Pitman estimation for ensembles and mixtures

dc.contributor.authorSrinivasan, Shankar S.en
dc.contributor.committeechairJensen, Donald R.en
dc.contributor.committeememberFoutz, Robert V.en
dc.contributor.committeememberMyers, Raymonden
dc.contributor.committeememberSmith, Eric P.en
dc.contributor.committeememberTerrell, George R.en
dc.contributor.departmentStatisticsen
dc.date.accessioned2014-03-14T21:17:43Zen
dc.date.adate2006-08-14en
dc.date.available2014-03-14T21:17:43Zen
dc.date.issued1995en
dc.date.rdate2006-08-14en
dc.date.sdate2006-08-14en
dc.description.abstractThis 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>โบ of positive definite matrices is partitioned into various equivalence classes. The classes ๐“<sub>๐‘ช</sub> are indexed through matrices ๐‘ช from a class ๐“’(n) comprising positive semidefinite (nร—n) matrices with one-dimensional subspace spanned by the unit vector ๐Ÿ<sub>n</sub>'=[1, 1, ..., 1]. Here ฮฃโˆˆ๐“<sub>๐‘ช</sub> has the structure ฮฃ(y) = ๐‘ช+ฮณ๐Ÿ<sub>n</sub>'+๐Ÿ<sub>n</sub>y'-ฮณฬ…๐Ÿ<sub>n</sub>๐Ÿ<sub>n</sub>', for some vector ฮณ such that ฮณ'๐‘ช<sub>๐“’(n)</sub>โปยนฮณ < ฮณฬ…, where ๐‘ช<sub>๐“’(n)</sub>โปยน is the Moore-Penrose inverse of ๐‘ช. Of particular interest is the class ฮ“(n) = ๐“<sub>๐</sub> with ๐ = [๐ˆ<sub>n</sub> - (1/n)๐Ÿ<sub>n</sub>๐Ÿ<sub>n</sub>']. In addition, the equivalence classes ฮ›(๐ฐ) in S<sub>n</sub>โบ are indexed through elements of ๐“ฆ(n) containing n-dimensional vectors ๐ฐ such that ฮฃ<sub>i=1</sub><sup>n</sup>w<sub>i</sub> = 1, where ๐ฐ'ฮฃ = c๐Ÿ<sub>n</sub>โ€™ for some scalar c>0. Of interest is the class ฮฉ(n) = ฮ›(nโปยน๐Ÿ<sub>n</sub>), containing equicorrelation matrices in the intersection ฮ“(n)โ‹‚ฮฉ(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>๐‘ช</sub> the estimator is obtained as a correction of Xฬ… taking the form ฮด<sub>ฮฃ</sub>(๐—) = Xฬ… -ฮณ'๐‘ชโปยน<sub>๐“’(n)</sub>๐—, with ฮณ as in the expansion for ฮฃ. This simplifies when ฮฃโˆˆฮ“(n) to ฮด<sub>ฮฃ</sub>(๐—) = Xฬ… -ฮณ'๐ž, where ๐ž 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 Sโบ<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>โบ, for which each conditional expectation for ฮฃ given ๐‘ช โˆˆ 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.degreePh. D.en
dc.format.extentviii, 119 leavesen
dc.format.mediumBTDen
dc.format.mimetypeapplication/pdfen
dc.identifier.otheretd-08142006-110114en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-08142006-110114/en
dc.identifier.urihttp://hdl.handle.net/10919/39155en
dc.language.isoenen
dc.publisherVirginia Techen
dc.relation.haspartLD5655.V856_1995.S677.pdfen
dc.relation.isformatofOCLC# 34347443en
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.lccLD5655.V856 1995.S677en
dc.titlePitman estimation for ensembles and mixturesen
dc.typeDissertationen
dc.type.dcmitypeTexten
thesis.degree.disciplineStatisticsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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