Optimal Sign Test for High Dimensional Location Parameters

TL;DR

This paper introduces a family of statistically optimal tests for high-dimensional location parameters, especially when data dimensions exceed sample sizes, using Le Cam's optimality principles and elliptical symmetry assumptions.

Contribution

It proposes a new uniformly optimal test for high-dimensional data based on asymptotic power maximization under elliptical symmetry.

Findings

Asymptotic normality of the proposed tests is established.

The tests are shown to be uniformly optimal across elliptical distributions.

Monte Carlo simulations confirm the theoretical optimality.

Abstract

This article concerns tests for location parameters in cases where the data dimension is larger than the sample size. We propose a family of tests based on the optimality arguments in Le Cam (1986) under elliptical symmetric. The asymptotic normality of these tests are established. By maximizing the asymptotic power function, we propose an uniformly optimal test for all elliptical symmetric distributions. The optimality is also confirmed by a Monte Carlo investigation.