Paired sample tests in infinite dimensional spaces

TL;DR

This paper develops and analyzes paired sample nonparametric tests in infinite dimensional spaces, demonstrating their asymptotic advantages over mean-based tests, especially under heavy-tailed distributions.

Contribution

It introduces spatial sign and signed rank tests for infinite dimensional data, deriving their asymptotic distributions and comparing their power to mean-based tests.

Findings

Proposed tests are asymptotically more powerful under heavy-tailed distributions.

Tests outperform mean-based methods for shrinking location shifts.

Simulation studies confirm theoretical advantages.

Abstract

The sign and the signed-rank tests for univariate data are perhaps the most popular nonparametric competitors of the t test for paired sample problems. These tests have been extended in various ways for multivariate data in finite dimensional spaces. These extensions include tests based on spatial signs and signed ranks, which have been studied extensively by Hannu Oja and his coauthors. They showed that these tests are asymptotically more powerful than Hotelling's $T^{2}$ test under several heavy tailed distributions. In this paper, we consider paired sample tests for data in infinite dimensional spaces based on notions of spatial sign and spatial signed rank in such spaces. We derive their asymptotic distributions under the null hypothesis and under sequences of shrinking location shift alternatives. We compare these tests with some mean based tests for infinite dimensional paired sample data. We show that for shrinking location shift alternatives, the proposed tests are asymptotically more powerful than the mean based tests for some heavy tailed distributions and even for some Gaussian distributions in infinite dimensional spaces. We also investigate the performance of different tests using some simulated data.