A central limit theorem applicable to robust regression estimators

Abstract

Consider a general linear model, Yi=x′iβ+Ri with R1, ..., Rn i.i.d., β∈Rp, and {x1, ..., xn} behaving like a random sample from a distribution in Rp. Let β̂ be a robust M-estimator of β. To obtain an asymptotic normal approximation for the distribution of β̂ requires a Central Limit Theorem for Wn = Σyiψ(Ri), where yi = (X′X)-1xi. When p→∞, previous results require p5 n→0, but here a strong normal approximation for the distribution of Wn in Rp is provided under the condition (plogn)3 2n→0. © 1987.