ESTIMATION OF SMOOTH FUNCTIONALS IN HIGH-DIMENSIONAL MODELS: BOOTSTRAP CHAINS AND GAUSSIAN APPROXIMATION

• 成果类型: Article
• 署名作者: Koltchinskii, Vladimir
• 署名单位: University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2197
• 发表日期: 2022
• 页码: 2386-2415
• 关键词: multivariate normal approximation; efficient estimation; integral functionals; Adaptive estimation; spectral gap; bounds; ORDER; CONVERGENCE; normality; distances

摘要

Let X-(n) be an observation sampled from a distribution P(theta)((n) )with an unknown parameter theta, theta being a vector in a Banach space E (most often, a high-dimensional space of dimension d). We study the problem of estimation of f (theta) for a functional f : E bar right arrow R of some smoothness s > 0 based on an observation X-(n) similar to P-theta((n)). Assuming that there exists an estimator (theta) over cap (n) = (theta) over cap (n) (X-(n)) of parameter theta such that root n((theta) over cap (n) - theta) is sufficiently close in distribution to a mean zero Gaussian random vector in E, we construct a functional g : E bar right arrow R such that g((theta) over cap (n)) is an asymptotically normal estimator of f(theta) with root n rate provided that s > 1/1-alpha and d <= n(alpha) for some alpha is an element of (0, 1). We also derive general upper bounds on Orlicz norm error rates for estimator g((theta) over cap) depending on smoothness s, dimension d, sample size n and the accuracy of normal approximation of root n((theta) over cap (n) - theta). In particular, this approach yields asymptotically efficient estimators in high-dimensional log-concave exponential models.