Hypothesis tests for structured rank correlation matrices

TL;DR

This paper develops hypothesis tests for structured correlation matrices based on Kendall's rank correlation, focusing on partial exchangeability, with theoretical analysis, scalable methods, and real data application.

Contribution

It introduces new tests for structured correlation matrices, especially under partial exchangeability, with improved computational efficiency and practical implementation strategies.

Findings

Tests perform well under simulations and local alternatives.

Simplified distributions under partial exchangeability improve computational efficiency.

Application to sea level data demonstrates practical utility.

Abstract

Joint modeling of a large number of variables often requires dimension reduction strategies that lead to structural assumptions of the underlying correlation matrix, such as equal pair-wise correlations within subsets of variables. The underlying correlation matrix is thus of interest for both model specification and model validation. In this paper, we develop tests of the hypothesis that the entries of the Kendall rank correlation matrix are linear combinations of a smaller number of parameters. The asymptotic behavior of the proposed test statistics is investigated both when the dimension is fixed and when it grows with the sample size. We pay special attention to the restricted hypothesis of partial exchangeability, which contains full exchangeability as a special case. We show that under partial exchangeability, the test statistics and their large-sample distributions simplify, which leads to computational advantages and better performance of the tests. We propose various scalable numerical strategies for implementation of the proposed procedures, investigate their behavior through simulations and power calculations under local alternatives, and demonstrate their use on a real dataset of mean sea levels at various geographical locations.