Estimation of Functionals of High-Dimensional Parameters of Statisical Models

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Estimation of low-dimensional features of high-dimensional parameters is an important subject in contemporary statistical analysis of complex, high-dimensional data. While information-theoretic limitations often make impossible the reliable estimation of the whole unknown parameter due to its high dimensionality, estimation of low-dimensional features could be done efficiently with much faster error rates, common in classical statistics. Such problems often occur in applications, in particular, when the unknown parameter is a large matrix such as the density matrix of a quantum system, and the features of interest are various spectral characteristics of such matrices. Despite the fact that these problems have been studied for many years, there are few general approaches to statistical estimation of functionals representing the features of interest. The main goal of this project is to study functional estimation problem in a general mathematical framework and to develop general estimation methods as well as a comprehensive theory showing how the error rates in functional estimation depend on the underlying properties of the target functional such as its smoothness. The project provides new opportunities for training graduate students in the areas of high-dimensional statistics, in particular, by developing graduate level courses and seminars. The main focus of the project is on the development of a higher order bias reduction method (bootstrap chain bias reduction) in estimation of smooth functionals of unknown high-dimensional parameter of statistical model. It is based on iterative bootstrap and it could be viewed as a method of approximate solution of certain integral equations on high-dimensional parameter spaces. In the case of high-dimensional Gaussian models, this method yields estimators of smooth functionals with optimal error rates. This research project will study the properties of such estimators for a variety of important high-dimensional statistical models, including log-concave models, models on manifolds, sparse models and density matrix estimation models in quantum statistics. The goal is to determine minimax optimal error rates in functional estimation and to study the phase transition between fast parametric and slow nonparametric rates depending on the degree of smoothness of the functional and complexity parameters of the problem. This requires solving a number of challenging analytic and probabilistic problems, including the study of approximation of bootstrap Markov chains by superpositions of independent stochastic processes (random homotopies), the development of high-dimensional normal approximation and coupling methods as well as of concentration bounds for statistical estimators. The project will result in much deeper understanding of functional estimation problems in high dimensions and in the development of a variety of new probabilistic tools in high-dimensional statistical inference.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.