On probabilities for separating sets of order statistics

TL;DR

This paper derives formulas for the probability that order statistics from two populations fall into specific intervals and applies these results to analyze the false discovery rate in multiple testing procedures.

Contribution

It introduces a method to compute probabilities involving order statistics from two different distributions and applies it to the Benjamini-Hochberg FDR procedure.

Findings

Formulas for probabilities of order statistics from two populations

Application to FDR control in multiple testing

Insights into the distribution of false rejections

Abstract

Consider a set of order statistics that arise from sorting samples from two different populations, each with their own, possibly different distribution function. The probability that these order statistics fall in disjoint, ordered intervals, and that of the smallest statistics, a certain number come from the first populations, are given in terms of the two distribution functions. The result is applied to computing the joint probability of the number of rejections and the number of false rejections for the Benjamini-Hochberg false discovery rate procedure.