Estimating E-Bayesian and Hierarchical Bayesian for R=P(X>Y) of the Weibull distribution

Authors
Shahram Yaghoobzade Shahrestani
Iman Makhdoom
Abstract
Introduction and Method:

Various approaches have been presented to estimate statistical distribution parameters. Bayesian estimation such as Emperical Bayesian (E-Bayesian) and Hierarchical Bayesian Estimation (HBE) based on prior distribution is an approach in which the error of posterior bayesian estimation reduces significantly using a proper prior for the parameter distribution. Sometimes the wide range of parameter space increases the error and inflates the comparison criteria. Therefore, defining the appropriate prior distribution and applying special conditions on the parameter space has an important role in reducing the comparison criteria. The HBE was first introduced by Lindly and smith (1972) and develped by Han (1977). Recently some estimation approaches have been introduced such as estimation of the exponential distribution parameter, estimating the binomial distribution parameter by Han (2011, 2009), parameter estimation and credible function of Boor 12 distribution based on increasingly second type of censorship samples by Jahin and Akasha (2011), Pascal distribution parameter estimation by Wang And Chen (2012). The use of HBE in data analysis was also demonstrated by several authors such as Mitch and Wickel (2009), Chair and Tingle (2010), Ando and Zelner (2010), Ossie and Ducer (2011) and Richard (2011).

Estimating the credible parameter or stress-resistance parameter, R = P (X> Y), which shows the efficiency of a system, is one of the important issues in inferential statistisl and is applicalble in in various sciences such as life expectancy theory, and mechanical reliability of a system or structures, missile engine and aircraft systems in engineering.

Many authors estimate the R parameter in the case when X and Y are independent random variables with the same distribution such as in bivariable exponential distribution by Avad et al. (1981), normal multivariate distribution by Gupta and Gupta (1990), in Bohr distribution 12 by Rokab and Kando (2005), in generalized exponential distribution by Kando and Gupta (2005), in the three-parameter eponential distribution by Rakb et al. (2008), in the generalized exponential distribution based on record samples by Baklizi (2008), in the Weibull distribution based on increasing censorship samples of the second type by Asgharzadeh et al. (2011), in the exponential distribution based on the increasingly second type of censorship samples by Sarakgloo et al. (2012), in the type 12 boron distribution based on the increasingly second type of censored samples by Liu and Tsai (2012), in the Lindley distribution by L-Motairi et al. (2013), and in the distribution of Lindley Towani examined by Gitani et al. (2015).

It is suggested to estimate the R parameter via different approaches regarding its application in various areas including survival analysis, weather forecast, and Reliability engineering. The current paper assumes a Weibull distribution with α and β as the parameters and utilizes E-Bayed and HBE to estimate the R parameter. Later, we estimate the R=P(X>Y) using E-Bayed and HBE where X~W(α.β1) and Y~W(α.β2) under squared error and entropy loss functions. Finally, the E-Bayed and HBE estimates are compared using a Monte Carlo simulation study.


Results and Discussion:



We estimated the E-Bayes and HBE estimates of R where the X and Y random variables followed Weibull distributions with the same shape parameter and different scalar under squared error and entropy loss functions. The results were compared by Monte Carlo simulation and it was revealed that the HBE estimate given squared error loss function and E-Bayes given entropy loss function outperform other approaches.

The simulation studies also showed that The HBE performs better than E-Bayes given the squared eror and entroy loss functions. It is strightforward to estimate R in exponential and Rayleigh distributions using the estimated results from Weibull where α=1 and α=2.
Keywords
Weibull distribution
E-Bayesian estimation
Hierarchical Bayesian estimation
Squared error loss function
General entropy loss function
Monte Carlo simulation
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Mathematical Research
Volume 7, Issue 4 - Serial Number 19
Spring 2021
Pages 912-931