Empirical Bayes estimators for Reliability Function of Inverse Rayleigh Distribution

Abstract

The Inverse Rayleigh distribution (IRD) used in various areas of statistics such as life testing and reliability and failure times of components tested, and healthy areas such as life times (in months) of bladder cancer patients. We used the classical method which is represented by the maximum likelihood estimation (MLE) and Bayes approach for the Reliability function for the Inverse Rayleigh distribution (IRD). We assumed that the scale parameter of the IRD is random variable, and we have two different informative prior distributions for the scale parameter of the IRD. So, we try to determine the suitable choice of informative prior distributions in bayes analysis for the Reliability function of the IRD at time , which are represented by Inverse Gamma - Inverse Chi-square, Inverse Gamma - Standard Levey, Inverse Chi-square - Standard Levy distribution as double prior distributions. We derived the posterior density for the scale parameter of the IRD under three different double informative priors and we derived bayes’ estimators for the Reliability function of IRD at timebased on Squared Error Loss Function (SELF).A simulation study performed to obtain the empirical results of this study, we assumed different cases for the true values of the scale parameter of an IRD under different double prior distributions for different sample sizes(n). A comparison is made between the results based on the minimum Mean Square Error (MSE) criterion. We concluded  that the best estimation for  the Reliability function when the true value for scale parameter is equal 0.5, under the double prior is the Inverse Gamma and the Inverse Chi-square distribution with  at time for sample sizes equal to 25, 50 and 100 and the maximum likelihood estimators (MLE’s) when the true value for scale parameter is equal 1.5 and 2.5 at time for small sample sizes. We recommend to use the double prior of the Inverse Gamma and the Inverse Chi-square distribution, to estimate reliability function using bayes analysis of the reliability function of IRD based on loss functions such as quadratic loss function and he general entropy loss function and the modified linear exponential (MLINEX) loss function to compare the accuracy of the different estimates.