Efficient Estimation of Smooth Functionals in Gaussian Shift Models
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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 Hölder smoothness s>0 such that _θ≤ 1 E_θ(T(X)-f(θ))^2 ≲(Σ∨ ( Eξ^2)^s)∧ 1, where Σ is the operator norm of Σ, and show that this mean squared error rate is minimax optimal (up to a logarithmic factor) at least in the case of standard Gaussian shift model (E= R^d equipped with the canonical Euclidean norm, ξ =σ Z, Z∼ N(0;I_d)). 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 Eξ^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).
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