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Extension of some important identities in shrinkage-pretest strategies

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Abstract

In this paper, we establish three identities which play a crucial role in deriving the asymptotic distributional risk function and the asymptotic distributional bias of a large class of estimators of a matrix parameter. In particular, we generalize the results in Judge and Bock (The statistical implication of pre-test and Stein-rule estimators in econometrics. North Holland, Amsterdam, 1978). The established results are useful in risk analysis of a class of Stein-rule type matrix estimators.

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Acknowledgments

The author would like to acknowledge the financial support received from Natural Sciences and Engineering Research Council of Canada. Also, the author is thankful to anonymous referees for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Windsor, 401 Sunset Avenue, Windsor, ON, N9B 3P4, Canada

    Sévérien Nkurunziza

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  1. Sévérien Nkurunziza
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Correspondence to Sévérien Nkurunziza.

Appendix

Appendix

1.1 Proofs of Theorems 3.1 and 3.2

Proof of Theorem 3.1

Since \(\varvec{\Xi }_{1}^{\frac{1}{2}}\varvec{\Upsilon }\varvec{\Xi }_{1}^{\frac{1}{2}}\) and \(\varvec{\Xi }_{2}^{\frac{1}{2}}\varvec{\Lambda }\varvec{\Xi }_{2}^{\frac{1}{2}}\) are symmetric idempotent matrices, there exist orthogonal matrices \(\varvec{Q}_{1}\) and \(\varvec{Q}_{2}\) such that

$$\begin{aligned} \varvec{Q}_{1}\varvec{\Xi }_{1}^{\frac{1}{2}}\varvec{\Upsilon }\varvec{\Xi }_{1}^{\frac{1}{2}}\varvec{Q}_{1}^{\prime }=\left( \begin{array}{cc} \varvec{I}_{p}&\varvec{0} \\ \varvec{0}&\varvec{0} \\ \end{array} \right)\quad { } \text{ and} \quad { } \varvec{Q}_{2}\varvec{\Xi }_{2}^{\frac{1}{2}}\varvec{\Lambda }\varvec{\Xi }_{2}^{\frac{1}{2}}\varvec{Q}_{2}^{\prime }=\left( \begin{array}{cc} \varvec{I}_{q_{1}}&\varvec{0} \\ \varvec{0}&\varvec{0} \\ \end{array} \right) . \end{aligned}$$
(5.1)

Moreover, let \(\varvec{V}=\varvec{Q}_{1}\varvec{\Xi }_{1}^{\frac{1}{2}}\varvec{X} \varvec{\Xi }_{2}^{\frac{1}{2}}\varvec{Q}^{\prime }_{2}\). Then, \(\text{ Vec}\left(\varvec{V}\right)=(\varvec{Q}_{1}\varvec{\Xi }_{1}^{\frac{1}{2}}\otimes \varvec{Q}_{2}\varvec{\Xi }_{2}^{\frac{1}{2}})\text{ Vec}\left(\varvec{X}\right)\) and hence,

$$\begin{aligned} \text{ Vec}\left(\varvec{V}\right)=\left( \begin{array}{c} \varvec{V}_{1} \\ \varvec{V}_{2} \\ \end{array} \right)\sim \fancyscript{N}_{q\times k}\left(\left( \begin{array}{c} \varvec{\mu }_{1} \\ \varvec{0} \\ \end{array} \right),\,\,\varvec{\Sigma }_{v}\right)\!, \end{aligned}$$
(5.2)

with

$$\begin{aligned} \varvec{\mu }_{1}&= \left[\varvec{I}_{pq},\,\varvec{0}\right]\, \text{ E}\left(\text{ Vec}\left(\varvec{V}\right)\right) =\left(\left[\varvec{I}_{p},\,\varvec{0}\right]\otimes \varvec{I}_{q}\right)\,\left(\varvec{Q}_{1}\varvec{\Xi }_{1}^{\frac{1}{2}} \otimes \varvec{Q}_{2}\varvec{\ Xi }_{2}^{\frac{1}{2}}\right) \text{ Vec}\left(\varvec{M}\right)\nonumber \\ \varvec{\Sigma }_{v}&= \left(\begin{array}{cc} \varvec{I}_{p}&\varvec{0} \\ \varvec{0}&\varvec{0} \\ \end{array} \right)\otimes \left(\begin{array}{cc} \varvec{I}_{q_{1}}&\varvec{0} \\ \varvec{0}&\varvec{0} \\ \end{array} \right). \end{aligned}$$
(5.3)

Therefore, from (5.2), \( \text{ trace}\left(\varvec{\Xi }_{2}\varvec{X}^{\prime }\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X}\right) =\text{ trace}(\varvec{V}^{\prime }\varvec{Q}_{1}\varvec{\Xi }_{1}^{\frac{1}{2}} \varvec{\Upsilon }\varvec{\Xi }_{1}^{\frac{1}{2}}\varvec{Q}_{1}^{\prime }\varvec{V}) =\varvec{V}_{1}^{\prime }\varvec{V}_{1}.\) Hence, \(\text{ Vec}\left(\text{ E}\left[\phi \left( \text{ trace}\left(\varvec{\Xi }_{2}\varvec{X}^{\prime }\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X}\right)\right)\varvec{X}\right]\right)= \text{ E}[\phi (\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,) (\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\otimes \) \( \varvec{\Xi }_{2}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{2})\text{ Vec}(\varvec{V})]\) and then,

$$\begin{aligned}&\text{ Vec}\left(\text{ E}\left[\phi \left( \text{ trace}\left(\varvec{\Xi }_{2}\varvec{X}^{\prime }\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X}\right)\right)\varvec{X}\right]\right) =\left(\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\otimes \varvec{\Xi }_{2}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{2}\right)\nonumber \\&\quad { } \times \left(\text{ E}\left[\phi \left(\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,\right)\varvec{V}_{1}\right],\,\text{ E}\left[\phi \left(\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,\right) \varvec{V}_{2}\right]\right)^{\prime }. \end{aligned}$$
(5.4)

Also, from (5.2) and (5.3), \(\text{ E}\left(\varvec{V}_{2}\right)=\varvec{0}\) and \(\text{ Var}\left(\varvec{V}_{2}\right)=\varvec{0}\), so \(\varvec{V}_{2}\) is \(\varvec{0}\) with probability one, and then, \(\text{ E}\left[\phi \left(\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,\right) \varvec{V}_{2}\right]=\varvec{0}\). Further, by using Theorem 1 in Judge and Bock (1978), we get

$$\begin{aligned} \text{ E}\left[\phi \left(\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,\right)\varvec{V}_{1}\right] =\varvec{\mu }_{1}\text{ E}\left(\phi \left(\chi ^{2}_{pq_{1}+2}\left(\varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1}\right)\right)\right)\!, \end{aligned}$$
(5.5)

where \(\varvec{\mu }_{1}\) is given by (5.3). Using (5.1)- (5.3), \(\varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1}=\text{ Vec}\left(\varvec{M}\right)^{\prime } \left(\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\otimes \varvec{\Xi }_{2}\right) \text{ Vec}\left(\varvec{M}\right)\), i.e.

$$\begin{aligned} \varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1} =\text{ trace}\left(\varvec{M}^{\prime }\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{M}\varvec{\Xi }_{2}\right) =\text{ trace}\left(\varvec{M}^{\prime }\varvec{\Xi }_{1}\varvec{M}\varvec{\Xi }_{2}\right)\!. \end{aligned}$$
(5.6)

Further, combining (5.4) and (5.5), we have

$$\begin{aligned} \text{ Vec}\left(\text{ E}\left[\phi \left( \text{ trace}\left(\varvec{\Xi }_{2}\varvec{X}^{\prime }\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X}\right)\right)\varvec{X}\right]\right) =\text{ E}\left(\phi \left(\chi ^{2}_{pq_{1}+2}\left(\varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1}\right)\right)\right)\text{ Vec}\left(\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{M}\right)\!, \end{aligned}$$

and then, \(\text{ Vec}\left(\! \text{ E}\left[\phi \left( \text{ trace}\left(\varvec{\Xi }_{2}\varvec{X}^{\prime }\varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X}\right)\right)\varvec{X}\right]\right)\!=\! \text{ Vec}\!\left(\!\text{ E}\left(\!\phi \left(\!\chi ^{2}_{pq_{1}\!+\!2}\left(\varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1}\right)\right)\right)\varvec{M}\right)\), this completes the proof. \(\square \)

Proof of Theorem 3.2

From the above computations, we have \(\phi ( \text{ trace}(\varvec{\Xi }_{2}\varvec{X}^{\prime }\varvec{\Xi }_{1} \varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X})) \,{=}\, \phi (\varvec{V}_{1}^{\prime }\varvec{V}_{1})\). Further, as in Theorem 3.1, we have \(\varvec{V}=\varvec{Q}_{1}\varvec{\Xi }_{1}^{\frac{1}{2}}\varvec{X}\varvec{\Xi }_{2}^{\frac{1}{2}}\varvec{Q}^{\prime }_{2}\) where \(\varvec{Q}_{1}\) and \(\varvec{Q}_{2}\) are the same as in (5.1). We have \(\varvec{X}=\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\varvec{V}\varvec{Q}_{2} \varvec{\Xi }_{2}^{-\frac{1}{2}}\) and hence,

$$\begin{aligned} \varvec{X}^{\prime }\varvec{A}\varvec{X}=\varvec{\Xi }_{2}^{-\frac{1}{2}}\varvec{Q}_{2}\varvec{V}^{\prime }\varvec{Q}_{1}\varvec{\Xi }_{1}^{-\frac{1}{2}} \varvec{A}\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\varvec{V}\varvec{Q}^{\prime }_{2} \varvec{\Xi }_{2}^{-\frac{1}{2}}. \end{aligned}$$
(5.7)

Therefore, \(\text{ E}\left[\phi \left( \text{ trace}\left(\varvec{\Xi }_{2}\varvec{X}^\prime \varvec{\Xi }_{1}\varvec{\Upsilon }\varvec{\Xi }_{1}\varvec{X}\right)\right) \text{ trace}\left(\varvec{X}^{\prime }\varvec{A}\varvec{X}\right)\right]=\text{ E}[\phi \left(\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,\right)\text{ trace}(\varvec{Q}_{1}\varvec{\Xi }_{1}^{-\frac{1}{2}} \varvec{A}\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\varvec{V}\varvec{Q}_{2} \varvec{\Xi }_{2}^{-1}\varvec{Q}^{\prime }_{2}\varvec{V}^{\prime })]\). Also, we have \(\text{ trace} (\varvec{Q}_{1}\varvec{\Xi }_{1}^{-\frac{1}{2}} \varvec{A}\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\varvec{V}\varvec{Q}_{2} \varvec{\Xi }_{2}^{-1}\varvec{Q}^{\prime }_{2}\varvec{V}^{\prime }) =\left(\text{ Vec}\left(\varvec{V}\right)\right)^{\prime } (\varvec{Q}_{1}\varvec{\Xi }_{1}^{-\frac{1}{2}} \varvec{A}\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\otimes \varvec{Q}_{2}\varvec{\Xi }_{2}^{-1}\varvec{Q}^{\prime }_{2}) \text{ Vec}\left(\varvec{V}\right)\). Further, let

$$\begin{aligned} \left(\varvec{Q}_{1}\varvec{\Xi }_{1}^{-\frac{1}{2}} \varvec{A}\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\otimes \varvec{Q}_{2}\varvec{\Xi }_{2}^{-1}\varvec{Q}^{\prime }_{2}\right)=\varvec{G}=\left( \begin{array}{cc} \varvec{G}_{11}&\varvec{G}_{12} \\ \varvec{G}_{21}&\varvec{G}_{22} \\ \end{array} \right). \end{aligned}$$
(5.8)

Therefore, \(\left(\text{ Vec}\left(\varvec{V}\right)\right)^{\prime } (\varvec{Q}_{1}\varvec{\Xi }_{1}^{-\frac{1}{2}} \varvec{A}\varvec{\Xi }_{1}^{-\frac{1}{2}}\varvec{Q}^{\prime }_{1}\otimes \varvec{Q}_{2}\varvec{\Xi }_{2}^{-1}\varvec{Q}^{\prime }_{2}) \left(\text{ Vec}\left(\varvec{V}\right)\right) =\varvec{V}_{1}^{\prime }\varvec{G}_{11}\varvec{V}_{1}. \) Further, using Theorem 2 in Judge and Bock (1978), we have

$$\begin{aligned} \text{ E}\left[\phi \left(\varvec{V}_{1}^{\prime }\varvec{V}_{1}\,\right)\varvec{V}_{1}^{\prime }\varvec{G}_{11}\varvec{V}_{1}\right]&= \text{ E}\left[\phi \left(\chi _{pq_{1}+2}^{2}\left( \text{ trace} \left(\varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1}\right)\right)\right)\right] \text{ trace}\left(\varvec{G}_{11}\right)\nonumber \\&\quad + \text{ E}\left[\phi \left(\chi _{pq_{1}+4}^{2}\left(\text{ trace} \left(\varvec{\mu }_{1}^{\prime }\varvec{\mu }_{1}\right)\right)\right)\right]\left(\varvec{\mu }_{1}^{\prime }\varvec{G}_{11}\varvec{\mu }_{1}\right)\!. \nonumber \\ \end{aligned}$$
(5.9)

where

$$\begin{aligned} \varvec{\mu }_{1}^{\prime }\varvec{G}_{11}\varvec{\mu }_{1}= \text{ trace}\left(\varvec{M}^{\prime }\varvec{A}\varvec{M}\right), \quad { } \text{ trace}\left(\varvec{G}_{11}\right) =\text{ trace}\left(\varvec{A}\varvec{\Upsilon }\right) \text{ trace}\left(\varvec{\Xi }_{2}^{-1}\right)\!, \qquad \end{aligned}$$
(5.10)

this completes the proof. \(\square \)

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Nkurunziza, S. Extension of some important identities in shrinkage-pretest strategies. Metrika 76, 937–947 (2013). https://doi.org/10.1007/s00184-012-0425-5

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