Multidimensional Nonparametric Density Estimates: Minimax Risk with Random Normalizing Factor

Abstract

We consider nonparametric minimax problem of multidimensional density estimation. Using the concept of random normalizing factor, by considering the plausible hypothesis of independence, we improve the accuracy of minimax estimation $n^{-\frac{\beta}{2\beta+d}}$: with prescribed confidence level $\alpha_{n}$, we show that the best possible attainable (random) rate is $\displaystyle{\big\{\sqrt{\log(2/\alpha_{n})}/n\big\}^{\frac{2\beta}{4\beta+d}}}$. We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski.