Hindawi Publishing Corporation
Journal of Probability and Statistics
Volume 2013, Article ID 364705, 7 pages
http://dx.doi.org/10.1155/2013/364705
Research Article
Bayesian and Non-Bayesian Inference for Survival Data
Using Generalised Exponential Distribution
Chris Bambey Guure and Samuel Bosomprah
Department of Biostatistics, School of Public Health, University of Ghana, Legon Accra, Ghana
Correspondence should be addressed to Chris Bambey Guure; chris@chrisguure.com
Received 29 April 2013; Revised 5 August 2013; Accepted 8 August 2013
Academic Editor: Zhidong Bai
Copyright © 2013 C. B. Guure and S. Bosomprah. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
A two-parameter lifetime distribution was introduced by Kundu and Gupta known as generalised exponential distribution. This
distribution has been touted to be an alternative to the well-known 2-parameter Weibull and gamma distributions. We seek to
determine the parameters and the survival function of this distribution. The survival function determines the probability that a
unit under investigation will survive beyond a certain specified time, say, (𝑥). We have employed different data sets to estimate the
parameters and see how well the distribution can be used to analyse survival data. A comparison is made about the estimators used
in this study. Standard errors of the estimators are determined and used for the comparisons. A simulation study is also carried out,
and the mean squared errors and absolute bias are obtained for the purpose of comparison.
1. Introduction increasing and decreasing failure rates depending on the
shape parameter.
As the need grows for conceptualization, formalization, and Some research has been done to compare MLE to that
abstraction in biology, so too does mathematics’ relevance of the Bayesian approach in estimating the survival function
to the field according to Fagerström et al. [1]. Mathematics and the parameters of the Weibull distribution which are
is particularly important for analysing and characterizing similar to the generalised exponential distribution. Amongst
random variation, for example, size and weight of individuals others, Sinha [9] determined the Bayes estimates of the
in populations, their sensitivity to chemicals, and time-to- reliability function and the hazard rate of the Weibull failure
event cases, such as the amount of time an individual needs time distribution by employing only squared error loss
to recover from illness. The frequency distribution of such function; Abdel-Wahid and Winterbottom [10] studied the
data is a major factor determining the type of statistical approximate Bayesian estimates for the Weibull reliability
analysis that can be validly carried out on any data set. Many function and hazard rate from censored data by employing
widely used statistical methods, such as ANOVA (analysis a new method that has the potential of reducing the number
of variance) and regression analysis, require that the data of terms in Lindley procedure; Guure and Ibrahim [11] did
be normally distributed, but only rarely is the frequency some studies onBayesian analysis of the survival function and
distribution of data tested when these techniques are used failure rate ofWeibull distributionwith censored data. Huang
(Limpert et al.) [2]. and Wu [12] considered Bayesian estimation and prediction
Gupta and Kundu [3] recently proposed the two- for Weibull model with progressive censoring. Similar work
parameter generalised exponential distribution (𝐺𝐸) as an can be seen in Guure et al. [13], Zellner [14], Guure et al. [15].
alternative to the lognormal, gamma, and Weibull distribu- Al-Aboud [16], Al-Athari [17], and Pandey et al. [18].
tions and did some studies on its properties. Some references Maximum likelihood method is one of the most popular
on 𝐺𝐸 distribution are Raqab [4], Raqab and Ahsanullah estimation techniques for many distributions. According
[5], Zheng [6], and Kundu and Gupta [7]. According to to the classical statistician, maximum likelihood has some
Gupta and Kundu [8], the two-parameter 𝐺𝐸(𝜃, 𝑝) can have outstanding properties. We have in this paper compared
2 Journal of Probability and Statistics
the maximum likelihood estimator and that of the Bayesian prior distribution is most often than not based on the type of
estimators to determine the best method that can be used prior information that is available to us. When we have little
to estimate the parameters of the generalised exponential or no information about the parameter, a noninformative
distribution. prior should be used else an informative prior. In analysing
The remaining part of the paper is arranged as follows. data from medical, engineering, or biological studies, it is
Section 2 contains materials and methods that deal with possible to obtain information with respect to similar studies
the derivative of the parameters under maximum likelihood in the past, and if that is even unattainable, information
estimator and that of the Bayesian estimators. Section 3 deals from an expert could be modelled to fit an appropriate prior
with analysis of some real data sets, followed by a simulation distribution. This can be referred to as prior elicitation.
study in Section 4. Results and discussion are in Section 5, We let the two unknown parameters take on the gamma
and then Section 6 is the conclusion. prior distributions by assuming that the hyper parameters are
all known and greater than zero, that is, 𝑎, 𝑏, 𝑐, 𝑑 > 0. The
2. Materials and Methods gamma prior is assumed for this distribution because boththe scale and shape parameters are greater than zero:
Let (𝑡 , . . . , 𝑡 ) be the set of 𝑛 random lifetimes with respect to
1 𝑛
the generalised exponential distribution, with 𝑝 and 𝜃 as the V 𝑏−1(𝜃) ∝ 𝜃 exp (−𝜃𝑎) , 𝜃, 𝑎, 𝑏 > 0,
parameters, where 𝜃 is the scale parameter and 𝑝 is the shape 1 (6)
parameter. The cumulative distribution function (𝑐𝑑𝑓), the V 𝑑−1(𝑝) ∝ 𝑝 exp (−𝑝𝑐) , 𝑝, 𝑐, 𝑑 > 0.
probability density function (𝑝𝑑𝑓), and the survival function 2
𝑆(𝑡) are given, respectively, as
Bayesian inference is based on the posterior distribution
exp 𝑝 (1) which is obtained by dividing the joint density function to the𝐹 (𝑡; 𝜃, 𝑝) = {1 − (−𝜃𝑡)} 𝜃, 𝑝, 𝑡 > 0,
marginal distribution function as given below:
the density function:
V (𝜃) V (𝑝) 𝐿 (𝑡 ; 𝜃, 𝑝)
1 2 𝑖
𝑓 (𝑡; 𝜃, 𝑝) = 𝑝𝜃{1 − exp 𝑝−1 ∗(−𝜃𝑡)} exp (−𝜃𝑡) , (2) 𝜋 (𝜃, 𝑝𝑡 ) ∝ .𝑖 ∞ ∞ (7)
∫ ∫ V (𝜃) V (𝑝) 𝐿 (𝑡 ; 𝜃, 𝑝) 𝑑𝜃𝑑𝑝
0 0 1 2 𝑖
and the survival function:
exp 𝑝 (3) Due to the complex nature of the posterior distribution𝑆 (𝑡; 𝜃, 𝑝) = 1 − {1 − (−𝜃𝑡)} . given in (7), Lindley approximation is employed in order to
Let the 𝐺𝐸 distribution with the shape parameter and the estimate the unknown parameters.𝑝
scale parameter be denoted by . The Bayes estimator is considered under two loss func-𝜃 𝐺𝐸(𝜃, 𝑝)
tions. Since in drawing conclusions about the survival or
duration of a living organism, an overestimation could be
2.1. Maximum Likelihood Estimation. Since (𝑡 , . . . , 𝑡 ) is the
1 𝑛 more detrimental to underestimation or vice versa, we have
set of 𝑛 random lifetimes from the generalised exponential considered both asymmetric (general entropy) loss function
distribution with parameters 𝜃 and 𝑝. and symmetric (squared error) loss function.
The likelihood function is
𝑛
𝑝−1 2.2.1. Lindley Approximation. Lindley [19] proposed an
𝐿 (𝑡 ; 𝜃, 𝑝) = ∏[𝑝𝜃{1 − exp (−𝜃𝑡)} exp (−𝜃𝑡)] .
𝑖 (4) approximation for a ratio of integral of the form
𝑖=1
To obtain the equations for the unknown parameters, we
take the log of (4) and partially different with respect to the ∫𝜔 (𝛼) exp {ℓ (𝛼)} 𝑑𝛼 , (8)
unknown parameters. By employing an iterative procedure ∫ V (𝛼) exp {ℓ (𝛼)} 𝑑𝛼
such as Newton-Raphson, the estimates of the scale and
shape parameters can easily be obtained. Readers can refer where ℓ(𝛼) is the log likelihood and 𝜔(𝛼), V(𝛼) are arbitrary
to Kundu and Gupta [7] for details. functions of 𝛼, V(𝛼) is the prior distribution for 𝛼, and 𝜔(𝛼) =
Therefore, the survival function can be obtained as
𝑢(𝛼) ⋅ V(𝛼) with 𝑢(𝛼) being some function of interest as seen
in (8).
𝑝
̂
𝑆 (𝑡) = 1 − {1 − exp (−𝜃𝑡)} , (5) The posterior expectation according to Sinha [9] is
where ̂(𝜃) and (𝑝) are the maximum likelihood estimates of
the parameters. ∫ 𝑢 (𝛼) exp {ℓ (𝛼) + 𝜌 (𝛼)} 𝑑𝛼𝐸 {𝑢 (𝛼) | 𝑥} = , (9)
∫ exp {ℓ (𝛼) + 𝜌 (𝛼)} 𝑑𝛼
2.2. Bayesian Inference. In Bayesian analysis, the parameter
of interest is always considered to be a random variable with where 𝜌(𝛼) = log{V(𝛼)} and ℓ(𝛼) represent the log-likelihood
a prior distribution. The prior distribution is the distribution function. Considering the Bayesian estimator under the
of the parameter before any data is observed.The selection of squared error loss function, which is the posterior mean, the
Journal of Probability and Statistics 3
posterior expectation can be approximated asymptotically 2.2.2. General Entropy Loss Function. The general entropy
with respect to the two parameters by (10): loss function is a generalisation of the entropy loss function.
The Bayes estimator ?̂? of 𝛼 under the general entropy loss
𝐵𝐺
1 is
̂
?̂? = 𝑢 (𝜃, 𝑝) + [(𝑢 𝜎 ) + (𝑢 𝜎 )] + 𝑢 𝜌 𝜎
11 11 22 22 1 1 11
2
(10) −1/𝑘−𝑘?̂? = [𝐸 (𝛼 )] , (15)
𝐵𝐺 𝛼
1 2 2
+ 𝑢 𝜌 𝜎 + [(ℓ 𝑢 𝜎 ) + (ℓ 𝑢 𝜎 )] ,
2 2 22 30 1 11 03 2 22
2 provided𝐸 (⋅) exists and is finite.The Bayes estimator for this
𝛼
loss function with respect to the parameters and the survival
𝜕𝑢
𝑢 = 𝜃, 𝑢 = = 1, 𝑢 = 0,
1 11 function are
𝜕𝜃
−𝑘 −𝑘
𝜕𝑢 𝐸 {[(𝜃) , (𝑝) ]}
𝑢 = 𝑝, 𝑢 = = 1, 𝑢 = 0,
2 22
𝜕𝑝
−𝑘 −𝑘
(11) ∬𝑢[(𝜃) , (𝑝) ] V (𝜃) V (𝑝) 𝐿 (𝑡 ; 𝜃, 𝑝) 𝑑𝜃 𝑑𝑝
𝜌 = ln V 1 2 𝑖(𝜃) + ln V (𝑝) ,
1 2 = ,
∬V (𝜃) V (𝑝) 𝐿 (𝑡 ; 𝜃, 𝑝) 𝑑𝜃 𝑑𝑝
1 2 𝑖
𝑏 − 1 𝑑 − 1
𝜌 = − 𝑎, 𝜌 = − 𝑐,
1 2 −𝑘
𝜃 𝑝 𝐸 {[(𝑆) ]}
−1 −1
−𝑘
𝜎 = (−ℓ ) , 𝜎 = (−ℓ ) . 𝑝
11 20 22 02 ∬𝑢[1 − {1 − exp (−𝜃𝑡)} ] V (𝜃)V (𝑝) 𝐿 (𝑡 ; 𝜃, 𝑝) 𝑑𝜃 𝑑𝑝
1 2 𝑖
= .
The second and third derivatives with respect to the scale and ∬V (𝜃) V (𝑝) 𝐿 (𝑡 ; 𝜃, 𝑝) 𝑑𝜃 𝑑𝑝1 2 𝑖
shape parameters are (16)
2 exp A similar Lindley approach is used for the general entropy𝑛𝑛 (𝑝 − 1) [(𝑡 ) (−𝜃𝑡 )]𝑖 𝑖 loss function as in the squared error loss function, with
ℓ = − −∑
20 2
𝜃 (1 − exp (−𝜃𝑡 ))
𝑖=1 𝑖
−𝑘 𝜕𝑢 −𝑘−1
𝑢 = (𝜃) , 𝑢 = = −𝑘(𝜃) ,
2 2 1
𝑛
(𝑝 − 1) [(𝑡 ) (exp (−𝜃𝑡 )) ] 𝜕𝜃
𝑖 𝑖
−∑ ,
2 2
𝑖=1 (1 − exp (−𝜃𝑡 )) 𝜕 𝑢𝑖 2 −𝑘−2
𝑢 = = − (−𝑘 − 𝑘) (𝜃) ,
11 2
𝜕(𝜃)
3
𝑛
2𝑛 (𝑝 − 1) [(𝑡 ) exp (−𝜃𝑡 )]𝑖 𝑖 (17)
ℓ = +∑
30 3 exp −𝑘 𝜕𝑢 −𝑘−1𝜃 (1 − (−𝜃𝑡 ))
𝑖 𝑢 = (𝑝) , 𝑢 = = −𝑘(𝑝) ,𝑖=1 2
𝜕𝑝
3 2
𝑛
3 (𝑝 − 1) [(𝑡 ) (exp (−𝜃𝑡 )) ] (12) 2
𝑖 𝑖 𝜕 𝑢 2 −𝑘−2
+∑
2 𝑢 = = − (−𝑘 − 𝑘) (𝑝) .
𝑖=1 (1 − exp 22 2(−𝜃𝑡 ))𝑖 𝜕(𝑝)
3 3
𝑛
2 (𝑝 − 1) [(𝑡 ) (exp (−𝜃𝑡 )) ]
𝑖 𝑖 For the general entropy loss function, the posterior expec-
+∑ ,
exp 3 tation according to Lindley [19] can be approximated by𝑖=1 (1 − (−𝜃𝑡 ))𝑖 using (18). The survival function is similarly obtained by
𝑛 substituting (14) into (18):
ℓ = − ,
02 2
𝑝
1
̂
?̂? = {𝑢 (𝜃, 𝑝) + [(𝑢 𝜎 ) + (𝑢 𝜎 )]
11 11 22 22
2𝑛 2
ℓ = .
03 3
𝑝
+ 𝑢 𝜌 𝜎 + 𝑢 𝜌 𝜎
1 1 11 2 2 22 (18)
To estimate the survival function, under the squared error −1/𝑘
1
loss function, we let 2 2+ [(ℓ 𝑢 𝜎 ) + (ℓ 𝑢 𝜎 )]} .30 1 11 03 2 22
2
𝑢 (𝑆) = 1 − {1 − exp 𝑝(−𝜃𝑡)} , (13) 3. Real Data Analysis
where 3.1. Example 1. We analyse two data sets in this section which
we have considered to be relatively small and moderate for
2
𝜕𝑢 (𝑆) 𝜕 𝑢 (𝑆) illustration and comparison purposes. The data is obtained
𝑢 = , 𝑢 = ,
1 11 2
𝜕𝜃 𝜕𝜃 from Lawless [20]. The data represent survival times for two
(14) groups of laboratory mice, all of which were exposed to a
2
𝜕𝑢 (𝑆) 𝜕 𝑢 (𝑆)
𝑢 = , 𝑢 = . fixed dose of radiation at an age of 5 to 6. The first group of
2 22 2
𝜕𝑝 𝜕𝑝 mice lived in a conventional lab environment, and the second
4 Journal of Probability and Statistics
group was kept in a germ-free environment. The cause of Table 1: Average parameters estimates and their corresponding
death for each mouse was assigned after autopsy to be one of standard errors (𝑛 = 72).
three things: thymic lymphoma (𝐶 ), reticulum cell sarcoma
1
(𝐶 ), or other causes (𝐶 ). The mice all died by the end of the par MLE BS BG (𝑘 = 0.6) BG (𝑘 = −0.6)
2 3
experiment, so there is no censoring. For the purpose of our 0.016962 0.016958 0.016773 0.016911̂𝜃
study, we have considered the first set of data from the control (0.001999) (0.001998) (0.001977) (0.001993)
groupwhich seem relatively small with 𝑛 = 22 and the second 2.474104 2.474104 2.446856 2.467235𝑝
which seem relatively moderate from the third set of data of (0.491576) (0.491576) (0.488365) (0.490766)
the germ-free group with 𝑛 = 38.
Using the iterative procedure suggested from the begin- Table 2: Standard errors of the survival function.
ning of this paper, theMLEs of ̂𝜃 and 𝑝 for the relatively small Est 𝑛 = 22 𝑛 = 38 𝑛 = 72
samples are 0.015024 and 36.935530, respectively, with their MLE 0.517012 0.467909 0.529970
corresponding standard errors as 0.003203 and 8.374682. BS 0.543902 0.471755 0.530072
The relatively moderate samples have ̂𝜃 = 0.003735 and BG (𝑘 = 0.6) 0.556815 0.480653 0.531440
𝑝 = 8.624091, with their corresponding standard errors as
0.000606 and 1.999012. BG (𝑘 = −0.6) 0.547363 0.474025 0.530418
Since we do not have any prior information on the hyper-
parameters, we assume 𝑎 = 𝑏 = 𝑐 = 𝑑 = 0.0001. This makes regimen 6.6 corresponds to 4.4 × 106 bacillary units per
the priors proper on ̂𝜃 and𝑝 and the corresponding posteriors 0.5mL (log 4.4 × 106 = 6.6). Corresponding to regimen 6.6,
also proper. there were 72 observations.
When we compute the Bayes estimators with squared Observing from Table 1, it is evident that the estima-
error loss of ̂𝜃 and 𝑝 under the relatively small sample tor with the smallest parameter estimate and having a
sizes, the following parameters estimates and standard errors corresponding smaller standard error is Bayesian with the
are obtained, respectively, 0.014605, 36.935530 and 0.003114, generalised entropy loss function. This occurred for both
8.374682. For the moderate samples, we have ̂ parameters with a positive loss parameter, that is, 0.6.𝜃 = 0.003716
with a standard error of 0.000603. The with a The importance of the survival function cannot be𝑝 = 8.624091
standard error of 1.999012. ignored; therefore, the correctness of its estimate is very
Computing the Bayes estimates of ̂ and and their crucial to both biological and medical studies. As clearly𝜃 𝑝
corresponding standard errors under the general entropy presented in Table 2, the estimator with the smallest stan-
loss functions with the loss parameter being , we have dard error under all the samples is the classical maximum0.6
0.014270, 35.630480 and 0.003042, 8.096443. With the loss likelihood estimator; therefore, MLE is preferred as a better
parameter being −0.6, we have 0.014517, 36.600361 and estimator to the others.
0.003095, 8.303224, respectively. Comparing all the estimators, it is clear from the resultsthat Bayes estimator under general entropy loss functionwith
Considering the Bayes estimates on the relatively mod- the loss parameter of +0.6 has the smallest standard error
erate samples of ̂𝜃 and 𝑝 and their corresponding standard and estimate for both the shape parameter (𝑝) and the scale
errors under the general entropy loss functions with the parameter ̂(𝜃).
loss parameter being 0.6, we have 0.003646, 8.445543 and
0.000591, 1.970048. With the loss parameter being −0.6, we
have 0.003698, 8.578749 and 0.00600, 1.991657, respectively. 4. Simulation Study
Observably, the Bayes estimator under squared error Since it is difficult to compare the performance of the estima-
loss for the shape parameter (𝑝) has the same estimate and tors theoretically and also to validate the real data employed
standard error as compared to that of the classical maximum in this paper, we have performed extensive simulations to
likelihood estimator but with the scale parameter ̂(𝜃), and compare the estimators through mean squared errors and
Bayes under squared error has a smaller standard error in absolute biases by employing different sample sizes with
juxtaposition to MLE. different parameter values.
We have considered in the simulation study a sample
3.2. Example 2. In this section, we analyse another real data size of 𝑛 = 25, 50, and 100, which is representative of
set which we have considered to be relatively large and small, moderate, and large data sets.The following steps were
complete for illustrative purpose. The data originates from employed to generate the data.
Bjerkedal [21]. The data represents the survival times of The generation of 𝐺𝐸(𝜃, 𝑝) is simple as stated in Gupta
guinea pigs injected with different doses of tubercle bacilli. and Kundu [8]. If 𝑈 follows a uniform distribution in the
It is known that guinea pigs have high susceptibility to that interval [0, 1], then ln (1/𝑝)𝑌 = (− (1 −𝑈 )/𝜃) follows 𝐺𝐸(𝜃, 𝑝).
of human tuberculosis, and this informs our decision to use Consequently, with a very good uniform random number
the data in our study. Here, our primary concern is with the generator, the generation of𝐺𝐸 randomdeviate is immediate.
animals in the same cage that were under the same regimen. A lifetime 𝑇 is generated from the sample sizes indicated
The regimennumber is the common logarithmof the number above from the 𝐺𝐸 distribution which represents failure of
of bacillary units in 0.5mL of challenge solution; that is, the product or unit. The values of the assumed actual shape
Journal of Probability and Statistics 5
Table 3: Average mean squared errors and absolute biases of ̂(𝜃).
𝑛 𝑝 = 0.8 𝑝 = 1.6 𝑝 = 2.2
ML 0.117259 (0.255219) 0.077502 (0.206616) 0.062024 (0.190155)
25 BS 0.117108 (0.255095) 0.076553 (0.205513) 0.060763 (0.188471)
BG (𝑘 = 0.6) 0.118176 (0.251456) 0.075759 (0.208706) 0.060024 (0.185077)
BG (𝑘 = −0.6) 0.108193 (0.246358) 0.075167 (0.203752) 0.059216 (0.179142)
ML 0.048628 (0.170036) 0.031950 (0.136181) 0.027519 (0.130563)
50 BS 0.048623 (0.170024) 0.031869 (0.136036) 0.027383 (0.130287)
BG (𝑘 = 0.6) 0.047058 (0.163792) 0.032612 (0.139630) 0.027723 (0.130121)
BG (𝑘 = −0.6) 0.045906 (0.163408) 0.033207 (0.159536) 0.027145 (0.127761)
ML 0.023911 (0.119557) 0.015292 (0.097709) 0.012631 (0.088584)
100 BS 0.023910 (0.119556) 0.015283 (0.097685) 0.012616 (0.088533)
BG (𝑘 = 0.6) 0.021514 (0.115652) 0.014627 (0.094909) 0.012650 (0.088935)
BG (𝑘 = −0.6) 0.020102 (0.113236) 0.014052 (0.092336) 0.013054 (0.089552)
Table 4: Average mean squared errors and absolute biases of (𝑝).
𝑛 𝑝 = 0.8 𝑝 = 1.6 𝑝 = 2.2
ML 0.076755 (0.191499) 0.490147 (0.476051) 0.743582 (0.661779)
25 BS 0.076755 (0.191499) 0.490147 (0.476051) 0.743582 (0.661779)
BG (𝑘 = 0.6) 0.065099 (0.188381) 0.553240 (0.462777) 0.910514 (0.683827)
BG (𝑘 = −0.6) 0.062020 (0.182668) 0.481165 (0.451018) 0.898282 (0.647155)
ML 0.025391 (0.118187) 0.147209 (0.283414) 0.364344 (0.445543)
50 BS 0.025391 (0.118187) 0.147209 (0.283414) 0.364344 (0.445543)
BG (𝑘 = 0.6) 0.027027 (0.123510) 0.149940 (0.288091) 0.368843 (0.438096)
BG (𝑘 = −0.6) 0.026932 (0.126065) 0.159536 (0.294393) 0.364682 (0.435072)
ML 0.011782 (0.085569) 0.060777 (0.186684) 0.130699 (0.278809)
100 BS 0.011782 (0.085569) 0.060777 (0.186684) 0.130699 (0.278809)
BG (𝑘 = 0.6) 0.012004 (0.084495) 0.058056 (0.187923) 0.141918 (0.286061)
BG (𝑘 = −0.6) 0.011583 (0.084047) 0.059122 (0.188084) 0.126457 (0.277587)
parameter (𝑝) of the𝐺𝐸 distribution were taken to be 0.8, 1.6, 5. Results and Discussion
and 2.2. The scale parameter (𝜃) was considered throughout
the paper to be one (4). In this study, our objective is to obtain the estimates of
We observed that the parameter estimates under the the parameters and to observe the performance of the
classical maximum likelihood method could not be obtained methods used for estimation. To examine the estimates of the
in close form, and we therefore employed Newton-Raphson parameters, we obtain the absolute biases and mean squared
iterative approach via the Hessian matrix. This can simply errors of the estimates under differentmethods of estimation.
be implemented in the programming language with the From Table 3, it is very clear that the most dominant𝑅
packagemaxLik. estimator that had the smallest mean squared errors vis-𝑎-
To compute the Bayes estimates, an assumption is made vis the absolute biases for the scale parameter ̂(𝜃) is Bayesian
such that and take, respectively, Gamma and under general entropy loss function. This is followed closely𝜃 𝑝 (𝑎, 𝑏)
Gamma(𝑐, 𝑑)priors.We set the hyperparameters to be 0.0001, by Bayes under squared error loss function. What has been
that is, , in order to obtain proper observed again is that, as the sample size increases, the mean𝑎 = 𝑏 = 𝑐 = 𝑑 = 0.0001
priors. This approach was suggested by Press and Tanur [22]. squared error of all the estimators decrease unswervingly.
Note that at this point the posterior distribution is also proper. This is simply an indication of how good and reliable the
The values of the loss parameter for the general entropy estimators are.
loss function are 𝑘 = ±0.6, which can be extended for other When we consider Table 4, which contains the mean
values of the loss parameter. Readers are referred to Calabria squared errors and the absolute biases of the estimated shape
and Pulcini [23] for the choice of the loss parameter values. parameter (𝑝), we noticed that the mean squared errors and
These were iterated 𝑅 = 1000 times. The mean squared error the absolute biases of the two estimators, that is, maximum
and the absolute bias values are determined and presented likelihood and Bayes under squared error loss function,
below for the purpose of comparison. have the same values for the estimated shape parameter.
6 Journal of Probability and Statistics
Table 5: Average mean squared errors and absolute biases of the survival function 𝑆(𝑡) at 𝑇 = 25, 50, and 100.
𝑝 = 0.8 𝑝 = 1.6 𝑝 = 2.2
ML 0.075982 (0.042809) 0.082151 (0.045894) 0.078309 (0.044532)
BS 0.075986 (0.042836) 0.082597 (0.046023) 0.078476 (0.044688)
BG (𝑘 = 0.6) 0.080981 (0.045294) 0.082265 (0.045688) 0.086399 (0.046789)
BG (𝑘 = −0.6) 0.077210 (0.042326) 0.083746 (0.044032) 0.083479 (0.043191)
ML 0.075706 (0.031059) 0.080699 (0.031094) 0.078253 (0.031479)
BS 0.075725 (0.031063) 0.080711 (0.031107) 0.078272 (0.031507)
BG (𝑘 = 0.6) 0.079284 (0.031347) 0.082140 (0.032464) 0.078831 (0.031191)
BG (𝑘 = −0.6) 0.075028 (0.031119) 0.078181 (0.031753) 0.080356 (0.032732)
ML 0.073820 (0.022000) 0.075437 (0.022445) 0.074257 (0.022533)
BS 0.073923 (0.022000) 0.075437 (0.022447) 0.077209 (0.022535)
BG (𝑘 = 0.6) 0.072898 (0.021704) 0.078784 (0.022582) 0.076435 (0.022308)
BG (𝑘 = −0.6) 0.070723 (0.022449) 0.075552 (0.022798) 0.076741 (0.022954)
ML: maximum likelihood, BG: general entropy, and BS: squared error.
This is expected in that the priors used for the Bayesian estimating it. For the survival function, maximum likelihood
analysis are noninformative. With regards to the survival performed better than the other estimators.
function, Bayes estimator under the general entropy loss
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