Minimax density estimation for growing dimension

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.