Empirical process of residuals for high-dimensional linear models

Abstract

We give a stochastic expansion for the empirical distribution function F̂n of residuals in a p-dimensional linear model. This expansion holds for p increasing with n. It shows that, for high-dimensional linear models, F̂n strongly depends on the chosen estimator θ̂ of the parameter θ of the linear model. In particular, if one uses an ML-estimator θ̂ML which is motivated by a wrongly specified error distribution function G, then F̂n is biased toward G. For p2/n → ∞ 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.