Estimation of Functionals of High-Dimensional and Infinite-Dimensional Parameters of Statistical Models

The mini-course will meet on Monday, April 1 and Wednesday, April 3rd from 1:30-3:00pm

This mini-course deals with a circle of problems related to estimation of real valued functionals of high-dimensional and infinite-dimensional parameters of statistical models.

In such problems, it is of interest to estimate one-dimensional features of a high-dimensional parameter represented by nonlinear functionals of certain degree of smoothness defined on the parameter space. The functionals of interest could be often estimated with faster convergence rates than the whole parameter (sometimes, even with parametric rates).

We will discuss some mathematical methods providing a way to develop estimators of functionals of high-dimensional parameters with optimal error rates in classes of functionals of some Hoelder smoothness and even to provide their efficient estimation with parametric rates when the smoothness is sufficiently large.

The main focus will be on functionals of unknown covariance operators in high-dimensional and infinite-dimensional Gaussian models, where the functionals of interest often capture their spectral properties. In particular, we will discuss the role of higher order bias reduction methods and concentration inequalities in these problems.

Vladimir Koltchinskii is a professor of Mathematics at the Georgia Institute of Technology.