Bayesian Estimation under Different Loss Functions Using Gamma Prior for the Case of Exponential Distribution

The Bayesian estimation approach is a non-classical estimation technique in statistical inference and is very useful in real world situation. The aim of this paper is to study the Bayes estimators of the parameter of exponential distribution under different loss functions and compared among them as well as with the classical estimator named maximum likelihood estimator (MLE). Since exponential distribution is the lifetime distribution, we have studied exponential distribution using gamma prior. Here the gamma prior is used as the prior distribution of exponential distribution for finding the Bayes estimator. In our study we also used different symmetric and asymmetric loss functions such as squared error loss function, quadratic loss function, modified linear exponential (MLINEX) loss function and non-linear exponential (NLINEX) loss function. We have used simulated data using R-coding to find out the mean squared error (MSE) of different loss functions and hence found that non-classical estimator is better than classical estimator. Finally, mean square error (MSE) of the estimators of different loss functions are presented graphically.


Introduction
The exponential distribution plays an important role in lifetime data analysis and waiting time or queuing problems [1].Many authors have developed inference procedures for exponential model.For example, Kulldorff devoted a large part of book to the estimation of the parameters of the exponential distribution [2] based on completely or partially grouped data.Sarhan found the empirical Bayes estimators of exponential model [3].Janeen explained the empirical Bayes estimators of the parameter of exponential distribution based on record values [4].To know more details the work of Balakrishnan et al. and Al-Hemyari of exponential distribution [4,5] can be seen.
Rahman et al. studied the bayes estimators under conjugate prior for power function distribution [7].We have studied this under gamma prior for exponential distribution to see the comparative situation.We can use the exponential distribution to estimate the stress-strength parameters and reliability of representing survival of head and neck cancer patients [8].
A continuous random variable X is said to have one parameter exponential distribution with parameter θ (θ>0) if its probability density function (pdf) is given by Roy [9].
(1) ; 0 Here we are interested to find the Bayes estimator of parameter θ under different loss functions.

Prior and Posterior density function of parameter
For Bayesian estimation we need to specify a prior distribution for the parameter.Consider a gamma prior for the parameter θ having density function [10] (2) 0 , , ; ) Then the posterior density function of θ for the given random sample X is given by Mood et al. [11] 

Different Estimators of Parameter θ
Here, Bayes estimator of θ for different loss functions along with maximum likelihood estimator has been determined.

Bayes estimator of parameter θ for squared error (SE) loss function
Here we have determined Bayes estimator of θ for squared error loss function [10] defined by .Here, the mean of posterior density function is is the Bayes estimator of θ under squared error loss function.

Bayes Estimator of θ for Quadratic Loss (QL) function
Let, the quadratic loss function is defined as [12]

L
Under this loss function the Bayes estimator of θ is obtained by solving the equation is the Bayes estimator of θ under quadratic loss function.

Bayes estimator of θ for MLINEX loss function
Let, the MLINEX loss function [12] is defined as For MLINEX loss function the Bayes estimator of θ is obtained by is the Bayes estimator of θ under MLINEX loss function.

Bayes Estimator of parameter θ for NLINEX Loss function
Let, the NLINEX loss function [13] of the form Here, D represents the estimation error i.e.,     D .For NLINEX loss function Bayes estimator [11] of θ is given by Where, (.) 11) and ( 12) in ( 10) we obtain, is the Bayes estimator of θ under NLINEX loss function.

Empirical Analysis
In order to compare estimators MLE In our study 6000 Samples have generated for each case.To obtain the variance of   , we have used the true (assume) value of the parameter θ under consideration.Again we have obtained the estimated value, MSE and Bias of the estimators by using Monte Carlo simulation method using R-Code from the exponential distribution.The results and their graphs (Using MS-Excel) are presented in Table 1.
From Table 1 we have observed that, the MSE of MLE   is very high for small sample size but decline sharply and become closer to other estimators with increase in sample size.Among Bayes estimators BQL   gives smaller value of MSE when sample size is small but for large sample they are almost identical (Fig. 1).are very close to each other for large sample (Fig. 3).Table 3.Estimated value and MSE of different estimators of θ for Exponential distribution where  = 1.5, β= 2.0, θ = 1.0 and c =2.0.It is clear from Table 6 that MSE of BSE   is smaller than other estimators (Fig. 6) and some cases it is very near to that of MSE of BNL   for different values of parameter θ.

Conclusion
From above analysis and graphical representation we have concluded that except for few cases Bayes estimator under Squared Error (SE) loss function and NLINEX loss function are better than other estimators in the study.We also concluded that, nonclassical estimator (the class of Bayes estimator under different loss functions) is better than classical estimator (MLE).We can apply Bayes estimator to calculate posterior probability on the basis of prior information.In real life Byes estimator can be used for forecasting insurance loss payments.
For squared error loss function Bayes estimator is the mean of posterior density function.From (3) posterior density function is a gamma distribution with parameter ) (


we have considered MSE of the estimators.The MSE of estimator   is defined as:

Table 3
represents the variation in the performance of the estimators for different sample size.More or less similar patterns are observed here as a previous table that is