Bayes Estimation under Conjugate Prior for the Case of Power Function Distribution

The Bayesian estimation approach is a non-classical device in the estimation part of statistical inference which is very useful in real world situation. The main objective of this paper is to study the Bayes estimators of the parameter of Power function distribution. In Bayesian estimation loss function, prior distribution and posterior distribution are the most important ingredients. In real life we try to minimize the loss and want to know some prior information about the problem to solve it accurately. The well known conjugate priors are considered for finding the Bayes estimator. In our study we have 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. The performance of the obtained estimators for different types of loss functions are then compared among themselves as well as with the classical maximum likelihood estimator (MLE). Mean Square Error (MSE) of the estimators are also computed and presented in graphs.


Introduction
The Power function distribution is often used to study the electrical component reliability [3]. A continuous random variable X is said to have Power function distribution if its probability density function is given by [2] f(x) = � (1) = 0, otherwise Where is the shape parameter, is the location parameter and is the scale parameter. We are interested to find Bayes estimator of shape parameter under different loss functions. Let = 0 and = 1 then the form of the density function becomes ( | )= −1 ; 0≤ ≤ 1, 0≤ ≤ ∞ (2)

Prior and Posterior Density Function of
For Bayesian estimation we need to specify a prior distribution for the parameter. Consider a Gamma prior for having pdf [6] g ( ) = − −1 ; , , > 0 (3) Now the Posterior density function of for the given random sample X is given by [4] , since prior and posterior distribution belongs to the same family hence the prior is conjugate prior.

Different Estimators of Parameter
In this section Bayes estimators of parameter for different loss functions along with maximum likelihood estimator has been determined.

Bayes Estimator of for Squared Error Loss Function
Here we have determined Bayes estimator of for squared error loss function defined as [6] L (t; ) =( − ) 2 (6) For squared error loss function Bayes estimator is the mean of posterior density function. From (4) posterior density function is a Gamma distribution with parameter ( + ) ( − ∑ ). Hence the mean of posterior density function is .Therefore ˆ= , is the Bayes estimator of under SE loss function.

Bayes Estimator of for Quadratic Loss Function
Now suppose the loss function is quadratic, which is defined as [5] L (t; ) = � − � 2 Under quadratic loss function Bayes estimator of is obtained by solving the following equation , is the Bayes estimator of under quadratic loss function

Bayes Estimator of for MLINEX Loss Function
Now let us consider the MLINEX loss function defined as [5] For MLINEX loss function Bayes estimator of is obtained from [5] Hence from (9) we get timator of under MLINEX loss function.

Bayes Estimator of for NLINEX Loss Function
Let us consider the following NLINEX loss function of the form [1] L (10) where D represents the estimation error .i.e. D=̂− ; For NLINEX loss function Bayes estimator of is [ Using (12) and (13) in (11) we get � /(c+2), is the Bayes estimator of under NLINEX loss function.

Empirical Study
To compare the estimators ̂, ̂, ̂, ̂ and ̂ we have considered MSE of the estimators. The MSE of an estimator is defined as       Table.3. shows the variation in the performance of the estimator for different sample size. More or less similar pattern are observed here as previous tables that is MSE of ̂ is higher than all other estimators. MSE of ̂ is least in the class of Bayes estimators. Also MSE of ̂& ̂a re very close to each other for large sample (figure 3)      The performance of the estimators for different values of parameter are also shown in the table 6 along with their graphical presentation.