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Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions by Ernst, Marie; Reinert, Gesine; Swan, Yvik is a scholarly article available to read on EtoBox.

What is Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions about?

In this paper, we present a minimal formalism for Stein operators which leads to different probabilistic representations of solutions to Stein equations. These in turn provide a wide family of Stein-Covariance identities which we put to use for revisiting the very classical topic of bounding the variance of functionals of random variables. Applying the Cauchy-Schwarz inequality yields first order upper and lower Klaassen-type variance bounds. A probabilistic representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder) leads to Papathanasiou-type variance expansions of arbitrary order. A matrix Cauchy-Schwarz inequality leads to Olkin-Shepp type covariance bounds. All results hold for univariate target distribution under very weak assumptions (in particular they hold for continuous and discrete distributions alike). Many concrete illustrations are provided.

Author

Ernst, Marie; Reinert, Gesine; Swan, Yvik

Published

2018

Language

EN