Asymptotic Behavior of the Empiric Distribution of M-Estimated Residuals from a Regression Model with Many Parameters

Consider a regression model Yᵢ=x'ᵢ+Rᵢ, i=1, , n, where Rᵢ are i. i. d. with c. d. f. , F; xᵢᵖ and ᵖ. Let be a M-estimator defined using kernel, ; let Fₙ (x) denote the empiric distribution of the residuals, Yᵢ-x'ᵢ and let F^ₙ be the empiric c. d. f. of the errors, Rᵢ. Under suitable smoothness conditions on, F, and the density F' = f and conditions requiring essentially that xᵢ behave like a random sample from some distribution in Rₚ, it is show that, for fixed x, n (Fₙ (x) - F^ₙ (x) -Hₙ (x) ) -Pn g (x) ₚ 0, where g (x) = af (x) (x) + bf' (x) and Hₙ (x) = (1/nd) f (x) ⁿ₈=₁ (Rᵢ) if the design has a constant term and Hₙ (x) vanishes otherwise. A tightness result shows that if p/ n c n (Fₙ (x) -F (x) ) converges weakly to a Gaussian process with drift given by the bias term cg (x), and covariance function strongly affected by Hₙ (x) and different from that for the usual Brownian bridge. In the course of the proof, an expansion for the fitted values, x'ᵢ, is obtained, with error Oₚ (p^11/4ln²n/n²) =oₚ (1/n) if p²/n is bounded.