Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring

To save time and cost for a parameter inference, the type-II hybrid censoring scheme has been broadly applied to collect one-component samples. In the current study, one of the essential parameters for comparing two distributions, that is, the stress–strength probability δ=Pr(X<Y), is investigated under a new proposed type-II hybrid censoring scheme that generates the type-II hybrid censored two-component sample from the bivariate normal distribution. The difficult issues occurred from extending the one-component type-II hybrid censored sample to a two-component type-II hybrid censored sample are keeping useful information from both components and the establishment of the corresponding likelihood function. To conquer these two drawbacks, the proposed type-II hybrid censoring scheme is addressed as follows. The observed values of the first component, X, of data pairs (X,Y) are recorded up to a random time τ=max{Xr:n,T}, where Xr:n is the rth ordered statistic among n items with r<n as two pre-specified positive integers and T is a pre-determined experimental time. The observed value from the other component variable Y is recorded only if it is the counterpart of X and also observed before time τ; otherwise, it is denoted as occurred or not at τ. Under the new proposed scheme, the likelihood function of the new bivariate censored data is derived to include the factors of double improper integrals to cover all possible cases without the loss of data information where any component is unobserved. A Monte Carlo Markov chain (MCMC) method is applied to find the Bayesian estimate of the bivariate distribution model parameters and the stress–strength probability, δ. An extensive simulation study is conducted to demonstrate the performance of the developed methods. Finally, the proposed methodologies are applied to a type-II hybrid censored sample generated from a bivariate normal distribution.