Optimal Global Rates of Convergence for Nonparametric Regression

1982/12/01 by Charles J. Stone · 18 citations
Mathematics · Engineering · #Numerical methods in inverse problems #Statistical Methods and Inference #Control Systems and Identification

paper · pdf · doi:10.1214/aos/1176345969

openalex publication_date 1982/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Consider a p-times differentiable unknown regression function θ of a d-dimensional measurement variable. Let T(θ) denote a derivative of θ of order m and set r = (p - m)/(2p + d). Let T_n denote an estimator of T(θ) based on a training sample of size n, and let ‖ T_n - T(θ)‖_q be the usual L^q norm of the restriction of T_n - T(θ) to a fixed compact set. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for ‖ T_n - T(θ)‖_q is n^-r if 0 < q < ∞; while (n^-1 log n)^r is the optimal rate if q = ∞.

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