Model-Free Statistical Inference on High-Dimensional Data

TL;DR

This paper introduces a new model-free statistical inference method for high-dimensional data, utilizing a reformulation via sufficient dimension reduction, a novel chi-squared test statistic, and a multiple testing procedure with theoretical guarantees.

Contribution

It develops a novel model-free inference framework for high-dimensional data, including a chi-squared test and multiple testing procedure with proven control of false discovery rate.

Findings

The proposed test statistic follows a chi-squared distribution asymptotically.

The multiple testing procedure effectively controls false discovery rate.

Simulation and real data analysis demonstrate the method's practical utility.

Abstract

This paper aims to develop an effective model-free inference procedure for high-dimensional data. We first reformulate the hypothesis testing problem via sufficient dimension reduction framework. With the aid of new reformulation, we propose a new test statistic and show that its asymptotic distribution is $\chi^2$ distribution whose degree of freedom does not depend on the unknown population distribution. We further conduct power analysis under local alternative hypotheses. In addition, we study how to control the false discovery rate of the proposed $\chi^2$ tests, which are correlated, to identify important predictors under a model-free framework. To this end, we propose a multiple testing procedure and establish its theoretical guarantees. Monte Carlo simulation studies are conducted to assess the performance of the proposed tests and an empirical analysis of a real-world data set is used to illustrate the proposed methodology.