Bayesian inference for Rayleigh distribution under hybrid censoring

Abstract

In this paper, and based on a hybrid censored sample from a Rayleigh distribution, Bayes estimators and highest posterior density credible intervals are obtained for the unknown parameter, and some lifetime parameters such as the reliability and hazard rate functions. Bayes estimators are obtained using squared error and linear–exponential loss functions. We also obtain the Bayes predictive estimator and %95 prediction interval for future observations. Finally, a numerical example is given to illustrate the application of the results and Monte Carlo simulations are performed to compare the performances of the different methods.

Appendix

Here we discuss the unimodality condition for the posterior density of $\theta$ given data. By considering the square root inverted gamma as a prior distribution for $\theta$, the posterior density of $\theta$ given data is

$$\pi (\theta \mid \underline{x}) \propto \theta ^{-2(b+d)-1} \exp \left\{ -\frac{1}{2\theta ^2}\left[ \sum _{i=1}^{d}x_i^2+(n-d)T_0^2+a\right] \right\} , \quad \theta >0.$$

We have $\pi (0\mid \underline{x})=\pi (\infty \mid \underline{x})=0$. To show that $\pi (\theta \mid \underline{x})$ is unimodal, it is enough to show it is log concave. We have

$$\frac{d}{d \theta } log \pi (\theta \mid \underline{x})=-\frac{2(b+d)+1}{\theta }+\frac{\sum _{i=1}^{d}x_i^2+(n-d)T_0^2+a}{\theta ^{3} },$$

and

$$\frac{d^{2} }{d \theta ^{2} } log \pi (\theta \mid \underline{x})=\frac{2(b+d)+1}{\theta ^{2} }-3\frac{\sum _{i=1}^{d}x_i^2+(n-d)T_0^2+a}{\theta ^{4} }.$$

So, the unimodality condition is summarized to $\frac{d^{2} }{d \theta ^{2} } log \pi (\theta \mid \underline{x})\le 0$, or

$$3\left\{ \sum _{i=1}^{d}x_i^2+(n-d)T_0^2+a\right\} \ge \left[ 2(d+b)+1\right] \theta ^2.$$

Therefore, when this condition achieves, we can calculate the HPD credible intervals.