Functional estimation in log-concave location families
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Let {P_θ:θ∈ℝ^d} be a log-concave location family with P_θ(dx)=e^-V(x-θ)dx, where V:ℝ^d↦ℝ is a known convex function and let X_1,…, X_n be i.i.d. r.v. sampled from distribution P_θ with an unknown location parameter θ. The goal is to estimate the value f(θ) of a smooth functional f:ℝ^d↦ℝ based on observations X_1,…, X_n. In the case when V is sufficiently smooth and f is a functional from a ball in a Hölder space C^s, we develop estimators of f(θ) with minimax optimal error rates measured by the L_2(ℙ_θ)-distance as well as by more general Orlicz norm distances. Moreover, we show that if d≤ n^α and s>1/1-α, then the resulting estimators are asymptotically efficient in Hájek-LeCam sense with the convergence rate √(n). This generalizes earlier results on estimation of smooth functionals in Gaussian shift models. The estimators have the form f_k(θ̂), where θ̂ is the maximum likelihood estimator and f_k: ℝ^d↦ℝ (with k depending on s) are functionals defined in terms of f and designed to provide a higher order bias reduction in functional estimation problem. The method of bias reduction is based on iterative parametric bootstrap and it has been successfully used before in the case of Gaussian models.
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