Statistically Efficient Estimation for Non-Smooth Probability Densities

We investigate statistical efficiency of estimators for non-smooth density functions. The density estimation problem appears in various situations, and it is intensively used in statistics and machine learning. The statistical efficiencies of estimators, i.e., their convergence rates, play a central role in advanced statistical analysis. Although estimators and their convergence rates for smooth density functions are well investigated in the literature, those for non-smooth density functions remain elusive despite their importance in application fields. In this paper, we propose new estimators for non-smooth density functions by employing the notion of Szemerédi partitions from graph theory. We derive convergence rates of the proposed estimators. One of them has the optimal convergence rate in minimax sense, and the other has slightly worse convergence rate but runs in polynomial time. Experimental results support the theoretical performance of our estimators.