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Minimization of $$\phi$$-divergences on sets of signed measures. (English) Zbl 1121.28004
Summary: We consider the minimization problem of $$\phi$$-divergences between a given probability measure $$P$$ and subsets $$\Omega$$ of the vector space $$\mathcal M_{\mathcal F}$$ of all signed measures which integrate a given class $$\mathcal F$$ of bounded or unbounded measurable functions. The vector space $$\mathcal M_{\mathcal F}$$ is endowed with the weak topology induced by the class $$\mathcal F \cup \mathcal B_b$$ where $$\mathcal B_b$$ is the class of all bounded measurable functions. We treat the problems of existence and characterization of the $$\phi$$-projections of $$P$$ on $$\Omega$$. We also consider the dual equality and the dual attainment problems when $$\Omega$$ is defined by linear constraints.

##### MSC:
 28A33 Spaces of measures, convergence of measures 46A20 Duality theory for topological vector spaces 62C20 Minimax procedures in statistical decision theory
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