Empirical process of residuals for high-dimensional linear models

Abstract

We give a stochastic expansion for the empirical distribution function $\hat{F}_n$ of residuals in a p-dimensional linear model. This expansion holds for p increasing with n. It shows that, for high-dimensional linear models, $\hat{F}_n$ strongly depends on the chosen estimator $\hat{\theta}$ of the parameter $\theta$ of the linear model. In particular, if one uses an ML-estimator $\hat{\theta}_{ML}$ which is ML motivated by a wrongly specified error distribution function G, then $\hat{F}_n$ is biased toward G. For p^2 / n \to \infty$, this bias effect is of larger order than the stochastic fluctuations of the empirical process. Hence, the statistical analysis may just reproduce the assumptions imposed.