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Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

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  • Brandwein, Ann Cohen

Abstract

For X one observation on a p-dimensional (p >= 4) spherically symmetric (s.s.) distribution about [theta], minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L([delta], [theta]) = ([delta] - [theta])' D([delta] - [theta]) where D is a known p - p positive definite matrix. For C a p - p known positive definite matrix, conditions are given under which estimators of the form [delta]a,r,C,D(X) = (I - (ar(X2)) D-1/2CD1/2 X-2)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X1, X2, ..., Xn are taken on a p-dimensional s.s. distribution about [theta], any spherically symmetric translation invariant estimator, [delta](X1, X2, ..., Xn), with have a s.s. distribution about [theta]. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.

Suggested Citation

  • Brandwein, Ann Cohen, 1979. "Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 579-588, December.
    • Handle: RePEc:eee:jmvana:v:9:y:1979:i:4:p:579-588
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    Cited by:

    1. Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E., 2015. "On improved shrinkage estimators for concave loss," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 241-246.
    2. Fourdrinier, Dominique & Strawderman, William E., 2008. "A unified and generalized set of shrinkage bounds on minimax Stein estimates," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2221-2233, November.
    3. Xu, Jian-Lun & Izmirlian, Grant, 2006. "Estimation of location parameters for spherically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 514-525, February.
    4. Shalabh & H. Toutenburg & C. Heumann, 2008. "Mean squared error matrix comparison of least aquares and Stein-rule estimators for regression coefficients under non-normal disturbances," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 285-298.
    5. Maruyama, Yazo & Takemura, Akimichi, 2008. "Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 50-73, January.
    6. Tatsuya Kubokawa & Éric Marchand & William E. Strawderman, 2014. "On Improved Shrinkage Estimators for Concave Loss," CIRJE F-Series CIRJE-F-936, CIRJE, Faculty of Economics, University of Tokyo.

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