An iterative technique for bounding derivatives of solutions of Stein equations
We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations Lf=h−Eh(Z), where L is a linear differential operator and Z is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one...
| Główni autorzy: | , , |
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| Format: | Journal article |
| Język: | English |
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Institute of Mathematical Statistics and Bernoulli Society
2017
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| _version_ | 1826314541984645120 |
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| author | Dӧbler, C Gaunt, RE Vollmer, SJ |
| author_facet | Dӧbler, C Gaunt, RE Vollmer, SJ |
| author_sort | Dӧbler, C |
| collection | OXFORD |
| description | We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations Lf=h−Eh(Z), where L is a linear differential operator and Z is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function h. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the solution of the general variance-gamma Stein equation. Finally, we present a connection between Stein equations and Poisson equations, from which we first recognised the importance of the iterative technique to Stein’s method. |
| first_indexed | 2024-03-07T04:25:17Z |
| format | Journal article |
| id | oxford-uuid:cc690057-55b4-4652-b670-650ccfba9c7c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:35:38Z |
| publishDate | 2017 |
| publisher | Institute of Mathematical Statistics and Bernoulli Society |
| record_format | dspace |
| spelling | oxford-uuid:cc690057-55b4-4652-b670-650ccfba9c7c2024-09-13T10:47:49ZAn iterative technique for bounding derivatives of solutions of Stein equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:cc690057-55b4-4652-b670-650ccfba9c7cEnglishSymplectic Elements at OxfordInstitute of Mathematical Statistics and Bernoulli Society2017Dӧbler, CGaunt, REVollmer, SJWe introduce a simple iterative technique for bounding derivatives of solutions of Stein equations Lf=h−Eh(Z), where L is a linear differential operator and Z is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function h. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the solution of the general variance-gamma Stein equation. Finally, we present a connection between Stein equations and Poisson equations, from which we first recognised the importance of the iterative technique to Stein’s method. |
| spellingShingle | Dӧbler, C Gaunt, RE Vollmer, SJ An iterative technique for bounding derivatives of solutions of Stein equations |
| title | An iterative technique for bounding derivatives of solutions of Stein equations |
| title_full | An iterative technique for bounding derivatives of solutions of Stein equations |
| title_fullStr | An iterative technique for bounding derivatives of solutions of Stein equations |
| title_full_unstemmed | An iterative technique for bounding derivatives of solutions of Stein equations |
| title_short | An iterative technique for bounding derivatives of solutions of Stein equations |
| title_sort | iterative technique for bounding derivatives of solutions of stein equations |
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