A Goodness-of-Fit Test of Errors in High-Dimensional Sparse Linear Regression

The underlying distributions of random errors play an essential role in statistical inferences of regression models. The goodness-of-fit test on errors is especially important in high-dimensional regression settings because error distributions can also influence model selections. In this paper, we consider the goodness-of-fit test on errors in high-dimensional linear regression models. Under suitable assumptions of model sparsity and a sure screening property, we show that the Bickel-Rosenblatt-type test statistic, based on residuals from the refitted cross-validation procedure, has an asymptotic normal distribution, both under the null hypothesis and a fixed alternative.