 77 
Table 4.12b: GP parameter estimate for Maximum Rainfall ................................................... 81 
Table 4.3c: GEV parameter estimate for Minimum Temperature ........................................... 85 
Table 4.4c: GEV parameter estimate for Maximum Temperature .......................................... 85 
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Table 4.5c: Exceedance Probabilities estimates for GEV fit for Temperature ........................ 88 
Table 4.6c: Return Periods estimates of GEV fit for Temperature.......................................... 89 
Table 4.7c: GEV parameter estimate for Maximum Rainfall .................................................. 91 
Table 4.8c: Exceedance Probabilities estimates for GEV fit Rainfall ..................................... 92 
Table 4.9c: Return Periods estimates of GEV fit for Rainfall ................................................. 92 
Table 4.10c: GP parameter estimate for maximum Temperature ............................................ 96 
Table 4.11c: GP parameter estimate for Minimum Temperature ............................................ 97 
Table 4.12c: GP parameter estimate for Maximum Rainfall ................................................. 101 
 
 
 
 
 
 
 
 
 
 
 
 
 
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LIST OF ABBREVIATIONS 
 
D.F …………………………………………………...Density Function  
EVI …………………………………………………..Extreme Value Index  
EVT ………………………………………………….Extreme Value Theory 
GEV………………………………………………….Generalised Extreme Value 
GEVD ……………………………………………….Generalised Extreme Value Distribution 
GP……………………………………………………Generalised Pareto 
GPD…………………………………………………..Generalised Pareto Distribution 
ML……………………………………………………Maximum Likelihood 
MLE………………………………………………….Maximum Likelihood Estimation 
PWM………………………………………………....Probability Weighted Moment 
   
 
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CHAPTER ONE 
INTRODUCTION 
Extreme weather events such as heavy precipitation episodes affect our society through their 
adverse impact on agriculture and economy (Rosenzweig, 2001). For example the flood in 
August 2016 in Northern Region of Ghana which lasted ‘only’ for two days arguably killed 
several people, caused damages of roughly 3 billion Ghana cedis, and destroyed several crops 
and homes of Ghanaian citizens. Thus, knowing how often such extreme weather conditions 
occurs is of practical significance for planning and mitigation purposes.  
“Extreme Value Theory (EVT)” is extensively applied in hydrology and climatology. Results 
obtained from EVT in engineering practice play an essential role e.g. in flood management 
and water resources design (Katz & Philippe, 2002). One of several assumptions of EVT is 
the stationarity of time series. However, climate has a distinct seasonal cycle and as a result 
temperature and precipitation data with respect to the seasonal cycle cannot be considered to 
be stationary. 
The seasonal evolution of extreme temperature and precipitation is in itself a subject of 
interest. Apart from its climatological significance, it can find practical use, for instance as 
input for flood models, amongst other parameters. The purpose of explicitly modeling the 
seasonal non-stationarity using daily temperature and precipitation data is to shed light on the 
seasonal behaviour of extreme temperature and extreme precipitation in Ghana. It is upon 
these reasons that this study would be focusing on applying the climate data in some selected 
regions in Ghana to EVT. 
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1.1 Background of the Study Area 
The first weather observations were unquestionably made in the 1980’s when the Aburi 
gardens was established though records from these times have not survived.  
The Ghana meteorological department was established in the year 1937 but under the 
ministry of communications in the year 1957 the Ghana Meteorological Service was 
established. 
The meteorological service was started with 14 stations, which had full range of 
meteorological instrumentation and equipment operated 24 hours a day measuring 
temperature, precipitation, wind and humidity among others with readings taken once or 
twice in a day. 
There was an upgrade in the late 1990’s to approximately 310 stations operating including 22 
synoptic stations across the country.  
Considering Ghana’s distance from the equator, its climate is relatively mild but tropical. 
There are usually two rainy seasons across the country: From April to June and September 
through November. In the northern parts of Ghana however, March and April experience 
squalls with occasional showers until August and September. Thereafter, heavy precipitation 
sets in.  
Available data from the meteorological service indicates that rainfall in Ghana varies from 83 
to 220cm per year.  
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In addition, temperature readings are highest during March while lower temperature readings 
are experienced in August. The lowest temperature to ever be recorded in Ghana is100C  
50F   but the average temperature readings range from 210C to320C  (70-90F). 
1.2 Research Problem and Justification 
Concerns about extreme weather events have increased due to the emergence of global 
warming. Evidences of increasing “extreme weather events” for example severe cyclones, 
droughts, floods, heat waves and cold waves over the past few decades have been found by 
many researchers (Mayooran and Laheetharan, 2014). 
The construction and design of certain projects, for example urban drainage systems, dams, 
water resources management and flood damage prevention require information on “spatial 
and temporal variability of extreme rainfall events” since they need a sufficient understanding 
of “extreme events of high return periods”.  
The “return levels” of interest in most cases is usually not in the range of available data and 
hence “extreme value theory” is useful since it focuses on extreme data points. However, 
based on “statistical analysis of maximum precipitation or maximum temperature records 
where the parameters of a selected frequency distribution” can be estimated using available 
sample data, the estimation of extreme temperature or extreme rainfall is accomplished. 
Event magnitudes corresponding to “return periods” less than or greater than those of the 
recorded events are then estimated using the fitted distribution (Mayooran and Laheetharan., 
2014). 
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Accurate estimation of low and high rainfall can help alleviate the damage caused by floods 
and droughts respectively in Ghana. In addition, accurate estimation of extreme (low or high) 
temperature and rainfall can help in various fields including health, agriculture among others.  
The “distribution of annual extreme rainfalls at a single site” have a number of probability 
models developed to describe them (Tao, Nguyen & Bourque, 2002). In engineering practice 
however, choosing a suitable model still remain a major difficulty since finding which 
distribution should be used for the “frequency analysis of extreme rainfalls” has no general 
agreement. However, these distributions fit well where the bulk of the datasets are 
concentrated. Large bias can occur in the tails of the distribution where little or no 
observations are found. “Applications of extreme value theory” to temperature and rainfall 
data in some regions in Ghana was investigated to help identify an appropriate tail 
distributions for rainfall and temperature in Ghana. 
1.3 Research Objectives 
1. To find the appropriate distribution for the tails of the distributions of rainfall and 
temperature 
2. To find out how extremes of rainfall relate to the extremes of temperature. 
3. To find the exceedance probabilities for selected levels of rainfall and temperature 
4. To obtain the return periods of extreme rainfall and temperature and their 
corresponding return levels  for rainfall and temperature 
1.4 Significance of the Study 
Inter-Governmental Panel on Climate Change (IPCC) in 2010 reported that, intensity and 
frequency of natural disasters will rise due to climate change. “Extreme weather events” such 
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as floods, droughts, salinization/ contamination of agricultural water and lands supplies lead 
to variations in rainfall patterns and temperature with a likely decrease in agricultural 
productivity in already fragile areas, especially in sub-Saharan Africa and declining water 
availability and quality in semi-arid and arid regions. 
In this thesis, we perform extreme value analysis of rainfall and temperature in Ghana to 
assess the return periods and exceedance probabilities of extreme weather conditions. 
The study adds to the existing literature and also enriches the knowledge in this field of 
statistics on the application of EVT in climatology, specifically to rainfall and temperature. 
1.5 Scope and Contributions of the Study 
A very important issue whenever one is confronted with a data set is to determine which 
distribution the data are likely to have come from. In practice however, many methods are 
centered on the “empirical distribution function” which can lead to misleading conclusions 
particularly when dealing with tails of distribution functions, such as income distributions. 
The main objective of this thesis was to apply tools from “Extreme Value Theory” to model 
extreme occurrences of rainfall and temperature in some selected regions in Ghana. The main 
contributions of this thesis are as follows: 
1. Data set for Greater Accra, Ashanti and Northern regions were assessed respectively 
to identify the extreme occurrence of temperature and rainfall readings. 
2. Identifying the distributions to which the temperature and rainfall data belongs to was 
considered for the selected regions. This was done by applying the data to the 
“Generalised Extreme value” and the “Generalised Pareto distributions”. 
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3. Forecasting was however done in other to ascertain the various extreme occurrences 
to be expected in the next 5, 10, 20, 50 and 100 years respectively. The probabilities 
of exceedances for these extreme occurrences were also estimated. 
4. The influence of the results in the field of Agriculture, Health and Environmental 
science (Global warming) and its impact on human life was extensively discussed. 
1.6 Outline of the Study 
In chapter 2, we provide an overview about the development of “extreme value theory” and 
we give an outline of the various applications of extreme value in climatology based on 
available literature. Chapter three is devoted to describing the Extreme Value methodology 
by considering the “Generalised Extreme Value” and the “Peak over Threshold” approach. In 
that chapter, we also introduce the various parameter estimation techniques and we review 
some methods of estimating the “Extreme Value Index” and choosing the threshold in the 
dataset above which threshold are fit to the “Generalised Pareto Distribution”. 
In chapter four, we shall apply the methods presented in chapter 3 to real data (Temperature 
and Rainfall data) by fitting it to the “Generalised Extreme Value distribution” and the 
“Generalised Pareto distribution” in our bid to find the appropriate distribution to describe the 
dataset. 
Chapter five presents the findings, conclusions and recommendations for interested bodies 
and future research. 
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CHAPTER TWO 
LITERATURE REVIEW 
This chapter reviews what other researchers have done on extreme rainfalls and extreme 
temperature by the scientific community that has served numerous reasons which include:  
estimation of life-threatening rainfalls for strategy resolutions; the evaluation of the rarity of 
observed rainfall and evaluating the approaches to evaluate strategy on rainfalls among 
others.  
2.1 The Concept of Extreme Value Theory  
“Extreme value theory entails the stochastic behaviour of the extreme values in a process”. 
Fisher and Tippett (1928) indicated that, the maximal behaviour can be defined for a single 
process by the “three extreme value distributions; Gumbel, Weibull and Frechet”.   
The “extreme value distributions” could be outlined back to the efforts by Bernoulli in 1709 
as indicated by Kotz and Nadajarah (2000). Fuller in 1914 undoubtedly introduced the first 
“application of extreme value distributions”. Subsequently, several researchers particularly 
brought out useful “applications of extreme value distributions” to climate data from diverse 
regions in the world. 
Nicolas Bernoulli deliberated on “the mean largest distance from the origin given n points 
lying randomly on a straight line of a fixed length t” (Kotz and Nadajarah, 2000). A paper by 
Von Bortkiewicz (1922), which considered the “distribution of range in random samples 
from a normal distribution” started the systematic development of the general theory. The 
relevance of the paper by Bortkiewicz is characterised in the fact that “the concept of 
distribution of largest value” was for the first time introduced (Kotz and Nadajarah, 2000).  
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Fisher and Tippet in 1928 obtained a key result on the possible “limit laws of the sample 
maximum” which apparently created the impression that EVT was something very distinct 
and rather special from “classical central limit theory” (Beirlant, Goegebeur and Teugels, 
2004). 
A detailed basis for the Extreme Value Theory (EVT) was however presented in 1943 by 
Gnedenko and provided “sufficient and essential conditions for the weak convergence of the 
extreme order statistics” (Kotz and Nadajarah, 2000). Gnedenko (1943) incorporated and 
formalised the ideas into the basic assumption in EVT termed as the “Extreme Value 
Condition” (Kotz & Nadajarah, 2000). The Extreme Value Condition provides a “Semi-
parametric model” solely for the tails of the distribution function. 
Doctoral dissertation by Haan in 1970, on “Regular Variation and its Application to the Weak 
Convergence of Sample Extremes”, is an important contribution to the theoretical 
development of EVT (Beirlant, Goegebeur & Teugels, 2004). 
Theoretical developments of the late 1930s and 1940s came with various papers which dealt 
with “practical applications of extreme value statistics in distributions of human lifetimes, 
flood analysis (Gumbel, 1941), material strength (Weibull, 1939) and Radioactive emission” 
as summarised by Kotz and Nadajarah (2000). 
The EVT is a blend of various applications concerning regular varying functions, 
sophisticated “mathematical results on point processes and natural occurrences (such as 
corrosion, rainfall, droughts, floods and wind gusts among others)”. Therefore, the first to 
show interest in the development of EVT were hydrologists, engineers and theoretical 
probabilists. This theory has recently attracted mainstream statisticians.  
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Founders of statistical theory and probability; Laplace, Fermat and Pascal, among others were 
too engaged with the “general behaviour of statistical masses” to be interested in rare 
“extreme values”.  
As early as 1709, Bernoulli worked on extreme value problems by discussing “the mean 
largest distance from the origin with a straight line of length t  having n  points on it at 
random” (Johnson, Kotz & Balakrishnan, 1995).  
In 1922, exploration of EVT and systematic study started in Germany. Bortkiewicz (1922) 
presented a paper which dealt with the “distribution of the range of random samples from the 
Gaussian distribution”. Bortkiewicz’s contribution was the introduction of the “concept of 
distribution of largest values”. However, the concept of “expected value of the largest 
member of a sample of observations” was introduced by another German, Von Mises, from 
the Gaussian distribution in the year later (Mises, 1923). Basically, he studied samples from 
the Gaussian distribution with particular interest in the asymptotic distribution of extreme 
values. 
A major first step in 1925 was initiated by (Tippet, 1925), when probabilities for various 
sample sizes and their corresponding tables of the largest values from a Gaussian distribution 
were presented along with the mean range of such samples. Frechet (1927) considered the 
“asymptotic distributions of largest values from a class of individual distributions” in the first 
paper he presented. A year later, the paper that is now regarded as the “basis of the 
asymptotic theory of extreme value distributions” was published by Fisher and Tippett 
(1928). They independently developed “Frechet’s asymptotic distribution and two others”. 
These three distributions were sufficient to define “extreme value distributions of all 
statistical distributions”. This result will be explored further in the subsequent chapter. Also, 
prior researcher’s encountered difficulties mainly because, they (Fisher and Tippet) 
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established that, “from Gaussian samples toward asymptote there is an extremely slow 
convergence of the distribution of the largest value". Some useful and simple sufficient 
conditions for the weak convergence of the largest order statistic to each of “the three types 
of limit distributions” were specified by Von Mises (1936). A few years later, Gnedenko 
(1943) provided a rigorous foundation for the EVT and sufficient and necessary conditions 
for the “weak convergence of the extreme order statistics”. The theoretical developments 
before the mid-1940s were followed by a number of publications concerning “applications of 
extreme value statistics”. The first to study the “application of EVT” was Gumbel (1941) 
with his first application being the consideration of the largest duration of life. The author 
showed in the result that, EVT could be used to understand the statistical distribution of 
floods. Extensively, extreme value procedures have been applied to breaking strength of 
structural materials, other meteorological phenomena and to the outlying observations of 
statistical problem.  
In 1958, an extensive bibliography of the developed literature was presented in Gumbel 
(1958). Several fields of application and theoretical developments have emerged since then 
and some of these recent developments are further discussed in the chapters to follow. 
2.2 Extreme Temperature 
A procedure for modeling of hot spells that is centered on “statistical theory of extreme 
values” has been suggested by Furrer, Kartz, Walter & Furrer (2010). The authors extended 
the “point process approach and geometric distribution to extreme value analysis” in order to 
model the duration, intensity and frequency of hot spells.“They modeled the length of hot 
spells by a geometric distribution whereas annual frequency of hot spells was fitted with a 
Poisson distribution. The authors demonstrated the excesses over some high threshold by a 
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conditional Generalised Pareto distribution, and the outcome can be used to account for the 
temporal dependence of maximum daily temperatures within a hot spell (Furrer et al., 2010)”.  
“In Coles (2001), statistical models for extreme values are presented. Exceedances over 
threshold can be modeled by Generalised Pareto (GP) distribution, and the declustering 
method” is used to model the dependence of exceedances. Block maxima, for which annual 
maxima or minima temperatures for an example, converges to Generalised Extreme Value 
(GEV) distribution. Return levels are obtained based on the estimation results of GP and 
GEV models.  However, for non-stationary sequences, trends are analysed in the GP and 
GEV models using the generalised linear models (GLM) (Coles, 2001)”.  
Igor, Rychlik & Jesper (2006) also considered the “Peak over Threshold approach to estimate 
quantiles, “return periods and probabilities of extreme events”.  
Bergstrom & Anders, (2002) reconstructed data in the period 1722-1998 for sea level air 
pressure and air temperature for Uppsala, Sweden“based on the raw daily meteorological 
observations from, printed monthly bulletins, computer records and hand written registers.”  
In addition, trend in annual minima and maxima temperatures in Uppsala, has been 
investigated by Ryden (2010). 
2.3 Extreme Rainfall 
Nadarajah (2005) and Nadarajah and Withers (2001) presented an analysis of rainfall data in 
over fourteen locations spread through West Central Florida and sixteen locations across New 
Zealand respectively using extreme value theory. “Weibull distribution” was found to be the 
suitable distribution for describing annual maximal rainfall in Japan (Hirose, 1994). 
Additionally, the future behaviour of the extremes of rainfall was described and fitted to the 
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generalised extreme value distribution for five locations in South Korea from 1961 to 2001 
(Nadajarah and Dongseok, 2007). Of the five locations considered, the most realistic model 
for four of them was the Gumbel distribution. Deka, Borah and Kakaty (2009) considered 
“five extreme value distributions and derived the best distribution to describe the annual 
series of maximum rainfall data of nine distantly located stations in north east India for the 
period 1966 to 2007.” 
Varathan, Perera & Nalin (2010) found the “Gumbel distribution to be the best fitting model 
for describing the annual maximum rainfall in Colombo district”.  
Rainfall data from 77 stations were analysed to determine the mean annual and inter- decadal 
rainfall variability and distribution in Ghana for the period 1981-2010 (Logah et al., 2013). 
They analysed the temporal variability of the decadal mean rainfall for the periods 1981-
1990, 1991-2000 and 2001- 2010. The authors established that, there is variation on high and 
low rainfall distributions in the country. They also found that, the mean decadal rainfall 
amounts in Ghana is on a gradual increase from 1981 to 2010 (Logah et al,. 2013). 
 Nkrumah, et al. (2014) used data from“six meteorological stations selected from three 
rainfall distribution zones to explore the nature of variability and extent in the annual rainfall 
and pattern of the raining seasons in Ghana” from 1990 to 2008. They concluded that, there is 
consistent downward trend in the rainfall pattern at all the stations.  
Several attempts have been made to study rainfall and temperature on the local level. 
However, none of the study reviewed used extreme value in estimating rainfall and 
temperature in Ghana. An attempt is made to study the tails of the “annual rainfall and 
temperature” data in some selected regions in Ghana.  
 
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CHAPTER THREE 
EXREME VALUE THEORY 
In order to draw information regarding particular measures in a “statistical distribution”, 
many statistical tools are available but this thesis looked to place emphasis on extreme values 
behaviour in the dataset under consideration. Assume the data are realisations of a sample 
Y ,Y ,Y ,...,Y  of m  “identical and independently distributed random variables” with the 1 2 3 m
ordered data denoted by Y1,m Y2,m  ...Ym,m . The properties about the “quantile function; 
Q  p  inf y : F  y  p” or a “distribution function; F  y  P Y  y ” are usually 
studied using the sampled data. 
Based on the foundation of the “law of large numbers, the sample mean,Y  is used as a 
consistent estimator of the expected value E(Y)”. Moreover, the “Central Limit Theorem 
(CLT) yields the asymptotic behaviour of the sample mean”. For sufficiently large sample 
sizes, this outcome can be used to find a “confidence interval” for E(Y), a condition that is 
essential when applying the central limit theorem. 
It can be just as essential in some cases to estimate tail probabilities. For example, supposing 
there can be hot spell if the temperature rises beyond 40oC , then it will be motivating to 
approximate the tail probability p  X  40 . The empirical distribution function defined by 
Fˆ
i
 y  , if ym yi,m , yi1,m   can however be used. Where yi,m  represent the ith m
ordered sample value. 
However, in situations where the mean EY   or the second moment EY 2  is infinite, both 
the classical theory and the central limit theorem are no longer applicable. In addition, 
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supposing we want to estimate p  P Y  y  , where y  ym,m  and p̂  which is the estimate 
results to a value 0. 
In what follows, we would introduce extreme value theory, which was the basis for our 
analysis of extreme values of temperature and rainfall in Ghana. Furthermore, the 
“Generalised Extreme Value theory” and the “Generalised Pareto distribution” would be 
introduced along with the parameter estimation procedures to be employed.  
3.1 Extreme Value Theorem 
Let Y1,Y2,...,Yn  be an order of “independent random variables” with a common “distribution 
function” F . The focus of the model is on the statistical behaviour of 
Mn  maxY1,Y2,...,Yn,         (3.1) 
The Yi  generally denotes “values of a process measured on an even time scale”, say the 
temperature recorded hourly within a day and the Mn  represents the daily maximum 
temperature. 
In principle, we can obtain the rigorous distribution of Mn  accurately for all values of n : 
n
PM n z  PY1  z,...,Yn  z PY1  z...PYn  zF z  (3.2) 
However, the “distribution function” of F  is usually unknown in practice and as a result very 
n
slight inconsistencies in the estimation of F  can result in significant inconsistencies for F . 
Standard statistical techniques are mostly used to estimate F  from the data observed. 
Therefore in EVT, F  is anticipated to be unknown and on the foundation of extreme values, 
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the focus is on estimated “families of models for F n  when n  , which can then be 
estimated”. 
Fisher and Tippett (1928) found the linear renormalisation of the variable M  so that the n
function F n  does not degenerate to a point mass and this is stated in the theorem below. 
The subsequent theorem gives a brief description of the “limit law for block maxima”, which 
is usually represented by M   where n  basically denote the block size. n
“The complete range of likely limit distributions for M *n  is given by extremal types theorem”. 
Theorem 3.2.1 (Fisher and Tippett (1928), Gnedenko (1943)). 
“If there exist sequences of constants bn  0  and an  such that 
PM  a  / b  zdn n n G  z   as n  , where G  is a non-degenerate distribution 
function, then G  belongs to one of the following families: 
   z  a 
I :G  z  expexp   ,   z  ;
   b 
 0, z  a,

II :G  z     z  a   (3.3) 
exp   , z  a
  
b  
     z  a  
exp    , z  a,
III :G  z    b    

 1 z  a,
For parameters b  0 , a  and b  0  for families II and III.” 
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These various distributions are jointly referred to as the “extreme value distribution”. 
“Gumbel, Frechet and Weibull families” represents type I, II and III respectively. The density 
function F  however is contained in the “domain of attraction” G  thus F D G  . 
Theorem 3.2.1 infers that, the corresponding “normalised variable M *n  has a limiting 
distribution that must be one of the three types of extreme value distributions”, if M  can be n
stabilised with appropriate sequences bn  and an  (Coles, 2001). 
3.2 Generalised Extreme Value Distribution 
The various types of limit arising in the extremal type theorem have a behaviour that appears 
to be of distinct forms. The Frechet distribution density, G , decays polynomially whilst the 
Gumbel distribution, F , decays exponentially.  
In applications, these various distributions give relatively different representations of 
“extreme value behaviour”. It used to be common in early applications to implement any of 
the “three families of extreme value theory” and approximate parameters of that distribution. 
Nevertheless, a weakness of this method is that you require a technique to select which of the 
three families is best suitable. In addition, once a domain of attraction is selected, successive 
inferences assume this selection as accurate and do not permit for the errors involved in such 
a selection, although these errors may be significant (Coles 2001). 
Jenkinson (1955) and Von mises (1930) proposed a three parameter model termed the 
Generalised Extreme Value model and the advantage of this is that, it is one family of model 
in which you are able to determine your domain of attraction based on the shape parameter. 
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The choice of the domain of attraction can be accurately made by allowing for the behaviour 
of the limit distribution G  at y  supz : F  z  1 , its upper end point. The “Weibull 
distribution” has a finite upper end point whereas the “Frechet and Gumbel family of 
distributions” both have an infinite upper end point (Coles, 2001). The three models can be 
jointly presented in one family of models referred to as the “generalised extreme value 
family” with distribution functions of the form 
 1
   z  

 
G  z  exp1   ,      (3.4) 
      
Defined on z :1  z      0  and with parameters satisfying 
   ,    and   0 . The parameters ,  and   are “location, shape and 
scale parameters” separately. The “three classes of extreme value distribution” are 
distinguished by the value of the shape parameter ( ). However, types II and III in Theorem 
3.2.1 correspond to values of   0and   0  respectively. In addition, the case of   0  is 
interpreted as the limit as   0 of the “GEV distribution” and its “distribution function” is 
   z   
G  z  exp exp  ,   z  .    (3.5) 
    
3.3 Asymptotic Models for Minima 
When looking at problems such as temperature variations, it is mostly of interest to research 
on the tail behaviour of the minimum temperature recorded especially when the interest is to 
study the possibility of a cold weather and these practicalities involve models for particularly 
small, rather than very large observations. 
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Consider the minimum M n  minY1,Y2 ,...,Yn , where the Y  denote a data point for the i
random variable Y . Supposing that Y  is identical and independently distributed, similar i
~
arguments relate to M n  as were applied to M , leading to a “limiting distribution” of an n
appropriately re-scaled variable. However, there is duality between minimum and maximum 
~ ~
such that M n  minY1 ,..., Yn with M n  maxY1,...,Yn , that is to say M n  M n  (Coles, 
2001). 
Therefore, for n  large, 
PM n  z  PM n  z  PM n  z 1 PM n  z   (3.6) 
 1 
  
  z    
P M n  z 1 exp 1   

 
   
     (3.7) 
 1 z     
P    M n  z 1 exp 1   
   


  
on z :1  z      0 , where ~   . 
This is termed as the “Generalised Extreme Value distribution for the minima”. 
Given that y1 , y2 ,..., ym  are observations from the “Generalised Extreme Value distribution 
for the minima, with parameters , ,  , this indicates fitting the GEV distribution for 
maxima to the data  y1,y2 ,...,ym ”. 
In some circumstances, “GEV distribution for minima” can be applied directly since 
modelling the “block minima” is suitable. Also, given data y1 , y2 ,..., ym  to model the 
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minima, the duality between the distributions for minima and maxima can be explored 
alternatively. 
Apart from the sign correction~̂  ˆ , the “maximum likelihood estimate” of the parameters 
of this distribution corresponds exactly to that of the required “Generalized Extreme Value 
(GEV) distribution for minima”. 
3.3.1 Parameter Estimation under the GEV 
Under the Generalised extreme value distribution, the “sample of size n  is divided into m  
blocks of size k , with n  mk  and k  sufficiently large”. In the Annual Maxima method 
generally, a block represents a period of one year and the k  being realisations per year. 
However, blocks can represent daily, weekly or monthly maxima. The largest observation in 
each block is carefully chosen to get a sample of m  “independent sample maxima”. Assume 
Y  to be the random variable denoting the maximum of a block with size k , then; 
M k  max Y1,...,Yk        (3.8) 
Bearing in mind m  blocks, we find a collection of m  sample maxima M1,k ,..., Mm,k  . 
Subsequently, we can find estimates of the “Generalized Extreme Value (GEV)” for the 
Extreme Value Index     , along with the location    and the scale    parameters. In 
literature, there are a number of estimation methods for the “Generalized Extreme Value 
Distribution (GEVD)” but we will consider the “Probability Weighted Moments (PWM)” and 
the “Maximum Likelihood (ML) method” in this subsection. 
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3.3.1.1 Maximum Likelihood Estimation 
The parameters of the “Generalized Extreme Value (GEV) distribution” were estimated by 
“Maximum Likelihood method”. Adaptability of the Maximum Likelihood to changes in 
model structure as compared to other parameter estimation techniques makes it preferable. 
Although, when the model is altered, the estimating equations change, the core procedure is 
fundamentally unaffected. The Maximum Likelihood (ML) also has “an expedient set of off-
the shelf large sample inference properties” (Coles, 2001). 
Let Y1,...,Ym   be a random sample of the random variable Y  taken from a “Generalized 
Extreme Value (GEV) distribution”. 
The “log-likelihood function” for   0  of an observed “random sample”  y1,..., ym   is given 
by 
1

 1  m  z m 
l , , | z ,..., z   m log  1  log 1 i
    zi   
1 m     1  ,         (3.9) 
   i1    i1   
Provided that 
z
1 i
 
 0, i 1,2,...,m  

However, when   0  the “log-likelihood function” becomes 
m  z m
l , , | z ,..., z   m log exp  i
   zi  
1 m    ,    (3.10) 
i1    i1 
The Maximum Likelihood estimators ˆ ,ˆ ,ˆ  for the unknown parameters , ,   was 
found by maximizing (3.10). 
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We get the likelihood system of equations which has no explicit solution by differentiating 
expression (3.10). Consequently, this system of equations must be solved by numerical 
methods or iteratively. 
The common regularity conditions fundamental to the “asymptotic properties of ML 
estimators” are not automatically applicable because the support of the GEV depends on 
unknown parameter values (Beirlant et al, 2004).  
According to Smith (1985) as cited in Coles (2001), researched thoroughly into this problem 
and his discoveries are briefly discussed as follow; 
 The “ML estimators” are not likely obtainable when   1  
 The “ML estimators” are regular, and therefore have a usual asymptotic properties 
when   0.5 . 
  Also, the “ML estimators” do not have the common asymptotic properties, but is 
obtainable when 1   0.5  
3.3.1.2 Probability Weighted Moments (PWM) 
“Probability-Weighted Moments (PWM) estimator” is also a very standard estimation 
method for the generalised extreme-value density function. PWM’s are generalisation of the 
usual moments of a “probability distribution”, which give increasing weight to the tail 
information.  Parameters of several distributions such as; the Weibull, Gumbel and logistic 
density functions can be appropriately estimated using the “probability-weighted moments”. 
However, the “Gumbel distribution” is an exceptional case of the “generalised extreme-value 
distribution”, thus indicating that the PWM may be significant in the “generalised extreme-
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value” d.f as well. Estimating parameters of the “generalised extreme value distribution” 
using the PWM is described in details by (Hosking, Wallis &Wood, 1985). 
Definition: The “probability-weighted moments of a random variable Y  with distribution 
function” F are the quantities: 
 p r sM p,r ,s  E Y F  y 1F  y  , p,r, s     (3.11)  
However in the context of “generalised extreme-value estimation”, we use “probability 
weighted moments” of the form 
r
  r  M1,r ,0  E Y F  y , r  0,1,2,3,...     (3.12)  
Given a “random sample” of size m  from the distribution F , estimation of    is most r
conveniently based on the ordered sample Y Y  ...Y  an “unbiased estimator” of  1:m 2:m m:m r
is given by the statistic 
1 m  i 1i  2i 3...i  r  
br    Y      mi1:m 
    (3.13) 
m i1  m1 m 2 m3 ... m r 
However, one can derive the PWMs for the generalised extreme value density function 
H , ,  (see Hosking et al., 1985 for proof) 
1    
r    1 r 1  1       (3.14) 
r 1   


 for  1 and   0 . 
For r  0,1,2  we have the following system of equations: 
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  3   
    1 1  ,   2     1 2    and  1 0
13
  0 1 0  (3.14a)   2 1  0 1 2
The PWM estimators ˆ  ˆ,ˆ,ˆ   of the parameters are the solutions of the equation (3.14a) 
when the moments   are replaced by their empirical counterparts. r
 
3.3.4 Estimation of other Parameters of Extreme Events 
In this subsection, we introduce the inference for return levels, return periods and exceedance 
probability. 
When considering “extreme values of a random variable”, say yp , such that there is a 
probability, p , that yp  is exceeded in any given year, the “return level” of an extreme event 
is mostly of concern. Conversely, the “return level” related with the “return period” 1 p  is 
the level that on the average is expected to be exceeded once every 1 p  years. However, yp   
is the “return level associated with the return period” 1 p  in extreme value theory.  
Assuming the “100-year return level” for temperature is found to be 50oC , then the 
probability of temperature surpassing 50oC  in any particular year is 0.01  (i.e. 1 100 ). 
The “rarity” of an “extreme event” is indicated by the quantity 1 p , generally termed as the 
waiting time or return period for an event. 
Let y1 p be “the extreme quantile with order 1 p of the generalised extreme value 
distribution” fundamental to the random variable Y , well-defined in (3.1), with p  
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sufficiently small. Inverting the “generalised extreme value distribution” will lead to 
estimates of extreme quantiles. That is, the return level is obtained as; 
  
   log 1 p 1, if   0
y G1 1 p | ,       (3.15) 1 p

   log  log 1 p , if   0
where , ,   is replaced by its equivalent maximum likelihood estimator. 
1
Extreme quantiles can be expressed in terms of the “tail quantile function” with t  : 
p
 1 
U    1 p | ,   y1 p    (3.16) 
 p 
Dealing with “Annual Maxima” in particular, the “T-period level, U T  ”,  is given by; 
1 1
T   ,  (3.17) 
1G u | ,  P X  u
where the denominator P X  u   is called exceedance probability. The mean value of a 
geometric random variable can also be used to interpret the return period.  
Let N  be the number of periods needed for the first time to exceed the level u . Therefore, u
N  is a “geometric random variable” having mean value 1 p , with pu  Pu u Nu  u  . 
Therefore, we have T  E Nu  . 
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If we are looking to obtain the quantiles of Y  and make inference about the population 
principal to the m  blocks of observations Y ,Y ,...,Y , with Y F , we then know the random 1 2 k
variable Y defined in (3.8) is distributed as 
FY  FM  F
k G     (3.18) k
Where k  is length of the block 
For the random variable Y , we denote by Y1 p  the quantile of order 1 p   of the density 
function F underlying the random variableY , i.e., a quantity such that F Y1 p  1 p . 
Using now relation (3.17), we have, F k Y1 p   G Y1 p | ,   and then 
  
   k log 1 p  1, if   0
Y G1
k
1 p  1 p | ,     (3.19) 
  k  log  log 1 p  , if   0
Again, estimates of (3.17) and (3.19) are obtained by replacing , ,   by its ML estimator. 
The right endpoint of the generalised extreme value distribution is finite if   0 . Hence, we 
can approximate the finite right endpoint of the underlying density function F  by 

yF U  Y1       (3.20) 

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3.4 Peaks-Over Threshold (POT) Method  
The “Peaks Over Threshold (POT) method”, considers the distribution of exceedances over a 
certain threshold. The interest in the use of the “peaks over threshold” method is estimating 
the “distribution function” F  of values of x above a certain threshold u . u
The “distribution function” F  is called the “conditional excess distribution function” and is u
defined as 
Fu x PY  u  x |Y  u 0 x  yF  u   (3.21) 
where Y  is a random variable, u is a threshold given, x  y u  are the excesses and yF    
is the right endpoint of F . However, we can verify that Fu  can be written in terms of F , 
that is 
Fu  x Fu Fy Fu
Fu x    (3.22) 
1 Fu 1 Fu
Generally, the estimation of F  in this interval generally poses no problem since the 
realizations of the random variable Y  lie mainly between 0 and u . However, the estimation 
of the portion Fu  might be difficult since we have very little observations in general within 
this area. EVT can prove very helpful at this point since it provides us with a dominant result 
about the conditional excess distribution function which is stated in theorem 3.4.1. 
Determining a threshold is the first consideration for the POT approach and data beyond the 
threshold are fitted to the “Generalised Pareto Distribution (GPD)”, which is based on the 
excesses above a threshold and it has an asymptotic justification as the “Extreme Value 
Theory”. As the sample size increases and the threshold u  gets large, the amounts by which 
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the observations exceed a threshold u  (termed exceedances) should approximately follow the 
“Generalised Pareto Distribution”. 
Let Y1 ,Y2 ,...  be a sequence of “identical and independently distributed random variables with 
a common distribution function F ” with u  being a sufficiently high threshold value. We 
regard an extreme value of observations as one exceeding the threshold and the description of 
its asymptotic behaviour as a conditional probability. Thus if the underlying distribution F  is 
known, then the “conditional distribution F ” can be obtained. However, as pointed out u
earlier, it is often the case that F  is unkown and as a result, an asymptotic distribution for 
high threshold is needed. Balkema and de Hann (1974) and Pickands (1975) provides 
analogous results to the Fisher-Tippet theorem as follows; 
Theorem 3.4.1 (Pickands (1975), Balkema and de Hann (1974)) 
For a large class of underlying “distribution function F ” the conditional excess “distribution 
function” PM n  z , for n  large, is well approximated by PM n  z  G  z  , n , 
where 
 1   
11 z  if   0
   

G ,  z      (3.23) 
  z

 1 e if   0


For z0,xF u  if   0  and z
0,   if   0 . 
  
G ,  z   is the “Generalised Pareto Distribution (GPD)”. 
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If x  is defined as x  u  z , then the “GPD” can be expressed as  
1
G   z 1 1 xu       (3.24) ,
Castillo and Hadi (1997) made the following remarks about the “Generalised Pareto 
Distribution” 
 When   0 , the GPD  ,   reduces to the “exponential distribution” with mean 
Exp   . 
 When  1, the GPD  ,   becomes “uniform”: U 0,  . 
 When   0 , the GPD  ,   reduces to the “Pareto distribution” of the second kind. 
The qualitative behaviour of the “Generalised Pareto distribution” is dominantly determined 
by the shape parameter   due to the duality between the “GEV and generalised Pareto 
families just as it is for the GEV distribution”. If   0  the distribution of excesses has an 
upper bound of u    (Beta type): If   0 , the distribution is “heavy tailed” and has no 
“upper limit” (Pareto type). The distribution is “light tailed” and unbounded if   0  
(Exponential type) which should again be interpreted by taking the limit   0 .  
From Theorem 3.4.1, if “block maxima” have an approximating distribution G  within the 
“generalised Pareto family”, the threshold excesses have a corresponding approximate 
distribution. Furthermore, “parameters of the generalised Pareto distribution” of threshold 
excesses are determined distinctively by those of the associated “GEV distribution of block 
maxima”. In particular, the parameter   in equation 3.6.1 is equal to that of the 
corresponding GEV distribution. Selecting a different, but still large, block size n  would 
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affect the values of the GEV parameters, but not those of the corresponding “generalised 
Pareto distribution of threshold excesses”. 
3.4.1 Parameter Estimation under the POT Method (Maximum Likelihood 
Estimation) 
From the Balkema and de Hann theorem, the exceedances over a sufficiently large threshold, 
u, can be characterized by the GPD. In the literature, several methods are presented for 
estimating the parameters of the GPD. These include; ML, PWM, Method of moments, 
elementary percentile method, among others. 
In this thesis, we make use of the ML and the PWM . In the case of ML, a likelihood function 
can be formed from the distribution (3.5) as 
  1  m   y 
m log u  1 log 11 ,   0
  i1 
l    ,  u u | y1,..., ym      (3.25) 
 1
m
m logu   yi ,   0
  u i1
 y
where 1 1  0, i 1,2,..., m  
 u
Generally, we prefer a reparametrization of the “log-likelihood function” in (3.25) for 
computational purposes. The main objective of this reparametrization, is to get ̂  explicitly 
as a function of ̂  as introduced by Davison (1984) is obtained numerically through (3.10.9), 
1 m
but after replacing ̂  by  log 1ˆ yi  .  
m i1
For   0 , ˆu Y  which is the classical case of the exponential distribution.  
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
Defining   , the “log-likelihood function” may well be rewritten as 
 u
 1  m
l  , | y1,..., ym   m log m log  1 log 1 yi     (3.26) 
   i1
where 1 y  0, i 1,2,...,m . Maximization of (3.26) with respect to   and   yields the i
“ML estimator ˆ,ˆ  ” of the parameters  ,  . 
3.4.2 Estimation of other Parameters of Extreme Events 
After obtaining estimates of the parameters   and  . Defining once again the “extreme 
quantile of order 1 p  ” of the GPD underlying the excesses Y  as Y1 p , with p sufficiently 
small, estimates of “extreme quantiles” can be obtained by inverting the GPD  
 ku
1   p
 1 , if   0
H 1 p | 0,u        (3.27) 

 u log p, if   0
where  , u   are replaced by its ML estimator. 
If   0 , the GPD has a finite endpoint and is given by 
 
UH   y
u
1      (3.28)  
Which can be estimated by replacing the  , u by its corresponding ML estimator. 
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However, with the “asymptotic normality of the ML estimators” as the basis, confidence 
intervals can be constructed for parameters of the GPD. The 1001 % confidence interval 
for   is 
2
  1 1 
CI   : log L    log L ˆ   (3.29) p p  
2
 
3.4.3 Threshold Selection 
It is still an unsolved problem as to the procedure for the choice of threshold u and it is not 
straightforward; indeed, a compromise has to be found between high values of u , where the 
bias of the estimators is smaller, and low values of u , where the variance is smaller. 
The “mean excess function” is a tool generally used to aid the choice of u  and also to 
determine the appropriateness of the “Generalised Pareto Distribution” model in practice. The 
mean excess function sometimes referred to as the “mean residual life function” of a random 
variable Y  is defined as 
M u   E X u |Y  u      (3.30) 
provided E Y   .  
Davison and Smith (1990) used the property that the linearity of the “mean excess function” 
illustrates the “Generalised Pareto Distribution” class to formulate a simple graphical check 
that data conform to a “Generalised Pareto Distribution model”.  
If the mean excess plot is close to linear for high values of the threshold then there is no 
indication against use of a GPD model. 
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Let Y be a positive random variable with finite first moment and “distribution function F ”. 
Then the “mean excess function” of Y for all u  0 is 
y
1 F
eu  E Y u |Y  u   F  ydy    (3.31) 
F (u) 
u
where yF  supy : F  y 1   
The “mean excess plot” is the set of points u,e u   for all u  0 . 
The empirical counterpart of the mean excess function based on the sample  y1, y2 , y3 ,..., yn   
n
 yi1u ,  yi 
is eˆ u  i1 u      (3.32) n
1u ,  yi 
i1
where 1   y 1 for y u,  0 otherwise u ,
Generally, the expression (3.32) is obtained by replacing the theoretical average by its 
empirical counterpart that is to say by averaging the data that are larger than u  and 
subtracting u  (Beirlant et al, 2004).  
The “mean excess plot” is estimated at the points u  xm p,m , p 1,2,...,m1, the  p 1  
largest observation and the mean residual function then takes the form 
1 p
eˆ ˆp,m  em x     xm j1,m  xm p,m , p 1,2,...,m   (3.6.3.d) m p :m p j1
If F D G     0  then it is easy to show that, 
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eln X ln u   E ln X  ln u | X  u   as u  
It can however be shown that the statistic eˆm,ln x ln xm p,m  with eˆm,ln x  denoting the empirical 
“mean excess function” of the log-transformed data, appears as the “Hill estimator” (Beirlant 
et al, 2004). 
3.4.4 Statistical Choice of GPD Models 
Exploratory data analysis is often used to test the “goodness-of-fit” of sample observations to 
specific target distributions. A few graphical tools have been extensively used to detect 
heavy-tailed behaviour or extremal behaviour in observed data 
In view of the likelihood of modeling any combination of the “extreme value model 
parameters” (such as Temperature or Rainfall) as functions of time or other covariates, there 
is a wide range of models to choose from, and selecting the best fitting model becomes an 
essential issue.  
We will employ the Quantile-Quantile plots, Probability-Probability plots, Mean Excess plots 
and Return Level plots to assess the quality of a fitted “Generalised Pareto model”. 
Assuming a threshold u , an estimated model Ĥ  and threshold excesses y   ... y1 k  , with 
ˆ  0 , the Quantile plot which is a plot of the empirical quantiles from the data against the 
theoretical quantiles of Ĥ  will consist of the pairs  
H 1 i k 1 , y   ,i 1,2,...,k    (3.33) i
1 ˆ
where Hˆ  y  u   ˆ
ˆ 
y 1  
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Again the probability plot will consist the pairs i k 1 , H  y  ;i 1,...,k  i
1
 y  ˆ
where Hˆ  y 11ˆ  , provided ˆ  0 . However, the plot is constructed by using 
 ˆ 
1
 z
   

1 e  in place of 11 z  . 
  
For the GP model to be reasonable for modelling excesses of u , both the “quantile and 
probability plots” should entail points that are approximately linear (Coles, 2001). Also, a 
“return level plot” which consist of the locus points m, yˆm   for large values of m , where 
ˆ ˆ
yˆ is the estimated mobservation return level yˆm  u  mˆ  1  which is modified if m
ˆ 
u 
ˆ  0 . 
3.5 Comparison of ML and PWM Estimation Methods for GEV 
Distribution 
Hosking et al. (1985) performed a simulation study of the small-sample properties (in 
particular their study was concentrated on sample sizes n 15,25,50,100  ) of the PWM 
estimators of the “generalised extreme-value distribution”, in comparison to ML and sextiles 
estimators. Sextiles estimators had been originally also used for the estimation of the 
parameters of “generalised extreme-value density functions”, still they have been proven to 
be inferior of the other methods, so they are not used anymore. As far as estimators of the 
parameter   are concerned, all three estimation methods are equivalent for n 100 , but for 
smaller sample sizes the “PWM estimator” has lower variance. Still, “PWM estimator” has in 
general larger bias that the other estimators, though the bias is small near the “critical value” 
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  0 . Similar results are obtained for the estimators of the scale and location parameters, 
though the differences in the variances are less pronounced. 
Another quantity of interest in extreme-value analysis are (large) quantiles. 
Estimators of quantiles can be simply obtaining by substituting the estimators of the 
parameters   to the quantile function of the generalised extreme-value distribution. So, when 
comparing (via simulation) the quantile estimators obtained by the three different estimation 
methods, Hosking et al. (1985) get the following findings. For n 100 , all methods are 
comparable. For small samples the upper quantiles obtained by the PWM method are rather 
biased, but they are still preferable to the ML estimators, which have very large biases and 
variances. All the methods are very inaccurate when estimating extreme quantiles in small 
samples with   0 .  
Another comparison of ML and PWM estimation methods was performed by Coles and 
Dixon (1999). They also compiled a simulation study to explore the small-sample properties 
of ML and PWM. Their results confirmed the results of Hosking et al. (1985), that is, for 
small sample sizes, the ML estimator is seen to be a poor competitor to the PWM estimator, 
in terms of both bias and mean square error. Still, while Hosking et al. (1985) were obviously 
in favor of the PWM estimators, Coles and Dixon (1999) share the view that as a general, all-
round procedure for extreme value modelling, likelihood-based methods are preferable to any 
other. Still, despite the many advantages of maximum likelihood mentioned in the previous 
section, poor performance in small samples remains a serious criticism, since it is not 
uncommon in practice to need to make inferences about extremes with very few data – the 
rarity of extreme events means that even long observational periods may lead to very few 
data that can be incorporated into an extreme value model. So, Coles and Dixon (1999) tried 
to examine more closely the comparison between ML and PWM for estimating parameters of 
the generalised extreme-value density function with small datasets by exploiting the 
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simulated distributions of the parameter estimates. Understanding more clearly the failings of 
ML, Coles and Dixon (1999) explore the possibility of correcting for such deficiencies. 
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CHAPTER FOUR 
RESULT AND DISCUSSION 
Daily minimum and maximum temperature as well as the rainfall readings for all the ten (10) 
regions in Ghana were obtained from the Ghana Meteorological Agency in Accra. The daily 
temperature and rainfall readings were employed in the study and it spanned the period 
January 1960 to December 2012. The study focused on three key regions which takes care of 
the Southern, Central and Northern sectors of the country. The selection was purely 
subjective. These regions were Greater Accra (Southern), Ashanti region (Central) and 
Northern region (Northern). This chapter presents the analysis of the data and it begins with 
basic statistical analysis involving graphical tools and basic statistics. In addition, the extreme 
value analysis of both temperature and rainfall for these three major regions in Ghana are 
presented. The Generalized Extreme Value Model and the Threshold Exceedances model 
were applied to the data. The chapter further explored the “left and right tails of the 
Generalized Pareto Distribution” via the data just as in the case of the “GEV distribution”. 
The R software was employed in the data analysis. 
4.1 Basic Statistics 
This section of the study presents the summary statistics and assesses the stationarity of the 
data. Extreme observations were explored through scatter plots. 
Table 4.1 presents the summary statistics of both the daily temperature and rainfall readings 
of the three selected regions.  
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Table 4.1: Summary statistics of Data 
Variable  Region Minimum 1st 3rd  Maximum Median Skewness 
Quartile Quartile 
Temperature Minimum Accra 17.5 20.6 21.95 25.50 21.20 0.17 
Ashanti 17.00 19.17 20.40 22.30 20.00 -0.56 
Northern 17.00 19.60 21.00 24.40 20.00 -0.28 
Maximum Accra 28.30 31.70 34.40 38.90 32.90 0.22 
Ashanti 28.30 31.70 34.40 38.90 32.90 0.22 
Northern 31.10 34.40 39.30 42.8 37.2 -0.09 
Rainfall Maximum Accra 0.00 6.925 43.630 243.90 29.30 2.02 
  Ashanti 0.00 6.00 21.30 167.90 9.10 2.36 
  Northern 0.00 2.40 46.70 143.00 27.60 0.91 
 
From Table 4.1, the minimum temperature in Greater Accra ranges from 17.5oC  to 25.5oC  
with a median temperature reading of 21.2oC . The corresponding maximum temperature for 
the region ranged from 28.3oC  to 38.9oC  with a median temperature reading of 32.9oC . 
Northern region recorded high temperatures as compared to Greater Accra and Ashanti. 
Interestingly, the temperatures recorded in Greater Accra and Ashanti regions were 
approximately the same. The data was approximately normally distributed based on the 
observed skewness.  
Also, the minimum rainfall readings observed in the three regions was 0.0mm. Greater Accra 
however recorded the highest rainfall of 243.9mm with Northern region recording the least 
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amount of rainfall. Again, the data on amount of rainfall was approximately normal. The 
scatter plots of the minimum and maximum daily temperature readings as well as that of the 
maximum rainfall observed in these regions are depicted in Appendix A.  
Furthermore, the stationarity of the data was tested using the “augmented Dickey-Fuller 
(ADF)”. In this test the null hypothesis that data is nonstationary (has a unit root) is tested. 
Both the undifferenced and the first difference of the temperature and rainfall data were 
tested. Table 4.2 present the stationarity test on the undifferenced series for temperature and 
rainfall.  
              Table 4.2: ADF Stationarity Test on Temperature and Rainfall Data 
 Temperature Rainfall 
Minimum Maximum Maximum 
Variable t – stat p – value  t – stat  p – value  t- stat p-value 
Accra -6.97 0.000 -0.68 0.000 -8.77 0.000 
Ashanti  -27.48 0.000 -0.68 0.000 -10.99 0.000 
Northern  -0.56 0.000 -1.00 0.000 -6.97 0.000 
 
From Table 4.2, the “null hypothesis of nonstationarity” was rejected for both the 
temperature and rainfall data for the selected regions (all p values are less than 5% level of 
significance).                              
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4.2 Greater Accra 
In this section of the study, the extreme value and the “Generalized Pareto models” were 
fitted to the temperature and rainfall data of Greater Accra. 
4.2.1 Fitting the Extreme Value Model  
This section addresses the “Generalized Extreme Value distribution (GEVd)” of the 
temperature and rainfall data in Greater Accra region. The Exponential QQ-plot was used to 
have a rough and quick confirmation of the plausible fit of the “GEVd” to the data.  
4.2.1.1 Temperature 
The exponential Q-Q plots of both maximum (right panel) and minimum (left panel) 
Temperature is presented in Figure 4.1a. The “QQ-plot” reflects the linear relationship 
between the theoretical quantiles of a “family of distributions” and the corresponding 
“theoretical quantiles” of the standard member of the involved family. 
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Figure 4.1a: Exponential Q-Q plot for minimum (Left panel) and maximum (right 
panel) Temperature 
From Figure 4.1a, the “Q-Q plots” deviates from linearity at the tails of the distribution. For 
the upper tails, it exhibits a non-linear pattern, pointing to an underlying distribution function 
with a lighter right tail than the proposed “exponential distribution”. This indicates that the 
distributions belong to the “Weibull domain of attraction”. 
Next we fit a GEV to the annual maximum and minimum temperature using the maximum 
likelihood. Tables 4.3a and 4.4a present the parameter estimates and their corresponding 
“standard errors” (in parentheses) for the minimum temperature and maximum temperature 
respectively.  
        
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 Table 4.3a: GEV Parameter Estimate for Minimum Temperature 
Parameter Estimate 95% Confidence Interval 
Location ()    -21.7443 (0.0464) (-21.8352, -21.6535) 
Scale ( )    1.0673 (0.0301) (1.0083, 1.1263) 
Shape ( )    -0.2315 (0.0123) (-0.0555, -0.2075) 
 
  
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            Table 4.4a: GEV Parameter estimate for Maximum Temperature 
Parameter Estimate 95% Confidence 
Interval 
Location ()     32.279(0.0825) (32.1172, 32.4407) 
Scale ( )    1.859 (0.059) (1.7435, 1.9748) 
Shape ( )    -0.212 (0.0107) (-0.2664, -0.1573) 
 
The shape parameter ( ), that is the “Extreme Value Index (EVI)”, is negative in both cases 
and this suggests that the “underlying distribution” belongs to the “Weibull family of 
distributions”. In addition, the confidence interval exclude zero (0) and hence the distribution 
indeed belongs to the Weibull family of distributions 
 
The “diagnostic plot”s for the GEV fit are presented in Figures 4.2a and 4.3a for minimum 
and maximum temperatures respectively.  
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 -24 -22 -20 -18
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 -26 -24 -22 -20 -18
Return Period z
 
Figure 4.2a: Diagnostic plot for the fitted GEV model for minimum Temperature 
From 4.2a, the linearity of the Q-Q and the PP plots indicate that, the model is valid. From 
the histogram, the density also appears to be consistent with the data points. We therefore 
conclude that the diagnostic plots indicate a good fit of the GEV model.  
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Return Level Model
-24 -20 0.0 0.6
f(z) Empirical
0.0 0.3 -24 -20
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Figure 4.3a: Diagnostic plot for the fitted GEV model for maximum Temperature 
From 4.3a, the Q-Q and PP plots appear to be linear indicating that, the model is valid. The 
histogram plots with density also appears to be approximately normal. Thus, the “GEV 
model” is appropriate for the data.  
The return periods and exceedance probabilities for the tails of the distribution was further 
explored. Table 4.5a presents the probabilities of exceedances for some selected 
temperatures. The exceedance probabilities for the minimum temperature is interpreted as, 
probability that, the average temperature falls below the absolute value of the temperature. 
 
 
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Table 4.5a: Exceedance Probabilities estimates for Temperature  
Maximum Temperature  
Temperature 34 35 36 37 38 39 
Exceedance Probs 0.3002 0.1593 0.0713 0.0257 0.0068 0.0010 
Min Temperature  
Temperature -22 -21 -20 -19 -18 -17 
Exceedance Probs 0.717 0.373 0.120 0.020 0.001 0.000 
Negative values are interpreted in absolute value terms 
From Table 4.5a, it was observed that, the probability of observing a temperature of 39 C  
was very small (0.0010). Thus, the maximum temperature within greater Accra region of 
Ghana is unlikely to exceed 39 C . However, the chances of observing a maximum 
temperature of 34 C  or more daily is about 0.3002. Furthermore, there was a probability of 
0.001 for the temperature to fall below 18 C  daily. The return periods for daily minimum 
and maximum temperature was then estimated and presented in Table 4.6a.  
The return periods measure how long, it is expected for a given temperature to be exceeded, 
once for a specified period (Salas and Fernandez, 1999). The return periods for the GEV 
model are in years. The periods considered in the study were 5, 10, 20, 50 and 100 years. 
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Table 4.6a: Return Periods estimates for Temperature 
Maximum Temperature 
Temperature 34.7 35.6 36.4 37.2 37.7 
Return Period (Years) 5 10 20 50 100 
Minimum Temperature 
Temp 22.3 22.8 23.2 23.7 24.0 
Return Period (Years) 5 10 20 50 100 
Negative values are interpreted in absolute value terms 
From Table 4.6a, a 5 year return period implies that, maximum temperature is expected to 
exceed 34.7 C  once every 5 years. Also, Greater Accra is expected to experience maximum 
temperature of 35.6 C , 36.4 C , 37.2 C , and 37.7 C  once every 10, 20, 50 and 100 years 
respectively. Furthermore, the 100 year return period for the minimum temperature means 
that, it is expected for the temperature to fall below 24 C  on the average once every 100 
years. 
4.2.1.2 Rainfall  
We fitted the extreme value model to the rainfall data in this section. The exponential Q-Q 
plot of maximum Rainfall for Greater Accra is presented in Figure 4.4a. 
 
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0 1 2 3 4 5
Theoretical Exponential
 
Figure 4.4a: Exponential Q-Q plot of maximum Rainfall 
From Figure 4.4a, the Q-Q plot for maximum rainfall present more linear pattern than that for 
temperature. The exponential distribution is therefore not appropriate parametric family to be 
fitted to the rainfall data. We therefore fitted a GEV model to the data. The estimated GEV 
model for the maximum rainfall with the associated confidence intervals for the parameters is 
presented in Table 4.7a.   
          
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Rainfall
60 80 100 120 140 160
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Table 4.7a: GEV Parameter estimate for Maximum Rainfall 
Parameter Estimate 95% Confidence Interval 
Location ()    62.3323 (1.3935) (59.6011, 65.0635) 
Scale ( )    12.1475 (1.2891) (9.6209, 14.641) 
Shape ( )    0.4514 (0.1172) (0.2216, 0.6811) 
  
From Table 4.7a, the shape parameter is positive this suggests that the underlying distribution 
belongs to the “Frechet family of distributions”. The diagnostics plots for assessing the model 
are presented in Figure 4.5a.  
 
Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200 250
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 60 80 100 140
Return Period z
 
Figure 4.5a: Diagnostic plot for the fitted GEV model for maximum rainfall  
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Return Level Model
100 500 0.0 0.6
f(z) Empirical
0.000 0.025 60 120
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From Figure 4.5a, the histogram plot shows that the residuals are rightly skewed. However, 
the PP plot is approximately linear whiles from the Q-Q plot, there is a deviation for larger 
values. The return level lot confirms that the model departures are not significant indicating 
the appropriateness of the GEV fit. Estimates for probabilities of exceedances for some 
selected rainfall readings were computed. This is presented in Table 4.8a. 
Table 4.8a: Exceedance Probabilities estimates for Rainfall 
Rainfall (X) in mm  
X 50 100 150 200 250 300 
P(X  x)  0.9795 0.1340 0.0396 0.0179 0.0100 0.0063 
 
From Table 4.8a, the chances of experiencing a rainfall amount of 50 mm annually was 
0.9795. Also, there was an observed chance of 0.134 of experiencing an amount of 100 mm 
rainfall reading annually.  The return periods was also estimated and presented in Table 4.9a. 
Table 4.9a: Return Periods estimates for Rainfall 
Maximum Rainfall in mm 
Rainfall 88.38 109.74 138.27 192.05 250.08 
Return Period (Years) 5 10 20 50 100 
 
From Table 4.9a, the 100 year return period for the maximum rainfall in Accra means that, on 
average it is expected to record a rainfall of 250.08 mm once every 100 years. 
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4.2.2 Fitting the Generalised Pareto (GP) Model 
 In this section, the generalized Pareto model was fitted to both the temperature and rainfall 
data. 
4.2.2.1 Temperature 
The daily temperatures are considered as realization of a random variable having a 
“distribution function F ”. The “peak-over threshold method” considers an event as extreme 
(here daily temperatures) if it exceeds a high threshold  .  The excesses over the threshold 
  are approximated with the “Generalized Pareto (GP) distribution”. The choice of   is a 
very important issue for this method. It is a trade-off between a low   with minimal variance 
and a high   with a minimal bias. In this section of the study, a generalized Pareto model 
was fitted to the minimum and maximum temperature in Accra. 
The Peak-Over Threshold Method; Threshold Selection 
The mean residual life plot also known as “mean excess plot” and a threshold stability plot 
was used to aid in the selection of an appropriate threshold. For the “mean excess plot”, we 
select   as the point to the right of which a linear pattern is observed. The second approach 
is to estimate the model at a range of thresholds and above a threshold o  at which the 
asymptotic properties for the GP model is valid; the parameter estimates should be 
approximately stable, thus approximately constant, Coles (2001). 
Figure 4.6a is the mean residual life plot for daily minimum (left panel) and maximum (right 
panel) temperature. 
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-24 -22 -20 -18 28 32 36
Threshold Threshold
 
Figure 4.6a: Mean excess plot for minimum (left panel) and maximum (right panel) 
Temperature  
From Figure 4.6a there is an observed linearity above   28 C  for the maximum average 
monthly temperature. There was also evidence of linearity above   26 C  for the 
minimum temperature. A threshold of 28 C  for the maximum temperature and 26 C for the 
minimum temperature seams feasible. The decreasing nature in both cases indicates that, they 
both have a tail lighter than the exponential distribution. Thus, they belong to the GP 
distribution with Weibull domain of attraction. If the “mean excess plot” is close to linearity 
for high threshold values, then the data conform to a GP distribution (Ghosh and Reinck 
2011). 
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Mean Excess
0 1 2 3 4
Mean Excess
0 1 2 3 4 5
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Figures 4.7a and 4.8a present the threshold stability plot of the maximum and minimum daily 
temperatures respectively. 
32.0 32.5 33.0 33.5 34.0 34.5 35.0
Threshold
32.0 32.5 33.0 33.5 34.0 34.5 35.0
Threshold
 
Figure 4.7a: Parameter estimates against threshold for maximum Temperature 
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Shape Modified Scale
-0.5 0 20
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Figure 4.8a: Parameter estimates against threshold for minimum Temperature 
From Figure 4.7a, the parameter estimates for the maximum temperature appear stable 
around   34 C . Also, from Figure 4.8a, the threshold stability plot shows stability of the 
parameter estimates around    21 C .    
From the mean threshold plot and the plot of the threshold stability, a threshold of  
  34.5 C  and   20.6 C  were selected for both the maximum and minimum 
temperatures respectively. 
Table 4.10a is the estimate of the GP for the maximum temperature for Accra. With the 
threshold of   34.5 C  there were 145 exceedances which constitutes a proportion of 
0.2294. 
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Table 4.10a: GP parameter estimate for maximum Temperature 
Parameter Estimate 95% Confidence Interval 
Scale ( )    1.5679 (0.1587) (1.2569, 1.8789) 
Shape ( )    -0.3059 (0.0632) (-0.4297, -0.1820) 
 
From Table 4.10a, the shape parameter is negative which implies that, the usual asymptotic 
properties are not met, (Smith, 1985).  
Table 4.11a is the estimate of the GP for the minimum temperature. With the threshold of 
  20.6 C  there were 98 exceedances which constitutes a proportion of 0.1612. 
 
Table 4.11a: GP Parameter Estimate for Minimum Temperature 
Parameter Estimate 95% Confidence Interval 
Scale ( )    0.8687 (0.1183) (0.6368, 1.1006) 
Shape ( )    -0.1636 (0.0900) (-0.3399, 0.0127) 
 
From Table 4.11a, the shape parameter for the minimum temperature is also negative.  
Figures 4.9a and 4.10a are the diagnostic plots of the fitted GP model for the maximum and 
minimum temperatures.  
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Figure 4.9a: Diagnostic plot for fitted GP for maximum Temperature 
From Figure 4.9a, the goodness-of-fit in the Q-Q and PP plots seems convincing. The 
confidence intervals on the return level plot suggest that the model departures are not 
significant. Thus, the fitted generalized Pareto distribution is appropriate. 
 
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Figure 4.10a: Diagnostic plot for fitted GP for minimum Temperature 
From Figure 4.10a, the Q-Q and PP plots is approximately linear as the deviations from the 
model is very small from the “confidence intervals” observed on the “return level plot”. Thus, 
the goodness-of-fit is satisfied and therefore the fitted generalised Pareto distribution is 
appropriate. 
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4.2.2.2 Rainfall 
Figure 4.11a is the mean residual life plot for maximum rainfall readings 
observed.
60 80 100 120 140 160
Threshold
 
 Figure 4.11a: Mean excess plot for maximum Rainfall 
From Figure 4.11a there is an observed non-linearity above   80mm. A threshold of    
  70  seems feasible. To settle on an appropriate threshold, the threshold stability plot was 
also examined. This is presented in Figure 4.12a  
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Mean Excess
0 10 20 30 40
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50 55 60 65 70 75 80
Threshold
50 55 60 65 70 75 80
Threshold
 
 Figure 4.12a: Parameter estimates against threshold for maximum Rainfall 
The threshold stability also suggested a threshold of   70 mm. Thus, from the mean 
threshold plot and the plot of the threshold stability, a threshold of   60 mm was selected. 
For this threshold, there were 77 exceedances which constitutes a proportion of 0.6814. Table 
4.13a is the estimate of the GP for the maximum rainfall. 
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Table 4.12a: GP parameter estimate for Maximum Rainfall 
Parameter Estimate 95% Confidence Interval 
Scale ( )    27.8372 (4.9216) (18.1911, 37.4833) 
Shape ( )    -0.0738 (0.1354) (-0.3392, 0.1917) 
 
From Table 4.12a, the shape parameter is negative and this is an indication that the normal 
heavy tailed Pareto distribution could have been considered. 
Figure 4.13a is the diagnostic plot of the fitted GP model for the maximum 
rainfall.
Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 60 80 120 160
Empirical Model
Density Plot Return Level Plot
60 80 100 140 0.01 0.05 0.20 1.00
Quantile Return Period
 
Figure 4.13a: Diagnostic plot for fitted GP for maximum Rainfall 
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Density Model
0.000 0.030 0.0 0.6
Return Level Empirical
50 150 50 150
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From Figure 4.13a, the Q-Q and PP plots is approximately linear as the deviations from the 
model is very small from the confidence intervals observed on the return level plot. Thus, the 
goodness-of-fit is satisfied and therefore the fitted generalized Pareto distribution is 
appropriate. 
4.2.3 General Comments 
Rainfall and temperature are important climatic factors for crop production in Ghana. Ghana, 
among other African countries’ vulnerability is heightened by the fact that most of the 
economies in this region rely mainly on rain-fed agriculture and natural resources, which are 
very sensitive to climate change and variability. This section of the study explored the 
extreme distribution of the temperature and rainfall occurrences in Greater Accra region of 
Ghana. The return periods of extreme occurrences and probability of exceeding such 
occurrences were also estimated. It was observed that the extreme occurrences of temperature 
in the region can be modelled using the “Weibull family of distributions” while the “Frechet 
family of distributions” can be used to model the extreme rainfall.  
A maximum temperature of 34.7cC is predicted to occur in the region once every five years. 
Furthermore, maximum temperatures of 35.6cC , 36.4cC ,37.2cC , and 37.7cC  were 
predicted to be experienced one every 10, 20, 50, and 100 years respectively. Also, a 
minimum temperature of 22.3cC was estimated to occur once every five years whiles a 
minimum temperature of 24cC was predicted to occur once every 100 years in the region. 
The probability of exceeding a maximum temperature of 39cC  was 0.001 which is very 
unlikely to be experienced in the region. However, there was a moderate likelihood of 
exceeding a maximum temperature of 34cC . Also, the likelihood of a minimum temperature 
falling below 18cC was very small. 
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The maximum average rainfall of Greater Accra recorded in the study period was 243.9 mm 
which was an extreme occurrence. An amount of 88.38 mm of rainfall reading was estimated 
for the region in the next five years and every five years after. Furthermore, a maximum 
amount of 250.1 mm of rainfall was also estimated for once every 100 years. In addition, the 
chances of observing a maximum amount of rainfall exceeding 100 mm was very small with 
a probability of 0.0.9795 of recording a rainfall of 50 mm. 
Climate affects humans’ basic needs, such as shelter, food, water and health. These needs are 
threatened by climate change with increased temperatures, changes in precipitation, and more 
frequent or intense extreme events. Among crop species, responses to temperatures differ 
throughout their life cycle and are primarily the phenological responses, that is, stages of 
plant development. For each species, a defined range of maximum and minimum 
temperatures form the boundaries of observable growth. Hatfield and Prueger (2011) 
established that increases of temperature may cause yield declines between 2.5% and 10% 
across a number of agronomic species throughout the 21st century. Interestingly, in the 
current study, average temperatures of Greater Accra is more likely to lie in the range 
24cC and 30cC , which favors the agro industry. However, there was indications of 
observing temperatures beyond this range though very unlikely. 
Furthermore, it has been demonstrated that weather is associated with changes in birth rates, 
and sperm counts, with outbreaks of bronchitis, pneumonia and influenza, and is related to 
other morbidity effects linked to high pollution levels and pollen concentrations The high 
birth rate and breeding of mosquitos in Accra can be attributed to high temperatures recorded 
in this region. Although rainfall is critical in providing suitable habitats for mosquitoes to 
breed (Craig et al., 1999). Temperature is a key driver of several of the essential mosquito 
and parasite life history traits that combine to determine transmission intensity, including 
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mosquito development rate, biting rate, and development rate and survival of the parasite 
within the mosquito (Mordecai et al., 2013). 
Greater Accra recently has been experiencing flooding every year due to several factors for 
which the amount of rainfall is the key factor. The current study revealed that the daily 
amount of rainfall recorded in the region ranged from 0 inches to 243.9mm. Extreme 
occurrences beyond this range was recorded though. Thus measures needs to be put in place 
to handle such occurrences in the near future.  
4.3 Ashanti Region 
In this section of the study, we focused on the extreme occurrence of temperature and rainfall 
readings in the Ashanti region. The extreme value and the Generalized Pareto models were 
fitted to the temperature and rainfall data observed.  
4.3.1 Fitting the Extreme Value Model 
The extreme behaviour of temperature and rainfall was explored through the Generalized 
Extreme Value distribution (GEVd). The Exponential QQ-plot was used to have a rough and 
quick confirmation of the plausible fit of the GEVd to the data.  
4.3.1.1 Temperature 
Figure 4.1b is the exponential Q-Q plots of both minimum (left panel) and maximum (right 
panel) Temperature.  
 
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0 2 4 6 0 2 4 6
Theoretical Exponential Theoretical Exponential
 
Figure 4.1b: Exponential Q-Q plot for minimum (left panel) and maximum (right panel 
Temperature 
From Figure 4.1b, the upper tail deviates from linearity pointing to an underlying distribution 
function with a “lighter right tail” than the “exponential distribution”. Thus the exponential 
family of distributions is not appropriate. In this vain, the “generalized extreme value 
distribution” was fitted to the data.  
Tables 4.3b and 4.4b present the parameter estimates and their corresponding “standard 
errors” (in parentheses) for the minimum temperature and maximum temperature 
respectively.  
            
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Temperature
17 18 19 20 21 22
Temperature
28 30 32 34 36 38
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 Table 4.3b: GEV Parameter estimates for Minimum Temperature 
Parameter Estimate 95% Confidence Interval 
Location ()    -20.0487 (0.0391) (-20.1613, -20.0081) 
Scale ( )    0.8454 (0.0272) (0.7921, 0.8987) 
Shape ( )    -0.1291 (0.0273) (-0.1826, -0.0755) 
 
 Table 4.4b: GEV Parameter estimates for Maximum Temperature 
Parameter Estimate 95% Confidence 
Interval 
Location ()    32.2790 (0.0825) (32.1172, 32.4407) 
Scale ( )    1.8592 (0.0590) (1.7435, 1.9748) 
Shape ( )    -0.2118 (0.0278) (-0.2664, -0.1573) 
 
The fitted generalised extreme distribution suggested that the extreme occurrences of both the 
average minimum and maximum temperatures observed in the region are in the Weibull 
family of distributions since   0 . Figures 4.2b present the diagnostic plot for the GEV fit 
for minimum temperature.    
 
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 -22 -21 -20 -19 -18 -17
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 -22 -21 -20 -19 -18 -17
Return Period z
 
Figure 4.2b: Diagnostic plot for the fitted GEV model for minimum Temperature  
From Figure 4.2b, the Q-Q and PP plots are approximately linear, an indication that the 
model is valid is. Thus the GEV model fit the data very well. The density plot shows that the 
normality of the residual is attained.  
 
Figure 4.3b shows the diagnostic plots for the GEV fit for maximum temperature.    
 
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Return Level Model
-22 -19 -16 0.0 0.6
f(z) Empirical
0.0 0.3 -22 -19
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 28 30 32 34 36 38
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 28 30 32 34 36 38
Return Period z
 
Figure 4.3b: Diagnostic plot for the fitted GEV model for maximum Temperature  
From Figure 4.3b, it was concluded that the “GEV model” is appropriate for the data due to 
normality of the residuals depicted by the density plot. Also, the Q-Q and PP plots are linear 
despite the slight deviations.  
After assessing and concluding on the appropriateness of the model, the “return periods” and 
“exceedance probabilities” for the tails of the distribution was estimated. The temperature 
experienced in the region ranged from 19.17 C  to 38.9 C  on a daily basis. The probabilities 
of exceedances for some selected temperatures is shown in Table 4.5b.  
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Return Level Model
28 34 0.0 0.6
f(z) Empirical
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Table 4.5b: Exceedance Probabilities estimates for GEV fit Temperature 
Maximum Temperature  
Temperature 34 35 36 37 38 39 
Exceedance Probs 0.3001 0.1593 0.0713 0.0258 0.0068 0.0011 
Minimum Temperature  
Temperature -22 -21 -20 -19 -18 -17 
Exceedance Probs 0.9993 0.9364 0.5951 0.2180 0.0501 0.0072 
 
From Table 4.5b, it was observed that, temperatures of 36 C  and above are very unlikely to 
be experienced in the region. Also, minimum temperatures of 22 C  to 20 C are likely to be 
observed in the region. We further estimated the return periods for average annual minimum 
and maximum temperature. Table 4.6b presents the return periods for various average 
temperatures. The periods considered in the study were 5, 10, 20, 50 and 100 years. 
Table 4.6b: Return Periods estimates of GEV fit for Temperature 
Maximum Temperature 
Temperature 34.66 35.61 36.38 37.22 37.74 
Return Period (Years) 5 10 20 50 100 
Minimum Temperature 
Temperature -18.93 -18.43 -18.00 -17.49 -17.15 
Return Period (Years) 5 10 20 50 100 
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Negative values are interpreted in absolute value terms 
 
From Table 4.6b, Ashanti region is expected to experience a maximum temperature of  
34.66 C once every five years with a minimum temperature of  18.93 C . Also, a minimum 
temperature of 17.15 C will be experienced in the region once every 100 years. 
4.3.1.2 Rainfall 
Figure 4.4b is the exponential Q-Q plot for maximum Rainfall. 
0 1 2 3 4 5
Theoretical Exponential
 
Figure 4.4b: Exponential Q-Q plot for maximum Rainfall 
From Figure 4.4b, the plot of the maximum rainfall exhibits a roughly linear pattern for the 
plotted points when the “exponential model” was proposed as a “parametric model” to be 
fitted to the data. However, the lower tail of the plot deviates from linearity. A more lighter 
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Rainfall
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tailed distribution will seem feasible as compared to the exponential distribution. A “GEV 
model” was fitted to the maximum rainfall data. The estimated GEV model with the 
associated confidence intervals for the parameters is presented in Table 4.7b.   
 Table 4.7b: GEV parameter estimate for Maximum Rainfall 
Parameter Estimate 95% Confidence 
Interval 
Location ()    63.8614 (1.2261) (59.6011, 65.0635) 
Scale ( )    12.2728 (1.0016) (9.6209, 14.6741) 
Shape ( )    0.2163 (0.0920) (0.2216, 0.6811) 
 
From Table 4.7b, the shape parameter is positive this suggests that the underlying distribution 
belongs to the “Frechet family of distribution”. The diagnostic plots is presented in Figure 
4.5b. 
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 60 80 120 160
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 60 80 100 140
Return Period z
 
Figure 4.5b: Diagnostic plot for the fitted GEV model for maximum Rainfall 
From Figure 4.5b, the PP plots is linear. However, the goodness-of-fit in the Q-Q plot seems 
unconvincing, but the confidence intervals on the return level plot suggest that the model 
departures are not large. Thus, the GEV model is valid. The exceedance probabilities 
exceedances for some selected rainfall readings was then estimated and presented in Table 
4.8b.  
Table 4.8b: Exceedance Probabilities estimates for GEV fit for Maximum Rainfall 
Rainfall (X) in mm  
X 50 100 150 200 250 300 
P(X  x)  0.9740 0.0974 0.0139 0.0035 0.0012 0.001 
 
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f(z) Empirical
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From Table 4.8b, there was 0.974 chance of experiencing a daily rainfall amount of 50 mm in 
the region. The maximum rainfall recorded by the region for over the period of the study was 
167.9 mm. The study estimated 0.002 likelihood of observing a daily rainfall amount of 100 
inches in the region. The return periods was also estimated. 
Table 4.9b presents the return periods for various average rainfall in Accra. 
Table 4.9b: Return Periods estimates of GEV fit for Maximum Rainfall 
Maximum Rainfall 
Rain 85.61 99.44 114.99 139.08 160.59 
Return Period (Years) 5 10 20 50 100 
 
From Table 4.9b, Ashanti region is expected to record 85.61 mm amount of rainfall readings 
daily once every 5 years. A 10 year return period for the maximum rainfall in the region was 
160.59 mm.  
4.3.2 Fitting the Generalized Pareto (GP) Model 
In this section, the generalized Pareto model was fitted to both the temperature and rainfall 
data. 
4.3.2.1 Temperature 
The generalized Pareto model was fitted to the average temperature. The mean residual life 
plot and a threshold stability plot was employed here as well in the selection of an appropriate 
threshold. Figure 4.6b is the mean residual life plot for monthly average minimum (left panel) 
and maximum (right panel) temperatures. 
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-22 -20 -18 28 32 36
Threshold Threshold
 
Figure 4.6b: Mean excess plot for minimum (left panel) and maximum (right panel) 
Temperature 
From Figure 4.6b there is an observed linearity above   34 C  for the maximum average 
monthly temperature. There was also evidence of linearity above   21 C  for the 
minimum temperature. A threshold of 34 C  for the maximum temperature and 21 C for the 
minimum temperature seems feasible. The decreasing nature in both cases indicates that, they 
both have a tail lighter than the exponential distribution. Thus, they both belong to the GP 
distribution with Weibull domain of attraction.  
 
Figures 4.7b and 4.8b present the threshold stability plot of the maximum and minimum 
average temperatures respectively. 
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Mean Excess
0.0 0.5 1.0 1.5 2.0 2.5
Mean Excess
0 1 2 3 4 5
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32.0 32.5 33.0 33.5 34.0 34.5 35.0
Threshold
32.0 32.5 33.0 33.5 34.0 34.5 35.0
Threshold
 
Figure 4.7b: Parameter estimates against threshold for maximum Temperature 
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-22.0 -21.5 -21.0 -20.5 -20.0
Threshold
-22.0 -21.5 -21.0 -20.5 -20.0
Threshold
 
Figure 4.8b: Parameter estimates against threshold for minimum Temperature 
From Figure 4.7b, the parameter estimates for the maximum temperature appear stable 
around   33.5 C . Also, from Figure 4.8b, the threshold stability plot shows stability of the 
parameter estimates around    21 C .    
From the mean threshold plot and the plot of the threshold stability, a threshold of  
  34.5 C  and  19.6 C  were selected for both the maximum and minimum temperatures 
respectively. 
Table 4.10b is the estimate of the GP for the maximum temperature. With the threshold of 
  34.5 C  there were 145 exceedances which constitutes a proportion of 0.2294. 
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-0.7 -12 0
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Table 4.10b: GP parameter estimate for maximum Temperature 
Parameter Estimate 95% Confidence Interval 
Scale ( )    1.5679 (0.1587) (1.2569, 1.8789) 
Shape ( )    -0.3059 (0.0632) (-0.4297, -0.1820) 
 
From Table 4.10b, the shape parameter is negative. This is an indication that the GP 
distribution has a “tail lighter” than the exponential distribution, thus, the GP distribution 
belongs to the “Weibull domain of attraction”. 
Table 4.11b is the estimate of the GP for the average minimum temperature. With the 
threshold of  19.6 C  there were 248 exceedances which constitutes a proportion of 
0.4306. 
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Table 4.11b: GP parameter estimate for Minimum Temperature 
Parameter Estimate 95% Confidence Interval 
Scale ( )    0.9964 (0.0966) (0.8071, 1.1857) 
Shape ( )    -0.2624 (0.0751) (-0.4096, -0.1152) 
 
From Table 4.11b, it was observed that the appropriate distribution for the extreme minimum 
temperature occurrence is the light tailed Pareto distribution. This as a result of the negative 
shape parameter.  
Figures 4.9b and 4.10b are the diagnostic plots of the fitted GP model for the maximum and 
minimum average temperatures.  
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 -19.5 -18.5 -17.5 -16.5
Empirical Model
Density Plot Return Level Plot
-19.5 -18.5 -17.5 0.005 0.050 0.500
Quantile Return Period
 
Figure 4.9b: Diagnostic plot for fitted GP for maximum Temperature 
From Figure 4.9b, the linearity of the Q-Q and PP plots suggest valid goodness-of-fit. 
Furthermore, confidence intervals on the return level plot is an indication that the model 
departures are not large. Thus, the fitted generalized Pareto distribution is appropriate. 
 
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Density Model
0.0 0.6 0.0 0.6
Return Level Empirical
-19.5 -17.0 -19.5 -17.0
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 35 36 37 38
Empirical Model
Density Plot Return Level Plot
35 36 37 38 39 0.01 0.05 0.20 1.00
Quantile Return Period
 
Figure 4.10b: Diagnostic plot for fitted GP for minimum Temperature 
From Figure 4.10b, the Q-Q and PP plots are approximately linear with small deviations as 
depicted by the return level plot. The fitted generalized Pareto distribution is thus appropriate.  
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Density Model
0.0 0.3 0.6 0.0 0.6
Return Level Empirical
35 38 35 37 39
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4.3.2.2 Rainfall 
Figure 4.11b is the mean residual life plot for monthly average maximum rainfall 
readings.
60 80 100 120 140 160
Threshold
 
                  Figure 4.11b: Mean excess plot for maximum Rainfall 
From Figure 4.11b the plot is decreasing with an observed non-linearity above   60 mm. A 
threshold of either   70  may be feasible. The threshold stability plot was also examined. 
This is presented in Figure 4.12b.  
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Mean Excess
0 10 20 30 40
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40 50 60 70 80 90 100
Threshold
40 50 60 70 80 90 100
Threshold
 
            Figure 4.12b: Parameter estimates against threshold for maximum Rainfall 
From Figure 4.12b, threshold of   70  inches was observed to be more feasible. Thus, from 
the mean threshold plot and the plot of the threshold stability, a threshold of   70  was 
selected. For this threshold, there were 74 exceedances which constitutes a proportion of 
0.5068. Table 4.12b is the estimate of the GP for the average maximum rainfall. 
Table 4.12b: GP parameter estimate for Maximum Rainfall 
Parameter Estimate 95% Confidence Interval 
Scale ( )    17.9901 (3.0549) (12.0026, 23.9777) 
Shape ( )    0.0025 (0.1238) (-0.2401, 0.2451) 
 
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Shape Modified Scale
-0.6 -50
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From Table 4.12b, the shape parameter is positive, which is also an indication that, the GP 
distribution has lighter tail than the exponential distribution. 
Figure 4.13b is the diagnostic plot of the fitted GP model for the maximum rainfall. 
Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 80 100 120 140 160
Empirical Model
Density Plot Return Level Plot
80 100 120 140 160 0.02 0.10 0.50 2.00
Quantile Return Period
 
         Figure 4.13b: Diagnostic plot for fitted GP for maximum Rainfall 
From Figure 4.13b, the Q-Q and PP plots had slight deviations from linearity. This is 
confirmed by the “confidence intervals” observed on the “return level plot”. Thus, the 
goodness-of-fit is satisfied and therefore the fitted generalized Pareto distribution is 
appropriate. 
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Density Model
0.00 0.04 0.0 0.6
Return Level Empirical
100 200 100 200
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4.3.3 General Comments 
The extreme distribution of the temperature and rainfall occurrences in Ashanti region was 
explored in this section of the study. Interestingly, the extreme occurrences of temperature 
and rainfall also follows the Weibull and Frechet family of distributions respectively.  
In this region, a maximum and minimum temperatures of 34.66cC and 19cC  was estimated 
to occur once every five years. Also, estimated temperatures of 37.74cC  and 17.15cC is 
predicted to occur in the region once every 100 years. Furthermore, the occurrence of 
maximum temperatures beyond 39cC  every year was very unlikely. There was an estimated 
probabilities of 0.0011 and 0.072 of observing a maximum temperature beyond 39cC  and 
below 17cC  respectively.  
The recorded maximum daily rainfall ever recorded over the study period was 169.9mm. It 
was estimated that a maximum daily rainfall amount of 85.61 mm will be recorded in the 
region every 5 years with an exceedance probability being small.  
4.4 Northern Region 
In this section of the study, the extreme value and the Generalized Pareto models were fitted 
to the extreme occurrence of temperature and rainfall readings in the Northern region.  
4.4.1 Fitting the Extreme Value Model 
The extreme occurrences of temperature and rainfall was explored through the Generalized 
Extreme Value distribution (GEVd). The Exponential QQ-plot was used to have a rough and 
quick confirmation of the plausible fit of the GEVd to the data.  
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4.4.1.1 Temperature 
Figure 4.1c is the exponential Q-Q plots of both minimum (left panel) and maximum (right 
panel) Temperature.  
0 1 2 3 4 5 6 7 0 2 4 6
Theoretical Exponential Theoretical Exponential
 
Figure 4.1c: Exponential Q-Q plot for minimum (left panel) and maximum (right panel) 
Temperature 
From Figure 4.1c, the upper tails of the plot for both the minimum and maximum 
temperatures deviates from linearity pointing to an underlying distribution function with a 
“lighter right tail” than the “exponential distribution”. The generalized extreme value 
distribution was fitted to the data. The GEV models for both monthly maximum and 
minimum temperature were fitted using maximum likelihood. Tables 4.3c and 4.4c present 
the parameter estimates and their corresponding standard errors (in parentheses) for the 
minimum temperature and maximum temperature respectively.  
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Temperature
18 20 22 24
Temperature
32 34 36 38 40 42
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 Table 4.3c: GEV parameter estimate for Minimum Temperature 
Parameter Estimate 95% Confidence 
Interval 
Location ()    -20.6998 (0.0610) (-20.8194, -20.5802) 
Scale ( )    1.2614 (0.0408) (1.1813, 1.3414) 
Shape ( )    -0.2244 (0.0242) (-0.2718, -0.1770) 
 
          Table 4.4c: GEV parameter estimate for Maximum Temperature 
Parameter Estimate 95% Confidence 
Interval 
Location ()    36.088 (0.1393) (35.8146, 36.3606) 
Scale ( )    3.262 (0.09851) (3.0686, 3.4547) 
Shape ( )    -0.474 (0.0169) (-0.5071, -0.4409) 
 
The shape parameter is negative for the distributions of both the minimum and maximum 
temperatures and this suggests that the underlying distribution belongs to the Weibull family 
of distributions.  
Figures 4.2c and 4.3c are the diagnostic plots for the GEV fit for both minimum and 
maximum temperatures respectively.    
 
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 -23 -21 -19 -17
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 -24 -22 -20 -18
Return Period z
 
Figure 4.2c: Diagnostic plot for the fitted GEV model for minimum Temperature 
From Figure 4.2c, deviations were observed from both Q-Q and PP plots. The deviations 
were significant though not that large as depicted by the return level plot. Also, the density 
plot shows that the residuals are approximately normal. Thus, the model is valid.   
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Return Level Model
-24 -20 -16 0.0 0.6
f(z) Empirical
0.0 0.3 -24 -20
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 30 35 40
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 10 20 30 40
Return Period z
 
Figure 4.3c: Diagnostic plot for the fitted GEV model for maximum Temperature in 
Northern 
From Figure 4.3c, the Q-Q plot shows some deviations at the lower tail but appears linear. 
The return level plot shows that these deviations are significant. However, on the average, the 
model is valid. The return periods and exceedance probabilities for the distributions was then 
estimated. Table 4.5c presents the probabilities of exceedances for some selected 
temperatures.  
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Return Level Model
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f(z) Empirical
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Table 4.5c: Exceedance Probabilities estimates for GEV fit for Temperature 
Maximum Temperature  
Temperature 37 38 39 40 41 42 
Exceedance Probs 0.5233 0.3954 0.2690 0.1562 0.0689 0.0159 
Minimum Temperature  
Temperature -22 -21 -20 -19 -18 -17 
Exceedance Probs 0.9201 0.7166 0.4248 0.1820 0.0527 0.0083 
 
From Table 4.5c, it was observed that, there was 0.5233 likelihood of observing a daily 
maximum temperature of 37 C  in the northern region. The chances of observing a 
temperature above 42 C  was very small. Also, there was 0.0083 chance of observing a 
minimum temperature of 17 C  and chances of observing temperature below this was 
unlikely. However, a minimum temperature of 22 C was very likely. Table 4.6b presents the 
return periods for various average temperatures. The periods considered in the study were 5, 
10, 20, 50 and 100 years. 
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Table 4.6c: Return Periods estimates of GEV fit for Temperature 
Maximum Temperature 
Temperature 39.5897 40.6015 41.2861 41.8873 42.1923 
Return Period (Years) 5 10 20 50 100 
Minimum Temperature 
Temperature -19.09 -18.47 -17.97 -17.42 -17.08 
Return Period (Years) 5 10 20 50 100 
Negative values are interpreted in absolute value terms 
From Table 4.6c, the region is expected to experience a maximum temperature of 
39.59 C once every five years with a minimum temperature of  19.1 C  on the average. 
Again, the region will experience a maximum temperature of 42 C  every 100 years.  
4.4.1.2 Rainfall 
The Exponential Q-Q plot of maximum Rainfall for is presented in Figure 4.4c. 
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0 1 2 3 4 5
Theoretical Exponential
 
Figure 4.4c: Exponential Q-Q plot for minimum (left panel) and maximum (right panel) 
Rainfall 
From Figure 4.4c, the plot of the maximum rainfall exhibits a roughly linear pattern for the 
plotted points, despite the slight deviation at the upper tail. The proposed exponential 
distribution thus appears to be inappropriate. The GEV model was rather fitted to the rainfall 
data. The estimated GEV model with the associated confidence intervals for the parameters is 
presented in Table 4.7c.   
  
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Rainfall
60 80 100 120 140
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 Table 4.7c: GEV parameter estimate for Maximum Rainfall 
Parameter Estimate 95% Confidence 
Interval 
Location ()    60.5174 (1.0690) (58.4222, 62.6125) 
Scale ( )    9.7901 (1.0033) (7.8236, 11.7565) 
Shape ( )    0.4774 (0.1191) (0.2440, 0.7109) 
 
From Table 4.7c, the shape parameter is positive this suggests that the underlying distribution 
belongs to the Frechet family of distribution. The diagnostic plot is presented in Figure 4.5c. 
Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200 250
Empirical Model
Return Level Plot Density Plot
1e-01 1e+01 1e+03 60 80 100 120 140
Return Period z
 
Figure 4.5c: Diagnostic plot for the fitted GEV model for maximum Temperature 
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Return Level Model
100 400 0.0 0.6
f(z) Empirical
0.00 0.03 60 120
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From Figure 4.5c, the return level plot indicate that the deviations depicted by both the Q-Q 
and PP plots is significant though not large. The goodness-of-fit therefore seems not 
convincing.  However, we went ahead to estimate the probabilities of exceedances for some 
selected rainfall readings. This is presented in Table 4.8c 
Table 4.8c: Exceedance Probabilities estimates for GEV fit Rainfall 
Rainfall (X) in mm  
X 50 100 150 200 250 300 
P(X  x)  0.9890 0.1002 0.0292 0.0134 0.0076 0.0049 
 
From Table 4.8c, there was 0.989 chance of experiencing a rainfall amount of 50 mm on the 
average.  The chances of recording an average amount of rainfall greater than 50 mm was 
very small. The return periods was also estimated. Table 4.9c presents the return periods for 
various average rainfall readings. 
Table 4.9c: Return Periods estimates of GEV fit for Rainfall 
Maximum Rainfall 
Rain 81.98 100.06 124.68 172.11 224.369 
Return Period (Years) 5 10 20 50 100 
 
From Table 4.9c, Northern region is expected to record 81.98 mm amount of rainfall readings 
once every 5 years. A 100 year return period for the maximum rainfall in the region is 
224.37mm.  
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4.3.2 Fitting the Generalized Pareto (GP) Model 
The generalized Pareto model was fitted to both the temperature and rainfall data in this 
section of the study 
4.3.2.1 Temperature 
The generalized Pareto model was fitted to the average temperature. The mean residual life 
plot and a threshold stability plot was employed also employed here as well in the selection 
of an appropriate threshold. Figure 4.6c is the mean residual life plot for monthly average 
minimum (left panel) and maximum (right panel) temperatures.  
 
 
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-24 -22 -20 -18 10 20 30 40
Threshold Threshold
 
Figure 4.6c: Mean excess plot for minimum (left panel) and maximum (right panel) 
Temperature 
From Figure 4.6c there is an observed linearity above   30 C  for the maximum average 
o
monthly temperature. There was also evidence of linearity above   21 C  for the 
minimum temperature. A threshold of 34 C  for the maximum temperature and 20 C for the 
minimum temperature seems feasible.  
Figures 4.7c and 4.8c present the threshold stability plot of the maximum and minimum 
average temperatures respectively. 
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Mean Excess
0 1 2 3 4
Mean Excess
0 5 10 15 20 25
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38.0 38.5 39.0 39.5 40.0 40.5 41.0
Threshold
38.0 38.5 39.0 39.5 40.0 40.5 41.0
Threshold
 
Figure 4.7c: Parameter estimates against threshold for maximum Temperature 
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-0.6 0 30
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-22.0 -21.5 -21.0 -20.5 -20.0
Threshold
-22.0 -21.5 -21.0 -20.5 -20.0
Threshold
 
Figure 4.8c: Parameter estimates against threshold for minimum Temperature 
From Figure 4.7c, the parameter estimates for the maximum temperature appear stable 
around   39 C . Also, from Figure 4.8c, the threshold stability plot shows stability of the 
parameter estimates around    21 C .    
From the mean threshold plot and the plot of the threshold stability, a threshold of  
  40.5 C  and   20.7 C  were selected for both the maximum and minimum 
temperatures respectively. 
Table 4.10c is the estimate of the GP for the average maximum temperature. With the 
threshold of   40.5 C  there were 65 exceedances which constitutes a proportion of 0.1055. 
Table 4.10c: GP parameter estimate for maximum Temperature 
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Parameter Estimate 95% Confidence Interval 
Scale ( )    0.8986 (0.1535) (0.5977, 1.1994) 
Shape ( )    -0.2793 (0.1225) (-0.5193, -0.0393) 
 
From Table 4.10c, the shape parameter is negative and this indicates that, the GP distribution 
has a tail lighter than the exponential distribution, thus, the GP distribution belongs to the 
Weibull domain of attraction.  
Table 4.11c is the estimate of the GP for the average minimum temperature. With the 
threshold of   20.7 C  there were 343 exceedances which constitutes a proportion of 
0.6765. 
 
Table 4.11c: GP parameter estimate for Minimum Temperature  
Parameter Estimate 95% Confidence Interval 
Scale ( )    1.3100 (0.1148) (1.0849, 1.5350) 
Shape ( )    -0.1771 (0.0697) (-0.3137, -0.0405) 
 
From Table 4.11c, the shape parameter is negative. The negativity of the shape indicates that, 
the GP distribution has a tail lighter than the exponential distribution, thus, the GP 
distribution belongs to the Weibull domain of attraction.  
Figures 4.9c and 4.10c are the diagnostic plots of the fitted GP model for the maximum and 
minimum average temperatures.  
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 40.5 41.0 41.5 42.0 42.5
Empirical Model
Density Plot Return Level Plot
41.0 41.5 42.0 42.5 43.0 0.02 0.10 0.50 2.00
Quantile Return Period
 
Figure 4.9c: Diagnostic plot for fitted GP for maximum Temperature 
From Figure 4.9c, the linearity of the Q-Q and PP plots suggest valid goodness-of-fit. 
Furthermore, confidence intervals on the return level plot is an indication that the model 
departures are not large. Thus, the fitted generalized Pareto distribution is appropriate. 
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Density Model
0.0 0.6 0.0 0.6
Return Level Empirical
40.5 42.5 40.5 42.5
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Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 -20 -19 -18 -17 -16
Empirical Model
Density Plot Return Level Plot
-20 -19 -18 -17 0.005 0.050 0.500
Quantile Return Period
 
Figure 4.10c: Diagnostic plot for fitted GP for minimum Temperature 
From Figure 4.10c, the linearity of the Q-Q and PP plots suggest valid goodness-of-fit. 
Furthermore, confidence intervals on the return level plot is an indication that the model 
departures are not large. Thus, the fitted generalized Pareto distribution is appropriate. 
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Density Model
0.0 0.4 0.0 0.6
Return Level Empirical
-20 -16 -20 -17
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4.3.2.2 Rainfall 
Figure 4.11b is the mean residual life plot for monthly average maximum rainfall 
readings.
60 80 100 120 140
Threshold
 
Figure 4.11c: Mean excess plot for maximum Rainfall 
From Figure 4.11c the plot is decreasing with an observed non-linearity above   70 inches. 
A threshold of either   70  may be feasible. The threshold stability plot was also examined. 
This is presented in Figure 4.12c.  
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Mean Excess
0 5 10 15 20 25
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40 50 60 70 80 90 100
Threshold
40 50 60 70 80 90 100
Threshold
 
Figure 4.12c: Parameter estimates against threshold for maximum Rainfall 
From Figure 4.12c, threshold of   70  inches was observed to be more feasible. Thus, from 
the mean threshold plot and the plot of the threshold stability, a threshold of   60  was 
selected. For this threshold, there were 81 exceedances which constitutes a proportion of 
0.6136. Table 4.12c is the estimate of the GP for the average maximum rainfall.  
Table 4.12c: GP parameter estimate for Maximum Rainfall  
Parameter Estimate 95% Confidence Interval 
Scale ( )    26.8341 (3.9725) (19.0481, 34.6201) 
Shape ( )    -0.2129 (0.1006) (-0.4101, -0.0157) 
 
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Shape Modified Scale
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From Table 4.12c, it was observed that the appropriate distribution for the extreme minimum 
temperature occurrence is the to heavy tailed Pareto distribution. This as a result of the 
positive shape parameter. Figure 4.13c is the diagnostic plot of the fitted GP model for the 
maximum rainfall.  
Probability Plot Quantile Plot
0.0 0.2 0.4 0.6 0.8 1.0 60 80 100 120 140
Empirical Model
Density Plot Return Level Plot
60 80 100 120 140 0.02 0.10 0.50 2.00
Quantile Return Period
 
Figure 4.13c: Diagnostic plot for fitted GP for maximum Rainfall 
From Figure 4.13c, the Q-Q and PP plots had slight deviations from linearity. This is 
confirmed by the confidence intervals observed on the return level plot. Thus, the fitted 
generalized Pareto distribution is appropriate. 
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Density Model
0.000 0.030 0.0 0.6
Return Level Empirical
60 120 60 120
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4.4.3 General Comments 
In this section, the extreme distribution of the temperature and rainfall occurrences in 
Northern region was explored. Again, the extreme occurrences of temperature and rainfall 
had the Weibull and Frechet family of distributions respectively.  
The minimum and maximum temperatures of this region was 19.6cC  and 42.8cC  
respectively. A maximum and minimum temperatures of 39.6oC and 19.1cC  was estimated 
to occur once every five years with probability of exceeding such temperatures as 0.1562 and 
0.1820 respectively. The exceedance for a maximum and minimum temperatures of   42cC  
and 17oC  were respectively 0.0159 and 0.0083. Furthermore, it was estimated that an 
amount of 81.98 mm of rainfall will be recorded in the region once every 5 years. Also, 
100.1, 124.68, and 172.11 mm were the estimated daily rainfall readings to be recorded in the 
once every 10, 20 and 50 years respectively. 
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CHAPTER FIVE 
CONCLUSIONS AND RECOMMENDATION 
In this chapter, we draw conclusions from the extreme value analysis of climate data (i.e 
Temperature and Rainfall) from previous chapters. We end this chapter with some 
recommendations and raises a few issues that call for further investigation. The aim of the 
study was to; 
1. find the appropriate distribution for the tails of the distributions of rainfall and 
temperature 
2. find out how extremes of rainfall relates to the extremes of temperature. 
3. find the exceedance probabilities for selected levels of rainfall and temperature 
4. obtain the return periods of extreme rainfall and temperatures and their corresponding 
return levels  for rainfall and temperature 
The study was based on secondary data obtained from the Ghana meteorological Agency 
office in Accra. Daily temperature and rainfall readings for the period January 1960 to 
December 2012 was obtained. The study focused on three key regions which takes care of the 
Southern (Greater Accra), Central (Ashanti Region) and Northern (Northern region) sectors 
of the country. 
Northern region recorded the highest temperature 42.8oC  as compared to 38.9oC  recorded 
by both Ashanti and Greater Accra Region. 
It was observed that the extreme occurrences of temperature and rainfall can be modelled 
using Weibull and Frechet family of distributions in Ghana. 
A maximum temperature of 34.7cC , 34.66oC , and 39.6oC  was predicted to occur in Accra, 
Ashanti and Northern regions respectively once every five years. The chances of exceeding a 
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temperature of  39cC  in Accra was estimated as 0.001 with likelihood of exceeding a 
minimum temperature of 22oC  being 0.014.  
Also, the likelihood of exceeding a maximum and minimum temperatures of 39cC  and 
17cC for Ashanti are 0.0011 and 0.072 respectively. In addition, the exceedance probabilities 
for maximum ( 42cC ) and minimum (17oC ) temperatures for Northern region were 0.0159 
and 0.0083 respectively.  
Furthermore, it was estimated that an amount of 88.38mm, 85.61mm, and 81.98mm will be 
recorded in Accra, Ashanti, and Northern regions once every 5 years respectively.  
RECOMMENDATIONS 
As a result of the findings of this study, the following are recommended: 
1. Combining time series with EVT for the case of applying EVT to climate data might 
be of interest. 
2. It may be of interest to look at non-stationary extremes 
3. Interested organizations can employ this technique as one of the means of forecasting 
extreme occurrences. 
4. Multivariate Extreme Value Analysis, which include other factors such as relative 
humidity, wind speed should be employed to further investigate this problem. 
5. Researchers are encouraged to look into other areas of EVT applications such as using 
Bayesian approach or extreme quantiles to further investigate this problem in Ghana. 
 
 
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APPENDIX A 
0 100 200 300 400 500 600
Days
 
Figure 1A: Scatter plot for Minimum Temperature in the Greater Accra Region 
 
0 100 200 300 400 500
Days
 
Figure 2A: Scatter plot for Minimum Temperature in the Ashanti Region 
 
 
113 
 
Ashanti Temperature (Min) Greater Accra Temperature (Min)
17 18 19 20 21 22 18 20 22 24
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0 100 200 300 400 500
Days
 
Figure 3A: Scatter plot for Minimum Temperature in the Northern Region 
 
0 100 200 300 400 500 600
Days
 
Figure 4A: Scatter plot for Maximum Temperature in the Greater Accra Region 
114 
 
Greater Accra Temperature (Max) Northern Temperature (Min)
28 30 32 34 36 38 18 20 22 24
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0 100 200 300 400 500 600
Days
 
Figure 5A: Scatter plot for Maximum Temperature in the Ashanti Region 
 
0 100 200 300 400 500 600
Days
 
Figure 6A: Scatter plot for Maximum Temperature in the Northern Region 
 
115 
 
Northern Temperature (Max) Ashanti Temperature (Max)
32 34 36 38 40 42 28 30 32 34 36 38
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0 100 200 300 400 500 600
Days
 
Figure 7A: Scatter plot for Maximum Rainfall in the Greater Accra Region 
 
0 500 1000 1500
Days
 
Figure 8A: Scatter plot for Maximum Rainfall in the Ashanti Region 
 
116 
 
Ashanti Rainfall (Max) Greater Accra Rainfall (Max)
0 50 100 150 0 50 100 150 200 250
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0 100 200 300 400 500 600
Days
 
Figure 8A: Scatter plot for Maximum Rainfall in the Northern Region 
 
 
 
 
117 
 
Northern Rainfall (Max)
0 20 40 60 80 120