A

B

C

D

E

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H

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J

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A

• advantage
  • There are indications, however, that our present estimator has some advantages and offers further imporment yet.
    Haff, L.R., Ann. Statist. vol19(1992), page 1163.

• also

  • Assume also that $S$, $ p \times p $, has a Wishart distribution with matrix $ \Sigma$, $ k -p - 1 > 0 $ and $ X $ and $ S $ are independent.
    Haff, L.R. Ann. Statist. vol 19(1992), page 1163.
• among others
  • Among others, Muirhead and Verathaworn(1983) used our method to estimate eigenvalues in a tow-population setting.
    Haff, L.R., Ann. Statist. vol19(1992), page 1168.
• analogue
  • Finally, the analogue of our variational method was worked out in the discrete setting by Alcaraz(1990), and minimax results were ottained for the estimatin of several Poisson means.
    Haff, L.R., Ann. Statist. vol19(1992), page 1168.
• appear
  • Stien's estimator appears intractable for irsk calculations.
    Haff, L.R., Ann. Statist. vol19(1992), page 1168.
• assume
  • Assume furhter that $ \Psi = ( \Psi _1,\,\Psi) $ in wichi $ \Psi _i in R ^{p _i }, i = 1,\,2$.
    Haff, L.R. Ann. Statist. vol19(1992), page 1169.

B     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• boy
  • 
    
        

C     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• comment
  • Now we comment on the derivation of the VFBE and also on the application of the result.
    Haff, L.R. Ann. Statist. vol19(1992), page 1164.
• compare
  • It is natural to compare our estimator $ \Sigma $ with that of Stein (1975), who obtained the rough estimates of $ \lambda $ by heuristic minimaization of the unbiased estimator of the risk function.
    Haff, L.R. Ann. Statist. vol19(1992), page 1167.
• comparison
  • More favouralbe comparisons were made under $L_2$ even though our won estimator takes on an ad hoc nature in this case.
    Haff, L.R. Ann. Statist. vol19(1992), page 1168.
• constaint
  • Hence, the natural constraint in (1.1) is $ \varphi _1 \ge \varphi _2 \ge \cdots \ge \varphi _p \ge 0$, where $ \varphi _i = \varphi _i ( l ), i = 1,\,2,\,\ldots,\,p$, are the feasible estimates.
    Haff, L.R. Ann. Statist. vol19(1992), page 1167.
  • Our constrained minimization is criterion dependent.
    Haff, L.R. Ann. Statist. vol19(1992), page 1168.

D     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• denote
  • Denote by $ (\Phi,\,\hat \Psi,\,L ) $ any of the preceeding estimation problems.
    Haff, L.R. Ann. Statist. vol19(1992), page 1164
    
  • In addition, denote by $ \Pi ( \Psi) $ the prior distribution of $ \Psi $.
    Haff, L.R. Ann. Statist. vol19(1992), page 1164

• entail

  • The proof entails routine calculus only; we omit the details.
    Haff, L.R., Ann. Statist. vol19(1992), page 1175.

E     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• either
  • Several workers have either used our basic approach or referenced the computational results since the first version of this article was submitted.
    Haff, L.R., Ann. Statist. vol19(1992), page 1168.

• emphasis

  • A certain emphasis is placed on the problem of estimating the covariance matrix.
    Haff, L.R., Ann. Statist. vol19(1992), page 1163.

• entail

  • The proof entails routine calculus only; we omit the details.
    Haff, L.R., Ann. Statist. vol19(1992), page 1175.

F     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• fall
  • 
    

• follow

  • The opening hours are as follows ....
    OALD.

G     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

H     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• high
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I     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• indication
  • There are indications, however, that our present estimator has some advantages and offers further imporment yet.
    Haff, L.R., Ann. Statist. vol19(1992), page 1163.
• intractable
  • Stien's estimator appears intractable for irsk calculations.
    Haff, L.R., Ann. Statist. vol19(1992), page 1168.

J     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• jail
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K     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

L     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• long
  • 
    

M     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• minimization
  • It is natural to compare our estimator $ \Sigma $ with that of Stein (1975), who obtained the rough estimates of $ \lambda $ by heuristic minimaization of the unbiased estimator of the risk function.
    Haff, L.R. Ann. Statist. vol19(1992), page 1167.
  
  
• Monte Carlo
  • Several Monte Carlo studies have indicated that Stein's estimator does remarkably well in spite of its ad hoc nature.
    Haff, L.R. Ann. Statist. vol19(1992), page 1167.
  
  

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• oil
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• omit

  • The proof entails routine calculus only; we omit the details.
    Haff, L.R., Ann. Statist. vol19(1992), page 1175.

P     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• property

  • The VFBF is used to obtain estimators that have good frequency properties relative to the usual estimators Haff, L.R. Ann. Statist. vol 19(1992), page 1163.
  • Fo that problem, our constrained optimaization provides an estimator with very good properties: Its eigenvalues are in the proper order, and they are not as distorted as those in the sample covariance matrix.
    Haff, L.R. Ann. Statist. vol 19(1992), page 1163.

Q     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• quick
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R     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• reduction
  • Following reduction by symmetry, we mostly replace $ f _\Pi ( A ) $ by a function that is not a formal density in the sense of (0.6).
    Haff, L.R. Ann. Statist. vol19(1992), page 1165.
    
  • The constrainted VFBE the performed significantly betetr that Steins's estimator ( in the statistical sense), but the reduction in risk relative to the usual estimator was only about 2 %.
    Haff, L.R. Ann. Statist. vol19(1992), page 1168.
• represent
  • For $ X \sim N _p (\theta,\,I) $ and $ \Pi ( \theta ) $a ( possilbly imporper ) prior distribution, the formal Bayes estimator can be represented by ..... in which $ \nabla = ( ... ) $.
    Haff, L.R. Ann. Statist. vol19(1992), page 1165.
• respectively
  • Then the risk and Bayes risk are given by $ A $ and $ B $, respectively.
    Haff, L.R. Ann. Statist. vol19(1992), page 1164.
• result
  • In addition to the work on estimation, some useful computational results are found in this article.
    Haff, L.R. Ann. Statist. vol19(1992), page 1168.
  
  
• role
  • The media play a a major role in influencing people's opinion.
    OALD.

• routine

  • The proof entails routine calculus only; we omit the details.
    Haff, L.R., Ann. Statist. vol19(1992), page 1175.

S     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• scheme
  • We develop a scheme for improving (0.11) and certain of its analogues.
    Haff, L.R., Ann. Statist. vol19(1992), page 1166.
• simplify
  
  • Typically, the VFBE is simplified if we assume that $ \Pi (\Psi) $ is symmetric in some sense.
    Haff, L.R., Ann. Statist. vol19(1992), page 1165.
  • Ub this way, we obtain a general representation of the formal Bayes rule that depends explictly on $ f _\Pi $.
    Haff, L.R., Ann. Statist. vol19(1992), page 1164.

• suffer

  • We assume that an estimator $ \hat \theta $ suffers a loss ....
    Haff, L.R., Ann. Statist. vol19(1992), page 1164.
  • Ub this way, we obtain a general representation of the formal Bayes rule that depends explictly on $ f _\Pi $.
    Haff, L.R., Ann. Statist. vol19(1992), page 1164.

T     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• tex
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U     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• usual
  • 
    
        

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• very
  • 
    

W     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• well-known
  • It is a well-known fact that caffeine is a stimulant.
    OALD.
  
  
• which
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• x-ray
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• young
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Z     A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Top

• zero
  •