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Monotonic minimax estimators of a 2×2 covariance matrix
Author(s) -
Perron François
Publication year - 1992
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315613
Subject(s) - wishart distribution , estimator , mathematics , monotonic function , invertible matrix , combinatorics , covariance matrix , estimation of covariance matrices , covariance , diagonal , statistics , pure mathematics , mathematical analysis , geometry , multivariate statistics
Let S : 2 × 2 have a nonsingular Wishart distribution with unknown matrix σ and n degrees of freedom. For estimating σ two families of mimmax estimators, with respect to the entropy loss, are presented. These estimators are of the form σ(S) = Rø(L)R t where R is orthogonal, L and Φ are diagonal, and RLR T = S. Conditions under which the components of Φ and L follow the same order relation [i.e., writing Φ = diag(Φ 1 ,Φ 2 ) and L = diag(l 1 ,/ 2 ) with l 1 ≥ l 2 , we have Φ 1 ≥ Φ 2 ] are established. Comparisons with Stein's estimators and other orthogonally invariant estimators are discussed.