Abstract
This paper takes into account the estimation for the unknown parameters of the Gompertz distribution from the frequentist and Bayesian view points by using both objective and subjective prior distributions. We first derive noninformative priors using formal rules, such as Jefreys prior and maximal data information prior (MDIP), based on Fisher information and entropy, respectively. We also propose a prior distribution that incorporate the expert’s knowledge about the issue under study. In this regard, we assume two independent gamma distributions for the parameters of the Gompertz distribution and it is employed for an elicitation process based on the predictive prior distribution by using Laplace approximation for integrals. We suppose that an expert can summarize his/her knowledge about the reliability of an item through statements of percentiles. We also present a set of priors proposed by Singpurwala assuming a truncated normal prior distribution for the median of distribution and a gamma prior for the scale parameter. Next, we investigate the effects of these priors in the posterior estimates of the parameters of the Gompertz distribution. The Bayes estimates are computed using Markov Chain Monte Carlo (MCMC) algorithm. An extensive numerical simulation is carried out to evaluate the performance of the maximum likelihood estimates and Bayes estimates based on bias, meansquared error and coverage probabilities. Finally, a real data set have been analyzed for illustrative purposes.
Key words
Gompertz distribution; objective prior; Jeffreys prior; subjective prior; maximal data information prior; elicitation
INTRODUCTION
Gompertz distribution was introduced in connection with human mortality and actuarial sciences by Benzamin Gompertz (^{1825}8 GOMPERTZ B. 1825. On the nature of the function expressive of the law of human mortality and on a new mode of determining the value of life contingencies. Philos Trans R Soc Lond 115: 513583.). 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 200228 WILLEKENS F. 2002. Gompertz in context: the Gompertz and related distributions. In: Tabeau E, Van den Berg JA and Heathcote C (Eds), Forecasting mortality in developed countries  insights from a statistical demographic and epidemiological perspective European studies of population. Dordrecht: Kluwer Academic Publishers 9: 105126.). 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 19795 ELANDTJOHNSON RC AND JOHNSON NL. 1979. Survival Models and Data Analysis. J Wiley & Sons: NY, 457 p.). In recent past, many authors have contributed to the studies of statistical methodology and characterization of this distribution; for example, Garg et al. (^{1970}7 GARG M, RAO B AND REDMOND C. 1970. Maximumlikelihood estimation of the parameters of the Gompertz survival function. J R Stat Soc Ser C Appl Stat 19: 152159.), Read (^{1983}23 READ CB. 1983. Gompertz Distribution. Encyclopedia of Statistical Sciences. J Wiley & Sons, NY.), Makany (^{1991}19 MAKANY RA. 1991. Theoretical basis of Gompertz’s curve. Biom J 33: 121128.), Rao and Damaraju (^{1992}22 RAO BR AND DAMARAJU CV. 1992. New better than used and other concepts for a class of life distribution. Biom J 34: 919935.), Franses (^{1994}6 FRANSES PH. 1994. Fitting a Gompertz curve. J Oper Res Soc 45: 109113.), Chen (^{1997}4 CHEN Z. 1997. Parameter estimation of the Gompertz population. Biom J 39: 117124.) and Wu and Lee (^{1999}30 WU JW AND LEE WC. 1999. Characterization of the mixtures of Gompertz distributions by conditional expectation of order statistics. Biom J 41: 371381.). Jaheen (^{2003a}11 JAHEEN ZF. 2003a. Prediction of Progressive Censored Data from the Gompertz Model. Commun Stat Simul Comput 32: 663676., b12 JAHEEN ZF. 2003b. A Bayesian analysis of record statistics from the Gompertz model. Appl Math Comput 145: 307320.) studied this distribution based on progressive typeII censoring and record values using Bayesian approach. Wu et al. (^{2003}32 WU SJ, CHANG CT AND TSAI TR. 2003. Point and interval estimations for the Gompertz distribution under progressive typeII censoring. Metron LXI: 403418.) derived the point and interval estimators for the parameters of the Gompertz distribution based on progressive type II censored samples. Wu et al. (^{2004}29 WU JW, HUNG WL AND TSAI CH. 2004. Estimation of parameters of the Gompertz distribution using the least squares method. Appl Math Comput 158: 133147.) used least squared method to estimate the parameters of the Gompertz distribution. Wu et al. (^{2006}31 WU JW AND TSENG HC. 2006. Statistical inference about the shape parameter of the Weibull distribution by upper record values. Stat Pap 48: 95129.) also studied this distribution under progressive censoring with binomial removals. Ismail (^{2010}9 ISMAIL AA. 2010. Bayes estimation of Gompertz distribution parameters and acceleration factor under partially accelerated life tests with typeI censoring. J Stat Comput Simul 80: 12531264.) obtained Bayes estimators under partially accelerated life tests with typeI censoring. Ismail (^{2011}10 ISMAIL AA. 2011. Planning stepstress life tests with typeII censored Data. Sci Res Essays 6: 40214028.) also discussed the point and interval estimations of a twoparameter Gompertz distribution under partially accelerated life tests with TypeII censoring. Asgharzadeh and Abdi (^{2011}1 ASGHARZADEH A AND ABDI M. 2011. Exact Confidence Intervals and Joint Confidence Regions for the Parameters of the Gompertz Distribution based on Records. Pak J Statist 271: 5564.) 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}14 KIANI K, ARASAN J AND MIDI H. 2012. Interval estimations for parameters of Gompertz model with timedependent covariate and right censored data. Sains Malays 414: 471480.) 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}24 SHANUBHOGUE A AND JAIN NR. 2013. Minimum Variance Unbiased Estimation in the Gompertz Distribution under Progressive Type II Censored Data with Binomial Removals. Int Sch Res Notices 2013: 17.) 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}17 LENART A. 2014. The moments of the Gompertz distribution and maximum likelihood estimation of its parameters. Scand Actuar J 3: 255277.) obtained moments of the Gompertz distribution and maximum likelihood estimators of its parameters. Lenart and Missov (^{2016}18 LENART A AND MISSOV TI. 2016. Goodnessoffit tests for the Gompertz distribution. Commun Stat Theory Methods 45: 29202937.) studied Goodnessoffit tests for the Gompertz distribution. Recently, Singh et al. (^{2016}25 SINGH N, YADAV KK AND RAJASEKHARAN R. 2016. ZAP1mediated modulation of triacylglycerol levels in yeast by transcriptional control of mitochondrial fatty acid biosynthesis. Mol Microbiol 1001: 5575.) 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 196713 JEFFREYS SIR HAROLD. 1967. Theory of probability. Oxford U Press: London, 470 p.), MDIP prior (Zellner 197733 ZELLNER A. 1977. Maximal Data Information Prior Distributions. In: Aykac A and Brumat C (Eds), New Methods in the applications of Bayesian Methods. NorthHolland Amsterdam, p. 211232., ^{1984}34 ZELLNER A. 1984. Maximal Data Information Prior Distributions. Basic Issues in Econometrics, U. of Chicago Press., ^{1990}35 ZELLNER A. 1990. Bayesian Methods and Entropy in Economics and Econometrics. In: Grandy Junior WT and Schick LH (Eds), Maximum Entropy and Bayesian Methods Dordrecht Netherlands: Kluwer Academic Publishers, p. 1731.), Tibshirani prior (Tibshirani 198927 TIBSHIRANI R. 1989. Noninformative Priors for One Parameters of Many. Biometrika 76: 604608.), reference prior (Bernardo 19792 BERNARDO JM. 1979. Reference Posterior Distributions for Bayesian Inference. J R Stat Soc Series B Stat Methodol 41: 113147.) 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 198521 MILGRAM M. 1985. The generalized integroexponential function. Math Comp 44: 443458.).
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 20013 CASELLA G AND BERGER RL. 2001. Statistical Inference. Cengage Learning, 660 p.). 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 200316 LAWLESS JF. 2003. Statistical Models and Methods for Lifetime Data. J Wiley & Sons: New Jersey, 664 p.), 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}26 SINGPURWALLA ND. 1988. An interactive PCbased procedure for reliability assessment incorporating expert opinion and survival data. J Am Stat Assoc 83: 4351.) 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
Average bias of the estimates of
Average bias of the estimates of
Average bias of the estimates of
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}15 KING M, BAILEY DM, GIBSON DG, PITHA JV AND MCCAY PB. 1979. Incidence and growth of mammary tumors induced by 7, l2Dimethylbenz antheacene as related to the dietary content of fat and antioxidant. J Natl Cancer Inst 63: 656664.):
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
Estimators and 95% confidence/credible intervals of
The posterior densities for the parameter c of the Gompertz distribution fitted by the data.
The posterior densities for the parameter λ of the Gompertz distribution fitted by the data.
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.
REFERENCES

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^{2}BERNARDO JM. 1979. Reference Posterior Distributions for Bayesian Inference. J R Stat Soc Series B Stat Methodol 41: 113147.

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^{4}CHEN Z. 1997. Parameter estimation of the Gompertz population. Biom J 39: 117124.

^{5}ELANDTJOHNSON RC AND JOHNSON NL. 1979. Survival Models and Data Analysis. J Wiley & Sons: NY, 457 p.

^{6}FRANSES PH. 1994. Fitting a Gompertz curve. J Oper Res Soc 45: 109113.

^{7}GARG M, RAO B AND REDMOND C. 1970. Maximumlikelihood estimation of the parameters of the Gompertz survival function. J R Stat Soc Ser C Appl Stat 19: 152159.

^{8}GOMPERTZ B. 1825. On the nature of the function expressive of the law of human mortality and on a new mode of determining the value of life contingencies. Philos Trans R Soc Lond 115: 513583.

^{9}ISMAIL AA. 2010. Bayes estimation of Gompertz distribution parameters and acceleration factor under partially accelerated life tests with typeI censoring. J Stat Comput Simul 80: 12531264.

^{10}ISMAIL AA. 2011. Planning stepstress life tests with typeII censored Data. Sci Res Essays 6: 40214028.

^{11}JAHEEN ZF. 2003a. Prediction of Progressive Censored Data from the Gompertz Model. Commun Stat Simul Comput 32: 663676.

^{12}JAHEEN ZF. 2003b. A Bayesian analysis of record statistics from the Gompertz model. Appl Math Comput 145: 307320.

^{13}JEFFREYS SIR HAROLD. 1967. Theory of probability. Oxford U Press: London, 470 p.

^{14}KIANI K, ARASAN J AND MIDI H. 2012. Interval estimations for parameters of Gompertz model with timedependent covariate and right censored data. Sains Malays 414: 471480.

^{15}KING M, BAILEY DM, GIBSON DG, PITHA JV AND MCCAY PB. 1979. Incidence and growth of mammary tumors induced by 7, l2Dimethylbenz antheacene as related to the dietary content of fat and antioxidant. J Natl Cancer Inst 63: 656664.

^{16}LAWLESS JF. 2003. Statistical Models and Methods for Lifetime Data. J Wiley & Sons: New Jersey, 664 p.

^{17}LENART A. 2014. The moments of the Gompertz distribution and maximum likelihood estimation of its parameters. Scand Actuar J 3: 255277.

^{18}LENART A AND MISSOV TI. 2016. Goodnessoffit tests for the Gompertz distribution. Commun Stat Theory Methods 45: 29202937.

^{19}MAKANY RA. 1991. Theoretical basis of Gompertz’s curve. Biom J 33: 121128.

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^{23}READ CB. 1983. Gompertz Distribution. Encyclopedia of Statistical Sciences. J Wiley & Sons, NY.

^{24}SHANUBHOGUE A AND JAIN NR. 2013. Minimum Variance Unbiased Estimation in the Gompertz Distribution under Progressive Type II Censored Data with Binomial Removals. Int Sch Res Notices 2013: 17.

^{25}SINGH N, YADAV KK AND RAJASEKHARAN R. 2016. ZAP1mediated modulation of triacylglycerol levels in yeast by transcriptional control of mitochondrial fatty acid biosynthesis. Mol Microbiol 1001: 5575.

^{26}SINGPURWALLA ND. 1988. An interactive PCbased procedure for reliability assessment incorporating expert opinion and survival data. J Am Stat Assoc 83: 4351.

^{27}TIBSHIRANI R. 1989. Noninformative Priors for One Parameters of Many. Biometrika 76: 604608.

^{28}WILLEKENS F. 2002. Gompertz in context: the Gompertz and related distributions. In: Tabeau E, Van den Berg JA and Heathcote C (Eds), Forecasting mortality in developed countries  insights from a statistical demographic and epidemiological perspective European studies of population. Dordrecht: Kluwer Academic Publishers 9: 105126.

^{29}WU JW, HUNG WL AND TSAI CH. 2004. Estimation of parameters of the Gompertz distribution using the least squares method. Appl Math Comput 158: 133147.

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^{31}WU JW AND TSENG HC. 2006. Statistical inference about the shape parameter of the Weibull distribution by upper record values. Stat Pap 48: 95129.

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^{34}ZELLNER A. 1984. Maximal Data Information Prior Distributions. Basic Issues in Econometrics, U. of Chicago Press.

^{35}ZELLNER A. 1990. Bayesian Methods and Entropy in Economics and Econometrics. In: Grandy Junior WT and Schick LH (Eds), Maximum Entropy and Bayesian Methods Dordrecht Netherlands: Kluwer Academic Publishers, p. 1731.

^{36}ZELLNER A AND MIN C. 1993. Bayesian analysis model selection and prediction. In: Grandy Junior WT and Milonni PW (Eds), Physics and Probability: Essays in honour of Edwin T Jaynes, Cambridge University Press, Cambridge, UK.
Publication Dates

Publication in this collection
Sept 2018
History

Received
29 Dec 2017 
Accepted
26 Mar 2018