INTRODUCTION
Gompertz distribution was introduced in connection with human mortality and actuarial sciences by Benzamin Gompertz (^{1825}). Right from the time of its introduction, this distribution has been receiving great attention from demographers and actuarist. This distribution is a generalization of the exponential distribution and is applied in various fields especially in reliability and life testing studies, actuarial science, epidemiological and biomedical studies. Gompertz distribution has some interesting relations with some of the wellknown distributions such as exponential, double exponential, Weibull, extreme value (Gumbel Distribution) or generalized logistic distribution (^{Willekens 2002}). An important characteristic of the Gompertz distribution is that it has an exponentially increasing failure rate for the life of the systems and is often used to model highly negatively skewed data in survival analysis (^{ElandtJohnson and Johnson 1979}). In recent past, many authors have contributed to the studies of statistical methodology and characterization of this distribution; for example, Garg et al. (^{1970}), Read (^{1983}), Makany (^{1991}), Rao and Damaraju (^{1992}), Franses (^{1994}), Chen (^{1997}) and Wu and Lee (^{1999}). Jaheen (^{2003a}, ^{b}) studied this distribution based on progressive typeII censoring and record values using Bayesian approach. Wu et al. (^{2003}) derived the point and interval estimators for the parameters of the Gompertz distribution based on progressive type II censored samples. Wu et al. (^{2004}) used least squared method to estimate the parameters of the Gompertz distribution. Wu et al. (^{2006}) also studied this distribution under progressive censoring with binomial removals. Ismail (^{2010}) obtained Bayes estimators under partially accelerated life tests with typeI censoring. Ismail (^{2011}) also discussed the point and interval estimations of a twoparameter Gompertz distribution under partially accelerated life tests with TypeII censoring. Asgharzadeh and Abdi (^{2011}) studied different types of exact confidence intervals and exact joint confidence regions for the parameters of the twoparameter Gompertz distribution based on record values. Kiani et al. (^{2012}) studied the performance of the Gompertz model with timedependent covariate in the presence of right censored data. Moreover, they compared the performance of the model under different censoring proportions (CP) and sample sizes. Shanubhogue and Jain (^{2013}) studied uniformly minimum variance unbiased estimation for the parameter of the Gompertz distribution based on progressively Type II censored data with binomial removals. Lenart (^{2014}) obtained moments of the Gompertz distribution and maximum likelihood estimators of its parameters. Lenart and Missov (^{2016}) studied Goodnessoffit tests for the Gompertz distribution. Recently, Singh et al. (^{2016}) studied different methods of estimation for the parameters of Gompertz distribution when the available data are in the form of fuzzy numbers. They also obtained Bayes estimators of the parameters under different symmetric and asymmetric loss functions.
In this paper, we present a Bayesian analysis when there is a limited prior knowledge about the parameter of interest. In this regard, it is important to use noninformative priors, however, it can be difficult to choose a prior distribution that represents this situation, because there is hardly any precise definition of the concept of noninformative prior. Nevertheless, we have many noninformative priors, for instance, Jeffreys prior (^{Jeffreys 1967}), MDIP prior (^{Zellner 1977}, ^{1984}, ^{1990}), Tibshirani prior (^{Tibshirani 1989}), reference prior (^{Bernardo 1979}) and many others which seemingly appropriate for a number of inference problems. It is to be noted that lack of enough information on the part of analysts often forces them to choose noninformative priors and this consideration ensures that the inferences are mostly data driven. In Bayesian analysis, many authors consider independent gamma priors for the estimation of parameters of the model, representing weak information as the use of a priori independence assumption simplifies the computations. Our main interest in the Bayesian analysis is to select a prior distribution that represents better dependence structure of the parameters in which the information regarding the parameters is not considered substantial as compared with information from the data. The focus is on the comparison of independent gamma prior, Jeffreys prior, maximal data information prior (MDIP), Singpurwalla’s prior and elicited prior. Jeffreys (1967) proposed a noninformative prior resulting from an argument based on the Fisher Information Measure and Zellner (1977, 1984) proposed an alternative prior, named maximal data information prior (MDIP) based on the Entropy Measure. The prior proposed by Singpurwalla (1988) for estimation of the parameters of Weibull distribution is also considered in this paper to estimate the parameters of Gompertz distribution.
There are many methods for eliciting parameters of prior distributions. In this paper, we also consider an elicitation method to specify the values of hyperparameters of the two gamma priors assigned to the parameters of the Gompertz distribution. The method requires the derivation of predictive prior distribution and it is assumed that the expert is able to provide some percentiles values. Thus, the main aim of this paper is to propose noninformative and informative prior distributions for the parameters
The paper is organized as follows. Some probability properties of the Gompertz distribution such as quantiles, moments, moment generating function are reviewed in Section 2. Section 3 describes the maximum likelihood estimation method. The Bayesian approach with proposed informative and noninformative priors is presented in section 4. In Section 5, simulation study is carried out to evaluate the performance of several estimation procedures along with coverage percentages is provided. The methodology developed in this paper and the usefulness of the Gompertz distribution is illustrated by using a real data example in Section 6. Finally, concluding remarks are provided in Section 7.
MODEL AND ITS BASIC PROPERTIES
A random variable X has the Gompertz distribution with parameters
and the corresponding c.d.f is given by
The basic tools for studying the ageing and reliability characteristics of the system are the hazard rate
Note that the hazard rate function is increasing function if
Figure 1a shows the shapes of the pdf of the Gompertz distribution for different values of the parameters
The quantile function
In particular, the median of the Gompertz distribution can be written as
If the random variable
On simplification, we get
where
and
is the generalized integroexponential function (^{Milgram 1985}).
The mean and variance of the random variable X of the Gompertz distribution are respectively, given by
and
Many of the interesting characteristics and features of a distribution can be obtained via its moment generating function and moments. Let X denote a random variable with the probability density function (1). By definition of moment generating function of X and using (1), we have
MAXIMUM LIKELIHOOD ESTIMATION
The method of maximum likelihood is the most frequently used method of parameter estimation (^{Casella and Berger 2001}). The success of the method stems no doubt from its many desirable properties including consistency, asymptotic efficiency, invariance property as well as its intuitive appeal. Let
For ease of notation, we denote the first partial derivatives of any function
we have
and
From (11) and (12), we find the MLE for
The MLE for "
The asymptotic distribution of the MLE
(^{Lawless 2003}), where
The derivatives in
Therefore, the above approach is used to derive the approximate
Here,
BAYESIAN ANALYSIS
In this section, we consider Bayesian inference of the unknown parameters of the
and
The hyperparameters
Thus, the joint posterior distribution is given by
The conditional distribution of
Similarly, the conditional distribution of
Note that the although the conditional
JEFFREYS PRIOR
A well known noninformative prior, which represents a situation with little a priori information on the parameters was introduced by Jeffreys (1967), also known as the Jeffreys rule. The Jeffreys prior has been widely used due to the invariance property for one to one transformations of the parameters. Since then Jeffreys prior has played an important role in Bayesian inference. This prior is derived from Fisher Information matrix
However,
where
In this way, from (15) and (24) the noninformative prior for (
Let us denote the prior (25) as "Jeffreys prior".
Thus, the corresponding posterior distribution is given by
Proposition 1: For the parameters of the Gompertz distribution, the posterior distribution given in (26) under Jeffreys prior
We need to prove that
Indeed,
The function
Therefore, from (27) we have
where
MAXIMAL DATA INFORMATION PRIOR (MDIP)
It is interesting to note that the data gives more information about the parameter than the information from the prior density, otherwise, there would not be justification for the realization of the experiment. Let
be a negative entropy of
which is the prior average information in the data density minus the information in the prior density.
The following theorem proposed by Zellner provides the formula for the MDIP prior.
Theorem: The MDIP prior is given by:
where
Proof. We have to maximize the function
Thus,
Therefore, the MDIP is a prior that leads to an emphasis on the information in the data density or likelihood function, that is, its information is weak in comparison with data information.
Zellner (1984) shows several interesting properties of MDIP and additional conditions that can also be imposed to the approach refleting given initial information.
Suppose that we do not have much prior information available about
where
Hence the MDIP prior is given by
Now combining the likelihood function given by
and the MDIP prior in (33), the posterior densitiy for the parameters
Proposition 2: For the parameters of the Gompertz distribution, the posterior distribution given in (35) under the corresponding MDIP prior
Proof. Indeed,
Now, we consider a substituition of variables in the integral above as
resulting in
where
Let us denote
where
Now consider
where
for 0
From (36) we have
which is not possible to obtain an analytical expression for this integral. However, the software Mathematica gives a convergence result of the integral.
PRIORS PROPOSED BY SINGPURWALLA
Singpurwalla (^{1988}) presented a procedure for the construction of the prior distribution with the use of expert opinion in order to estimate the parameters
Our aim is to derive the prior
Consider the median of
where
A gamma prior distribution is chosen to model the uncertainty about
with the parameters
Thus, we can determine the conditional prior distribution
Finally, the joint prior for
where the vector of parameters
ELICITED PRIOR
In this Section, we provide a methodology that permits the experts to use their knowledges about the reliability of an item through statements of percentiles. This method requires the derivation of prior predictive distribution for elicitation. Suppose that joint prior
for a fixed mission time
In order to elicit the four hyperparameters
for a given pth percentile elicited from the expert where
By considering Gompertz distribution, the reliability function
and assuming a joint prior
where
Using (42), (43) and (44), the probability in (42) becomes
Let
that is,
Since it is not possible to obtain a closed form for the integral (47), one possibility to work around this problem is to use the Laplace approximation.
Assuming
by expanding
where
We can write (47) as
Thus, the function
By applying Laplace approximation to the integral in (47) we have
where
We suppose that an expert can summarize his/her knowledge about the reliability of an item through statements of percentiles. Thus, we ask for
expert’s information in the form of four distinct percentiles
The nonlinear system composed by the equation (52) under the four pair of values
SIMULATIONS
In this section, we perform a simulation study to examine the behavior of the proposed methods under different conditions. We considered three different sample sizes;
To investigate the convergence of the MCMC sampling via MH algorithm, we have used the GelmanRubin multiple sequence diagnostics. For computation, we have used R package coda. For each case of “
For the informative gamma priors, the elicited percentiles provided by the expert and the corresponding elicited values of the hyperparameters have been found to be: for Table I, we have
From the simulation results, we reach to the following conclusions: 1. With increase in sample size, biases and MSEs of the estimators decrease for given values of
Method 











MLE  0.8476  0.3080  0.2077  0.0196  0.0088  0.0060 
(1.4217)  (0.1556)  (0.0685)  (0.0009)  (0.0001)  (6.0e05)  
Gamma prior  0.6607  0.2935  0.1997  0.0363  0.0097  0.0062 
(0.6696)  (0.1378)  (0.0649)  (0.0042)  (0.0002)  (6.9e05)  
Jeffrey’s prior  0.5204  0.2907  0.1978  0.0360  0.0096  0.0061 
(0.4204)  (0.1341)  (0.0613)  (0.0037)  (0.0001)  (6.8e05)  
MDIP  0.5639  0.2749  0.1913  0.0573  0.0114  0.0068 
(0.4732)  (0.1178)  (0.0570)  (0.0063)  (0.0002)  (8.4e05)  
Singpurwalla’s prior  0.4687  0.2898  0.1780  0.0248  0.0091  0.0058 
(0.3387)  (0.1327)  (0.0494)  (0.0015)  (0.0001)  (6.2e05)  
Elicited prior  0.2146  0.1822  0.1528  0.0040  0.0045  0.0040 
(0.0680)  (0.0538)  (0.0367)  (2.5e05)  (3.2e05)  (2.5e05) 
Method 











MLE  2.8437  0.9839  0.6354  1.0200  0.4439  0.3116 
(15.2918)  (1.5513)  (0.6406)  (1.8551)  (0.3017)  (0.1536)  
Gamma prior  2.5298  1.0271  0.6519  1.2956  0.5039  0.3304 
(10.2579)  (1.6401)  (0.6689)  (2.8684)  (0.4233)  (0.1806)  
Jeffrey’s prior  2.2830  1.0219  0.6508  1.1710  0.5007  0.3308 
(8.5106)  (1.6199)  (0.6672)  (2.4183)  (0.4134)  (0.1805)  
MDIP  1.7570  0.8304  0.6015  0.6181  0.4049  0.3135 
(6.6596)  (1.1011)  (0.5643)  (0.6339)  (0.2464)  (0.1575)  
Singpurwalla’s prior  2.5279  0.9786  0.6323  0.7457  0.4248  0.3042 
(12.1530)  (1.5391)  (0.6358)  (0.8266)  (0.2741)  (0.1458)  
Elicited prior  0.5518  0.4103  0.3436  0.1301  0.1467  0.1445 
(0.4509)  (0.2713)  (0.1920)  (0.0255)  (0.0326)  (0.0324) 
Method 











MLE  1.2171  0.4331  0.2967  0.4908  0.2139  0.1575 
(2.9378)  (0.3166)  (0.1414)  (0.4166)  (0.0720)  (0.0393)  
Gamma prior  1.0368  0.4557  0.3030  0.5919  0.24807  0.1659 
(1.8688)  (0.3422)  (0.1449)  (0.5873)  (0.1055)  (0.0460)  
Jeffrey’s prior  0.9578  0.4536  0.3020  0.5465  0.2466  0.1661 
(1.6174)  (0.3380)  (0.1443)  (0.5124)  (0.1039)  (0.0461)  
MDIP  0.7924  0.3410  0.2636  0.2828  0.1757  0.1454 
(1.3917)  (0.2110)  (0.1102)  (0.1328)  (0.0482)  (0.0331)  
Singpurwalla’s prior  1.0689  0.4324  0.2965  0.3593  0.2054  0.1543 
(2.2494)  (0.3157)  (0.1415)  (0.1916)  (0.0653)  (0.0374)  
Elicited prior  0.2306  0.1922  0.1674  0.1071  0.0932  0.0858 
(0.0664)  (0.0539)  (0.0432)  (0.0171)  (0.0130)  (0.0112) 
Method 











MLE  0.95  0.95  0.95  0.73  0.87  0.91 
Gamma prior  0.91  0.92  0.92  0.91  0.92  0.92 
Jeffrey’s prior  0.97  0.96  0.96  0.97  0.96  0.96 
MDIP  0.94  0.95  0.96  0.93  0.95  0.96 
Singpurwalla’s prior  0.97  0.96  0.98  0.98  0.96  0.98 
Elicited prior  1.00  0.98  0.98  1.00  0.99  0.98 
AN EXAMPLE WITH LITERATURE DATA
In this section, we use a real data set to illustrate the proposed estimation methods discussed in the previous sections.
Let us consider the following data set provided in King et al. (^{1979}):
112, 68, 84, 109, 153, 143, 60, 70, 98, 164, 63, 63, 77, 91, 91, 66, 70, 77, 63, 66, 66, 94, 101, 105, 108, 112, 115, 126, 161, 178.
These data represent the numbers of tumordays of 30 rats fed with unsaturated diet. Chen (1997) and Asgharzadeh and Abdi (2011) used the Gompertz distribution for these data set in order to obtain exact confidence intervals and joint confidence regions for the parameters based on two different statistical analysis. Let us also assume the Gompertz distribution with density (1) fitted to the data and to compare the performance of the methods discussed in this paper.
For a Bayesian analysis, we assume independent Gamma prior distributions for the parameters
The marginal posterior distributions for the parameters
Method 

95% CI 

95% CI 

MLE  0.0241  (0.0160, 0.0322)  0.0016  (0.0002, 0.0031) 
Gamma Prior  0.0232  (0.0150, 0.0312)  0.0019  (0.0007, 0.0038) 
Jeffrey’s prior  0.0234  (0.0152, 0.0318)  0.0018  (0.0007, 0.0038) 
MDIP  0.0226  (0.0147, 0.0304)  0.0020  (0.0008, 0.0045) 
Singpurwalla’s Prior  0.0242  (0.0165, 0.0317)  0.0016  (0.0006, 0.0033) 
CONCLUSIONS
In this paper, we have considered estimation of the parameters of the Gompertz distribution using frequentist and Bayesian methods. In Bayesian methods, we have consider objective priors (Jeffreys and MDIP), gamma prior, Singpurwalla’s prior and Elicited prior. We have performed an extensive simulation study to compare these methods. From the simulation study regarding the bias, MSE and CP we observe that in general the MDIP provides best results for both parameters and in some cases, with MDIP and Jeffreys priors the results are quite similar. The real data application shows the same situation. It is worth remembering that both forms result from formal procedures for representing absence of information, that is, they are noninformative. The commonly assumption used of independent gamma priors and the priors proposed by Singpurwalla do not present as good results as the objective priors. The independent gamma priors are generally used in situations where no objective priors are possible to obtain or provide improper posterior distributions and mainly due to computational ease. Elicited prior produces much smaller bias and MSE than using the other assumed priors and also provides an overcoverage probability than their counterparts. Hence, we can conclude that, in the situation of the absence of information, the MDIP prior is more indicate for a Bayesian estimation of the twoparameter Gompertz distribution. On the other hand, in the situation where we have available expert’s information, the Elicited prior will perform the best estimators.