Statistical Inference for Ultrahigh Dimensional Location Parameter Based on Spatial Median

TL;DR

This paper develops statistical inference methods for the spatial median in ultrahigh dimensional settings, enabling confidence intervals, hypothesis testing, and multiple testing with FDR control, supported by theoretical guarantees and practical algorithms.

Contribution

It introduces a novel Bahadur representation and Gaussian approximation for the sample spatial median in ultrahigh dimensions, along with a bootstrap method for inference.

Findings

Valid inference when dimension grows exponentially with sample size

Effective construction of confidence intervals and tests in high dimensions

Successful application to genomic microarray data

Abstract

Motivated by the widely used geometric median-of-means estimator in machine learning, this paper studies statistical inference for ultrahigh dimensionality location parameter based on the sample spatial median under a general multivariate model, including simultaneous confidence intervals construction, global tests, and multiple testing with false discovery rate control. To achieve these goals, we derive a novel Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term, and establish Gaussian approximation for the sample spatial median over the class of hyperrectangles. In addition, a multiplier bootstrap algorithm is proposed to approximate the distribution of the sample spatial median. The approximations are valid when the dimension diverges at an exponentially rate of the sample size, which facilitates the application of the spatial median in the ultrahigh dimensional region. The proposed approaches are further illustrated by simulations and analysis of a genomic dataset from a microarray study.