A Supremum-Norm Based Test for the Equality of Several Covariance Functions

TL;DR

This paper introduces a new supremum-norm based statistical test for comparing multiple covariance functions in functional data, demonstrating improved power over existing methods through theoretical analysis and simulations.

Contribution

It proposes a novel supremum-norm based test statistic for equality of covariance functions, with asymptotic properties, bootstrap approximations, and superior performance in correlated data.

Findings

Test is more powerful than existing methods in simulations.

The test is root-n consistent under local alternatives.

Bootstrap methods effectively approximate the null distribution.

Abstract

In this paper, we propose a new test for the equality of several covariance functions for functional data. Its test statistic is taken as the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function, hoping to obtain a more powerful test than some existing tests for the same testing problem. The asymptotic random expression of this test statistic under the null hypothesis is obtained. To approximate the null distribution of the proposed test statistic, we describe a parametric bootstrap method and a non-parametric bootstrap method. The asymptotic random expression of the proposed test is also studied under a local alternative and it is shown that the proposed test is root-$n$ consistent. Intensive simulation studies are conducted to demonstrate the finite sample performance of the proposed test and it turns out that the proposed test is indeed more powerful than some existing tests when functional data are highly correlated. The proposed test is illustrated with three real data examples.