The Asymptotic Properties of the One-Sample Spatial Rank Methods

For a set of $p$-variate data points $\boldsymbol y_1,\ldots,\boldsymbol y_n$, there are several versions of multivariate median and related multivariate sign test proposed and studied in the literature. In this paper we consider the asymptotic properties of the multivariate extension of the Hodges-Lehmann (HL) estimator, the spatial HL-estimator, and the related test statistic. The asymptotic behavior of the spatial HL-estimator and the related test statistic when $n$ tends to infinity are collected, reviewed, and proved, some for the first time though being used already for a longer time. We also derive the limiting behavior of the HL-estimator when both the sample size $n$ and the dimension $p$ tend to infinity.

💡 Research Summary

This paper provides a comprehensive treatment of the asymptotic behavior of the multivariate spatial Hodges‑Lehmann (HL) estimator and its associated signed‑rank test statistic. Beginning with a review of the spatial median, the authors recall that the spatial median µ̂_SM solves the estimating equation Σ_i u(y_i‑µ)=0, where u(y)=y/‖y‖ for non‑zero y. Under mild regularity conditions (continuous, bounded density near the origin and uniqueness of the median), Theorem 1 establishes almost‑sure convergence of µ̂_SM to the true median, while Theorem 2 shows √n‑consistency and asymptotic normality with covariance A⁻¹BA⁻¹, where A=E