First order covariance inequalities via Stein's method
We propose probabilistic representations for inverse Stein operators (i.e., solutions to Stein equations) under general conditions; in particular, we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and nonuniform Stein factors (i.e., bounds on soluti...
| Main Authors: | , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Bernoulli Society for Mathematical Statistics and Probability
2020
|
| _version_ | 1797053965241679872 |
|---|---|
| author | Ernst, M Reinert, G Swan, Y |
| author_facet | Ernst, M Reinert, G Swan, Y |
| author_sort | Ernst, M |
| collection | OXFORD |
| description | We propose probabilistic representations for inverse Stein operators (i.e., solutions to Stein equations) under general conditions; in particular, we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and nonuniform Stein factors (i.e., bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases and expressed in terms of objects familiar within the context of Stein’s method. Applications of the Cauchy–Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed. |
| first_indexed | 2024-03-06T18:50:58Z |
| format | Journal article |
| id | oxford-uuid:1031ef42-fa37-4f26-874d-9e0e37c99ad0 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:50:58Z |
| publishDate | 2020 |
| publisher | Bernoulli Society for Mathematical Statistics and Probability |
| record_format | dspace |
| spelling | oxford-uuid:1031ef42-fa37-4f26-874d-9e0e37c99ad02022-03-26T09:55:15ZFirst order covariance inequalities via Stein's methodJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:1031ef42-fa37-4f26-874d-9e0e37c99ad0EnglishSymplectic Elements at OxfordBernoulli Society for Mathematical Statistics and Probability2020Ernst, MReinert, GSwan, YWe propose probabilistic representations for inverse Stein operators (i.e., solutions to Stein equations) under general conditions; in particular, we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and nonuniform Stein factors (i.e., bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases and expressed in terms of objects familiar within the context of Stein’s method. Applications of the Cauchy–Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed. |
| spellingShingle | Ernst, M Reinert, G Swan, Y First order covariance inequalities via Stein's method |
| title | First order covariance inequalities via Stein's method |
| title_full | First order covariance inequalities via Stein's method |
| title_fullStr | First order covariance inequalities via Stein's method |
| title_full_unstemmed | First order covariance inequalities via Stein's method |
| title_short | First order covariance inequalities via Stein's method |
| title_sort | first order covariance inequalities via stein s method |
| work_keys_str_mv | AT ernstm firstordercovarianceinequalitiesviasteinsmethod AT reinertg firstordercovarianceinequalitiesviasteinsmethod AT swany firstordercovarianceinequalitiesviasteinsmethod |