Model-Free Statistical Inference on High-Dimensional Data

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.

Keywords: False discovery rate control; Marginal coordinate hypothesis; Orthogonality; Sufficient dimension reduction.

Fig. 1

The plot of empirical size and power with respect to $h$ when the Lasso penalty is used. “t1” and “t2” represent randomly selected inactive variables.