Efficient Estimation of Smooth Functionals of High-Dimensional Parameters: Two Models and Two Estimation Methods

VLADIMIR KOLTCHINSKII – GEORGIA TECH 

ABSTRACT

We will discuss a problem of estimation of smooth functionals of high-dimensional parameters of statistical models. Such functionals often represent some nonlinear low-dimensional features of high-dimensional parameters (such as, for instance, spectral characteristics of unknown covariance matrix) and it is of interest to develop methods of their statistical estimation with better error rates than for the unknown parameter itself. We will focus on two particular statistical models, high-dimensional log-concave location families and infinite-dimensional Gaussian models with unknown covariance operators, and describe two different estimation methods with optimal error rates of estimation of H\”older smooth functionals of parameters of these models. In the case of functionals of location parameter of high-dimensional log-concave distribution, the estimation method is based on iterative bias reduction via Neumann series (joint work with Martin Wahl). In the case of functionals of covariance, the method is based on bias reduction via linear aggregation of plug-in estimators with different sample sizes. We will also discuss the last method in the case of functionals of covariance that are not H\”older smooth (such as trace functionals).

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