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In this article, an explicit form of the stress-strength reliability R = P(X < Y ) is introduced when X and Y are independent random variables belonging to Marshall-Olkin extended Weibull family. Also a characterization of the parent distributions associated with R is presented. Based on Type-II progressive censoring with fixed and random number of removals, maximum likelihood and Bayesian estimators of the parameter R are obtained. Two distributions for the random number of removals are considered, namely discrete uniform and binomial distributions. Using informative and non-informative priors, the Bayesian estimation is discussed under two different loss functions: the squared error loss function (SELF) and linear exponential loss functions (LINEX). A numerical illustration is performed to highlight the theoretical results that are obtained. Also a real data example is provided.

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