ESTIMATION OF SMOOTH FUNCTIONALS IN NORMAL MODELS: BIAS REDUCTION AND ASYMPTOTIC EFFICIENCY

• 成果类型: Article
• 署名作者: Koltchinskii, Vladimir; Zhilova, Mayya
• 署名单位: University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS2047
• 发表日期: 2021
• 页码: 2577-2610
• 关键词: Concentration Inequalities; integral functionals; spectral projectors; Adaptive estimation; derivatives; bounds; forms; ORDER; sharp

摘要

Let X1,..., X-n be i.i.d. random variables sampled from a normal distribution N(mu, Sigma) in R-d with unknown parameter theta = (mu, Sigma) is an element of Theta := R-d x C-+(d), where C-+(d) is the cone of positively definite covariance operators in R-d. Given a smooth functional f : Theta 1 bar right arrow R-1, the goal is to estimate f (theta) based on X-1,.., X-n. Let Theta(a; d) := R-d x {Sigma is an element of C-+(d) : sigma(Sigma) subset of [1/a, a]}, a >= 1, where sigma(Sigma) is the spectrum of covariance Sigma. Let (theta) over cap := ((mu) over cap, (Sigma) over cap), where (mu) over cap is the sample mean and (Sigma) over cap is the sample covariance, based on the observations X-1,..., X-n. For an arbitrary functional f is an element of C-s(Theta), S=k+1+rho,k >= 0, rho is an element of (0, 1], we define a functional f(k) : Theta bar right arrow R such that sup theta is an element of Theta(a;d)parallel to f(k)((theta)over cap>) - f(theta)parallel to L-2(P-theta)less than or similar to s,beta parallel to f parallel to C-s(Theta)[(a/root n boolean OR a(beta s)(root d/n)(s))boolean AND 1], where beta = 1 for k = 0 and beta > s - 1 is arbitrary for k >= 1. This error rate is minimax optimal and similar bounds hold for more general loss functions. If d = d(n) <= n(alpha) for some alpha is an element of (0, 1) and s >= 1/1-alpha , the rate becomes 0(n(-1/2)). Moreover, for s > 1/1-alpha the estimator f(k) ((theta)over cap) is shown to be asymptotically efficient. The crucial part of the construction of estimator f(k)((theta)over cap) is a bias reduction method studied in the paper for more general statistical models than normal.