Abstract
dc:description.abstractThis dissertation considers minimal risk equivariant (MRE) estimation of a location scalar μ in ensembles and mixtures of translation families having structured dispersion matrices Σ. The principal focus is the preservation of Pitman's solutions across classes of distributions. To these ends the cone S<sub>n</sub>⁺ of positive definite matrices is partitioned into various equivalence classes. The classes 𝓝<sub>𝑪</sub> are indexed through matrices 𝑪 from a class 𝓒(n) comprising positive semidefinite (n×n) matrices with one-dimensional subspace spanned by the unit vector 𝟏<sub>n</sub>'=[1, 1, ..., 1]. Here Σ∈𝓝<sub>𝑪</sub> has the structure Σ(y) = 𝑪+γ𝟏<sub>n</sub>'+𝟏<sub>n</sub>y'-γ̅𝟏<sub>n</sub>𝟏<sub>n</sub>', for some vector γ such that γ'𝑪<sub>𝓒(n)</sub>⁻¹γ < γ̅, where 𝑪<sub>𝓒(n)</sub>⁻¹ is the Moore-Penrose inverse of 𝑪. Of particular interest is the class Γ(n) = 𝓝<sub>𝐁</sub> with 𝐁 = [𝐈<sub>n</sub> - (1/n)𝟏<sub>n</sub>𝟏<sub>n</sub>']. In addition, the equivalence classes Λ(𝐰) in S<sub>n</sub>⁺ are indexed through elements of 𝓦(n) containing n-dimensional vectors 𝐰 such that Σ<sub>i=1</sub><sup>n</sup>w<sub>i</sub> = 1, where 𝐰'Σ = c𝟏<sub>n</sub>’ for some scalar c>0. Of interest is the class Ω(n) = Λ(n⁻¹𝟏<sub>n</sub>), containing equicorrelation matrices in the intersection Γ(n)⋂Ω(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 μ. For elliptical random vectors 𝐗 the Pitman estimator continues to take the generalized least squares form. Further, ensembles of elliptically symmetric distributions having dispersion matrices in Ω(n) preserve the equivariant admissibility of the sample average X̅ under squared error loss. For dispersion matrices Σ in each class 𝓝<sub>𝑪</sub> the estimator is obtained as a correction of X̅ taking the form δ<sub>Σ</sub>(𝐗) = X̅ -γ'𝑪⁻¹<sub>𝓒(n)</sub>𝐗, with γ as in the expansion for Σ. This simplifies when Σ∈Γ(n) to δ<sub>Σ</sub>(𝐗) = X̅ -γ'𝐞, where 𝐞 is the vector of residuals {e<sub>i</sub> = x<sub>i</sub>-x̅; i = 1, 2, ..., n}. These results carry over to dispersion mixtures of elliptically symmetric distributions when the mixing measure 𝐆 is restricted to the corresponding subsets of S⁺<sub>n</sub>. The estimators are now given through a dispersion matrix Ψ which is the expectation of Σ over 𝐆. For mixing measures over S<sub>n</sub>⁺, for which each conditional expectation for Σ given 𝑪 ∈ 𝓒(n) is in Ω(n), X̅ is the Pitman estimator for μ 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.
Degree
thesis:*- Name thesis:degree_name
- Ph. D.
- Level thesis:degree_level
- doctoral
- Discipline thesis:degree_discipline
- Statistics
- Department dc:contributor.department
- Statistics
- Grantor dc:publisher
- Virginia Tech
- Year dc:date.issued
- 1995
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Srinivasan, Shankar S.
- Chair dc:contributor.committeechair
-
- Jensen, Donald R.
- Committee members dc:contributor.committeemember
-
- Foutz, Robert V.
- Myers, Raymond
- Smith, Eric P.
- Terrell, George R.
Rights
dc:rights- Statement dc:rights
-
- In Copyright
- Licence dc:rights.uri
- Language dc:language.iso
- en
Identifiers
dc:identifier.*- Dc Identifier Other
- etd-08142006-110114
- OAI identifier oai:identifier
- oai:vtechworks.lib.vt.edu:10919/39155