arXiv subjects

A. Juditsky

Publications and source records attributed to A. Juditsky.

4 recordsLinked to original sources

arXiv2012

On deconvolution of distribution functions

The subject of this paper is the problem of nonparametric estimation of a continuous distribution function from observations with measurement errors. We study minimax complexity of this problem when unknown distribution has a density belonging to the Sobolev class, and the error density is ordinary smooth. We develop rate optimal estimators based on direct inversion of empirical characteristic function. We also derive minimax affine estimators of the distribution function which are given by an explicit convex optimization problem. Adaptive versions of these estimators are proposed, and some numerical results demonstrating good practical behavior of the developed procedures are presented.

math.ST↗

arXiv2008

Learning by mirror averaging

Given a finite collection of estimators or classifiers, we study the problem of model selection type aggregation, that is, we construct a new estimator or classifier, called aggregate, which is nearly as good as the best among them with respect to a given risk criterion. We define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original nonlinear one and constitutes a special case of the mirror averaging algorithm. We show that the aggregate satisfies sharp oracle inequalities under some general assumptions. The results are applied to several problems including regression, classification and density estimation.

math.ST↗

arXiv2005

Moderate deviation principle for ergodic Markov chain. Lipschitz summands

For ${1/2}<α<1$, we propose the MDP analysis for family $$ S^α_n=\frac{1}{n^α}\sum_{i=1}^nH(X_{i-1}), n\ge 1, $$ where $(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in \mathbb{R}^d$, when the spectrum of operator $P_x$ is continuous. The vector-valued function $H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is required. The main helpful tools in our approach are Poisson's equation and Stochastic Exponential; the first enables to replace the original family by $\frac{1}{n^α}M_n$ with a martingale $M_n$ while the second to avoid the direct Laplace transform analysis.

math.PR↗

arXiv2004

Moderate deviation principle for exponentially ergodic Markov chain

For ${1/2}<α<1$, we propose the MDP analysis for family $$ S^α_n=\frac{1}{n^α}\sum_{i=1}^nH(X_{i-1}), n\ge 1, $$ where $(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in \mathbb{R}^d$, when the spectrum of operator $P_x$ is continuous. The vector-valued function $H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is required. The main helpful tools in our approach are Poisson equation and Stochastic Exponential; the first enables to replace the original family by $\frac{1}{n^α}M_n$ with a martingale $M_n$ while the second to avoid the direct Laplace transform analysis.

math.PR↗