Asymptotic Behavior of M Estimators of p Regression Parameters when p² / n is Large; II. Normal Approximation

In a general linear model, Y = Xβ + R with Y and $R n$-dimensional, X a n × p matrix, and β p-dimensional, let β 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 χ²ₚ.