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Maximum d’entropie et problème des moments. (Maximum entropy and the moment problem). (French) Zbl 0788.62007
In addition to the usual linear constraints (moment conditions) in a maximum entropy problem, i.e., maximize the entropy $$S(g)$$ subject to $\int^ 1_ 0 \varphi(x)g(x)dx=c\quad\text{with given }\varphi\in C^ k[0,1]\quad\text{and}\quad c\in{\mathbf R}^ k,\tag{1}$ the authors consider nonlinear constraints on the probability density function $$g: [0,1]\to {\mathbf R}$$. In particular, solutions of the above maximum entropy problem are obtained if in addition to (1) nonlinear constraints of the type $- \infty\leq a< g(x)< b\leq +\infty\quad\text{for all } 0\leq x\leq 1,\quad\text{or }\int^ 1_ 0 g(x)^ 2 dx<1$ are considered. Several consequences of these results for the classical moment problem and various principles of optimization, e.g., least squares, are discussed.

##### MSC:
 62B10 Statistical aspects of information-theoretic topics 62A01 Foundations and philosophical topics in statistics 44A60 Moment problems 41A29 Approximation with constraints
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