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

2009
*
Journal of Multivariate Analysis
*

This paper addresses

doi:10.1016/j.jmva.2009.04.009
fatcat:lcup766hvjfwriqlw4u7zyabfy
*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. ...##
###
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

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. ...

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

1975
*
Journal of Multivariate Analysis
*

Lin and Tsai [9] have obtained

doi:10.1016/0047-259x(75)90057-3
fatcat:s24x6efnkzfytivysix6x66dem
*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. ...##
###
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

2011
*
Journal of Multivariate Analysis
*

We investigate

doi:10.1016/j.jmva.2011.04.008
fatcat:6wzprvd5anfpbhhzerrohq4opu
*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. ...##
###
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

2007
*
Journal of Multivariate Analysis
*

This paper is concerned

doi:10.1016/j.jmva.2007.04.009
fatcat:nwnsyd7hjfhgvisz3j6qshxpzu
*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. ...##
###
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

2000
*
Annals of Statistics
*

We then present a class

doi:10.1214/aos/1016218229
fatcat:lyl5s6jyqbbgfggxc6s462uaya
*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. ...
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