Differentiability of t-functionals of location and scatter
The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix Σ of an elliptically symmetric t distribution on ℝd with degrees of freedom ν>1 extends to an M-...
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| Format: | Article |
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Institute of Mathematical Statistics
2011
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| Online Access: | http://hdl.handle.net/1721.1/63108 https://orcid.org/0000-0002-6195-4161 |
| _version_ | 1811091726729740288 |
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| author | Dudley, Richard M. Sidenko, Sergiy Wang, Zuoqin |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Dudley, Richard M. Sidenko, Sergiy Wang, Zuoqin |
| author_sort | Dudley, Richard M. |
| collection | MIT |
| description | The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix Σ of an elliptically symmetric t distribution on ℝd with degrees of freedom ν>1 extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P putting not too much mass in hyperplanes of dimension <d, as shown for empirical measures by Kent and Tyler [Ann. Statist. 19 (1991) 2102–2119]. It will be seen here that (μ, Σ) is analytic on U for the bounded Lipschitz norm, or for d=1 for the sup norm on distribution functions. For k=1, 2, …, and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (μ, Σ) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (μn, Σn). In dimension d=1 only, the tν functional (μ, σ) extends to be defined and weakly continuous at all P. |
| first_indexed | 2024-09-23T15:07:01Z |
| format | Article |
| id | mit-1721.1/63108 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:07:01Z |
| publishDate | 2011 |
| publisher | Institute of Mathematical Statistics |
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| spelling | mit-1721.1/631082022-09-29T12:47:59Z Differentiability of t-functionals of location and scatter Dudley, Richard M. Sidenko, Sergiy Wang, Zuoqin Massachusetts Institute of Technology. Department of Mathematics Dudley, Richard M. Dudley, Richard M. The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix Σ of an elliptically symmetric t distribution on ℝd with degrees of freedom ν>1 extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P putting not too much mass in hyperplanes of dimension <d, as shown for empirical measures by Kent and Tyler [Ann. Statist. 19 (1991) 2102–2119]. It will be seen here that (μ, Σ) is analytic on U for the bounded Lipschitz norm, or for d=1 for the sup norm on distribution functions. For k=1, 2, …, and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (μ, Σ) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (μn, Σn). In dimension d=1 only, the tν functional (μ, σ) extends to be defined and weakly continuous at all P. National Science Foundation (U.S.) (Grant No. DMS-0103821) National Science Foundation (U.S.) (Grant No. DMS-0504859) 2011-05-25T14:47:31Z 2011-05-25T14:47:31Z 2011-06 Article http://purl.org/eprint/type/JournalArticle 0090-5364 http://hdl.handle.net/1721.1/63108 Dudley, R. M., Sergiy Sidenko and Zuoqin Wang. "Differentiability of t-functionals of location and scatter." Ann. Statist. 37.2 (2009): 939-960. https://orcid.org/0000-0002-6195-4161 en_US http://dx.doi.org/10.1214/08-AOS592 Annals of Statistics Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Institute of Mathematical Statistics Prof. Dudley via Michael Noga |
| spellingShingle | Dudley, Richard M. Sidenko, Sergiy Wang, Zuoqin Differentiability of t-functionals of location and scatter |
| title | Differentiability of t-functionals of location and scatter |
| title_full | Differentiability of t-functionals of location and scatter |
| title_fullStr | Differentiability of t-functionals of location and scatter |
| title_full_unstemmed | Differentiability of t-functionals of location and scatter |
| title_short | Differentiability of t-functionals of location and scatter |
| title_sort | differentiability of t functionals of location and scatter |
| url | http://hdl.handle.net/1721.1/63108 https://orcid.org/0000-0002-6195-4161 |
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