Asymptotic relative efficiency of the Kendall and Spearman correlation statistics

TL;DR

This paper establishes conditions under which the asymptotic relative efficiency of Kendall and Spearman correlation tests equals one, providing practical criteria and illustrating their application across various dependence models.

Contribution

It derives broad, easy-to-verify conditions for the asymptotic efficiency of correlation tests, extending Pitman's ARE concept to more general and applicable scenarios.

Findings

Conditions for ARE=1 are characterized in terms of model smoothness and nondegeneracy.

Easy-to-apply sufficient conditions are provided and illustrated on known models.

A generalized version of Pitman's ARE is developed with broader applicability.

Abstract

A necessary and suffcient condition for Pitman's asymptotic relative effciency (ARE) of the Kendall and Spearman correlation statistics for the independence test to be 1 is given, in terms of certain smoothness and nondegeneracy properties of the model. Corresponding easy to use and broadly applicable sufficient conditions are obtained, which are then illustrated on several known models of dependence. Effects of the presence or absence of the smoothness and/or nondegeneracy parts of the mentioned necessary and suffcient condition are demonstrated using certain specially constructed dependence models. A more general (than usual) version of Pitman's ARE is developed, with broader and easier to check conditions of applicability. This version of the ARE, which is then used in the rest of the paper, may also be of value elsewhere.