Bayesian Estimation for Two Parameters of Gamma Distribution under Generalized Weighted Loss Function

This paper deals with, Bayesian estimation of the parameters of Gamma distribution under Generalized Weighted loss function, based on Gamma and Exponential priors for the shape and scale parameters, respectively. Moment, Maximum likelihood estimators and Lindley’s approximation have been used effectively in Bayesian estimation. Based on Monte Carlo simulation method, those estimators are compared in terms of the mean squared errors (MSE’s).


Introduction
Gamma distribution is extremely important in statistical inferential problems. It is widely used in reliability analysis and as a conjugate prior in Bayesian statistics.
It is a good alternative to the popular Weibull distribution; also, it is a flexible distribution that commonly offers a good fit to any variable such as in environmental, meteorology, climatology and other physical situation [1]. Therefore, the successful estimation of two unknown parameters of Gamma distribution will be very important. Even though the theoretical and practical sides of Gamma distribution, remain poorly studied. The probability density function of the Gamma distribution is defined as follows ) is the lower incomplete gamma function. Therefore, the reliability function for ( ) is If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion And the reliability function will be: [2] R( ) ∑ ( ) The maximum likelihood estimators are obtained through Newton-Raphson method as follows [3,4] * + * + Where, The initial value for MLE for and are the moment estimators which are given by [

Bayesian Estimation
In this section, we obtain some Bayes estimators for and based on the maximum likelihood estimators ̂ and ̂ as the initial values for and respectively.

Posterior Density Functions Using Gamma and Exponential Priors
To estimate α and β parameters for Gamma distribution, we assume that α has a prior π 1 (·), which follows Gamma(a,b). Also, we assume that, the prior on β is π 2 (·) and the density function of π 2 (·) is Exponential and it is independent of π 1 (·).
Where, a, b, c are known parameters. The marginal p.d.f. of ( ) is given by The joint posterior density functions of  and  is defined as follows

Bayes Estimators under Generalized weighted loss function
k is an positive integer number and c is a constant. The Risk function is defined as Hence, Bayes estimator using Generalized Weighted Loss function is given as follows [6] In general, Where u(α,β) be any function for α and β. Therefore,

ii) Bayesian estimation for β under Generalized Weighted loss function
Similarly, some Bayesian estimators for β can be obtained under Generalized Weighted loss function by assuming that, k = 1, 2 and c = 0, 1, 2 as follows When K=1 and c = 0 the equation (4) To find ( | ), assume that Where ̂ ̂ are the maximum likelihood estimators for and respectively, When k=1 and c=2 the equation (4) becomes as follows: To obtain ( | ) , assume that, After substituting (21) and (23) into (22) we get When k=2 and c=0 the equation (4) becomes as follows To derive ( | ), Let, ( ) then,

Simulation study
In this section, Monte -Carlo simulation is employed to compare the performance of different estimates, for unknown shape and scale parameters of gamma distribution based on the mean squared errors (MSE's), where, , I is the number of replications.
We generated I = 3000 samples of size n = 20, 30, 50, and 100 to represent small, moderate and large sample sizes from Gamma distribution with α = 2, 3 and β=0.5, 1. The values of α's prior parameters are chosen as a = 3 , b=3 and for β prior parameter is c =4. QB language has been employed for Monte-Carlo simulation study.

The results and discussions
The results for Monte-Carlo simulation study are summarized in Tables-(1-8), which can be summarized by the following points 1.It is clear that, the results for (expected values and MSE ) at = 0.5 are the same as the corresponding results when = 1,we can clarify the reason easily, according to moment method we have is a random sample from a Gamma distribution defined by (1), and each observation say x is generated by the following equation Where, is a random number followed uniform distribution with (0,1), i.e., ( ) And, according to Monte-Carlo simulation, e = ( ) is a random number from Exponential distribution with parameter . Therefore, is a random variable follows the Gamma distribution defined by (1). After substitute (28) into (29) yields, α ∑ ∑ ( ) Therefore, will be canceled.
Recall that, the moment estimator for α is the initial value for the corresponding MLE and then, Bayesian estimators for α are depending on MLE for α. Therefore the results for expected values and MSE's for α are the same.