Computing the nonnull asymptotic variance and the asymptotic relative efficiency of Spearman's rank correlation

Abstract

Over the past century, Spearman's rank correlation, ρ s, has become one of the most commonly used nonparametric statistics, yet much remains unknown about its finite and asymptotic behavior. This paper presents a method for computing the asymptotic variance of the point estimate, ρ ̂ s , in terms of expectations of the joint and marginal distribution functions, for any underlying bivariate distribution that satisfies minimal regularity conditions. Also presented are numerical results for certain bivariate distributions of interest in order to demonstrate that distributions with similar values of Pearson's or Spearman's correlations can yield surprisingly different values for the asymptotic variance of ρ ̂ s . In particular, these results emphasize that one should not use certain standard procedures for hypothesis testing and confidence interval construction that assume bivariate normality without first checking this distributional assumption. Finally, these numerical results are used to compute the asymptotic relative efficiency of Spearman's rank correlation compared to Pearson's correlation.