Pitman estimation for ensembles and mixtures
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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 Snโบ of positive definite matrices is partitioned into various equivalence classes. The classes ๐๐ช are indexed through matrices ๐ช from a class ๐(n) comprising positive semidefinite (nรn) matrices with one-dimensional subspace spanned by the unit vector ๐n'=[1, 1, ..., 1]. Here ฮฃโ๐๐ช has the structure ฮฃ(y) = ๐ช+ฮณ๐n'+๐ny'-ฮณฬ ๐n๐n', for some vector ฮณ such that ฮณ'๐ช๐(n)โปยนฮณ < ฮณฬ , where ๐ช๐(n)โปยน is the Moore-Penrose inverse of ๐ช. Of particular interest is the class ฮ(n) = ๐๐ with ๐ = [๐n - (1/n)๐n๐n']. In addition, the equivalence classes ฮ(๐ฐ) in Snโบ are indexed through elements of ๐ฆ(n) containing n-dimensional vectors ๐ฐ such that ฮฃi=1nwi = 1, where ๐ฐ'ฮฃ = c๐nโ for some scalar c>0. Of interest is the class ฮฉ(n) = ฮ(nโปยน๐n), 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 ๐๐ช the estimator is obtained as a correction of Xฬ taking the form ฮดฮฃ(๐) = Xฬ -ฮณ'๐ชโปยน๐(n)๐, with ฮณ as in the expansion for ฮฃ. This simplifies when ฮฃโฮ(n) to ฮดฮฃ(๐) = Xฬ -ฮณ'๐, where ๐ is the vector of residuals {ei = xi-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 Sโบn. The estimators are now given through a dispersion matrix ฮจ which is the expectation of ฮฃ over ๐. For mixing measures over Snโบ, 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.