Analysis of the progressively Type-II right censored data with dependent random removals

In the progressive Type-II censoring with random removal, choosing a distribution for removing the number of unfailed units which is independent from the lifetime distribution, may be a serious challenge and lead to an unreliable inference. Two approaches based on the time distance between consecutive failure times are proposed to determine the removal numbers. Then, the joint probability mass functions of random removals are derived where the underlying distribution of the lifetime data is assumed to be an exponential. Furthermore, The expected test time is investigated under these two approaches and is subsequently considered as an optimality criterion for identifying progressive Type-II censoring schemes. Some statistical inference results are also obtained by using Monte Carlo simulations. The empirical results indicate that the proposed methods work perfectly well in the sense that, reduce the expected total time on the test. Finally, a numerical example, based on a real data set, is presented and analysed to illustrate the proposed approaches.