Efficient estimation of linear functionals of principal components

Stochastics Seminar
Thursday, January 26, 2017 - 15:05
1 hour (actually 50 minutes)
Georgia Tech
We study the problem of estimation of a linear functional of the eigenvector of covariance operator that corresponds to its largest eigenvalue (the first principal component) based on i.i.d. sample of centered Gaussian observations with this covariance. The problem is studied in a dimension-free framework with its complexity being characterized by so called "effective rank" of the true covariance. In this framework, we establish a minimax lower bound on the mean squared error of estimation of linear functional and construct an asymptotically normal estimator for which the bound is attained. The standard "naive" estimator (the linear functional of the empirical principal component) is suboptimal in this problem. The talk is based on a joint work with Richard Nickl.