An extremal problem with applications to testing multivariate independence

TL;DR

This paper addresses extremal problems related to testing multivariate independence, providing a unified approach to weak convergence of empirical processes and analyzing the optimality of nonparametric independence tests.

Contribution

It introduces a class of extremal problems linked to independence testing and offers a unified method for establishing weak convergence in this context.

Findings

Unified approach to weak convergence for empirical processes

Characterization of local asymptotic optimality domains for tests

Connection between extremal problems and independence testing

Abstract

Some problems of statistics can be reduced to extremal problems of minimizing functionals of smooth functions defined on the cube $[0,1]^m$, $m\geq 2$. In this paper, we study a class of extremal problems that is closely connected to the problem of testing multivariate independence. By solving the extremal problem, we provide a unified approach to establishing weak convergence for a wide class of empirical processes which emerge in connection with testing independence. The use of our result is also illustrated by describing the domain of local asymptotic optimality of some nonparametric tests of independence.