A LACK-OF-FIT TEST WITH SCREENING IN SUFFICIENT DIMENSION REDUCTION

摘要

Researchers often need to infer how the conditional mean of a response varies with the predictors. Sufficient dimension-reduction techniques reduce the dimension by identifying a minimal set of linear combinations of the original predictors, without loss of information. This study tests whether a given small number of linear combinations of the original ultrahigh-dimensional covariates is sufficient to characterize the conditional mean of the response. We first introduce a novel consistent lack-of-fit test statistic for the case when the dimensionality of the covariates is moderate. The proposed test is shown to be n-consistent under the null hypothesis, and root-n-consistent under the alternative hypothesis. A bootstrap procedure is developed to approximate the p-values, and the consistency of the test is studied theoretically. To deal with the ultrahigh dimensionality, we introduce a two-stage lack-of-fit test with screening (LOFTS) procedure, based on a data-splitting strategy. The data are randomly partitioned into two equal halves. In the first stage, we apply the martingale difference correlation-based screening to one half of the data, and select a moderate set of covariates. In the second stage, we perform the proposed test, based on the selected covariates, using the second half of the data. The data-splitting strategy is crucial to eliminate the effect of spurious correlations and to prevent an increase in the type-I error rates. We also demonstrate the effectiveness of our two-stage test procedure by means of comprehensive simulations and a real-data application.