Functional estimation in high-dimensional and infinite-dimensional models
Abstract
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. In particular, if P=\Pθ: θ∈ \ is an identifiable statistical model with parameter set ⊂ E, one can consider the mapping θ(P)=θ for P∈ P, P=Pθ, resulting in a problem of estimation of f(θ) based on i.i.d. observations X1,…, Xn Pθ, θ∈ . Given a smooth functional f and estimators θn(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 prove upper bounds on the Lp-errors of estimator Tf(X1,…, Xn) under certain moment assumptions on the base estimators θn. We study the performance of estimators Tf(X1,…, Xn) in several concrete problems, showing their minimax optimality and asymptotic efficiency. In particular, this includes 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.
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