Bayesian Estimators of the parameter and Reliability Function of Inverse Rayleigh Distribution"A comparison study”

In this paper, Bayesian estimator for the parameter and reliability function of inverse Rayleigh distribution (IRD) were obtained Under three types of loss function, namely, square error loss function (SELF), Modified Square error loss function (MSELF) and Precautionary loss function (PLF), taking into consideration the informative and noninformative prior. The performance of such estimators was assessed on the basis of mean square error (MSE) criterion by performing a Monte Carlo simulation technique.


Introduction
In reliability studies, most of the life time distributions used are characterized by a monotone failure rate, however, one parameter inverse Rayleigh distribution has also been used as a failure time distribution. Recently, many researchers interested in studying Inverse Rayleigh distribution in various aspects, for example: Mukherjee and Saran (1984) demonstrated that for a given parameter the distribution is increasing (or decreasing) failure rate according as the distribution variety is less than or more than 1.069543/√

Inverse Rayleigh Distribution (IRD)
The continuous random variable t is said to have inverse Rayleigh distribution with scale parameter if it has the probability density function , exp , 0, 0 1 The corresponding cumulative distribution function is , exp , 0, 0 2 Therefore, the reliably function of IRD is given by A variance and higher order moments do not exist for this distribution, moreover, it can be shown that IRD is special case of inverse Weibull distribution with parameters , when 2 [7].

Prior information's
A convenient choice of priors is indispensable for Bayesian analysis. Many researchers choose priors on the basis of their subjective beliefs and knowledge's. However, if enough information about parameter is presented, we should use informative priors; otherwise, it is better to employ vague or non-informative priors. In this paper, we consider the general rule developed by Jeffrey (1961) to obtain the non -informative prior. He established that the single unknown parameter which is regarded as a random variable follows such a distribution that is proportional to the square root of Fisher information , that is ∝ where denotes the prior information. Equivalently, Where C is a constant of proportionality and, It follows that: ,

Posterior density of inverse Rayleigh parameter based on Jeffrey's prior information
From equation (1), we have:

It follows that
From Bayesian perspective, the posterior density denoted by ℎ | can be determined by combining the prior distribution with the likelihood function | , as follows: Let = ( , , … … be a random sample drawn from inverse Rayleigh distribution, then the likelihood function is: Let ℎ | denote the posterior density based on Jeffrey's prior distribution for inverse Rayleigh parameter , then by substituting equations (7) and (9) into equation (8) with simplification, we get: The posterior density function in equation (10) is recognized as the density of gamma distribution, that is:

Posterior density of inverse Rayleigh parameter based on exponential prior distribution
Assuming that the inverse Rayleigh parameter follows exponential prior distribution with parameter, that is: Where denotes the exponential prior distribution of the inverse Rayleigh parameter .
By substituting equations (9) and (12) into equation (8) with simplification, we get: Where ℎ | denotes the posterior density based on exponential prior distribution and .
From equation (13), it can be easily noted that | distributed as gamma with parameter 1, . It follows that:

Types of loss functions
From Bayesian viewpoint, [8]. The essential step in the estimation and prediction problems represented by choosing the loss function. In fact, there is no specific analytical procedure to determine the suitable loss function to be employed. In most of studies concerning Bayesian estimation problem, the researchers consider the underlying loss function to be squared error loss function (SELF) which is symmetric in nature. However, in many cases, using the squared error loss function is not appropriate, especially in those cases where the losses are not symmetric. Accordingly, in order to make the statistical inferences more practical and applicable, we often need to choose an asymmetric loss function.
In this paper, we consider both symmetric and asymmetric loss functions for better realization of Bayesian analysis. In particular, the following loss functions have been considered assuming that is an estimate of , and L θ , θ symbolizes the loss function.
i) The Squared error loss function (SELF) is defined as: , ii) Modified Squared error loss function (MSELF) is defined as: Iii\) Precautionary loss function (PLF) is defined as: ,

Bayesian Estimation
The Bayes estimator of the parameter is the value of that minimizes the posterior expectation known as the risk function and denotes , , that is: Where ℎ | is the posterior density of |

Bayes estimator of the parameter of IRD under SELF
In general, if SELF is chosen, then according to equation (18), we have: It follows that: with respect to and setting the resultant derivative equal to zero, we get: By Solving for , we obtain the Bayes estimator of under SELF denoted by From equation (11) and on the basis of non-informative prior, the Bayes estimator of inverse Rayleigh parameter denoted as is given by.
Where denotes the Bayes estimate of based on Jeffrey's prior information. If the inverse Rayleigh parameter follows the exponential prior distribution, then by equation (14), we conclude that Where denotes the Bayes estimate of based on exponential prior distribution.

Bayes estimator of the Inverse Rayleigh parameter under MSELF
By substituting , given in equation (16) If k is positive integer, it is well known that On the basis of Jeffrey's prior information and by substituting ℎ | by ℎ | given in equation (10), then evaluating the integral in (23); it can easily be shown Therefore, the Bayes estimator given in equation (22) can be simplified to be: If r=1 we get If r=2 we get If r=3 then we obtain, On the basis of exponential prior distribution and by replacing ℎ | in equation (23) by ℎ | given in equation (13), we get: Where Therefore, the Bayes estimator given in equation (22) can be simplified to be: For r=1 we get: For r=2 we get: For r=3 we get: On the basis of Jeffrey's prior information, the Bayes estimator of the inverse Rayleigh parameter can be obtained by assuming that k=2 in equation (24) to get | , then substituting it into equation (34), so we obtain, On the same manner, by putting k=2 in equation (29), then substituting) it into equation (34), we get the Bayes estimator of inverse Rayleigh parameter under precautionary loss function and exponential prior distribution , denoted as and given by: (36)

Bayes estimator of the reliability function R(t) of IRD under SELF
Let us assume that k is any positive integer, then, | = ℎ | For IRD and according to equation (3), we have, For Jeffrey's prior information, we substitute from ℎ | in equation (37) by ℎ | given in equation (10), then in order to evaluate the integral in equation (37), we develop the following formula: The integral in equation (37) should be reduced to:

Bayes estimator of the reliability function R(t) of IRD under precautionary loss function
From equation (34), it is clear that the estimator of the reliability function R(t) under precautionary loss function is given by: | For Jeffrey's prior information, putting k=2 in equation (39), we get: On the basis of exponential prior distribution, putting k=2 in equation (40), we get:

Simulation Study
In our simulation study, L=2000 sample of size n=10, 50, 100 and 200 were generated in order to represent, small, moderate, large and very large sample sizes from inverse Rayleigh distribution with two values of the scale parameter ( 0.5 , 1.5 (.The) scale parameter λ of exponential prior was chosen to be (λ=0.5, λ=1) and assumed the values of r in modified square error loss function to be r=1, r=2 and r=3(. The) criterion mean square error (MSE) was employed to compare the performance of different methods of estimation of the scale parameter and reliability function of IRD where The results are presented in the following tables.

Simulation Results and Conclusions
From our simulation study, we conclude that the performance of the Bayes estimators under square error loss function are the best compared to other estimators in all cases that are included in our study, followed by the estimators under Precautionary loss function in the in the cases presented in Table 2, 3,6. The estimators under modified square error loss function when (r=1) presented in Table 1, 5, 7.