Dependence modeling for order statistics and non-parametric estimation.

Summary In this thesis, we consider the joint law modeling of order statistics, i.e. random vectors with almost surely ordered components. The first part is dedicated to the probabilistic modeling of order statistics of maximum entropy with fixed marginals. Since the marginals are fixed, the characterization of the joint distribution amounts to considering the associated copula. In Chapter 2, we present an auxiliary result on maximum entropy copulas with fixed diagonal. A necessary and sufficient condition is given for the existence of such a copula, as well as an explicit formula for its density and entropy. The solution of the entropy maximization problem for order statistics with fixed marginals is presented in Chapter 3. Explicit formulas for its copula and joint density are given. In the second part of the thesis, we study the problem of non-parametric estimation of maximum entropy densities of order statistics in Kullback-Leibler distance. Chapter 5 describes an aggregation method for probability and spectral densities, based on a convex combination of its logarithms, and shows non-asymptotic optimal bounds in deviation. In Chapter 6, we propose an adaptive method based on an exponential log-additive model to estimate the considered densities, and we show that it reaches the known minimax speeds. The application of this method to estimate the dimensions of defects is presented in Chapter 7.