Bayesian and Non-Bayesian Inference for Shape Parameter and Reliability Function of Basic Gompertz Distribution

In this paper, some estimators of the unknown shape parameter and reliability function of Basic Gompertz distribution (BGD) have been obtained, such as MLE, UMVUE, and MINMSE, in addition to estimating Bayesian estimators under Scale invariant squared error loss function assuming informative prior represented by Gamma distribution and non-informative prior by using Jefferys prior. Using Monte Carlo simulation method, these estimators of the shape parameter and R(t), have been compared based on mean squared errors and integrated mean squared, respectively


Introduction:
The Gompertz distribution (GD) was originally introduced by Gompertz in1825 (1).This distribution is used in model survival times, modeling human mortality and actuarial tables. It has many real life applications, especially in medical and actuarial studies. Due to its complicated form, it has not received enough attention in the past. However, recently, this distribution has received considerable attention from actuaries and demographers. The probability density function of the (GD) is given by (2): f(t ; φ) = φ exp [ct + φ (1 − e ct )] ; t ≥ 0 , c, φ > 0 Where φ is the shape parameter and c is the scale parameter of the Gompertz distribution. In the current paper, it will be assumed c=1 which is a special case of Gompertz distribution known as Basic Gompertz distribution with the following probability density function (3): f(t ; φ) = φe t+φ(1−e t ) ; t ≥ 0, φ > 0 … (1) The corresponding cumulative distribution function F(t) is given by ; t ≥ 0 … (2) Accordingly, R(t) is given by R(t) = F(t ) ̅̅̅̅̅̅ = exp[ φ(1 − e t )] ; t ≥ 0

Non-Bayes Estimators of the Shape Parameter Maximum likelihood Estimator (MLE)
Assume that, t = t 1 , t 2 ,... ,t n be the set of n random lifetimes from the Basic Gompertz distribution defined by equation (1), the likelihood function for the sample observation will be as follows (4): L (t 1 , t 2 , . . . , t n ; φ) = π i=1 n f(t i ; φ) By letting, φ ln L (t 1 , t 2 , . . . , t n ; φ) = 0, the MLE of φ becomes ) Based on the invariant property of the MLE, the MLE for R(t) will be as follows

Posterior Density Functions Using Jeffreys Prior
Supposing φ to have non-informative prior presented after using Jeffreys prior information g 2 (φ) which is signified by (8): Where I(φ) stands for Fisher information designated as follows (9): Hence, After substitution into equation (11) yields, After substituting in equation (10), the posterior density function based on Jeffreys prior π 2 (φ|t 1 , … , t n ) is The posterior density is then realized as the Gamma distribution density, i.e.

Bayes Estimator under Scale Invariant Squared Error Loss Function
The Scale invariant squared error loss function (SISELF) is a continuous and non-negative symmetric loss function. It has been discussed by De Groot (1970) (10), and is defined as: Based on (SISELF), risk function R(φ, φ) can be derived as Therefore, Bayesian estimator under (SISELF), that minimizes the risk function, as follows

Bayes Estimation under (SISELF) with Jefferys Prior
The Bayes estimator for the shape parameter φ under Jefferys prior can be obtained using equation (12) as follows: After substituting into (12), Bayesian estimation for the shape parameter of BGD under (SISELF) with Jefferys prior, denoted by φ BJ is This is equivalent to φ . Now, Bayesian estimation for R(t) under (SISELF) can be obtained using equation (14), as follows

Simulation study
To compare the behavior of the different estimators of φ and R(t) Monte-Carlo simulation has been employed. The process has been repeated 5000 times (L=5000) with different sample sizes (n= 15, 50, and100). The values of φ have chosen as φ= 0.5 and 3. Two values of the parameters of Gamma prior are chosen as γ=0.8 and 3, δ=0.5 and 3. All estimators for φ derived previously are evaluated based on their mean squared errors (MSE's), where, The integrated mean squared error (IMSE) has been employed to compare the behavior of the Bayesian estimation for R(t). IMSE is an important global measure and it more accurate than MSE, where, Where =1, 2,…, , the random limits of . In this paper, we chose t = 0.1, 0.2, 0.3, 0.4, 0.5.

Estimating the shape parameter
The results of estimating the shape parameter of Basic Gompertz distribution φ including expected values (EXP) and mean squared errors (MSE's) are tabulated in Tables 1-4. The discussion of the results can be summarized as follows:  The comparison between MSE's of the three non-Bayesian estimators (MLE, UMVUE and MinMSE) for the shape parameter φ showed that (theoretically and empirically), the performance of (MinMSE) is the best estimator, followed by UMVUE. In other words,  Generally, the MSE's of all estimators of the shape parameter φ increase with the increase of the shape parameter value.

Estimating the Reliability function
The discussion of the results IMSE's of all estimators of R(t) were tabulated in Tables (5-8) and the dissection can be expressed by the following important point:  According to the results for Non-Bayesian methods, when the shape parameter less than 1, the performance of MinMSE (approximated) is the best estimator for R(t), followed by UMVUE (approximated). In other words, MSE(MinMSE) ≤ MSE(UMVUE) ≤ MSE(MLE) (see Table 5). While the UMVUE(approximated) is the best estimator for R(t), followed by MLE when the value of the shape parameter φ greater than 1(see Table 6).  The Bayes estimator under Scale invariant squared error loss function with Gamma prior is the best estimator for R(t) in comparison to others, such that the value of the shape parameter of Gamma prior γ should be greater than (1) for all cases while the value of the scale parameter of Gamma prior should be greater than 1 if the shape parameter of Basic Gompertz distribution is less than 1 and vice versa (see Tables 7-8).  The IMSE's of all estimators of R(t) increase with the increase of the shape parameter value.