Functional estimation in log-concave location families

TL;DR

This paper develops minimax optimal estimators for smooth functionals of the unknown location parameter in log-concave families, achieving asymptotic efficiency and bias reduction, extending Gaussian shift model results.

Contribution

It introduces new bias-reduction estimators for functional estimation in log-concave families, generalizing Gaussian model techniques to broader settings.

Findings

Achieves minimax optimal error rates in $L_2$ and Orlicz norms.

Establishes asymptotic efficiency under certain smoothness and dimensionality conditions.

Extends bias reduction methods to non-Gaussian log-concave models.

Abstract

Let $\{P_{\theta}:\theta \in {\mathbb R}^d\}$ be a log-concave location family with $P_{\theta}(dx)=e^{-V(x-\theta)}dx,$ where $V:{\mathbb R}^d\mapsto {\mathbb R}$ is a known convex function and let $X_1,\dots, X_n$ be i.i.d. r.v. sampled from distribution $P_{\theta}$ with an unknown location parameter $\theta.$ The goal is to estimate the value $f(\theta)$ of a smooth functional $f:{\mathbb R}^d\mapsto {\mathbb R}$ based on observations $X_1,\dots, X_n.$ In the case when $V$ is sufficiently smooth and $f$ is a functional from a ball in a H\"older space $C^s,$ we develop estimators of $f(\theta)$ with minimax optimal error rates measured by the $L_2({\mathbb P}_{\theta})$-distance as well as by more general Orlicz norm distances. Moreover, we show that if $d\leq n^{\alpha}$ and $s>\frac{1}{1-\alpha},$ then the resulting estimators are asymptotically efficient in H\'ajek-LeCam sense with the convergence rate $\sqrt{n}.$ This generalizes earlier results on estimation of smooth functionals in Gaussian shift models. The estimators have the form $f_k(\hat \theta),$ where $\hat \theta$ is the maximum likelihood estimator and $f_k: {\mathbb R}^d\mapsto {\mathbb R}$ (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.