Solution to Dalal and Mallows conjecture on monotone property of the joint distribution of order statistics

Abstract

The joint distributions of order statistics play a critical role in constructing some testing procedures e.g. establishing a stopping rule of testing software faults and determining the critical values of step-up multiple test procedures. Despite the well development of those procedures, whether the set of critical values derived by the joint distribution of order statistics has monotone property remains an open question in most procedures. Dalal and Mallows (Ann. Appl. Probab. 2 (1992) 752–765) provided a theorem to show the monotone property in one particular testing procedure and conjectured the property in the other procedure based on an extensive numerical study. In this paper, we would state and prove a more general theorem such that their theorem and the conjecture are just two special cases.