Conditional Mean and Quantile Dependence Testing in High Dimension

• Activity Number: 631 - Recent Developments in Statistical Inference Using Distance Correlation and Related Dependence Metrics
• Type: Invited
• Date/Time: Thursday, August 3, 2017 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Xianyang Zhang* and Shun Yao and Xiaofeng Shao
• Companies: Texas A&M University and University of Illinois at Urbana-Champaign and University of Illinois, At Urbana-Champaign
• Keywords: Large-p-small-n ; Martingale difference divergence ; Simultaneous test ; U-statistics

Abstract

Motivated by applications in biological science, we propose a novel test to assess the conditional mean dependence of a response variable on a large number of covariates. Our procedure is built on the martingale difference divergence recently proposed in Shao and Zhang (2014), and it is able to detect certain type of departure from the null hypothesis of conditional mean independence without making any specific model assumptions. Theoretically, we establish the asymptotic normality of the proposed test statistic under suitable assumption on the eigenvalues of a Hermitian operator, which is constructed based on the characteristic function of the covariates. To account for heterogeneity within the data, we further develop a testing procedure for conditional quantile independence at a given quantile level and provide an asymptotic justification. Empirically, our test of conditional mean independence delivers comparable results to the competitor, which was constructed under the linear model framework, when the underlying model is linear. It significantly outperforms the competitor when the conditional mean admits a nonlinear form.