Estimation of the mean vector in a singular multivariate normal distribution

Hisayuki Tsukuma and Tatsuya Kubokawa

Journal of Multivariate Analysis, 2015, vol. 140, issue C, 245-258

Abstract: This paper addresses the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. The maximum likelihood estimator is shown to be minimax relative to a quadratic loss weighted by the Moore–Penrose inverse of the covariance matrix. An unbiased risk estimator relative to the weighted quadratic loss is provided for a Baranchik type class of shrinkage estimators. Based on the unbiased risk estimator, a sufficient condition for the minimaxity is expressed not only as a differential inequality, but also as an integral inequality. Also, generalized Bayes minimax estimators are established by using an interesting structure of singular multivariate normal distribution.

Keywords: Empirical Bayes method; Generalized Bayes estimator; Inadmissibility; Minimaxity; Moore–Penrose inverse; Pseudo-Wishart distribution; Quadratic loss; Shrinkage estimator; Statistical decision theory (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:jmvana:v:140:y:2015:i:c:p:245-258

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DOI: 10.1016/j.jmva.2015.05.016

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