Symmetrisation of a class of two-sample tests by mutually considering depth ranks including functional spaces

TL;DR

This paper enhances a two-sample test based on statistical depth functions, making it symmetric and applicable to functional data, with proven asymptotic properties and demonstrated effectiveness through simulations and real data analysis.

Contribution

It introduces a symmetrized test statistic based on depth ranks, improving power and extending applicability to functional data.

Findings

The proposed test is asymptotically valid under the null hypothesis.

Simulation studies show improved power over existing methods.

Real data analysis demonstrates practical applicability to temperature curves.

Abstract

Statistical depth functions provide measures of the outlyingness, or centrality, of the elements of a space with respect to a distribution. It is a nonparametric concept applicable to spaces of any dimension, for instance, multivariate and functional. Liu and Singh (1993) presented a multivariate two-sample test based on depth-ranks. We dedicate this paper to improving the power of the associated test statistic and incorporating its applicability to functional data. In doing so, we obtain a more natural test statistic that is symmetric in both samples. We derive the null asymptotic of the proposed test statistic, also proving the validity of the testing procedure for functional data. Finally, the finite sample performance of the test for functional data is illustrated by means of a simulation study and a real data analysis on annual temperature curves of ocean drifters is executed.