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Title: Differential Stein operators for multivariate distributions and applications Authors:  Yvik Swan - Universite de Liege (Belgium) [presenting]
Gesine Reinert - Oxford University (United Kingdom)
Guillaume Mijoule - University of Liege (Belgium)
Abstract: After introducing the concept of ``directional Stein operator'', we discuss several types of gradient and divergence based differential Stein operators for multivariate random vectors with absolutely continuous densities $p$ on $\mathbb{R}^d$, $d \ge 1$. In particular, we provide minimal conditions on $p$ to guarantee the existence of a score function and a Stein kernel; this leads to probabilistic integration by parts formulas which generalize Stein's famous Gaussian covariance lemma. We illustrate the operators and identities on the family of elliptical distributions (particularly the Gaussian, multivariate Student, and power exponential distributions), hereby providing new tractable operators which moreover bear nice interpretations. We introduce a new family of kernelized Stein discrepancies. Several applications are outlined: aside from the habitual Stein-type measures of discrepancies, we also discuss problems of goodness-of-fit testing.