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Estimation des moindres carrés d'une densité discrète sous contrainte de k-monotonie et bornes de risque. Application à l'estimation du nombre d'espèces dans une population.

Abstract : This thesis belongs to the field of nonparametric density estimation under shape constraint. The densities are discrete and the form is k-monotonicity, k>1, which is a generalization of convexity. The integer k is an indicator for the hollow's degree of a convex function. This thesis is composed of three parts, an introduction, a conclusion and an appendix.Introduction :The introduction is structured in three chapters. First Chapter is a state of the art of the topic of density estimation under shape constraint. The second chapter of the introduction is a synthesis of the thesis, available in French and in English. Finally Chapter 3 is a short chapter which summarizes the notations and the classical mathematical results used in the manuscript.Part I : Estimation of a discrete distribution under k-monotonicityconstraintTwo least-square estimators of a discrete distribution p* under constraint of k-monotonicity are proposed. Their characterisation is based on the decomposition on a spline basis of k-monotone sequences, and on the properties of their primitives. Their statistical properties are studied, and in particular their quality of estimation is measured in terms of the quadratic error. They are proved to converge at the parametric rate. An algorithm derived from the support reduction algorithm is implemented in the R-package pkmon. A simulation study illustrates the properties of the estimators. This piece of works, which constitutes Part I of the manuscript, has been published in ElectronicJournal of Statistics (Giguelay, 2017).Part II : Calculation of risks boundsIn the first chapter of Part II, a methodology for calculating riskbounds of the least-square estimator is given. These bounds are adaptive in that they depend on a compromise between the distance of p* on the frontier of the set of k-monotone densities with finite support, and the complexity (linked to the spline decomposition) of densities belonging to this set that are closed to p*. The methodology based on the variational formula of the risk proposed by Chatterjee (2014) is generalized to the framework of discrete k-monotone densities. Then the bracketting entropies of the relevant functionnal space are calculating, leading to control the empirical process involved in the quadratic risk. Optimality of the risk bound is discussed in comparaison with the results previously obtained in the continuous case and for the gaussian regression framework. In the second chapter of Part II, several results concerningbracketting entropies of spaces of k-monotone sequences are presented.Part III : Estimating the number of species in a population and tests of k-monotonicityThe last part deals with the problem of estimating the number ofpresent species in a given area at a given time, based on theabundances of species that have been observed. A definition of ak-monotone abundance distribution is proposed. It allows to relatethe probability of observing zero species to the truncated abundancedistribution. Two approaches are proposed. The first one is based on the Least-Squares estimator under constraint of k-monotonicity, the second oneis based on the empirical distribution. Both estimators are comparedusing a simulation study. Because the estimator of the number ofspecies depends on the value of the degree of monotonicity k, we proposea procedure for choosing this parameter, based on nested testingprocedures. The asymptotic levels and power of the testing procedureare calculated, and the behaviour of the method in practical cases isassessed on the basis of a simulation study.
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Jade Giguelay. Estimation des moindres carrés d'une densité discrète sous contrainte de k-monotonie et bornes de risque. Application à l'estimation du nombre d'espèces dans une population.. Théorie [stat.TH]. Université Paris-Saclay, 2017. Français. ⟨NNT : 2017SACLS248⟩. ⟨tel-02110252⟩

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