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An Alternative Matrix Skew-Normal Random Matrix and Some Properties

Abstract

We propose an alternative skew-normal random matrix, which is an extension of the multivariate skew-normal vector parameterized in Vernic (A Stiint Univ Ovidius Constanta. 13, 83–96 2005, Insur. Math. Econ.38, 413–426 2006). We define the density function and then derive and apply the corresponding moment generating function to determine the mean matrix, covariance matrix, and third and fourth moments of the new skew-normal random matrix. Additionally, we derive eight marginal and two conditional density functions and provide necessary and sufficient conditions such that two pairs of sub-matrices are independent. Finally, we derive the moment generating function for a skew-normal random matrix-based quadratic form and show its relationship to the moment generating function of the noncentral Wishart and central Wishart random matrices.

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Acknowledgements

The authors wish to thank two anonymous referees for their comments that markedly improved the paper.

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Correspondence to Joshua D. Patrick.

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Appendices

Appendix 1

Proof of Theorem 2.

Proof.

Let \({\mathbf {X}} \sim SNM^{V}_{m,n}\left ({\mathbf {M}}, {\boldsymbol {{\Sigma }}}, {\boldsymbol {{\Psi }}}, {\mathbf {Q}}, \delta _{0}\right )\) and let MX (T) be defined as in (3.3), where \({\mathbf {T}} \in \mathbb {R}_{m \times n}\). First, we derive E (X). By taking the derivative of (3.3) with respect to T, we have

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} \frac{\partial M_{{\mathbf{X}}} \left( {\mathbf{T}} \right)}{\partial {\mathbf{T}}} &=& \frac{1}{{\Phi}\left( \delta_{0}\right)} etr \left\{ {\mathbf{T}}^{\prime}{\mathbf{M}} + \frac{1}{2}{\boldsymbol{{\Sigma}}}{\mathbf{T}}{\boldsymbol{{\Psi}}}{\mathbf{T}}^{\prime} \right\} \left[ \phi\left( \delta_{0} + tr \left( {\mathbf{Q}}^{\prime}{\mathbf{T}}\right)\right)\right]{\mathbf{Q}} \\ & &+ \frac{1}{{\Phi}\left( \delta_{0}\right)} etr \left\{ {\mathbf{T}}^{\prime} {\mathbf{M}} + \frac{1}{2}{\boldsymbol{{\Sigma}}} {\mathbf{T}}{\boldsymbol{{\Psi}}}{\mathbf{T}}^{\prime}\right\}\left[{\Phi}\left( \delta_{0} + tr\left( {\mathbf{Q}}^{\prime}{\mathbf{T}}\right)\right)\right]\left[ {\mathbf{M}} + \left( {\boldsymbol{{\Sigma}}}{\mathbf{T}}{\boldsymbol{{\Psi}}} \right)\right]. \end{array}\ \end{array} $$
(A.1)

By evaluating (A.1) at T = 0, we have

$$ E\left( \mathbf{X}\right) = {\mathbf{M}} + \frac{\phi\left( \delta_{0}\right)}{{\Phi}\left( \delta_{0}\right)}{\mathbf{Q}}. $$
(A.2)

Let

$$\begin{array}{@{}rcl@{}} {\mathbf{A}}_{1} :=& \left[\left( vec {\mathbf{T}} \right)^{\prime}\left( vec {\mathbf{M}} \right) + \frac{1}{2}\left( vec {\mathbf{T}} \right)^{\prime}\left( {\boldsymbol{{\Sigma}}} \otimes {\boldsymbol{{\Psi}}}\right) \left( vec {\mathbf{T}}\right)\right], \end{array} $$
(A.3)
$$\begin{array}{@{}rcl@{}} {\mathbf{A}}_{2} :=& \left[\delta_{0} + \left( vec {\mathbf{Q}}\right)^{\prime} \left( vec {\mathbf{T}}\right)\right]. \end{array} $$
(A.4)

The second moment matrix, used to determine the covariance matrix of \({\mathbf {X}} \sim SNM^{V}_{m,n}\left ({\mathbf {M}}, {\boldsymbol {{\Sigma }}}, {\boldsymbol {{\Psi }}}, {\mathbf {Q}}, \delta _{0}\right )\), is

$$\begin{array}{@{}rcl@{}} \frac{\partial^{2} M_{{\mathbf{X}}} \left( {\mathbf{T}}\right)}{\partial \left( vec {\mathbf{T}} \right)\partial\left( vec {\mathbf{T}} \right)^{\prime}} &= & \frac{1}{{\Phi}\left( \delta_{0}\right)} \left\{ [\exp \left( {\mathbf{A}}_{1} \right)] [\phi^{\prime} \left( \mathbf{A}_{2}\right)]\left( vec {\mathbf{Q}}\right)\left( vec {\mathbf{Q}}\right)^{\prime} + \left( vec {\mathbf{Q}}\right) [\phi \left( {\mathbf{A}}_{2}\right)] \right.\\ &&\times[\exp \left( \mathbf{{A}}_{1} \right)] \left[\left( vec {\mathbf{M}}\right)^{\prime} + \left( vec {\mathbf{T}}\right)^{\prime}\left( {\boldsymbol{{\Sigma}}} \otimes {\boldsymbol{{\Psi}}}\right)\right]\\ && + \exp\left( {\mathbf{A}}_{1} \right) \left\{ {\Phi}\left( \mathbf{{A}}_{2}\right)\left( {\boldsymbol{{\Sigma}}} \otimes {\boldsymbol{{\Psi}}}\right) \right. + \left[ \left( vec {\mathbf{M}}\right) + \left( {\boldsymbol{{\Sigma}}} \otimes {\boldsymbol{{\Psi}}}\right) \left( vec {\mathbf{T}}\right)\right]\\ && \left. \times [\phi \left( {\mathbf{A}}_{2} \right)] \left( vec {\mathbf{Q}}\right)^{\prime}\right\} +{\Phi}({\mathbf{A}}_{2}) \left[\left( vec\mathbf{M} \right)+\left( \boldsymbol{{\Sigma}}\otimes \boldsymbol{{\Psi}}\right)\left( vec\mathbf{T}\right)\right]\\ && \left.\times [\exp ({\mathbf{A}}_{1})] \left[\left( vec {\mathbf{M}}\right)^{\prime} + \left( vec {\mathbf{T}}\right)^{\prime}\left( {\boldsymbol{{\Sigma}}} \otimes {\boldsymbol{{\Psi}}}\right)\right] \right\}, \end{array} $$

where ϕ (⋅) and Φ (⋅) are the standard normal pdf and CDF, respectively. We then have

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} \left.\frac{\partial^{2} M_{{\mathbf{X}}} \left( {\mathbf{T}}\right)}{\partial \left( vec {\mathbf{T}} \right)\partial\left( vec {\mathbf{T}} \right)^{\prime}}\right|_{{\mathbf{T}}={\mathbf{0}}} &= & \frac{\phi\left( \delta_{0}\right)}{{\Phi}\left( \delta_{0}\right)} \left( vec {\mathbf{Q}} \right)\left( vec {\mathbf{M}} \right)^{\prime}+ \left( {\boldsymbol{{\Sigma}}} \otimes {\boldsymbol{{\Psi}}}\right) \\ & &+ \frac{\phi^{\prime}\left( \delta_{0}\right)}{{\Phi}\left( \delta_{0}\right)}\left( vec {\mathbf{M}} \right)\left( vec {\mathbf{Q}} \right)^{\prime} + \left( vec {\mathbf{M}} \right)\left( vec {\mathbf{M}} \right)^{\prime}. \end{array} \end{array} $$
(A.5)

We then rewrite (A.2) as

$$ E\left[ \left( vec{\mathbf{X}} \right)\right] = \frac{\phi\left( \delta_{0}\right)}{{\Phi}\left( \delta_{0}\right)}\left( vec {\mathbf{Q}} \right) + \left( vec {\mathbf{M}} \right). $$
(A.6)

By subtracting E (vecX)E (vecX) from (A.5), where E (X) is given in (A.6), we have that (3.4) holds.

Next, in addition to A1 and A2 defined in (A.3) and (A.4), let

$$\begin{array}{@{}rcl@{}} \mathbf{A}_{3}&:= & \left( vec\mathbf{M}\right)^{\prime}+\left( vec\mathbf{T}\right)^{\prime}\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right), \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \mathbf{A}_{4}&:= & \left( vec\mathbf{M}\right)+\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\left( vec\mathbf{T}\right). \end{array} $$

To obtain the third moment, we first calculate

$$\begin{array}{@{}rcl@{}} &&\frac{\partial^{3}M_{\mathbf{X}}\left( \mathbf{T}\right)}{\partial\left( vec\mathbf{T}\right)\partial\left( vec\mathbf{T}\right)^{\prime}\partial\left( vec\mathbf{T}\right)} \end{array} $$
(A.7)
$$\begin{array}{@{}rcl@{}} &&\phantom{}= \frac{\exp\left( \mathbf{A}_{1}\right)}{{\Phi}\left( \delta_{0}\right)}\left\{{\Phi}\left( \mathbf{A}_{2}\right)\left[3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\mathbf{A}_{4}+\mathbf{A}_{4}\mathbf{A}_{3}\mathbf{A}_{4}\right]\right.\\ &&\phantom{=} +\phi\left( \mathbf{A}_{2}\right)\left[3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\left( vec\mathbf{Q}\right)+\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\mathbf{A}_{4}+\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\mathbf{A}_{4}+\mathbf{A}_{4}\mathbf{A}_{3}\left( vec\mathbf{Q}\right)\right]\\ &&\phantom{=}+\phi^{\prime}\left( \mathbf{A}_{2}\right)\left[\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\mathbf{A}_{4}+\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\left( vec\mathbf{Q}\right)+\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\left( vec\mathbf{Q}\right)\right] \end{array} $$
(A.8)
$$\begin{array}{@{}rcl@{}} &&\left. \phantom{=} +\phi^{\prime\prime}\left( \mathbf{A}_{2}\right)\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\left( vec\mathbf{Q}\right) \right\}. \end{array} $$

Evaluating (A.8) at T = 0 yields (3.5).

To determine the fourth moment, we obtain

$$\begin{array}{@{}rcl@{}} &&\frac{\partial^{4}M_{\mathbf{X}}\left( \mathbf{T}\right)}{\partial\left( vec\mathbf{T}\right)\partial\left( vec\mathbf{T}\right)^{\prime}\partial\left( vec\mathbf{T}\right) \partial\left( vec\mathbf{T}\right)^{\prime}} \end{array} $$
(A.9)
$$\begin{array}{@{}rcl@{}} &&\phantom{}= \frac{\exp\left( \mathbf{A}_{1}\right)}{{\Phi}\left( \delta_{0}\right)}\left\{{\Phi}\left( \mathbf{A}_{2}\right)\left[3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right) \left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)+ 3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\mathbf{A}_{4}\mathbf{A}_{3}+ 3\mathbf{A}_{4} \mathbf{A}_{3}\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\right.\right.\\ &&\left.\phantom{=}+\mathbf{A}_{4}\mathbf{A}_{3}\mathbf{A}_{4}\mathbf{A}_{3}\right]+\phi\left( \mathbf{A}_{2}\right)\left[2\left( vec\mathbf{Q}\right)\mathbf{A}_{3} \left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)+ 4\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\right. \\ &&\phantom{=}+ 3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\left( vec\mathbf{Q}\right)\mathbf{A}_{3}+ 3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}} \right)\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}+\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\mathbf{A}_{4}\mathbf{A}_{3}\\ &&\left.\phantom{=}+\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\mathbf{A}_{4}\mathbf{A}_{3}+\mathbf{A}_{4}\mathbf{A}_{3}\mathbf{A}_{4} \left( vec\mathbf{Q}\right)^{\prime}+\mathbf{A}_{4}\mathbf{A}_{3}\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\right]\\ &&\phantom{=}+\phi^{\prime}\left( \mathbf{A}_{2}\right)\left[2\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right) +\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\mathbf{A}_{4}\mathbf{A}_{3}\right. \end{array} $$
(A.10)
$$\begin{array}{@{}rcl@{}} &&\phantom{=} + 3\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}+\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\mathbf{A}_{4} \left( vec\mathbf{Q}\right)^{\prime}+\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\\ &&\phantom{=} +\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}+\mathbf{A}_{4}\mathbf{A}_{3}\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q} \right)^{\prime}+\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\left( {\boldsymbol{{\Sigma}}}\otimes{\boldsymbol{{\Psi}}}\right)\\ &&\left.\phantom{=}+\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\right]+\phi^{\prime\prime}\left( \mathbf{A}_{2}\right) \left[\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\right.\\ &&\left.\phantom{=} +\left( vec\mathbf{Q}\right)\mathbf{A}_{3}\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}+ \mathbf{A}_{4}\left( vec\mathbf{Q}\right)^{\prime}\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\right.\\ && \left.\left. \phantom{=}+\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\right]+\phi^{\prime\prime\prime} \left( \mathbf{A}_{2}\right)\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime}\left( vec\mathbf{Q}\right)\left( vec\mathbf{Q}\right)^{\prime} \right\} . \end{array} $$

Evaluating (A.10) at T = 0 gives us (3.6). □

Appendix 2

We have used the following properties of the Kronecker product, trace operator, and vec operator in our proofs.

Let \(\mathbf {A}\in \mathbb {R}_{m\times n}\), \(\mathbf {B}\in \mathbb {R}_{n\times p}\), \(\mathbf {C}\in \mathbb {R}_{p\times q}\), and \(\mathbf {D}\in \mathbb {R}_{q\times m}\). Then

$$\begin{array}{@{}rcl@{}} vec\left( \mathbf{A}\mathbf{B}\mathbf{C}\right) &=&\left( \mathbf{C}^{\prime}\otimes\mathbf{A}\right)vec\mathbf{B},\\ tr\left( \mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D}\right) &=&\left( vec\left( \mathbf{A}^{\prime}\right)\right)^{\prime}\left( \mathbf{D}^{\prime}\otimes\mathbf{B}\right)vec\mathbf{C}, \\ \left( \mathbf{A}\otimes\mathbf{B}\right)^{\prime} &=&\mathbf{A}^{\prime}\otimes\mathbf{B}^{\prime},\\ \left( \mathbf{A}\otimes\mathbf{B}\right)^{-1} &=&\mathbf{A}^{-1}\otimes\mathbf{B}^{-1},\\ \left( \mathbf{A}\otimes\mathbf{C}\right)\left( \mathbf{B}\otimes\mathbf{D}\right)&=& \mathbf{A}\mathbf{B}\otimes\mathbf{C}\mathbf{D}. \end{array} $$

If \(\mathbf {A}\in \mathbb {R}_{m\times n}\) and \(\mathbf {B}\in \mathbb {R}_{n\times m}\), then

$$\begin{array}{@{}rcl@{}} tr\left( \mathbf{A}\mathbf{B}\right) =tr\left( \mathbf{B}\mathbf{A}\right). \end{array} $$

If \(\mathbf {A},\mathbf {B}\in \mathbb {R}_{m\times n}\), then

$$\begin{array}{@{}rcl@{}} tr\left( \mathbf{A}^{\prime}\mathbf{B}\right)=\left\{ vec\left( \mathbf{A}\right)\right\}^{\prime}vec\left( \mathbf{B}\right). \end{array} $$

If \(\mathbf {A}\in \mathbb {R}_{m\times m}\) and \(\mathbf {B}\in \mathbb {R}_{n\times n}\), then

$$\begin{array}{@{}rcl@{}} tr\left( \mathbf{A}\otimes\mathbf{B}\right)=tr\left( \mathbf{A}\right)tr\left( \mathbf{B}\right). \end{array} $$

One can find proofs of these properties in Schott (2016).

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Young, P.D., Patrick, J.D., Ramey, J.A. et al. An Alternative Matrix Skew-Normal Random Matrix and Some Properties. Sankhya A 82, 28–49 (2020). https://doi.org/10.1007/s13171-019-00165-4

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Keywords and phrases.

  • Moment generating function
  • Conditional distributions
  • Marginal distributions
  • Matrix quadratic form.

AMS (2000) subject classification.

  • Primary 62E15
  • Secondary 60E05