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.

翻译：对于一组$p$维数据点$\boldsymbol y_1,\ldots,\boldsymbol y_n$，文献中已提出并研究了多种版本的多元中位数及其相关的多元符号检验。本文考虑Hodges-Lehmann（HL）估计量的多元扩展、空间HL估计量及其相关检验统计量的渐近性质。我们收集、综述并证明了当$n$趋于无穷大时空间HL估计量及相关检验统计量的渐近行为，其中部分结论尽管已被长期使用，但系首次得到证明。此外，我们还推导了当样本容量$n$和维度$p$同时趋于无穷大时HL估计量的极限行为。