Probability and Statistics Seminar: Efficient estimation of smooth functionals in high and infinite dimensions

Vladimir Koltchinskii, Georgia Tech

Title: Efficient estimation of smooth functionals in high and infinite dimensions

Abstract: I will discuss a problem of estimation of smooth functionals of parameters of high-dimensional and infinite-dimensional statistical models, representing their nonlinear low-dimensional features. The main goal in these problems is to understand the impact of the smoothness of the functional on the quality of statistical estimation and, in particular, to study the phase transition from slow error rates to faster classical $\sqrt{n}$-rates of the CLT when the smoothness increases. The focus will be on two specific models, high-dimensional log-concave location models and infinite-dimensional Gaussian models with unknown covariance operator, for which different estimation methods provide minimax optimal mean squared error rates under H\”older smoothness of the functionals. One of the methods is based on iterative bias reduction via Neumann series and another one is based on linear aggregation of plug-in estimators with different sample sizes. The proofs of the optimal error rates for such estimators are largely based on concentration inequalities. I will also discuss the last estimation method in the case of functionals lacking H\”older smoothness (such as the trace of operator function $f(\Sigma)$ for a smooth function $f$ in the real line).