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Shrinkage Estimation for Mean and Covariance Matrices

Part of the SpringerBriefs in Statistics Book Series

DOI: 10.1007/978-981-15-1596-5

ISBN: 978-981-15-1595-8

Publisher: Springer Singapore - Springer Journals

Apr 17, 2020

Available

Decision-Theoretic Approach to Estimation

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 1 –5

[Statistical decision theory has been studied from around the 1940s and the researchers have already been producing many remarkable results. In the field of decision-theoretic estimation, the most surprising result is the inadmissibility of the sample mean vector in estimation of a mean vector of...
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Matrix Algebra

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 7 –12

[Matrix algebra is an important step in mathematical treatment of shrinkage estimation for matrix parameters, and in particular the Moore-Penrose inverse and some matrix decompositions are required for defining matricial shrinkage estimators. This chapter first explains the notation used in this...
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Matrix-Variate Distributions

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 13 –26

[This chapter provides the definition and some useful properties of a matrix-variate normal distribution, nonsingular and singular Wishart distributions and other related distributions. The matrix-variate distributions considered here are based on the multivariate (vector-valued) normal...
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Multivariate Linear Model and Group Invariance

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 27 –33

[Multivariate linear model is a multivariate generalization for the dimension of response variable in traditional multiple linear regression model. This chapter provides some fundamental properties in terms of the multivariate linear model and the corresponding canonical form. The group...
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A Generalized Stein Identity and Matrix Differential Operators

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 35 –43

[In shrinkage estimation, the Stein (1973, 1981) identity is known as an integration by parts formula for deriving unbiased risk estimates. It is a simple but very powerful mathematical tool and has contributed significantly to the development of shrinkage estimation. This chapter provides a...
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Estimation of the Mean Matrix

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 45 –74

[This chapter introduces a unified approach to high- and low-dimensional cases for matricial shrinkage estimation of a normal mean matrix with unknown covariance matrix. A historical background is briefly explained, and matricial shrinkage estimators are motivated from an empirical Bayes method....
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Estimation of the Covariance Matrix

Tsukuma, Hisayuki; Kubokawa, Tatsuya

Pages: 75 –110

[This chapter addresses decision-theoretic estimation of an error covarianceError covariance matrix in a multivariate linear model relative to a Stein-type entropy loss. With a unified treatment for high and low dimensions, some important improving methods of the best scale and the best...

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