Conditional independence of blocked ordered data

Abstract

In this paper, we prove that blocks of ordered data formed by some conditioning events are mutually independent. We establish this result by considering the usual order statistics, progressively censored order statistics, and concomitants of order statistics.

On a Correction of the Scale MLE for a Two-Parameter Exponential Distribution Under Progressive Type-I Censoring

“…This property causes a lot of problems in inference and distribution theory so that most results are obtained conditional on the assumption that at least one observation is available. Moreover, from a mathematical point of view, the (conditional) distribution theory is complicated and only a few results have been established in this direction (see, e.g., Balakrishnan 2007;Iliopoulos and Balakrishnan 2009;Balakrishnan et al, 2011;Balakrishnan and Cramer 2014).…”

“…, k}. These random variables have also been considered in Iliopoulos and Balakrishnan (2009) who used them to prove a conditional block independence result. Notice that M = k j =1 D j .…”

Expected termination times of progressively Type‐I censored step‐stress accelerated life tests under continuous and interval inspections

“…It is assumed that without changes in the failure mechanism, the lifetimes of test units follow an exponential distribution at each stress level, along with the accelerated failure time (AFT) model for the effect of changing stress. This results in the conditional block independence of the ordered failure time data as discussed by Balakrishnan and Cramer () and Iliopoulos and Balakrishnan (). Allowing the intermediate censoring to take place at each stress change time point (viz., τ i , i =1,2,…, k ), two different modes of failure inspections are considered: continuous inspection, where the exact failure times are observed, and interval inspection, where the exact failure times are not available but only the number of failures that occurred.…”

“…Of course, f i ( t ) and F i ( t ) are as given in and , respectively. It is worth mentioning that, under the assumption of exponentiality, the AFT model coincides with the cumulative exposure model, which produces the conditional block independence of the ordered failure time data as discussed by Iliopoulos and Balakrishnan () and Balakrishnan and Cramer (). This is a critical property for deriving the expected termination time of a k ‐level step‐stress ALT under progressive Type‐I censoring, as shown in the following sections.…”

One- and Two-Sample Bayesian Prediction Intervals Based on Type-I Hybrid Censored Data

“…For a proof of this result as well as some generalizations of this result, one may refer to Iliopoulos and Balakrishnan (2009).…”