On the joint distribution of placement statistics under progressive censoring and applications to precedence test

In reliability and survival analysis, comparison of two or more populations is an important problem. For example, while comparing a treatment group with a control group, one may be interested in determining whether the observations in the treatment group have a longer lifetime than those from the control group, that is, whether the treatment is effective or not. In such a study, it would be extremely valuable to make a decision based on early failures. In this paper, we consider independent progressively Type-II censored random samples from two populations with cumulative distribution function's (cdf) F(·)$F(·)$ and G(·)$G(·)$ respectively, and discuss a precedence test for testing the equality of the two distributions based on placements. For this purpose, we derive the joint distribution of the first l   placement statistics from the progressively censored sample from F(·)$F(·)$. We then derive the exact null distribution of the precedence test statistic which is simply the sum of the placements. We provide the rejection regions for fixed levels of significance and various sample sizes and different progressive censoring schemes.