Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation

Abstract

In a general linear model, Y = Xβ + R with Y and R n-dimensional, X a n × p matrix, and β p-dimensional, let $\hat\beta$ be an M estimator of β satisfying 0 = ∑ xiψ(yi - x'iβ). Let p → ∞ such that (p log n)3/2 /n → 0. Then maxi|x'i(β̂ - β)| → P 0, and it is possible to find a uniform normal approximation for the distribution of β̂ under which arbitrary linear combinations a'n (β̂ - β) are asymptotically normal (when appropriately normalized) and (β̂ - β)'(X'X)(β̂ - β) is approximately χ2 p.