A review on dimension-reduction based tests for regressions

Abstract

Curse of dimensionality is a big obstacle for constructing efficient goodness-of-fit tests for regression models with large or moderate number of covariates. To alleviate this difficulty, numerous efforts have been devoted in the last two decades. This review intends to collect and comment on the developments in this aspect. To make the paper self-contained, basic ideas on goodness-of-fit testing for regression models are also briefly reviewed, and the main classes of methods and their advantages and disadvantages are presented. Further, the difficulty caused by the dimensionality (number of covariates) is then discussed. The relevant dimension reduction methodologies are presented. Further, as a dedication to Stute's 70th birthday, we also include a section to summarize his great contributions other than the results in dimension reduction-based tests.