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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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