Pitman estimation for ensembles and mixtures
dc.contributor.author | Srinivasan, Shankar S. | en |
dc.contributor.committeechair | Jensen, Donald R. | en |
dc.contributor.committeemember | Foutz, Robert V. | en |
dc.contributor.committeemember | Myers, Raymond | en |
dc.contributor.committeemember | Smith, Eric P. | en |
dc.contributor.committeemember | Terrell, George R. | en |
dc.contributor.department | Statistics | en |
dc.date.accessioned | 2014-03-14T21:17:43Z | en |
dc.date.adate | 2006-08-14 | en |
dc.date.available | 2014-03-14T21:17:43Z | en |
dc.date.issued | 1995 | en |
dc.date.rdate | 2006-08-14 | en |
dc.date.sdate | 2006-08-14 | en |
dc.description.abstract | This dissertation considers minimal risk equivariant (MRE) estimation of a location scalar ฮผ in ensembles and mixtures of translation families having structured dispersion matrices ฮฃ. The principal focus is the preservation of Pitman's solutions across classes of distributions. To these ends the cone S<sub>n</sub>โบ of positive definite matrices is partitioned into various equivalence classes. The classes ๐<sub>๐ช</sub> are indexed through matrices ๐ช from a class ๐(n) comprising positive semidefinite (nรn) matrices with one-dimensional subspace spanned by the unit vector ๐<sub>n</sub>'=[1, 1, ..., 1]. Here ฮฃโ๐<sub>๐ช</sub> has the structure ฮฃ(y) = ๐ช+ฮณ๐<sub>n</sub>'+๐<sub>n</sub>y'-ฮณฬ ๐<sub>n</sub>๐<sub>n</sub>', for some vector ฮณ such that ฮณ'๐ช<sub>๐(n)</sub>โปยนฮณ < ฮณฬ , where ๐ช<sub>๐(n)</sub>โปยน is the Moore-Penrose inverse of ๐ช. Of particular interest is the class ฮ(n) = ๐<sub>๐</sub> with ๐ = [๐<sub>n</sub> - (1/n)๐<sub>n</sub>๐<sub>n</sub>']. In addition, the equivalence classes ฮ(๐ฐ) in S<sub>n</sub>โบ are indexed through elements of ๐ฆ(n) containing n-dimensional vectors ๐ฐ such that ฮฃ<sub>i=1</sub><sup>n</sup>w<sub>i</sub> = 1, where ๐ฐ'ฮฃ = c๐<sub>n</sub>โ for some scalar c>0. Of interest is the class ฮฉ(n) = ฮ(nโปยน๐<sub>n</sub>), containing equicorrelation matrices in the intersection ฮ(n)โฮฉ(n). Ensembles of elliptically contoured distributions having dispersion matrices in the foregoing classes, and mixtures over these, are considered further with regard to Pitman estimation of ฮผ. For elliptical random vectors ๐ the Pitman estimator continues to take the generalized least squares form. Further, ensembles of elliptically symmetric distributions having dispersion matrices in ฮฉ(n) preserve the equivariant admissibility of the sample average Xฬ under squared error loss. For dispersion matrices ฮฃ in each class ๐<sub>๐ช</sub> the estimator is obtained as a correction of Xฬ taking the form ฮด<sub>ฮฃ</sub>(๐) = Xฬ -ฮณ'๐ชโปยน<sub>๐(n)</sub>๐, with ฮณ as in the expansion for ฮฃ. This simplifies when ฮฃโฮ(n) to ฮด<sub>ฮฃ</sub>(๐) = Xฬ -ฮณ'๐, where ๐ is the vector of residuals {e<sub>i</sub> = x<sub>i</sub>-xฬ ; i = 1, 2, ..., n}. These results carry over to dispersion mixtures of elliptically symmetric distributions when the mixing measure ๐ is restricted to the corresponding subsets of Sโบ<sub>n</sub>. The estimators are now given through a dispersion matrix ฮจ which is the expectation of ฮฃ over ๐. For mixing measures over S<sub>n</sub>โบ, for which each conditional expectation for ฮฃ given ๐ช โ ๐(n) is in ฮฉ(n), Xฬ is the Pitman estimator for ฮผ for the corresponding mixture distribution. Similar results apply for each linear estimator. In both elliptical ensembles and mixtures over these, the Pitman estimator is shown to be linear and unbiased. | en |
dc.description.degree | Ph. D. | en |
dc.format.extent | viii, 119 leaves | en |
dc.format.medium | BTD | en |
dc.format.mimetype | application/pdf | en |
dc.identifier.other | etd-08142006-110114 | en |
dc.identifier.sourceurl | http://scholar.lib.vt.edu/theses/available/etd-08142006-110114/ | en |
dc.identifier.uri | http://hdl.handle.net/10919/39155 | en |
dc.language.iso | en | en |
dc.publisher | Virginia Tech | en |
dc.relation.haspart | LD5655.V856_1995.S677.pdf | en |
dc.relation.isformatof | OCLC# 34347443 | en |
dc.rights | In Copyright | en |
dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
dc.subject.lcc | LD5655.V856 1995.S677 | en |
dc.title | Pitman estimation for ensembles and mixtures | en |
dc.type | Dissertation | en |
dc.type.dcmitype | Text | en |
thesis.degree.discipline | Statistics | en |
thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
thesis.degree.level | doctoral | en |
thesis.degree.name | Ph. D. | en |
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