Locally most powerful rank tests of independence for copula models

Abstract

A formula is given for the statistic that provides the locally most powerful rank test of independence against alternatives expressed by copula models. The Savage, Spearman and van der Waerden statistics are seen to be optimal in special cases of interest. The asymptotic relative efficiency (ARE) of any linear rank procedure with respect to the optimal test is expressed as a squared correlation in which the bivariate dependence structure of the data only enters through the copula. In contrast, the margins are shown to influence the ARE of the best rank test, compared to the standard test of independence based on Pearson’s correlation. An extensive simulation study is used to assess the effect of both the margins and the dependence structure on the power of several parametric and nonparametric procedures in small samples from a variety of bivariate distributions.

Keywords:

• Asymptotic relative efficiency
• Blest’s coefficient
• Copula
• Dependence
• Kendall’s tau
• Log-rank test
• Pearson’s correlation
• Power
• Savage statistic
• Spearman’s rho
• Test of independence
• van der Waerden statistic

Acknowledgements

This work was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and from the Fonds québécois de la recherche sur la nature et les technologies.