Generalized van Trees inequality: Local minimax bounds for non-smooth functionals and irregular statistical models

TL;DR

This paper introduces a generalized van Trees inequality that provides sharp, non-asymptotic minimax lower bounds for estimating non-smooth functionals in irregular statistical models, extending classical efficiency theory.

Contribution

It develops new minimax lower bounds applicable to non-smooth functionals and irregular models, overcoming limitations of existing asymptotic and regularity-dependent results.

Findings

Derived non-asymptotic minimax lower bounds with sharp constants

Extended efficiency theory to non-smooth and irregular models

Applied bounds to density estimation and differentiable parameters

Abstract

In a decision-theoretic framework, the minimax lower bound provides the worst-case performance of estimators relative to a given class of statistical models. For parametric and semiparametric models, the H\'{a}jek--Le Cam local asymptotic minimax (LAM) theorem provides the sharp local asymptotic lower bound. Despite its relative generality, this result comes with limitations as it only applies to the estimation of differentiable functionals under regular statistical models. On the other hand, minimax lower bound techniques such as Fano's or Assoud's are applicable in more general settings but are not sharp enough to imply the LAM theorem. To address this gap, we provide new non-asymptotic minimax lower bounds under minimal regularity assumptions, which imply sharp asymptotic constants. The proposed lower bounds do not require the differentiability of functionals or regularity of statistical models, extending the efficiency theory to broader situations where standard results fail. The use of the new lower bounds is illustrated through the local minimax lower bound constants for estimating the density at a point and directionally differentiable parameters.