2015/03/06 by David L. Donoho, Andrea Montanari, Donoho, David L. +1 · 2 citations
Mathematics · #62C20 #62G35 #62J05 #Advanced Statistical Methods and Models #FOS: Mathematics #Statistical Distribution Estimation and Applications #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1503.02106
openalex publication_date 2015/03/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A half century ago, Huber evaluated the minimax asymptotic variance in scalar location estimation, minψmax_F ∈ \cal Fε V(ψ, F) = (1)/(I(Fε^*)) , where V(ψ,F) denotes the asymptotic variance of the (M)-estimator for location with score function ψ, and I(Fε^*) is the minimal Fisher information min_\cal Fε I(F) over the class of ε-Contaminated Normal distributions. 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. Then Vm ≠ 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 ∈ \cal Fε. Suppose that I(Fε^*) ⋅ m > 1. Then minλmax_F ∈ \cal Fε Vm(ψλ, F) = (1)/(I(Fε^*) - 1/m) . Strikingly, if I(Fε^*) ⋅ m ≤ 1, then the minimax asymptotic variance is +∞. The breakdown point is where the Fisher information per parameter equals unity.