The Asymptotic Efficiency of Daniels’s Generalized Correlation Coefficients

Abstract

SUMMARY The asymptotic efficiency of the generalized correlation coefficient devised by Daniels, used as a test of association in the bivariate function H(x, y) = F(x) G(y) {1 + αA(x) B(y)}, is shown to depend in a simple way on the score-matrices used to define the coefficient. Two coefficients for which the corresponding row sums of the two score-matrices, written in a standard form, are proportional are shown to have the same asymptotic efficiency. In particular, this is shown to be true for Spearman’s rank correlation coefficient and Kendall’s alternative coefficient, τ. For alternatives of the above type, maximum efficiency can be obtained within Daniels’s family of coefficients; for a more general class of alternatives a weighted combination of Daniels’s coefficients is necessary to attain maximum asymptotic efficiency.