Efficient estimation of smooth functionals in Gaussian shift models

Abstract

We study a problem of estimation of smooth functionals of parameter θ of Gaussian shift model X=θ+ξ,θ∈E, where E is a separable Banach space and X is an observation of unknown vector θ in Gaussian noise ξ with zero mean and known covariance operator Σ. In particular, we develop estimators T(X) of f(θ) for functionals f:E↦R of Holder smoothness s>0 such that sup‖θ‖≤1Eθ(T(X)−f(θ))2≲(‖Σ‖∨(E‖ξ‖2)s)∧1, where ‖Σ‖ is the operator norm of Σ, and show that this mean squared error rate is minimax optimal at least in the case of standard finite-dimensional Gaussian shift model (E=Rd equipped with the canonical Euclidean norm, ξ=σZ, Z∼N(0;Id)). Moreover, we determine a sharp threshold on the smoothness s of functional f such that, for all s above the threshold, f(θ) can be estimated efficiently with a mean squared error rate of the order ‖Σ‖ in a “small noise” setting (that is, when E‖ξ‖2 is small). The construction of efficient estimators is crucially based on a “bootstrap chain” method of bias reduction. The results could be applied to a variety of special high-dimensional and infinite-dimensional Gaussian models (for vector, matrix and functional data).