Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions
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 var...
| Asıl Yazarlar: | , , |
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| Materyal Türü: | Working paper |
| Baskı/Yayın Bilgisi: |
2018
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| _version_ | 1826281778990546944 |
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| author | Ernst, M Reinert, G Swan, Y |
| author_facet | Ernst, M Reinert, G Swan, Y |
| author_sort | Ernst, M |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T00:33:56Z |
| format | Working paper |
| id | oxford-uuid:80beee6f-931e-41c6-b8a5-cdb0c64a3a3f |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:33:56Z |
| publishDate | 2018 |
| record_format | dspace |
| spelling | oxford-uuid:80beee6f-931e-41c6-b8a5-cdb0c64a3a3f2022-03-26T21:25:29ZStein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributionsWorking paperhttp://purl.org/coar/resource_type/c_8042uuid:80beee6f-931e-41c6-b8a5-cdb0c64a3a3fSymplectic Elements at Oxford2018Ernst, MReinert, GSwan, YIn 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. |
| spellingShingle | Ernst, M Reinert, G Swan, Y Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions |
| title | Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions |
| title_full | Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions |
| title_fullStr | Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions |
| title_full_unstemmed | Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions |
| title_short | Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions |
| title_sort | stein type covariance identities klaassen papathanasiou and olkin shepp type bounds for arbitrary target distributions |
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