We present two related contributions of independent interest: (1) high-probability finite sample rates for k-NN density estimation, and (2) practical mode estimators – based on k-NN – which attain minimax-optimal rates under surprisingly general distributional conditions.

This paper concerns the estimation of a function at a point in nonparametric heteroscedastic regression models with Gaussian noise or noise having unknown distribution. In those cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk.

Minimax state estimation for linear descriptor systems 01.05.04 – system analysis and optimal decision theory Author's Summary of the dissertation for the degree of the Candidate of Science (physics and mathematics)

This paper studies sparse density estimation via l1 penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a g...

We explain some basic theoretical issues regarding nonparametric statistics applied to inverse problems. Simple examples are used to present classical concepts such as the white noise model, risk estimation, minimax risk, model selection and optimal rates of convergence, as well as more recent concepts such as adaptive estimation, oracle inequalities, modern model selection methods, Stein’s unb...

A simulated annealing method is presented for Maximum A Posteriori state estimation. Specifically, I consider the problems of smoothing and filtering. A minimax entropy interpretation is proposed to motivate the method.

We refine the general methodology in [1] for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions with support size S comparable with the number of observations n. Specifically, we determine the “smooth” and “non-smooth” regimes based on the confidence set and the smo...