Inference of R = P(Y < X) for two-parameter Rayleigh distribution based on progressively censored samples

Abstract

Based on independent progressively Type-II censored samples from two-parameter Rayleigh distributions with the same location parameter but different scale parameters, the UMVUE and maximum likelihood estimator of R = P (Y < X) are obtained. Also the exact, asymptotic and bootstrap confidence intervals for R are evaluated. Using Gibbs sampling, the Bayes estimator and corresponding credible interval for R are obtained too. Applying Monte Carlo simulations, we compare the performances of the different estimation methods. Finally we make use of simulated data and two real data sets to show the competitive performance of our method. 1 further details on progressive censoring schemes and relevant references, the reader is referred to the book by Balakrishnan and Aggarwala [4].In reliability analysis, a general problem of interest is inference of the stress-strength parameter R = P (Y < X). The stress Y and the strength X are treated as independent random variables. The system fails when the applied stress is greater than the strength. Estimation of the stress-strength parameter has received considerable attention in the statistical literature. These studies started with the pioneering work of Birnbaum [5]. Since then many studies have been accomplished on the estimation and inference of the stress-strength parameter, from the frequentist and Bayesian point of view, by imposing different classes of distributions. The monograph by Kotz et al. [15] provided a comprehensive review of the development of this model till 2003. Further recent work on the stress-strength model can be fined in Kundu and Gupta [17, 18], Raqab and Kundu [23], Krishnamoorthy et al. [16], Raqab et al. [24], Kundu and Raqab [19], Panahi and Asadi [22], Lio and Tsai [21] and Babayi et al. [2].Based on progressively Type-II censored samples, this paper deals with inference for the stress-strength reliability R = P (Y < X) when X and Y are two independent two-parameter Rayleigh distributions with different scale parameters but having the same location parameter. This distribution was originally proposed by Khan et al. [14]. Statistical inference about this distribution was studied by Dey et al. [9]. In the rest of the paper, a two-parameter Rayleigh distribution with the pdf (1) is denoted by tR(µ, λ).In this paper, we study the estimation of the stress-strength parameter R = P (Y < X) when X and Y are independent two-parameter Rayleigh random variables, with common location parameter µ and scale parameters λ > 0 and α > 0, respectively. The probability density functions (pdfs) of X and Y for x > µ and y > µ are;

Reliability Inference for the Multicomponent System Based on Progressively Type II Censored Samples from Generalized Pareto Distributions

“…The stress-strength parameter, denoted by δ = P(Y < X), of a single-component system has been well studied. Kohansal and Rezakhah [3] investigated δ based on progressively type II censored samples. Lio and Tsai [11] studied δ based on progressive first failure-censoring samples from the Burr-XII distributions.…”

“…where α > 0 is the shape parameter and λ > 0 is the scale parameter. The GP distribution with Equations (2) and (3) has been used to investigate δ for a single-component system based on random samples (see Rezaei et al [2]) and progressively type II censored samples (see Rezaei et al [12]). In this study, the reliability of an MSS system is investigated based on the GP distribution defined by Equations (2) and (3).…”

“…Rezaei et al [2] investigated δ utilizing random samples from two independent generalized Pareto (GP) distributions. Kohansal and Rezakhah [3] worked on δ with progressively censored samples from the independent two-parameter Rayleigh distribution, and Akgül andŞenglu [4] considered δ based on two type II right censoring ranked set samples from two independent Weibull distributions.…”

Inference for two‐parameter Rayleigh competing risks data under generalized progressive hybrid censoring

“…Due to the presence of location parameter, the two‐parameter distribution can be used more effectively to analyse real‐life data than the one‐parameter Rayleigh distribution. We refer to Ali Mousa and Al‐Sagheer, 35 Khan et al., 36 Khan, 37 Dey et al., 38,39 Kohansal and Rezakhah, 40 Fundi et al 41 . and Mkolesia et al 42 .…”