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Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

Hisayuki Tsukuma
2009 Journal of Multivariate Analysis  
This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models.  ...  The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given  ...  Acknowledgments The author thanks Professor Tatsuya Kubokawa for his helpful comments and suggestions. The author is also grateful to the referee for his/her careful reading and valuable comments.  ... 
doi:10.1016/j.jmva.2009.04.009 fatcat:lcup766hvjfwriqlw4u7zyabfy

Page 1015 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
The author studies the classical problem of estimating the mul- tivariate normal mean when the covariance matrix is unknown.  ...  Gessaman (Omaha, NE) 95b:62008 62C12 62C20 62H12 Shieh, Gwowen (RC-NCT-MN; Taipei) Empirical Bayes minimax estimators of matrix normal means for arbitrary quadratic loss and unknown covariance matrix.  ... 

Page 7204 of Mathematical Reviews Vol. , Issue 87m [page]

1987 Mathematical Reviews  
Let X be a k-variate normal random variable with unknown mean @ and covariance matrix © = J;, the identity matrix.  ...  Summary: “We consider a p-variate normal population with mean p and covariance matrix ©.  ... 

Page 276 of Mathematical Reviews Vol. 52, Issue 1 [page]

1976 Mathematical Reviews  
Another application of the result is to the following problem. Let X be a random p xn matrix, n2p, with unknown mean é and with coordinates X;, independently normally distributed with variance 1.  ...  ., 1956; MR 18, 948] proved that the sample mean vector, from a multivariate normal distribution with mean vector ¢ and covariance matrix /, is an inadmissible estimator for ¢ when the loss function is  ... 

Restricted Risk Bayes Linear State Estimation

Yoav Levinbook, Tan F. Wong
2009 IEEE Transactions on Information Theory  
The problem of state estimation with stochastic uncertainties in the initial state, model noise, and measurement noise is considered using the restricted risk Bayes approach.  ...  This method is illustrated with a target tracking example and a wireless channel tracking example for which the Bayes, minimax, and restricted risk Bayes estimators are derived and their performance is  ...  These figures compare the normalized histograms of for the Bayes, restricted risk Bayes, and minimax estimators with , and , respectively.  ... 
doi:10.1109/tit.2009.2027551 fatcat:tq3w34437jevnhslskjjdvu3gm

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Khursheed Alam
1975 Journal of Multivariate Analysis  
Lin and Tsai [9] have obtained generalized Bayes minimax estimators for unknown covariance matrix.  ...  Strawderman [l 1, 121 has obtained Bayes minimax estimators for the case of known covariance matrix withp 2 5 and for the case of common unknown variances.  ... 
doi:10.1016/0047-259x(75)90057-3 fatcat:s24x6efnkzfytivysix6x66dem

Page 2628 of Mathematical Reviews Vol. , Issue 82f [page]

1982 Mathematical Reviews  
Let X have a p-variate normal distribution with unknown mean vector @ and nonsingular covariance matrix 2.  ...  The estimator is the generalized Bayes estimator corresponding to a “prior” which, conditional on A, is p-normal with mean p (the prior mean) and covariance matrix B(A)=A"'c— = (more about c later) and  ... 

Page 5608 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
Assume we have a random sample of size n from a normal popu- lation with unknown mean é and unknown standard deviation co. Consider the problem of estimating the noncentrality parameter 6 = (&/a)?  ...  {For the entire collection see MR 93e:62002.} 93j:62015 62C15 62H12 Perron, F. (3-MTRL) Minimax estimators of a covariance matrix. (English summary) J. Multivariate Anal. 43 (1992), no. 1, 16-28.  ... 

Bayes minimax estimation of the multivariate normal mean vector for the case of common unknown variance

S. Zinodiny, W.E. Strawderman, A. Parsian
2011 Journal of Multivariate Analysis  
We investigate the problem of estimating the mean vector θ of a multivariate normal distribution with covariance matrix σ 2 I p , when σ 2 is unknown, and where the loss function is ‖δ−θ ‖ 2 σ 2 .  ...  We find a large class of (proper and generalized) Bayes minimax estimators of θ , and show that the result of [8] is a special case of our result.  ...  The third author's research was supported by a grant of the Research Council of the University of Tehran.  ... 
doi:10.1016/j.jmva.2011.04.008 fatcat:6wzprvd5anfpbhhzerrohq4opu

Page 5375 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
Consider the problem of estimating the common mean yu of two normal populations with unknown variances a? and o} under the quadratic loss (fi — u)?/a?.  ...  A family of minimax estimators with smaller risk than the sample mean in the first population is given, out of which admissible minimax estimators are developed.  ... 

Methods for improvement in estimation of a normal mean matrix

Hisayuki Tsukuma, Tatsuya Kubokawa
2007 Journal of Multivariate Analysis  
This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss.  ...  It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators.  ...  Acknowledgments The research of the first author was supported in part by Grant-in-Aid for Scientific Research No. 1610018.  ... 
doi:10.1016/j.jmva.2007.04.009 fatcat:nwnsyd7hjfhgvisz3j6qshxpzu

Page 7407 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
Consider a k-dimensional and normally distributed vector y with mean vector » and known covariance matrix VJ,, V >0.  ...  The results obtained are valid for general quadratic loss and possibly singular covariance matrix.  ... 

Page 3906 of Mathematical Reviews Vol. , Issue 92g [page]

1992 Mathematical Reviews  
Francoise Garcia-Brouaye (Gif-sur-Yvette) 929:62011 62C20 62H12 Konno, Yoshihiko Families of minimax estimators of matrix of normal means with unknown covariance matrix. J. Japan Statist.  ...  Summary: “Wright (1983) defined the class of QR-predictors for general linear models with diagonal covariance matrices.  ... 

Page 1704 of Mathematical Reviews Vol. 48, Issue 5 [page]

1974 Mathematical Reviews  
Reinhardt (London) Lin, Pi Erh; Tsai, Hui Liang 9905 Generalized Bayes minimax estimators of the multi- variate normal mean with unknown covariance matrix. Ann. Statist. 1 (1973), 142-145.  ...  Authors’ summary: “Let X be a p-variate (p23) vector normally distributed with mean @ and covariance matrix x, positive definite but unknown.  ... 

Bayesian aspects of some nonparametric problems

Linda H. Zhao
2000 Annals of Statistics  
We then present a class of priors whose Bayes procedures attain the optimal minimax rate of convergence.  ...  We then present a class of priors whose Bayes procedures attain the optimal minimax rate of convergence.  ...  The author would like to thank all the referees and especially the Associate Editor for many very useful suggestions.  ... 
doi:10.1214/aos/1016218229 fatcat:lyl5s6jyqbbgfggxc6s462uaya
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