Goodness-of-fit tests for high dimensional linear models

摘要

Given the complexity in dealing with high dimensional data where the number of variables may exceed the number of observations, there are methods available for fitting models to such high dimensional data. The articles provides a framework for constructing goodness-of-fit tests in both low and high dimensional linear models. An approach is introduced for creating diagnostic measures or goodness-of-fit tests that are sensitive to different sorts of departures from standard high dimensional linear models. Since the measures are derived from examining residuals, the proposed tests are named residual prediction tests. Wwhile simulations are used to obtain the critical values, some numerical studies are presented to demonstrate the use of parametric bootstrap in dealing with high dimensional data. It is observed that the proposed approach is the first methodology for deriving confirmatory statistical conclusions to test for a broad range of deviations from a high dimensional linear model. The proposed method can be used to test for significance of groups or individual variables as special cases. The residual prediction tests are found to be appropriate for testing diverse model misspecifications as heterogeneity and non-linearity.