Minimax bounds for Besov classes in density estimation

Minimax lower bounds for estimation

Minimax Density Estimation for Growing Dimension

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 t...

Risk Bounds for Density Estimation

Motivation: Li and Barron [6, 7] give a greedy method for estimating densities by a finite mixture. The trade-off between approximation and estimation error is present in their risk bound in terms of the optimal number of components used for density estimation. Using techniques from Empirical Process theory [11] and results from geometry in Banach Spaces [5], we improve the bounds and show that...

Learning sub-Gaussian classes : Upper and minimax bounds

Most the results contained in this note have been presented at the SMF meeting, which took place in May 2011; the rest have been obtained shortly after the time of the meeting. The question we study has to do with the optimality of Empirical Risk Minimization as a learning procedure in a convex class – when the problem is subgaussian. Subgaussian learning problems are a natural object because t...

Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation

We provide a detailed study of the estimation of probability distributions— discrete and continuous—in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental trade-offs between privacy and convergence rate, as well as providing tools to allow movement along ...