On shrinkage estimators in matrix variate elliptical models

Abstract

This paper derives the risk functions of a class of shrinkage estimators for the mean parameter matrix of a matrix variate elliptically contoured distribution. It is showed that the positive rule shrinkage estimator outperformed the shrinkage and unrestricted (maximum likelihood) estimators. To illustrate the findings of the paper, the relative risk functions for different degrees of freedoms are given for a multivariate t distribution. Shrinkage estimators for the matrix variate regression model under matrix normal, matrix t or Pearson VII error distributions would be special cases of this paper.

Appendix

Here we propose some necessary tools for mathematical computations with matrices. For the purpose of this appendix we precisely take matrix variate elliptically contoured (MEC) distributions into consideration.

Let $\varvec{X}$ be an $n\times p$ random matrix, which can be expressed in terms of its elements, column and rows as

$$\begin{aligned} \varvec{X}=(x_{ij})=(\varvec{x}_1,\ldots ,\varvec{x}_p)=(\varvec{x}_{(1)},\ldots ,\varvec{x}_{(n)})'. \end{aligned}$$

(6.1)

Here $\varvec{x}_{(1)},\ldots ,\varvec{x}_{(n)}$ can be regarded as a sample of size $n$ from a $p$-dimensional population.

As pointed by Fang and Li (1999), there are many ways (not completely different) to define elliptical matrix distributions. Four classes of elliptical matrix distributions are defined and discussed by Dawid (1977) and Anderson and Fang (1990). For the purpose of this appendix we specifically consider the following situations.

Definition 6.1

The $n\times p$ random matrix $\varvec{X}$ has a MEC distribution if its density has the form

$$\begin{aligned} g(\varvec{X})=d_{n,p}|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}f\left\{ \varvec{\Omega }^{-1}(\varvec{X}-\varvec{1}\varvec{\mu }')'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{1}\varvec{\mu }')\right\} , \end{aligned}$$

(6.2)

where $\varvec{\mu }\in \mathbb {R}^p,\,\varvec{\Omega }$ and $\varvec{\Sigma }$ are $p\times p$ and $n\times n$ positive definite matrices. This distribution is denoted by $\varvec{X}\sim \mathcal {E}_{n,p}(\varvec{\mu },\varvec{\Sigma }\otimes \varvec{\Omega },f)$. For notational convenience we may also use $\varvec{X}\sim \mathcal {E}_{n,p}(\varvec{\mu },\varvec{\Sigma },\varvec{\Omega },f)$ where needed.

Definition 6.1 imposes the condition $f(\varvec{A}\varvec{B})=f(\varvec{B}\varvec{A})$ on the density generator $f$ for any $p\times p$ positive definite symmetric matrices $\varvec{A}$ and $\varvec{B}$. Throughout, without loss of generality, we take $\mathop {\mathrm{tr}}\nolimits$ operation from the argument of $f(.)$.

Some examples of MEC distributions are given below.

• (i) Matrix Variate Normal (MN) Distribution $\varvec{X}\in \mathbb {R}^{n\times p}$ has MN distribution, with mean $\varvec{M}$, row and column covariance matrices $\varvec{\Omega }$ and $\varvec{\Sigma }$, respectively denoted by $\varvec{X}\sim MN_{n,p}(\varvec{M},\varvec{\Sigma },\varvec{\Omega })$, if its pdf is given by

  $$\begin{aligned} f(\varvec{X})=\frac{|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}}{(2\pi )^{\frac{np}{2}}}\; \exp \left\{ -\frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left[ \varvec{\Omega }^{-1}(\varvec{X}-\varvec{M})'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{M})\right] \right\} . \end{aligned}$$

• (ii) Matrix Variate Student-t (MT) Distribution $\varvec{X}\in \mathbb {R}^{n\times p}$ has MT distribution, with mean $\varvec{M}$, row and column scale matrices $\varvec{\Omega }$ and $\varvec{\Sigma }$, respectively and $\nu$ d.f. denoted by $\varvec{X}\sim MT_{n,p}(\varvec{M},\varvec{\Sigma },\varvec{\Omega },\nu )$, if its pdf is given by

  $$\begin{aligned} f(\varvec{X})=\frac{|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}}{g_{n,p}}\; \left| \varvec{I}_n+\varvec{\Omega }^{-1}(\varvec{X}-\varvec{M})'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{M})\right| ^{-\frac{n+p+\nu -1}{2}}, \end{aligned}$$

  where

  $$\begin{aligned} g_{n,p}=\frac{(\nu \pi )^{\frac{np}{2}}\Gamma _p\left( \frac{\nu +p-1}{2}\right) }{\Gamma _p\left( \frac{\nu +n+p-1}{2}\right) }. \end{aligned}$$

• (iii) Matrix Variate Pearson Type-VII (MPVII) Distribution $\varvec{X}\in \mathbb {R}^{n\times p}$ has MPVII distribution, with mean $\varvec{M}$, row and column scale matrices $\varvec{\Omega }$ and $\varvec{\Sigma }$, respectively and parameters $m$ and $q$ denoted by $\varvec{X}\sim MPVII_{n,p}(\varvec{M},\varvec{\Sigma },\varvec{\Omega },m,q)$, if its pdf is given by

  $$\begin{aligned} f(\varvec{X})=\frac{|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}}{h_{n,p,m,q}}\; \left| \varvec{I}_n+\frac{1}{q}\varvec{\Omega }^{-1}(\varvec{X}-\varvec{M})'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{M})\right| ^{-m}, \end{aligned}$$

  where

  $$\begin{aligned} h_{n,p,m,q}=\frac{(q\pi )^{\frac{np}{2}}\Gamma _p\left( m-\frac{n}{2}\right) }{\Gamma _p\left( m\right) }. \end{aligned}$$

• (iv) Matrix Variate Power Exponential (MPE) Distribution $\varvec{X}\in \mathbb {R}^{n\times p}$ has MPE distribution, with mean $\varvec{M}$, row and column scale matrices $\varvec{\Omega }$ and $\varvec{\Sigma }$, respectively and parameters $r$ and $s$ denoted by $\varvec{X}\sim MPE_{n,p}(\varvec{M},\varvec{\Sigma },\varvec{\Omega },r,s)$, if its pdf is given by

  $$\begin{aligned} f(\varvec{X})=\frac{|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}}{j_{n,p,r,s}}\; \exp \left\{ -\frac{r}{2}\left( \mathop {\mathrm{tr}}\nolimits \left[ \varvec{\Omega }^{-1}(\varvec{X}-\varvec{M})'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{M})\right] \right) ^s\right\} , \end{aligned}$$

  where

  $$\begin{aligned} j_{n,p,r,s}=\frac{(2\pi )^{\frac{np}{2}}\Gamma \left( \frac{np}{2s}\right) }{s\Gamma \left( \frac{np}{2}\right) r^{\frac{np}{2s}}}. \end{aligned}$$

• (v) Matrix Variate Laplace (ML) Distribution $\varvec{X}\in \mathbb {R}^{n\times p}$ has ML distribution, with mean $\varvec{M}$, row and column scale matrices $\varvec{\Omega }$ and $\varvec{\Sigma }$, respectively denoted by $\varvec{X}\sim ML_{n,p}(\varvec{M},\varvec{\Sigma },\varvec{\Omega })$, if its pdf is given by

  $$\begin{aligned} f(\varvec{X})=\frac{|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}}{l_{n,p}}\; \exp \left\{ -\frac{\sqrt{2}}{2}\left( \mathop {\mathrm{tr}}\nolimits \left[ \varvec{\Omega }^{-1}(\varvec{X}-\varvec{M})'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{M})\right] \right) ^{\frac{1}{2}}\right\} , \end{aligned}$$

  where

  $$\begin{aligned} i_{n,p}=\frac{2\pi ^{\frac{np}{2}}\Gamma \left( np\right) }{\Gamma \left( \frac{np}{2}\right) }. \end{aligned}$$

• (vi) Matrix Variate Kotz-Type (MK) Distribution $\varvec{X}\in \mathbb {R}^{n\times p}$ has MK distribution, with mean $\varvec{M}$, row and column scale matrices $\varvec{\Omega }$ and $\varvec{\Sigma }$, respectively and parameter $s$ denoted by $\varvec{X}\sim MK_{n,p}(\varvec{M},\varvec{\Sigma },\varvec{\Omega },s)$, if its pdf is given by

  $$\begin{aligned} f(\varvec{X})=\frac{|\varvec{\Omega }|^{-\frac{n}{2}}|\varvec{\Sigma }|^{-\frac{p}{2}}}{t_{n,p,r}}\; \exp \left\{ -\frac{r}{2}\left( \mathop {\mathrm{tr}}\nolimits \left[ \varvec{\Omega }^{-1}(\varvec{X}-\varvec{M})'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{M})\right] \right) \right\} , \end{aligned}$$

  where

  $$\begin{aligned} t_{n,p,r}=\frac{(2\pi )^{\frac{np}{2}}\Gamma \left( \frac{np}{2}\right) }{\Gamma \left( \frac{np}{2}\right) r^{\frac{np}{2}}}. \end{aligned}$$

Theorem 6.1

Let $\varvec{X}\sim \mathcal {E}_{n,p}(\varvec{\mu },\varvec{\Sigma }\otimes \varvec{\Omega }, f)$ where the p.d.f. $g(\varvec{X})$ of $\varvec{X}$ is defined by

$$\begin{aligned} g(\varvec{X}) = |\varvec{\Sigma }|^{-\frac{p}{2}}|\varvec{\Omega }|^{-\frac{n}{2}} h\left[ \mathop {\mathrm{tr}}\nolimits \varvec{\Omega }^{-1}(\varvec{X}-\varvec{\mu })'\varvec{\Sigma }^{-1}(\varvec{X}-\varvec{\mu })\right] . \end{aligned}$$

If $h(t),\,t\in [0,\infty )$ has the inverse Laplace transform (denoted by $\mathcal {L}^{-1}[h(t)]$), then we have

$$\begin{aligned} g(\varvec{X})=\int \limits _{\mathbb {R}^+} w(z) f_{\mathcal {N}\left( \varvec{\mu },z^{-1}\varvec{\Sigma }\otimes \varvec{\Omega }\right) }(\varvec{X})\, d z, \end{aligned}$$

(6.3)

where $f_{\mathcal {N}\left( \varvec{\mu },z^{-1}\varvec{\Sigma }\otimes \varvec{\Omega }\right) }(\varvec{X})$ stands for the p.d.f. of the $n\times p$ matrix $\varvec{X}$ distributed as matrix variate normal with the mean matrix $\varvec{\mu }$ and the covariance matrix $z^{-1}\varvec{\Sigma }\otimes \varvec{\Omega }$, and $w(z)$ is the weight function given by

$$\begin{aligned} w(z)=(2\pi )^{{np}/2}z^{-{np}/2}\mathcal {L}^{-1}[h(2t)]. \end{aligned}$$

(6.4)

For the proof we refer the readers to Theorem 4.2.1 of Gupta and Varga (1995).

Remark 6.1

An important issue raises when considering Theorem 6.1, is the mixture representation given by (1.3) and the form of the weighting function $w(.)$. Note that $w(.)$ is not always nonnegative [(see examples in Chu (1973), Provost and Cheong (2002) and Arashi et al. (2012b)]. This makes a difference with respect to that of the class of multivariate scale mixtures of matrix normal distributions.

Remark 6.2

It is worthwhile to consider that for the special case for which $\varvec{\Sigma }$ and $\varvec{\Omega }$ are not invertible, i.e. singular matrix elliptical contoured distribution, one may get similar result, as stated in the above, by the method proposed in Díaz-García et al. (2002). Further discussion on how to get similar result to Theorem 6.1 using rank decomposition is provided in Arashi and Nadarajah (2012).

Remark 6.3

In Theorem 6.1 it is stated that $f$ is the density of $N_{n,p}(\varvec{\mu },z^{-1}\varvec{\Sigma }\otimes \varvec{\Omega })$. It is realized from the proof that $f$ can even has one of the following densities

• (1) $\mathcal {N}_{n,p}(\varvec{\mu },z^{-1}\varvec{\Sigma },\varvec{\Omega })$ or

• (2) $\mathcal {N}_{n,p}(\varvec{\mu },\varvec{\Sigma },z^{-1}\varvec{\Omega })$ or

• (3) $\mathcal {N}_{n,p}(\varvec{\mu },z^{-\frac{1}{2}}\varvec{\Sigma },z^{-\frac{1}{2}}\varvec{\Omega })$. This fact enables us to adopt each representation whenever is needed for practical use.