The Asymptotic Properties of the One-Sample Spatial Rank Methods

TL;DR

This paper investigates the asymptotic properties of multivariate median estimators and sign tests, focusing on the spatial Hodges-Lehmann estimator and its behavior as sample size and dimension grow large.

Contribution

It provides new theoretical results on the asymptotic distribution of the spatial HL-estimator and related test statistic, including cases where both sample size and dimension increase.

Findings

Asymptotic distribution of the spatial HL-estimator is derived.

Behavior of the HL-estimator when both sample size and dimension tend to infinity.

Some results are established for the first time.

Abstract

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.