Minimization of divergences on sets of signed measures

Abstract

We consider the minimization problem of $\varphi$-divergences between a given probability measure $P$ and subsets $\Omega$ of the vector space ${\mathcal{M}}_{\mathcal{F}}$ of all signed finite 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 $\varphi$-projections of $P$ on $\Omega$. We consider also the dual equality and the dual attainment problems when $\Omega$ is defined by linear constraints.

References in corpus (1)

• Estimation and tests for models satisfying linear constraints with unknown parameter

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