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Best equivariant estimators of a Cholesky decomposition. (English) Zbl 0629.62057

Every positive definite matrix \(\Sigma\) has a unique Cholesky decomposition \(\Sigma =\theta \theta '\), where \(\theta\) is lower triangular with positive diagonal elements. Suppose that S has a Wishart distribution with mean \(n\Sigma\) and that S has the Cholesky decomposition \(S=XX'\). We show, for a variety of loss functions, that XD, where D is diagonal, is a best equivariant estimator of \(\theta\). Explicit expressions for D are provided.

MSC:

62H12 Estimation in multivariate analysis
15A23 Factorization of matrices
62H10 Multivariate distribution of statistics
62A01 Foundations and philosophical topics in statistics
15B52 Random matrices (algebraic aspects)
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