Minimax density estimation for growing dimension

TL;DR

This paper investigates the limits of density estimation as data dimension increases with sample size, establishing minimax rates and demonstrating that kernel estimators can achieve these bounds.

Contribution

It provides the first non-asymptotic minimax lower bounds for high-dimensional density estimation with growing dimension and shows kernel estimators attain these optimal rates.

Findings

Established non-asymptotic minimax lower bounds for growing dimension

Kernel density estimators achieve the minimax rates

Derived the maximum growth rate of dimension for consistent estimation

Abstract

This paper presents minimax rates for density estimation when the data dimension $d$ is allowed to grow with the number of observations $n$ rather than remaining fixed as in previous analyses. We prove a non-asymptotic lower bound which gives the worst-case rate over standard classes of smooth densities, and we show that kernel density estimators achieve this rate. We also give oracle choices for the bandwidth and derive the fastest rate $d$ can grow with $n$ to maintain estimation consistency.