A sufficient criterion for control of generalised error rates in multiple testing

TL;DR

This paper introduces a new criterion for controlling generalized error rates in multiple testing scenarios with dependent p-values, enabling optimization of testing procedures through linear programming.

Contribution

It provides a sufficient criterion based on matrices for controlling error rates under dependence, allowing for improved power via linear programming solutions.

Findings

Criterion applicable to arbitrarily dependent p-values

Potential to enhance existing procedures' power

Framework for optimizing critical constants

Abstract

Based on the work of Romano and Shaikh (2006) and Lehmann and Romano (2005) we give a sufficient criterion for controlling generalised error rates for arbitrarily dependent p-values. This criterion is formulated in terms of matrices associated with the corresponding error rates and thus it is possible to view the corresponding critical constants as solutions of sets of certain linear inequalities. This property can in some cases be used to improve the power of existing procedures by finding optimal solutions to an associated linear programming problem.