Pitman estimation for ensembles and mixtures
| Autor: | Srinivasan, Shankar S. |
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| Jazyk: | angličtina |
| Rok vydání: | 1995 |
| Předmět: | |
| Druh dokumentu: | Dissertation |
| Popis: | This 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 Sn⁺ of positive definite matrices is partitioned into various equivalence classes. The classes 𝓝𝑪 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 𝟏n'=[1, 1, ..., 1]. Here Σ∈𝓝𝑪 has the structure Σ(y) = 𝑪+γ𝟏n'+𝟏ny'-γ̅𝟏n𝟏n', for some vector γ such that γ'𝑪𝓒(n)⁻¹γ < γ̅, where 𝑪𝓒(n)⁻¹ is the Moore-Penrose inverse of 𝑪. Of particular interest is the class Γ(n) = 𝓝𝐁 with 𝐁 = [𝐈n - (1/n)𝟏n𝟏n']. In addition, the equivalence classes Λ(𝐰) in Sn⁺ are indexed through elements of 𝓦(n) containing n-dimensional vectors 𝐰 such that Σi=1nwi = 1, where 𝐰'Σ = c𝟏n’ for some scalar c>0. Of interest is the class Ω(n) = Λ(n⁻¹𝟏n), 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 𝓝𝑪 the estimator is obtained as a correction of X̅ taking the form δΣ(𝐗) = X̅ -γ'𝑪⁻¹𝓒(n)𝐗, with γ as in the expansion for Σ. This simplifies when Σ∈Γ(n) to δΣ(𝐗) = X̅ -γ'𝐞, where 𝐞 is the vector of residuals {ei = xi-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⁺n. The estimators are now given through a dispersion matrix Ψ which is the expectation of Σ over 𝐆. For mixing measures over Sn⁺, 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. Ph. D. |
| Databáze: | Networked Digital Library of Theses & Dissertations |
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