Minimax optimal estimators for general additive functional estimation

TL;DR

This paper develops minimax optimal estimators for a broad class of additive functionals of sparse Gaussian mean vectors, extending previous results and linking the estimation rate to polynomial approximation properties.

Contribution

It generalizes earlier work to a wider class of functionals and constructs estimators that achieve the optimal minimax rate based on polynomial approximation.

Findings

Optimal minimax rate depends on polynomial approximation rate.

Constructed estimators achieve the minimax rate.

Extended previous results to more general functionals.

Abstract

In this paper, we observe a sparse mean vector through Gaussian noise and we aim at estimating some additive functional of the mean in the minimax sense. More precisely, we generalize the results of (Collier et al., 2017, 2019) to a very large class of functionals. The optimal minimax rate is shown to depend on the polynomial approximation rate of the marginal functional, and optimal estimators achieving this rate are built.