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Functional $K$ Sample Problem via Multivariate Optimal Measure Transport-Based Permutation Test

Šárka Hudecová, Daniel Hlubinka, Zdeněk Hlávka

Abstract

The null hypothesis of equality of distributions of functional data coming from $K$ samples is considered. The proposed test statistic is multivariate and its components are based on pairwise Cramér von Mises comparisons of empirical characteristic functionals. The significance of the test statistic is evaluated via the novel multivariate permutation test, where the final single $p$-value is computed using the discrete optimal measure transport. The methodology is illustrated by real data on cumulative intraday returns of Bitcoin.

Functional $K$ Sample Problem via Multivariate Optimal Measure Transport-Based Permutation Test

Abstract

The null hypothesis of equality of distributions of functional data coming from samples is considered. The proposed test statistic is multivariate and its components are based on pairwise Cramér von Mises comparisons of empirical characteristic functionals. The significance of the test statistic is evaluated via the novel multivariate permutation test, where the final single -value is computed using the discrete optimal measure transport. The methodology is illustrated by real data on cumulative intraday returns of Bitcoin.

Paper Structure

This paper contains 4 sections, 9 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multivariate Test Statistic for the $K$ Sample Problem
  3. Permutation Test Based on the Optimal Measure Transport
  4. Real Data Application

Figures (2)

  • Figure 1: Logarithmic cumulative intraday returns for Bitcoin in 2022: Mondays (52 curves), Wednesdays (52 curves), and Saturdays (53 curves).
  • Figure 2: Left panel: The test statistic $\boldsymbol{T}^\text{inv}$ (larger square) together with $B$ permutation replicas (small circles). All values multiplied by $10^4$. Right panel: The grid $\mathcal{G}$ with highlighted point $F^*(\boldsymbol{T}_0^{\text{inv}})$.

Theorems & Definitions (1)

  • remark 1