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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.

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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