Asymptotics of the two-stage spatial sign correlation

TL;DR

This paper proves the asymptotic normality of the two-stage spatial sign correlation estimator, providing theoretical validation and deriving variance-stabilizing transformations for robust bivariate correlation analysis.

Contribution

It offers the first formal proof of the asymptotic distribution of the two-stage spatial sign correlation and derives its asymptotic variance for elliptical distributions.

Findings

Asymptotic normality of the two-stage estimator is established.

Derived a variance-stabilizing transformation similar to Fisher's z-transform.

Numerical comparison of confidence interval coverage probabilities.

Abstract

The spatial sign correlation (D\"urre, Vogel and Fried, 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. D\"urre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation, similar to Fisher's z-transform, and numerically compare the small-sample coverage probabilities of several confidence intervals.