Testing distributional equality for functional random variables

TL;DR

This paper introduces a nonparametric permutation test for assessing distributional equality of functional data in a Hilbert space, with proven consistency, efficiency, and demonstrated superior performance through simulations and real data analysis.

Contribution

It develops a novel energy-based measure and a permutation test for two-sample problems involving functional data, with theoretical properties and efficiency analysis.

Findings

The test is consistent under general conditions.

It is statistically efficient in the Pitman sense.

Simulation studies show superior performance over existing methods.

Abstract

In this article, we present a nonparametric method for the general two-sample problem involving functional random variables modelled as elements of a separable Hilbert space ${\cal H}$. First, we present a general recipe based on linear projections to construct a measure of dissimilarity between two probability distributions on ${\cal H}$. In particular, we consider a measure based on the energy statistic and present some of its nice theoretical properties. A plug-in estimator of this measure is used as the test statistic to construct a general two-sample test. Large sample distribution of this statistic is derived both under null and alternative hypotheses. However, since the quantiles of the limiting null distribution are analytically intractable, the test is calibrated using the permutation method. We prove the large sample consistency of the resulting permutation test under fairly general assumptions. We also study the efficiency of the proposed test by establishing a new local asymptotic normality result for functional random variables. Using that result, we derive the asymptotic distribution of the permuted test statistic and the asymptotic power of the permutation test under local contiguous alternatives. This establishes that the permutation test is statistically efficient in the Pitman sense. Extensive simulation studies are carried out and a real data set is analyzed to compare the performance of our proposed test with some state-of-the-art methods.