Empirical process of residuals for high-dimensional linear models - Mammen, Enno (The annals of statistics, 1996)
We give a stochastic expansion for the empirical distribution function F^nF^n\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, F^nF^n\hat{F}_n strongly depends on the chosen estimator θ^θ^\hat{θ} of the parameter θθθ of the linear model. In particular, if one uses an ML-estimator θ^MLθ^ML\hat{θ}_{ML} which is ML motivated by a wrongly specified error distribution function G, then F^nF^n\hat{F}_n is biased toward G. For p^2 / 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.