Estimation of stress–strength reliability for Marshall–Olkin distributions based on progressively Type-II censored samples

We are mainly interested in estimating the stress–strength parameter, say $ \mathcal {R} $ R, when the parent distribution follows the well-known Marshall–Olkin model and the accessible data has the form of the progressively Type-II censored samples. In this case, the stress–strength parameter is free of the base distribution employed in the Marshall–Olkin model. Thus, we use the exponential distribution for simplicity. The maximum likelihood methods as well as some Bayesian approaches are used for the estimation purpose. The corresponding estimators of the latter approach are obtained by using Lindley's approximation and Gibbs sampling methods since the Bayesian estimator of $ \mathcal {R} $ R cannot be obtained as an explicit form. Moreover, some confidence intervals of various types are derived for $ \mathcal {R} $ R and then compared via a Monte Carlo simulation. Finally, the survival times of head and neck cancer patients are analyzed by two therapies for illustrating purposes.