Adaptive non-asymptotic confidence balls in density estimation

We build confidence balls for the common density $s$ of a real valued sample $X_1,...,X_n$. We use resampling methods to estimate the projection of $s$ onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all $n\geq2$ and the balls are adaptive over a collection of linear spaces

An Oracle Approach for Interaction Neighborhood Estimation in Random Fields

We consider the problem of interaction neighborhood estimation from the partial observation of a finite number of realizations of a random field. We introduce a model selection rule to choose estimators of conditional probabilities among natural candidates. Our main result is an oracle inequality satisfied by the resulting estimator. We use then this selection rule in a two-step procedure to evaluate the interacting neighborhoods. The selection rule selects a small prior set of possible interacting points and a cutting step remove from this prior set the irrelevant points. We also prove that the Ising models satisfy the assumptions of the main theorems, without restrictions on the temperature, on the structure of the interacting graph or on the range of the interactions. It provides therefore a large class of applications for our results. We give a computationally efficient procedure in these models. We finally show the practical efficiency of our approach in a simulation study.Comment: 36 pages, 10 figure

The number of potential winners in Bradley-Terry model in random environment

We consider a Bradley-Terry model in random environment where each player faces each other once. More precisely the strengths of the players are assumed to be random and we study the influence of their distributions on the asymptotic number of potential winners.First we prove that under mild assumptions, mainly on their moments, if the strengths are unbounded, the asymptotic probability that the best player wins is 1. We also exhibit a sufficient convexity condition to obtain the same result when the strengths are bounded. When this last condition fails, the number of potential winners grows at a rate depending on the tail of the distribution of strengths. We also study the minimal strength required for an additional player to win in this last case