  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.
