Functional estimation in high-dimensional and infinite-dimensional models

Let P be a family of probability measures on a measurable space (S,A). Given a Banach space E, a functional f:E↦R and a mapping θ:P↦E, our goal is to estimate f(θ(P)) based on i.i.d. observations X1,…,Xn∼P,P∈P. Given a smooth functional f and estimators θˆn(X1,…,Xn),n≥1 of θ(P), we use these estimators, the sample split and the Taylor expansion of f(θ(P)) of a proper order to construct estimators Tf(X1,…,Xn) of f(θ(P)). For these estimators and for a functional f of smoothness s≥1, we derive upper bounds on the Lp-errors of Tf(X1,…,Xn) with optimal dependence on sample size n, on the dimension or other complexity characteristics of parameter and on degree s of smoothness of the functional, and also study asymptotic normality and asymptotic efficiency of these estimators. The examples include functional estimation in high-dimensional models with many low dimensional components, functional estimation in high-dimensional exponential families and estimation of functionals of covariance operators in infinite-dimensional subgaussian models.