Méthode du maximum d'entropie sur la moyenne et applications

Gamboa, Fabrice

Directeur de thèse : Dacunha-Castelle, Didier

Membres du jury : Dacunha-Castelle, Didier ; Azencott, Robert ; Comets, Francis ; Demoment, Guy ; Guyon, Xavier ; Navaza, Jorge

Thèses d'Orsay, no. 248 (1989) , 145 p.

Résumé

An explicit solution for the problem of probability reconstruction when only the averages of random variables are known is given by the maximum entropy method. We use this method to reconstruct a function constrained to a convex set $𝒞$ (no linear constraint) using a finite number of its generalized moments (linear constraint). A sequence of entropy maximization problems is considered. The nth problem consists in the reconstruction of a probability distribution on $𝒞_{n}$ , the projection of $𝒞$ on $ℝ^{n}$ , whose mean satisfies a constraint approximating the initial linear constraint (generalized moments). When n approaches infinity this gives a solution for the initial problem as the limit of the sequence of means of maximum entropy distributions on $𝒞_{n}$ . We call this technique the maximum entropy method on the mean (M.E.M) because linear constraints are only on the mean of the distribution to be reconstructed. We mainly study the case where $𝒞$ is a band of continuous functions. We find a reconstruction family, each element of this family only depends of referenced measures used for the sequence of entropy problems. We show that the M.E.M method is equivalent to a concav criteria maximization. We then use the M.E.M method to construct a numerically computable criteria to solve generalized moments problem on a bounded band of continuous functions. In the last chapter we discuss statistical applications of the method.

Classification : 62B10, 62A99
Keywords: Maximum of entropy, Functionna.1 estimation, Generalized moments, Moment problems, Large deviations

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     title = {M\'ethode du maximum d'entropie sur la moyenne et applications},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud Centre d'Orsay},
     number = {248},
     year = {1989},
     language = {fr},
     url = {https://www.numdam.org/item/BJHTUP11_1989__0248__P0_0/}
}

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Gamboa, Fabrice. Méthode du maximum d'entropie sur la moyenne et applications. Thèses d'Orsay, no. 248 (1989), 145 p. http://numdam.org/item/BJHTUP11_1989__0248__P0_0/

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