Asymptotic local efficiency of Cram\'{e}r--von Mises tests for multivariate independence

TL;DR

This paper analyzes the asymptotic local efficiency of Cramér--von Mises tests for multivariate independence, providing a theoretical framework for their limiting distribution and efficiency comparisons under various copula alternatives.

Contribution

It offers a new representation of the limiting distribution and efficiency of Cramér--von Mises based tests, extending previous work with a focus on local asymptotic behavior.

Findings

Derived the limiting distribution of the test statistics.

Compared local power and efficiency under different copula models.

Provided insights into the test's performance under contiguous alternatives.

Abstract

Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and R\'{e}millard [Test 13 (2004) 335--369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cram\'{e}r--von Mises statistics derived from a M\"{o}bius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives.