Minimax bounds for estimation of normal mixtures

TL;DR

This paper establishes minimax convergence rates for estimating normal mixture densities on the real line, introducing novel Fourier and Hermite polynomial techniques to handle the complexities of the problem.

Contribution

It introduces new Fourier and Hermite polynomial methods to determine minimax rates for normal mixture density estimation under squared error and Hellinger losses.

Findings

Minimax optimal rate slightly exceeds parametric rate

New techniques for lower bounds under Hellinger loss

Addresses difficulties in standard minimax methods for mixtures

Abstract

This paper deals with minimax rates of convergence for estimation of density functions on the real line. The densities are assumed to be location mixtures of normals, a global regularity requirement that creates subtle difficulties for the application of standard minimax lower bound methods. Using novel Fourier and Hermite polynomial techniques, we determine the minimax optimal rate - slightly larger than the parametric rate - under squared error loss. For Hellinger loss, we provide a minimax lower bound using ideas modified from the squared error loss case.