Optimal estimation of functionals of high-dimensional mean and covariance matrix

2019/08/20 by Jianqing Fan, Fan, Jianqing, Haolei Weng +3
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Bayesian Methods and Mixture Models #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1908.07460

openalex publication_date 2019/08/20 · openalex created_date 2019/08/29 · openalex updated_date 2026/07/28

Abstract

Motivated by portfolio allocation and linear discriminant analysis, we consider estimating a functional \mathbfμ^T \mathbfΣ^-1 \mathbfμ involving both the mean vector \mathbfμ and covariance matrix \mathbfΣ. We study the minimax estimation of the functional in the high-dimensional setting where \mathbfΣ^-1 \mathbfμ is sparse. Akin to past works on functional estimation, we show that the optimal rate for estimating the functional undergoes a phase transition between regular parametric rate and some form of high-dimensional estimation rate. We further show that the optimal rate is attained by a carefully designed plug-in estimator based on de-biasing, while a family of naive plug-in estimators are proved to fall short. We further generalize the estimation problem and techniques that allow robust inputs of mean and covariance matrix estimators. Extensive numerical experiments lend further supports to our theoretical results.

Citations

• On the distribution of the largest eigenvalue in principal components analysis
• Nearly unbiased variable selection under minimax concave penalty
• A test for the mean vector with fewer observations than the dimension under non-normality
• On Nonparametric Estimation of the Value of a Linear Functional in Gaussian White Noise
• Nonquadratic estimators of a quadratic functional
• A test for the mean vector with fewer observations than the dimension
• A direct approach to sparse discriminant analysis in ultra-high dimensions
• Minimax estimation of linear and quadratic functionals on sparsity classes

Related

• Factor-guided estimation of large covariance matrix function with conditional functional sparsity
• Minimax estimation of functionals in sparse vector model with correlated observations
• Computationally Efficient Robust Estimation of Sparse Functionals
• Minimax optimal estimators for general additive functional estimation
• Optimal rates of convergence for covariance matrix estimation