Variance Breakdown of Huber ( M )-estimators : n / p → m ∈ ( 1 , ∞ )

Huber’s gross-errors contamination model considers the class Fε of all noise distributions F = (1 − ε)Φ + εH, with Φ standard normal, ε ∈ (0, 1) the contamination fraction, and H the contaminating distribution. A half century ago, Huber evaluated the minimax asymptotic variance in scalar location estimation, min ψ max F∈Fε V (ψ, F ) = 1 I(F ∗ ε ) (1) where V (ψ,F ) denotes the asymptotic variance of the (M)-estimator for location with score function ψ, and I(F ∗ ε ) is the minimal Fisher information minFε I(F ). We consider the linear regression model Y = Xθ0 +W , Wi ∼i.i.d. F , and iid Normal predictors Xi,j , working in the high-dimensional-limit asymptotic where the number n of observations and p of variables both grow large, while n/p→ m ∈ (1,∞); hence m plays the role of ‘asymptotic number of observations per parameter estimated’. Let Vm(ψ,F ) denote the per-coordinate asymptotic variance of the (M)-estimator of regression in the n/p → m regime [EKBBL13, DM13, Kar13]. Then Vm 6= V ; however Vm → V as m→∞. In this paper we evaluate the minimax asymptotic variance of the Huber (M)-estimate. The statistician minimizes over the family (ψλ)λ>0 of all tunings of Huber (M)-estimates of regression, and Nature maximizes over gross-error contaminations F ∈ Fε. Suppose that I(F ∗ ε )·m > 1. Then min λ max F∈Fε Vm(ψλ, F ) = 1 I(F ∗ ε )− 1/m . (2) Of course, the RHS of (2) is strictly bigger than the RHS of (1). Strikingly, if I(F ∗ ε ) ·m ≤ 1, then min λ max F∈Fε Vm(ψλ, F ) =∞. In short, the asymptotic variance of the Huber estimator breaks down at a critical ratio of observations per parameter. Classically, for the minimax (M)-estimator of location, no such breakdown occurs [DH83]. However, under this paper’s n/p→ m asymptotic, the breakdown point is where the Fisher information per parameter equals unity: ε∗ ≡ εm(Minimax Huber-(M) Estimate) = inf{ε : m · I(F ∗ ε ) ≥ 1}. ∗Department of Statistics, Stanford University †Department of Electrical Engineering and Department of Statistics, Stanford University