Pitman estimation for ensembles and mixtures

Abstract

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 S_n⁺ of positive definite matrices is partitioned into various equivalence classes. The classes 𝓝_𝑪 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_n'=[1, 1, ..., 1]. Here Σ∈𝓝_𝑪 has the structure Σ(y) = 𝑪+γ𝟏_n'+𝟏_ny'-γ̅𝟏_n𝟏_n', for some vector γ such that γ'𝑪_𝓒(n)⁻¹γ < γ̅, where 𝑪_𝓒(n)⁻¹ is the Moore-Penrose inverse of 𝑪. Of particular interest is the class Γ(n) = 𝓝_𝐁 with 𝐁 = [𝐈_n - (1/n)𝟏_n𝟏_n']. In addition, the equivalence classes Λ(𝐰) in S_n⁺ are indexed through elements of 𝓦(n) containing n-dimensional vectors 𝐰 such that Σ_i=1ⁿw_i = 1, where 𝐰'Σ = c𝟏_n’ for some scalar c>0. Of interest is the class Ω(n) = Λ(n⁻¹𝟏_n), 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_𝑪 the estimator is obtained as a correction of X̅ taking the form δ_Σ(𝐗) = X̅ -γ'𝑪⁻¹_𝓒(n)𝐗, with γ as in the expansion for Σ. This simplifies when Σ∈Γ(n) to δ_Σ(𝐗) = X̅ -γ'𝐞, where 𝐞 is the vector of residuals {e_i = x_i-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 S_n⁺, 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.