A multivariate modified skew-normal distribution

Abstract

We introduce a multivariate version of the modified skew-normal distribution, which contains the multivariate normal distribution as a special case. Unlike the Azzalini multivariate skew-normal distribution, this new distribution has a nonsingular Fisher information matrix when the skewness parameters are all zero, and its profile log-likelihood of the skewness parameters is always a non-monotonic function. We study some basic properties of the proposed family of distributions and present an expectation-maximization (EM) algorithm for parameter estimation that we validate through simulation studies. Finally, we apply the proposed model to the univariate frontier data and to a trivariate wind speed data, and compare its performance with the Azzalini skew-normal model.

Appendices

Appendix A

We discuss the benefits of using the alternative parameterization for defining the $\mathcal{S}\mathcal{N}$ distribution compared to the $\mathcal {ASN}$ distribution. We start with a preliminary stochastic representation of the $\mathcal{S}\mathcal{N}$ distribution, from which we can derive most of its main basic properties.

Proposition 10

(Stochastic representation of $\mathcal{S}\mathcal{N}$ distribution) If $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$, then $\varvec{X} \,{\buildrel d \over =}\, \varvec{\xi }+T \varvec{\eta }+ \varvec{V}$, where T and $\varvec{V}$ are independently distributed, with half-normal T denoted by $T \sim \mathcal{H}\mathcal{N}(0,1)$, and $\varvec{V} \sim {\mathcal {N}}_p(\varvec{0},\varvec{\Psi })$.

Proof

Let $\tilde{\varvec{X}} = \varvec{\xi }+T \varvec{\eta }+ \varvec{V}$. Since the conditional mpdf of $\tilde{\varvec{X}}| T = t$ is $f_{\tilde{\varvec{X}}|T = t}(\varvec{x}) = \phi _p(\varvec{x};\varvec{\xi }+ t \varvec{\eta },\varvec{\Psi })$ for $t>0$, and T has marginal density $f_T(t) = 2\phi (t)I_{(t>0)}$, then for the mpdf of $\tilde{\varvec{X}}$ we have

$$\begin{aligned} \begin{aligned} f_{\tilde{\varvec{X}}}( \varvec{x})&= \int _{0}^{\infty } f_{\tilde{\varvec{X}}|T=t}(\varvec{x}) f_T(t) d t = 2 \int _{0}^{\infty } \phi _p(\varvec{x};\varvec{\xi }+ t \varvec{\eta },\varvec{\Psi }) \phi (t)d t \\&= 2 \phi _p\left( \varvec{x}; \varvec{\xi }, \varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top \right) \int _{0}^{\infty }\phi \{ t; \varvec{\eta }^\top (\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top )^{-1}(\varvec{x} -\varvec{\xi }), \\&\quad 1 - \varvec{\eta }^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )^{-1}\varvec{\eta }\} d t\\&= 2\phi _p(\varvec{x}; \varvec{\xi }, \varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) \Phi \Bigg \{ \dfrac{\varvec{\eta }^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )^{-1}(\varvec{x} - \varvec{\xi })}{\sqrt{1 - \varvec{\eta }^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )^{-1} \varvec{\eta }}} \Bigg \}, \quad \varvec{x} \in {\mathbb {R}}^p, \end{aligned} \end{aligned}$$

(15)

where we have used the identity (see Lemma 2 in Arellano-Valle et al. (2005))

$$\begin{aligned} \phi _p(\varvec{x},\varvec{\xi }+ t \varvec{\eta },\varvec{\Psi }) \phi (t)&= \phi _p(\varvec{x}; \varvec{\xi }, \varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) \\&\quad \times \phi \{ t; \varvec{\eta }^\top (\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top )^{-1}(\varvec{x} -\varvec{\xi }), 1 - \varvec{\eta }^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )^{-1}\varvec{\eta }\}. \end{aligned}$$

Finally, using the following result:

$$\begin{aligned} (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )^{-1}= & {} \varvec{\Psi }^{-1} - \dfrac{\varvec{\Psi }^{-1} \varvec{\eta }\varvec{\eta }^\top \varvec{\Psi }^{-1}}{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }} \Rightarrow \\ \dfrac{\varvec{\eta }^\top (\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top )^{-1}(\varvec{x} -\varvec{\xi }) }{\sqrt{1-\varvec{\eta }^\top (\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top )^{-1} \varvec{\eta }}}= & {} \dfrac{\varvec{\eta }^\top \Psi ^{-1}(\varvec{x}- \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}}, \end{aligned}$$

it can be easily established that $\tilde{\varvec{X}} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$. $\square$

Further basic properties As immediate consequences of the above stochastic representation of a random vector $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$, we have the following basic properties:

1) Expectation and covariance: The mean vector and covariance matrix of $\varvec{X}$ are

$$\begin{aligned} {\mathbb {E}}(\varvec{X}) = \varvec{\xi }+ \sqrt{\frac{2}{\pi }} \varvec{\eta }\quad and \quad {\mathbb {V}}(\varvec{X}) = \varvec{\Psi }+ \left( 1-\frac{2}{\pi }\right) \varvec{\eta }\varvec{\eta }^\top . \end{aligned}$$

(16)

2) Distribution of an affine transformation: For any fixed vector $\varvec{a} \in {\mathbb {R}}^q$ and any fixed matrix $\varvec{B} \in {\mathbb {R}}^{q \times p}$ of full row rank and $q \le p$, we have $\varvec{a} + \varvec{B} \varvec{X} \,{\buildrel d \over =}\, \varvec{a} + \varvec{B} \varvec{\xi }+ T \varvec{B} \varvec{\eta }+ \varvec{B} \varvec{V}\sim \mathcal{S}\mathcal{N}_q(\varvec{a} + \varvec{B} \varvec{\xi }, \varvec{B} \varvec{\Psi }\varvec{B}^\top , \varvec{B} \varvec{\eta })$, since, by assumption, T and $\varvec{V}$ are independently distributed, with $T \sim \mathcal{H}\mathcal{N}(0,1)$ and $\varvec{V} \sim {\mathcal {N}}_p(\varvec{0},\varvec{\Psi })$.

3) Marginal distributions: Partition now $\varvec{X}$ in two sub-vectors of sizes $p_1$ and $p_2$ such that $p_1+p_2 = p$, with corresponding partitions of the parameters in blocks of matching sizes, as follows

$$\begin{aligned} \varvec{X} = \begin{pmatrix} \varvec{X}_1\\ \varvec{X}_2 \end{pmatrix},\quad \varvec{\xi }= \begin{pmatrix} \varvec{\xi }_1 \\ \varvec{\xi }_2 \end{pmatrix},\quad \varvec{\Psi }= \begin{pmatrix} \varvec{\Psi }_{11} &{} \varvec{\Psi }_{12}\\ \varvec{\Psi }_{21} &{} \varvec{\Psi }_{22} \end{pmatrix},\quad \varvec{\eta }= \begin{pmatrix} \varvec{\eta }_1 \\ \varvec{\eta }_2 \end{pmatrix}. \end{aligned}$$

(17)

Thus, by using property 2) with $\varvec{a}=\varvec{0}$ and $\varvec{B} = (I _{p_1}, \varvec{0})$ for $\varvec{X}_1$ and $\varvec{B} = (\varvec{0},I _{p_2} )$ for $\varvec{X}_2$ it follows for their respective marginals that $\varvec{X}_1 \sim \mathcal{S}\mathcal{N}_{p_1} (\varvec{\xi }_1,\varvec{\Psi }_{11}, \varvec{\eta }_1)$ and $\varvec{X}_2 \sim \mathcal{S}\mathcal{N}_{p_2} (\varvec{\xi }_2,\varvec{\Psi }_{22}, \varvec{\eta }_2)$.

4) Moment generating function: The multivariate moment generating function (mmgf) of the $\mathcal{S}\mathcal{N}$ distribution can be derived in closed form. We present the mmgf of the of $\mathcal{S}\mathcal{N}$ distribution in the next proposition.

Proposition 11

The mmgf of $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ is

$$\begin{aligned} M_{\varvec{X}}(\varvec{t})&=2\exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{t}^\top \varvec{\Omega }\varvec{t}\right) \Phi (\varvec{\eta }^\top \varvec{t}),\quad \varvec{t} \in {\mathbb {R}}^p, \end{aligned}$$

where $\varvec{\Omega }= \varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top$.

Proof

The mpdf of $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ in (15) can be rewritten as

$$\begin{aligned} f_{\varvec{X}} (\varvec{x}) = \dfrac{2}{|\varvec{\Omega }|^{1/2}} \phi _p\left\{ \varvec{\Omega }^{-1/2}(\varvec{x} -\varvec{\xi })\right\} \Phi \left\{ \varvec{\gamma }^\top \varvec{\Omega }^{-1/2}(\varvec{x}-\varvec{\xi }) \right\} , \quad \varvec{x}\in {\mathbb {R}}^p, \end{aligned}$$

(18)

where $\phi _p(\varvec{z})=\phi _p(\varvec{z}; \varvec{0}, I _p)$, $\varvec{\Omega }= \varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top$, $\varvec{\gamma }= (1- \varvec{\eta }^\top \varvec{\Omega }^{-1} \varvec{\eta })^{-1/2}\varvec{\Omega }^{-1/2} \varvec{\eta }$, and, conversely, we have $\varvec{\eta }=(1 + \varvec{\gamma }^\top \varvec{\gamma })^{-1/2}\varvec{\Omega }^{1/2}\varvec{\gamma }$ and $\varvec{\Psi }= \varvec{\Omega }-(1+\varvec{\gamma }^\top \varvec{\gamma })^{-1}\varvec{\Omega }^{1/2} \varvec{\gamma }\varvec{\gamma }^\top \varvec{\Omega }^{1/2}$. From Eq. (18) and by using the change of variable $\varvec{z}=\varvec{\Omega }^{-1/2}(\varvec{x}-\varvec{\xi })$ we have that the mmgf of $\varvec{X}$, $M_{\varvec{X}}(\varvec{t})={\mathbb {E}}\{\exp (\varvec{t}^\top \varvec{X}) \}$, is given by

$$\begin{aligned} M_{\varvec{X}}(\varvec{t})&=\int _{{\mathbb {R}}^p} \exp (\varvec{t}^\top \varvec{x}) 2\phi _p(\varvec{x};\varvec{\xi },\varvec{\Omega })\Phi \{\varvec{\gamma }^\top \varvec{\Omega }^{-1/2}(\varvec{x}-\varvec{\xi })\} \text {d} \varvec{x}\\&=2 \exp (\varvec{t}^\top \varvec{\xi }) \int _{{\mathbb {R}}^p} \exp (\varvec{s}^\top \varvec{z}) \phi _p(\varvec{z};\varvec{0},{\textbf {I}}_p)\Phi \{ \varvec{\gamma }^\top \varvec{z}\} \text {d} \varvec{z},\quad (\varvec{s}=\varvec{\Omega }^{1/2} \varvec{t})\\&=2 \exp (\varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{s}^\top \varvec{s}) \int _{{\mathbb {R}}^p}\phi _p(\varvec{z};\varvec{s},{\textbf {I}}_p)\Phi \{ \varvec{\gamma }^\top \varvec{z}\} \text {d} \varvec{z}\\&=2 \exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{s}^\top \varvec{s}\right) {\mathbb {E}}\{\Phi (\varvec{\gamma }^\top \varvec{Z})\},\quad \varvec{\gamma }^\top \varvec{Z}\sim {\mathcal {N}}_1(\varvec{\gamma }^\top \varvec{s},\varvec{\gamma }^\top \varvec{\gamma })\\&=2 \exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{s}^\top \varvec{s}\right) \Phi \left( \frac{\varvec{\gamma }^\top \varvec{s}}{\sqrt{1+ \varvec{\gamma }^\top \varvec{\gamma }}}\right) ,\\&=2\exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{t}^\top \varvec{\Omega }\varvec{t}\right) \Phi (\varvec{\eta }^\top \varvec{t}). \end{aligned}$$

$\square$

5) Cumulative distribution function: In the next proposition we present the exact functional form of the multivariate cumulative distribution function (mcdf) of the $\mathcal{S}\mathcal{N}$ distribution.

Proposition 12

The mcdf of $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ is

$$\begin{aligned} F_{\varvec{X}}(\varvec{x}) = 2 \Phi _{p+1}(\varvec{x}_*;\varvec{\xi }_*,\varvec{\Omega }_{*}), \quad \text {with} \quad \varvec{x}_* = (\varvec{x}^\top ,0)^\top , \quad \varvec{x} \in {\mathbb {R}}^p, \end{aligned}$$

where, $\varvec{\xi }_* = (\varvec{\xi }^\top ,0)^\top$ and $\varvec{\Omega }_* = \begin{pmatrix}\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top &{} -\varvec{\eta }\\ -\varvec{\eta }^\top &{} 1 \end{pmatrix}$.

Proof

The mpdf of $\varvec{X}$ is

$$\begin{aligned} f_{\varvec{X}}(\varvec{z})&= 2 \phi _p (\varvec{z};\varvec{\xi },\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top ) \Phi \left\{ \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}}\right\} \\&= 2 \phi _p (\varvec{z};\varvec{\xi },\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top ) \int _{-\infty }^{\frac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}}} \phi (u;0,1) \text {d} u\\&=2 \phi _p (\varvec{z};\varvec{\xi },\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top ) \int _{-\infty }^{0} \phi \left( z_0+ \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}};0,1\right) \text {d} z_0\\&=2 \phi _p (\varvec{z};\varvec{\xi },\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top ) \int _{-\infty }^{0} \phi \left( z_0;- \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}},1\right) \text {d} z_0 \end{aligned}$$

where we use the change of variable $z_0 = u - \varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })/\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}$. Now, from the marginal-conditional factorization of a $(p+1)$-variate normal mpdf we can write

$$\begin{aligned} \phi _p(\varvec{z};\varvec{\xi },\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) \phi \left( z_0;- \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}},1\right) = \phi _{p+1}(\varvec{z}_*;\varvec{\xi }_*,\varvec{\Omega }_{**}), \end{aligned}$$

where $\varvec{z}_* = (\varvec{z}^\top ,z_0)^\top$ and $\varvec{\Omega }_{**} = \begin{pmatrix}\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top &{} - \sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }} \varvec{\eta }\\ - \sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }} \varvec{\eta }^\top &{} 1 \end{pmatrix}$. Moreover, we have $\varvec{\Omega }_{**} = \varvec{D}_{*} \varvec{\Omega }_{*} \varvec{D}_*$, with $D_* = \text {diag} \left( {\textbf {I}}_p,\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }} \right)$. Then, the mcdf of $\varvec{X}$ is

$$\begin{aligned} F_{\varvec{X}} (\varvec{x})&= 2 \int _{(- \varvec{\infty }, \varvec{x}]} \phi _p (\varvec{z};\varvec{\xi },\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top ) \Phi \left\{ \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}}\right\} \text {d} \varvec{z}\\&= 2 \int _{(- \varvec{\infty }, \varvec{x}]} \int _{-\infty }^0 \phi _p (\varvec{z};\varvec{\xi },\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top ) \phi \left( z_0; -\dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{z} - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1}\varvec{\eta }}} ,1\right) \text {d}z_0 \text {d} \varvec{z}\\&= 2 \int _{(- \varvec{\infty }, \varvec{x}_*]} \phi _{p+1}(\varvec{z}_*;\varvec{\xi }_*,\varvec{\Omega }_{**}) \text {d} \varvec{z}_*\\&= 2 \Phi _{p+1}(\varvec{x}_*;\varvec{\xi }_*,\varvec{\Omega }_{**}),\quad \text {with} \quad \varvec{x}_* = (\varvec{x}^\top ,0)^\top , \quad \varvec{x} \in {\mathbb {R}}^p. \end{aligned}$$

But, since $\varvec{x}_* - \varvec{\xi }_* = ((\varvec{x} - \varvec{\xi })^\top , 0)^\top$, then $\varvec{D}_*^{-1} (\varvec{x}_* - \varvec{\xi }_*) = \varvec{x}_* - \varvec{\xi }_*$ and so

$$\begin{aligned} \Phi _{p+1} (\varvec{x}_*;\varvec{\xi }_*,\varvec{\Omega }_{**})= & {} \Phi _{p+1} (\varvec{x}_*;\varvec{\xi }_*,\varvec{D}_* \varvec{\Omega }_{*} \varvec{D}_*) = \Phi _{p+1}(\varvec{D}_*^{-1}(\varvec{x}_* - \varvec{\xi }_*);\varvec{0}, \varvec{\Omega }_*)\\= & {} \Phi _{p+1}(\varvec{x}_*;\varvec{\xi }_*, \varvec{\Omega }_*). \end{aligned}$$

$\square$

Behavior of the likelihood function In the following two results, we show that the alternative parameterization used in (3) for defining the $\mathcal{S}\mathcal{N}$ mpdf fixes the problem of the infinite maximum likelihood estimate of the skewness parameter, unlike the original parameterization used in (2) for defining the $\mathcal {ASN}$ mpdf, but it also fails to resolve the problem of the singular Fisher information matrix. In fact, let $\varvec{x}_1,\ldots ,\varvec{x}_n$ be an observed random sample from $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$. The corresponding likelihood function for $(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$ is

$$\begin{aligned} L(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta }) = \prod _{i=1}^n 2 \phi _p\left( \varvec{x}_i;\varvec{\xi },\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top \right) \Phi \bigg \{ \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{x}_i - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}}\bigg \}. \end{aligned}$$

(19)

For fixed $\varvec{\xi }$ and $\varvec{\Psi }$, (19) becomes the profile likelihood function of $\varvec{\eta }$, which we denote by $L(\varvec{\eta })$, $\varvec{\eta }\in {\mathbb {R}}^p$.

Proposition 13

The maximum likelihood estimator of the skewness parameter $\varvec{\eta }$ of the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$ family is always finite.

Proof

We find the limit of $L(\varvec{\eta })$ when some (or all) components of $\varvec{\eta }$ tend to $+\infty$ or $-\infty$. For this, first note from the Cauchy–Schwarz inequality that

$$\begin{aligned}&\dfrac{ |\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{x}_i - \varvec{\xi }) |}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }} } \\ {}&\le \sqrt{\dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}}\,\sqrt{(\varvec{x}_i-\varvec{\xi })^\top \varvec{\Psi }^{-1}(\varvec{x}_i-\varvec{\xi })} \le \sqrt{(\varvec{x}_i-\varvec{\xi })^\top \varvec{\Psi }^{-1}(\varvec{x}_i-\varvec{\xi })},\quad \forall \,\varvec{\eta }\in {\mathbb {R}}^p, \end{aligned}$$

for $i=1,\ldots ,n$. From this inequality, it clearly follows, for all $\varvec{\eta }\in {\mathbb {R}}^p$ and each $i=1,\ldots ,n$, that

$$\begin{aligned} \phi _p (\varvec{x}_i;\varvec{\xi },\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )&= \dfrac{\exp \bigg \{ -\dfrac{1}{2} (\varvec{x}_i -\varvec{\xi })^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top )^{-1} (\varvec{x}_i- \varvec{\xi }) \bigg \}}{(2 \pi )^{p/2} |\varvec{\Psi }|^{1/2}\sqrt{1 + \varvec{\eta }^ \top \varvec{\Psi }^{-1} \varvec{\eta }}} \\&= \dfrac{\exp \bigg \{ -\dfrac{1}{2} (\varvec{x}_i - \varvec{\xi })^\top \varvec{\Psi }^{-1} (\varvec{x}_i -\varvec{\xi }) \bigg \}}{(2 \pi )^{p/2} |\varvec{\Psi }|^{1/2}}\,\,\\&\quad \times \dfrac{\exp \bigg [ - \dfrac{1}{2} \dfrac{ \{ \varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{x}_i - \varvec{\xi })\}^2}{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }} \bigg ]}{\sqrt{1 + \varvec{\eta }^ \top \varvec{\Psi }^{-1} \varvec{\eta }}}\\&\le \dfrac{1}{(2 \pi )^{p/2} |\varvec{\Psi }|^{1/2}\sqrt{1 + \varvec{\eta }^ \top \varvec{\Psi }^{-1} \varvec{\eta }}}, \end{aligned}$$

and

$$\begin{aligned} 0\le \Phi \bigg \{-\sqrt{(\varvec{x}_i - \varvec{\xi })^\top \varvec{\Psi }^{-1} (\varvec{x}_i -\varvec{\xi })} \bigg \} \le \Phi \bigg \{ \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{x}_i - \varvec{\xi })}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}}\bigg \}\le \Phi \bigg \{\sqrt{(\varvec{x}_i - \varvec{\xi })^\top \varvec{\Psi }^{-1} (\varvec{x}_i -\varvec{\xi })} \bigg \} \le 1. \end{aligned}$$

These results hold whatever fixed value of $(\varvec{\xi },\varvec{\Psi })$. Thus, noting also that whenever some (or all) components of $\varvec{\eta }$ tend to $\pm \infty$, then $\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }\rightarrow \infty ,$ we can easily deduce from the first of the previous inequalities that $\phi _p (\varvec{x}_i;\varvec{\xi },\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) \rightarrow 0$ for each $i=1,\ldots ,n$ as some (or all) components of $\varvec{\eta }$ tend to $\pm \infty$. Thus, now taking into account the second inequality, we find that $L(\varvec{\eta }) \rightarrow 0$ whenever some (or all) components of $\varvec{\eta }$ tend to $\pm \infty$ and whatever fixed value of $(\varvec{\xi },\varvec{\Psi })$. This result leads us to the conclusion that $L(\varvec{\eta })$ is not a monotonically increasing or decreasing function of any of the components of $\varvec{\eta }$. This means that the profile likelihood of the skewness parameter $\varvec{\eta }$ is always maximized at a finite point for the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$ family.$\square$

Remark Proposition 13 shows that the MLE of the skewness parameter of the $\mathcal{S}\mathcal{N}$ distribution is always finite for the new parameterization. It was not the case for the $\mathcal {ASN}$ parameterization. So, if we use the new parameterization for a particular data, then we will get a finite MLE of the skewness parameter, whereas we might get infinite MLE for the skewness parameter in the $\mathcal {ASN}$ parameterization. Because these two parameterizations are a one-to-one transformation of each other, due to the invariance property of the MLE, if we transform back the MLE from the new-parameterization to the $\mathcal {ASN}$ parameterization, then we will get back the old results. In other words, although the MLE of the skewness parameter $\varvec{\eta }$ is always finite, it may sometimes correspond to an MLE of the skewness parameter $\varvec{\alpha }$ that is infinite.

In what follows, we use the notation of Magnus and Neudecker (1979) related to the Kronecker product and matrix vectorization. For instance, let $\text{ vech }(\varvec{\Psi })$ be the $p(p + 1)/2$-subvector of $\text{ vec }(\varvec{\Psi })$, where only upper-diagonal entries of $\varvec{\Psi }$ are considered. Also, let $\varvec{K}_p$ be the $p^2 \times p^2$ commutation matrix, i.e., $\varvec{K}_p \text{ vec }(\varvec{A})=\text{ vec }(\varvec{A}^\top )$ for any $p\times q$ matrix $\varvec{A}$, and let $\varvec{D}_p$ be the $p^2\times p(p+1)/2$ duplication matrix, i.e., $\varvec{D}_p \text{ vech }(\varvec{A}) = \text{ vec }(\varvec{A})$ for any $p\times p$ symmetric matrix $\varvec{A}$.

Proposition 14

The Fisher information matrix of the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$ family is singular when the skewness parameter $\varvec{\eta }$ is set to zero.

Proof

The score vector and Fisher information matrix for the $\mathcal {ASN}_p(\varvec{\xi }, \varvec{\Omega }, \varvec{\alpha })$ family are derived by Arellano-Valle and Azzalini (2008) in terms of the reparametrization $(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })$, where $\varvec{\lambda }=\varvec{\omega }^{-1}\varvec{\alpha }$. These author also showed that the Fisher information matrix $\varvec{I}(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })$ is singular at $\varvec{\lambda }=\varvec{0}$:

$$\begin{aligned} \varvec{I}(\varvec{\xi },\varvec{\Omega },\varvec{0})= \begin{bmatrix}\varvec{\omega }^{-1}&{}\varvec{0}&{}\sqrt{\frac{2}{\pi }}I _p\\ \varvec{0}&{}\frac{1}{2}\varvec{D}_p^\top (\varvec{\Omega }^{-1}\otimes \varvec{\Omega }^{-1})\varvec{D}_p&{}\varvec{0}\\ \sqrt{\frac{2}{\pi }}I _p&{}\varvec{0}&{}\frac{2}{\pi }\varvec{\Omega }\end{bmatrix}. \end{aligned}$$

Since the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$ family corresponds to a reparametrization of the $\mathcal {ASN}_p(\varvec{\xi }, \varvec{\Omega }, \varvec{\alpha })$ family, its Fisher information matrix becomes $\varvec{I}(\varvec{\xi },\varvec{\Psi },\varvec{\eta })=\varvec{J}(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })^\top \varvec{I}(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })\varvec{J}(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })$, where $\varvec{J}(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })$ denotes the Jacobian matrix of the transformation from $(\varvec{\xi },vech (\varvec{\Omega }),\varvec{\lambda })$ to $(\varvec{\xi },vech (\varvec{\Psi }),\varvec{\eta })$. Thus, since the inverse transformation from $(\varvec{\xi },vech (\varvec{\Psi }),\varvec{\eta })$ to $(\varvec{\xi },vech (\varvec{\Omega }),\varvec{\lambda })$ turns out to be $\varvec{\xi }=\varvec{\xi },$ $\varvec{\Omega }=\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^{\top }$ and $\varvec{\lambda }= (1 + \varvec{\eta }\varvec{\Psi }^{-1} \varvec{\eta })^{-1/2}\varvec{\Psi }^{-1} \varvec{\eta },$ for the Jacobian matrix we have

$$\begin{aligned} \varvec{J}(\varvec{\xi },\varvec{\Omega },\varvec{\lambda })=\begin{bmatrix}\frac{\partial \varvec{\xi }}{\partial \varvec{\xi }^\top }&{} \frac{\partial vec (\varvec{\Omega })}{\partial \varvec{\xi }^\top }&{} \frac{\partial \varvec{\lambda }}{\partial \varvec{\xi }^\top }\\ \frac{\partial \varvec{\xi }}{\partial vec (\varvec{\Psi })^\top } &{} \frac{\partial vec (\varvec{\Omega })}{\partial vec (\varvec{\Psi })^\top } &{} \frac{\partial \varvec{\lambda }}{\partial vec (\varvec{\Psi })^\top }\\ \frac{\partial \varvec{\xi }}{\partial \varvec{\eta }^\top } &{} \frac{\partial vec (\varvec{\Omega })}{\partial \varvec{\eta }^\top } &{}\frac{\partial \varvec{\lambda }}{\partial \varvec{\eta }^\top } \end{bmatrix}=\begin{bmatrix}I _p &{} \varvec{0} &{} \varvec{0}\\ \varvec{0} &{} I _{p(p+1)/2} &{} \varvec{J}_{23}\\ \varvec{0} &{} \varvec{J}_{32} &{} \varvec{J}_{33}\end{bmatrix}, \end{aligned}$$

where $\varvec{J}_{23}=\varvec{D}_p^+(I _p\otimes \varvec{\eta }+\varvec{\eta }\otimes I _p)$, with $\varvec{D}_p^+=(\varvec{D}_p^\top \varvec{D}_p)^{-1}\varvec{D}_p^\top$, $\varvec{J}_{32}=(1 + \varvec{\eta }\varvec{\Psi }^{-1} \varvec{\eta })^{-1/2}\{\frac{1}{2}(\varvec{\eta }^{\top }\varvec{\Psi }^{-1}\otimes \varvec{\Psi }^{-1}\varvec{\eta }\varvec{\eta }^{\top }\varvec{\Psi }^{-1})-(\varvec{\eta }^{\top }\varvec{\Psi }^{-1}\otimes \varvec{\Psi }^{-1}\varvec{)}\}\varvec{D}_p$ and $\varvec{J}_{33}=(1 + \varvec{\eta }\varvec{\Psi }^{-1} \varvec{\eta })^{-1/2}(\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^{\top })^{-1}$. When $\varvec{\eta }=\varvec{0}$ we have that $\varvec{\lambda }=\varvec{0}$, $\varvec{\Omega }=\varvec{\Psi }$ and the Jacobian matrix $\varvec{J}(\varvec{\xi },\varvec{\Omega },\varvec{0})=diag (I _p,I _{p(p+1)/2},\varvec{\Psi }^{-1})$. Hence, at $\varvec{\eta }=\varvec{0}$, the Fisher information matrix $\varvec{I}(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ of the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$ family becomes

$$\begin{aligned} \varvec{I}(\varvec{\xi },\varvec{\Psi },\varvec{0})=\varvec{I}(\varvec{\xi },\varvec{\Omega },\varvec{0})= \begin{bmatrix}\varvec{\Psi }^{-1}&{}\varvec{0}&{}\sqrt{\frac{2}{\pi }}\varvec{\Psi }^{-1}\\ \varvec{0}&{}\frac{1}{2}\varvec{D}_p^\top (\varvec{\Psi }^{-1}\otimes \varvec{\Psi }^{-1})\varvec{D}_p&{}\varvec{0}\\ \sqrt{\frac{2}{\pi }}\varvec{\Psi }^{-1}&{}\varvec{0}&{}\frac{2}{\pi }\varvec{\Psi }^{-1} \end{bmatrix}, \end{aligned}$$

which is clearly singular. $\square$

Remark Since $\varvec{I}(\varvec{\xi },\varvec{\Omega },\varvec{0})$ is singular, in the proof that $\varvec{I}(\varvec{\xi },\varvec{\Psi },\varvec{0})$ is also singular it is enough to prove that the Jacobian matrix $\varvec{J}(\varvec{\xi },\varvec{\Omega },\varvec{0})$ is finite (in the matrix sense).

Obviously, the singularity of the Fisher information matrix of the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$ family when $\varvec{\eta }= \varvec{0}$ is due to the fact that the score vectors corresponding to the location vector $\varvec{\xi }$ and the skewness vector $\varvec{\eta }$ are linearly dependent at $\varvec{\eta }= \varvec{0}$. In fact, the score vector of $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$ for $(\varvec{\xi }^\top ,\text{ vech }(\varvec{\Psi })^\top ,\varvec{\eta }^\top )^\top$, at $\varvec{\eta }= \varvec{0}$, becomes

$$\begin{aligned} \varvec{l}_{\varvec{\xi },\varvec{\Psi },\varvec{0}}(\varvec{X})&= \left( \varvec{\Psi }^{-1} (\varvec{X}- \varvec{\xi }), \dfrac{1}{2} \varvec{D}_p^\top (\varvec{\Psi }\otimes \varvec{\Psi })^{-1} \text{ vec }\left\{ (\varvec{X}- \varvec{\xi })(\varvec{X}- \varvec{\xi })^\top - \varvec{\Psi }\right\} , \right. \nonumber \\&\left. \sqrt{\dfrac{2}{\pi }} \varvec{\Psi }^{-1}(\varvec{X}-\varvec{\xi })\right) ^\top . \end{aligned}$$

(20)

It is evident from (20) that, at $\varvec{\eta }= \varvec{0}$, the score vectors of $\varvec{\xi }$ and $\varvec{\eta }$ are linearly related. Consequently, the Fisher information matrix, which is the covariance matrix of the score vector, is singular at $\varvec{\eta }= \varvec{0}$. The score vector in (20) can be obtained from a direct differentiation of the $\mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$ log-likelihood function, or from the results in Arellano-Valle and Azzalini (2008) as well as from Ley and Paindaveine (2010) since the $\mathcal{S}\mathcal{N}$ family belongs to the generalized skew-normal family (see Genton and Loperfido (2005)). This last fact can be easily verified by the form of the mpdf of $\varvec{X} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ in Equation (18), and from there it is clear that $\mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ is a generalized skew-normal distribution with location parameter $\varvec{\xi }$, dispersion matrix $\varvec{\Omega }$, density generator $\phi _p (\varvec{z})$, and skewing function $\pi (\varvec{z}): {\mathbb {R}}^p \rightarrow [0,1]$ with $\pi (\varvec{z}) = \Phi \left( \varvec{\gamma }^\top \varvec{z} \right) .$

Appendix B

1.1 Proof of Proposition 1

Let $\tilde{\varvec{X}} = \varvec{\xi }+ \dfrac{1}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\,\varvec{\Omega }\varvec{Y}\,T + \varvec{U}$. Since, by assumption, T is independent of $(\varvec{U}^\top ,\varvec{Y}^\top )^\top$, it is then clear that, conditionally on $\varvec{Y} = \varvec{y}$, $\tilde{\varvec{X}}$ has the same distribution as $\varvec{\xi }+ \dfrac{1}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\,\varvec{\Omega }\varvec{y}\,T + \varvec{U}_{\varvec{y}}$, where $\varvec{U}_{\varvec{y}} \,{\buildrel d \over =}\, \varvec{U}| \varvec{Y} = \varvec{y} \sim {\mathcal {N}}_p \left( \varvec{0}, (\varvec{\Omega }^{-1} + \varvec{y} \varvec{y} ^\top )^{-1} \right)$, independent of T. By Proposition 10, this means $\tilde{\varvec{X}} | \varvec{Y} = \varvec{y} \sim \mathcal{S}\mathcal{N}_p \left( \varvec{\xi }, (\varvec{\Omega }^{-1} + \varvec{y} \varvec{y}^\top )^{-1},\dfrac{1}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\,\varvec{\Omega }\varvec{y} \right)$, and hence, by Proposition 10, the mpdf of $\tilde{\varvec{X}}| \varvec{Y} = \varvec{y}$ becomes $f_{\tilde{\varvec{X}} | \varvec{Y} = \varvec{y}}(\varvec{x}) = 2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \Phi \{ \varvec{y}^\top (\varvec{x}- \varvec{\xi }) \}.$ Thus, since the mpdf of $\tilde{\varvec{X}}$ is $f_{\tilde{\varvec{X}}}(\varvec{x}) = \int _{{\mathbb {R}}^p} f_{\tilde{\varvec{X}} | \varvec{Y}}(\varvec{x}|\varvec{y}) f_{\varvec{Y}} (\varvec{y}) d \varvec{y}$, we have

$$\begin{aligned} f_{\tilde{\varvec{X}}}(\varvec{x})&=2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \int _{{\mathbb {R}}^p} \Phi \{ \varvec{y}^\top (\varvec{x}- \varvec{\xi }) \} \phi _p\left( \varvec{y}; \varvec{\lambda },\varvec{\Omega }^{-1} \right) d y\\&=2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) {\mathbb {E}}[\Phi \{ \varvec{Y}^\top (\varvec{x}- \varvec{\xi }) \} ]. \end{aligned}$$

Now, using the fact that, if $X \sim {\mathcal {N}}(\mu , \sigma ^2)$, then ${\mathbb {E}} \{ \Phi (X) \}=\Phi \left( \dfrac{\mu }{\sqrt{1+\sigma ^2}}\right)$, we get the above result. $\square$

1.2 Proof of Corollary 1

Since, by assumption, $\varvec{X} \sim \mathcal {MSN}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$, we have by Proposition 1 that $\varvec{X}$ can be represented as ${\varvec{X}} | \varvec{Y} = \varvec{y} \sim \mathcal{S}\mathcal{N}_p \left( \varvec{\xi }, (\varvec{\Omega }^{-1} + \varvec{y} \varvec{y}^\top )^{-1},\dfrac{1}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\,\varvec{\Omega }\varvec{y} \right)$ and $\varvec{Y} \sim {\mathcal {N}}_p(\varvec{\lambda },\varvec{\Omega }^{-1})$. Combining this statement with the stochastic representation of the $\mathcal{S}\mathcal{N}$ distribution as given in Proposition 10, the mpdf of $\varvec{X}$ can be expressed as

$$\begin{aligned} f_{\varvec{X}} (\varvec{x})&= 2 \displaystyle {\int }_{{\mathbb {R}}^p} \displaystyle {\int }_{0} ^{\infty } \phi _p \left( \varvec{x};\varvec{\xi }+ \dfrac{1}{\sqrt{1+\varvec{y}^\top {\varvec{\Omega }} \varvec{y}}}\,\varvec{\Omega }\varvec{y}\, t, (\varvec{\Omega }^{-1} + \varvec{y} \varvec{y}^\top )^{-1} \right) \\&\times \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi (t) d t d \varvec{y}. \end{aligned}$$

Now, using Lemma 2 in Arellano-Valle et al. (2005) and that $(\varvec{\Omega }^{-1} + \varvec{y} \varvec{y}^\top )^{-1} = \varvec{\Omega }- \dfrac{1}{1+\varvec{y}^\top {\varvec{\Omega }} \varvec{y}}\,\varvec{\Omega }\varvec{y} \varvec{y} ^\top \varvec{\Omega }$, we have the identity given by

$$\begin{aligned}&\phi _p \left( \varvec{x};\varvec{\xi }+ \dfrac{1}{\sqrt{1+\varvec{y}^\top {\varvec{\Omega }} \varvec{y}}}\,\varvec{\Omega }\varvec{y}\, t, \varvec{\Omega }- \dfrac{1}{1+\varvec{y}^\top {\varvec{\Omega }} \varvec{y}}\,\varvec{\Omega }\varvec{y} \varvec{y} ^\top \varvec{\Omega }\right) \phi (t)\\&\quad = \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \phi \bigg \{ t;\dfrac{\varvec{y}^\top (\varvec{x}- \varvec{\xi })}{\sqrt{1+\varvec{y}^\top {\varvec{\Omega }}\varvec{y}}},\dfrac{1}{1+\varvec{y}^\top {\varvec{\Omega }} \varvec{y}} \bigg \}. \end{aligned}$$

Thus, we have

$$\begin{aligned} f_{\varvec{X}} (\varvec{x}) = 2 \int _{{\mathbb {R}}^p} \int _0 ^{\infty } \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi \bigg \{ t;\dfrac{\varvec{y}^\top (\varvec{x}- \varvec{\xi })}{\sqrt{1+\varvec{y}^\top {\varvec{\Omega }}\varvec{y}}},\dfrac{1}{1+\varvec{y}^\top {\varvec{\Omega }} \varvec{y}} \bigg \} d t d \varvec{y}. \end{aligned}$$

Considering the transformation $w = \sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}\,t$, we have, for the joint mpdf of $(\varvec{X}, \varvec{Y},W)$:

$$\begin{aligned} f_{\varvec{X},\varvec{Y},W}(\varvec{x},\varvec{y},w)&= 2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi \{ w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1 \}, \nonumber \\&\varvec{x} \in {\mathbb {R}}^p,\varvec{y} \in {\mathbb {R}}^p, w>0. \end{aligned}$$

(21)

Again, using Lemma 2 in Arellano-Valle et al. (2005), we have

$$\begin{aligned}&\phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega })\phi \{ w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1 \}=\phi _p \bigg \{ \varvec{x}; \varvec{\xi }+ w \dfrac{1}{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}\,\varvec{\Omega }\varvec{y}, (\varvec{\Omega }^{-1} + \varvec{y} \varvec{y}^\top )^{-1} \bigg \}\\&\quad \times \phi ( w;0,1+\varvec{y}^\top \varvec{\Omega }\varvec{y} ), \end{aligned}$$

and using this result in (21), we get

$$\begin{aligned} f_{\varvec{X},\varvec{Y},W}(\varvec{x},\varvec{y},w)&= 2 \phi _p \bigg \{ \varvec{x}; \varvec{\xi }+ w \dfrac{1}{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}\,\varvec{\Omega }\varvec{y}, (\varvec{\Omega }^{-1} + \varvec{y} \varvec{y}^\top )^{-1} \bigg \}\nonumber \\&\times \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi ( w;0,1+\varvec{y}^\top \varvec{\Omega }\varvec{y} ) \end{aligned}$$

(22)

for $\varvec{x} \in {\mathbb {R}}^p$, $\varvec{y} \in {\mathbb {R}}^p$, and $w>0$. The rest of the proof is trivial from (22). $\square$

1.3 Proof of Corollary 2

From Eq. (21), we get that the joint mpdf of $(\varvec{X},\varvec{Y})$ is given by

$$\begin{aligned} f_{\varvec{X},\varvec{Y}}(\varvec{x},\varvec{y})&= 2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \int _{0} ^ \infty \phi \{ w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1 \} d w\\&= 2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \Phi \{ \varvec{y}^\top (\varvec{x}-\varvec{\xi })\}, \quad \varvec{x},\varvec{y} \in {\mathbb {R}}^p. \end{aligned}$$

From this mpdf it follows that the marginal mpdf of $\varvec{X}$ and the conditional mpdf of $\varvec{Y}|\varvec{X}=\varvec{x}$ are:

$$\begin{aligned}&f_{\varvec{X}}(\varvec{x}) = 2 \phi _p(\varvec{x}; \varvec{\xi },\varvec{\Omega }) \Phi \bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}, \quad \varvec{x} \in {\mathbb {R}}^p, \\ {}&f_{\varvec{Y}|\varvec{X} = \varvec{x}}(\varvec{y}) \\ {}&\quad = \dfrac{1}{\Phi \bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \} }\phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \Phi \{ \varvec{y}^\top (\varvec{x}-\varvec{\xi })\},\quad \varvec{y} \in {\mathbb {R}}^p. \end{aligned}$$

Also, the conditional multivariate moment generating function of $\varvec{Y}|\varvec{X}=\varvec{x}$ is

$$\begin{aligned} M_{\varvec{Y}|\varvec{X}=\varvec{x}}(\varvec{t})&= {\mathbb {E}}\{\exp (\varvec{t}^\top \varvec{Y})|\varvec{X}=\varvec{x}\} \\&= \int _{{\mathbb {R}}^p} \dfrac{\exp (\varvec{t}^\top \varvec{y})}{\Phi \bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}}\phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \Phi \{ \varvec{y}^\top (\varvec{x}-\varvec{\xi }) \} d \varvec{y}\\&= \dfrac{\exp \left( -\dfrac{1}{2}\varvec{\lambda }^\top \varvec{\Omega }\varvec{\lambda }\right) }{\Phi \bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}} \\&\quad \times \int _{{\mathbb {R}}^p} \dfrac{1}{(\sqrt{2 \pi })^p\{\det (\varvec{\Omega }^{-1}) \}^{1/2}} \exp [-\dfrac{1}{2}\{ \varvec{y}^\top \varvec{\Omega }\varvec{y} -2 (\varvec{\Omega }\varvec{\lambda }+\varvec{t})^\top \varvec{y} \}]\\&\quad \times \Phi \{ \varvec{y}^\top (\varvec{x} -\varvec{\xi }) \} d \varvec{y}\\&= \dfrac{\exp \bigg (\varvec{\lambda }^\top \varvec{t} +\dfrac{1}{2} \varvec{t}^\top \varvec{\Omega }^{-1} \varvec{t}\bigg )}{\Phi \bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}}\\&\quad \times \Phi \bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1} \varvec{t}}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}, \quad \varvec{t} \in {\mathbb {R}}^p, \end{aligned}$$

where the last step follows from Lemma 5.3 in Azzalini and Capitanio (2014). Thus, we have

$$\begin{aligned} {\mathbb {E}}(\varvec{Y}|\varvec{X}=\varvec{x})&= \dfrac{\partial }{\partial \varvec{t}} M_{\varvec{Y}|\varvec{X}=\varvec{x}}(\varvec{t}) \bigg |_{\varvec{t} =\varvec{0}}\\&= \varvec{\lambda }+ W_{\phi }\bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}\\&\quad \times \dfrac{1}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}}\,\varvec{\Omega }^{-1}(\varvec{x} - \varvec{\xi }), \\&{\mathbb {E}}(\varvec{Y} \varvec{Y}^\top |\varvec{X}= \varvec{x}) = \dfrac{\partial ^2}{\partial \varvec{t} \partial \varvec{t}^\top } M_{\varvec{Y}|\varvec{X}=\varvec{x}}(\varvec{t}) \bigg |_{\varvec{t} =\varvec{0}}\\&= \varvec{\lambda }\varvec{\lambda }^\top + \varvec{\Omega }^{-1} + W_{\phi }\bigg \{ \dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}\\&\quad \times \dfrac{1}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}}\\&\quad \times \bigg \{\varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi }) \varvec{\lambda }^\top + \varvec{\lambda }(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1}\\&\quad -\dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}\,\varvec{\Omega }^{-1}(\varvec{x} - \varvec{\xi })(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} \bigg \}. \end{aligned}$$

Now, since ${\mathbb {E}}(W\varvec{Y}|\varvec{X} =\varvec{x}) = {\mathbb {E}}\{ W {\mathbb {E}}(\varvec{Y}|W,\varvec{X}=\varvec{x})|\varvec{X}=\varvec{x}\}$, then for the evaluation of this quantity we need the conditional mpdfs of $\varvec{Y}|W=w,\varvec{X}=\varvec{x}$ and $W|\varvec{X} = \varvec{x}$. Again, from (21):

$$\begin{aligned} f_{W|\varvec{X} = \varvec{x}}(w)&\propto \int _{{\mathbb {R}}^p} \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi \{w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1\} d \varvec{y} , \quad w>0, \end{aligned}$$

where, from Lemma 2 in Arellano-Valle et al. (2005), the product $\phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi \{w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1\}$ is equal to $\phi \{ w;(\varvec{x} - \varvec{\xi })^\top \varvec{\lambda },1+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi }) \} \phi _p[\varvec{y};\varvec{\lambda }+ \varvec{\Lambda }(\varvec{x}-\varvec{\xi }) \{ w-\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi }) \},\varvec{\Lambda }]$, where $\varvec{\Lambda }= \{\varvec{\Omega }+(\varvec{x}- \varvec{\xi })(\varvec{x}- \varvec{\xi })^\top \}^{-1}$. Thus,

$$\begin{aligned} f_{W|\varvec{X} = \varvec{x}}(w) \propto \phi \{ w;(\varvec{x} - \varvec{\xi })^\top \varvec{\lambda },1+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi }) \}, \quad w>0. \end{aligned}$$

That is, $W|\varvec{X}=\varvec{x} \sim \mathcal{T}\mathcal{N} \{ (\varvec{x} - \varvec{\xi })^\top \varvec{\lambda },1+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi });(0,\infty ) \}$, and hence

$$\begin{aligned} {\mathbb {E}}(W|\varvec{X}=\varvec{x})&= (\varvec{x} - \varvec{\xi })^\top \varvec{\lambda }+\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}\,\\&\quad \times W_{\phi }\bigg \{\dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \},\\ {\mathbb {E}}(W^2|\varvec{X}=\varvec{x})&= \{(\varvec{x} - \varvec{\xi })^\top \varvec{\lambda }\}^2 + \{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })\} \\&\quad +(\varvec{x} - \varvec{\xi })^\top \varvec{\lambda }\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}\,\\&\quad \times W_{\phi }\bigg \{\dfrac{\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{\sqrt{1+(\varvec{x} - \varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}} \bigg \}. \end{aligned}$$

Furthermore, from Eq. (21), we have $f_{\varvec{Y}|\varvec{X} = \varvec{x},W = w}(\varvec{y}) \propto \phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi \{w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1\}$, where again, from Lemma 2 in Arellano-Valle et al. (2005), the product $\phi _p(\varvec{y};\varvec{\lambda },{\varvec{\Omega }}^{-1}) \phi \{w;\varvec{y}^\top (\varvec{x}- \varvec{\xi }),1\}$ is equal to $\phi \{w;(\varvec{x} - \varvec{\xi })^\top \varvec{\lambda },1+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })\}\phi _p[\varvec{y};\varvec{\lambda }+ \varvec{\Lambda }(\varvec{x}-\varvec{\xi })\{w-\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })\},\varvec{\Lambda }]$, with $\varvec{\Lambda }= \{\varvec{\Omega }+(\varvec{x}- \varvec{\xi })(\varvec{x}- \varvec{\xi })^\top \}^{-1}$. Therefore,

$$\begin{aligned} f_{\varvec{Y}|\varvec{X} = \varvec{x},W = w}(\varvec{y}) = \phi _p[\varvec{y};\varvec{\lambda }+ \varvec{\Lambda }(\varvec{x}-\varvec{\xi })\{w-\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })\},\varvec{\Lambda }], \quad \varvec{y} \in {\mathbb {R}}^p. \end{aligned}$$

That is, $\varvec{Y}|\varvec{X} = \varvec{x},W = w\sim {\mathcal {N}}_p(\varvec{\lambda }+ \varvec{\Lambda }(\varvec{x}-\varvec{\xi })\{w-\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })\},\varvec{\Lambda })$. Hence,

$$\begin{aligned} {\mathbb {E}}(W\varvec{Y}|\varvec{X} =\varvec{x})&= {\mathbb {E}}\{W {\mathbb {E}}(\varvec{Y}|W,\varvec{X}=\varvec{x})|\varvec{X}=\varvec{x}\} \\&= {\mathbb {E}}\left[ W\left\{ \varvec{\lambda }+ \dfrac{W-\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi })}{1+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{x}-\varvec{\xi })}\,\varvec{\Omega }^{-1}(\varvec{x}-\varvec{\xi }) \right\} \left| \right. \varvec{X}=\varvec{x}\right] \\&={\mathbb {E}}(W|\varvec{X}=\varvec{x})\varvec{\lambda }\\&\quad +\dfrac{\{{\mathbb {E}}(W^2|\varvec{X}=\varvec{x}) -\varvec{\lambda }^\top (\varvec{x}-\varvec{\xi }){\mathbb {E}}(W|\varvec{X}=\varvec{x})\}}{1+(\varvec{x}-\varvec{\xi })^\top \varvec{\Omega }^{-1}(\varvec{x}-\varvec{\xi })}\,\varvec{\Omega }^{-1}(\varvec{x}-\varvec{\xi }). \end{aligned}$$

$\square$

1.4 Proof of Proposition 2

From Proposition 1, we have that $\varvec{X} \sim \mathcal {MSN}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$ can be represented as ${\varvec{X}} | \varvec{Y} = \varvec{y} \sim \mathcal{S}\mathcal{N}_p \left( \varvec{\xi }, \varvec{\Omega }- \dfrac{1}{1 + \varvec{y}^\top \varvec{\Omega }\varvec{y}}\,\varvec{\Omega }\varvec{y} \varvec{y}^\top \varvec{\Omega }, \dfrac{1}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\,\varvec{\Omega }\varvec{y} \right)$ and $\varvec{Y} \sim {\mathcal {N}}_p(\varvec{\lambda },\varvec{\Omega }^{-1})$. Now, using the results in Eq. (16), we get ${\mathbb {E}}(\varvec{X}|\varvec{Y} = \varvec{y}) = \varvec{\xi }+ \sqrt{\dfrac{2}{\pi }}\dfrac{1}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\,\varvec{\Omega }\varvec{y}$ and ${\mathbb {V}}ar (\varvec{X}|\varvec{Y} = \varvec{y}) = \varvec{\Omega }- \dfrac{2}{\pi }\dfrac{1}{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}\,\varvec{\Omega }\varvec{y} \varvec{y} ^\top \varvec{\Omega }$, where $\varvec{Y} \sim {\mathcal {N}}_p\left( \varvec{\lambda },\varvec{\Omega }^{-1}\right)$, so that

$$\begin{aligned} {\mathbb {E}}(\varvec{X}) = {\mathbb {E}}\{ {\mathbb {E}}(\varvec{X}|\varvec{Y}) \}= \varvec{\xi }+ \sqrt{\dfrac{2}{\pi }} {\mathbb {E}} \left( \dfrac{1}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\,\varvec{\Omega }\varvec{Y} \right) \end{aligned}$$

and

$$\begin{aligned} {\mathbb {V}}ar (\varvec{X})&= {\mathbb {E}}\{ {\mathbb {V}}ar (\varvec{X}|\varvec{Y})\} + {\mathbb {V}}ar \{ {\mathbb {E}}(\varvec{X}| \varvec{Y}) \} \\ {}&= \varvec{\Omega }- \dfrac{2}{\pi }{\mathbb {E}}\left( \dfrac{1}{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}\,\varvec{\Omega }\varvec{Y} \varvec{Y} ^\top \varvec{\Omega }\right) + \dfrac{2}{\pi } {\mathbb {V}}ar \left( \dfrac{1}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}} \,\varvec{\Omega }\varvec{Y}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} {\mathbb {V}}ar (\varvec{X}) = \varvec{\Omega }- \dfrac{2}{\pi } {\mathbb {E}} \left( \dfrac{}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\,\varvec{\Omega }\varvec{Y} \right) {\mathbb {E}} \left( \dfrac{1}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\,\varvec{\Omega }\varvec{Y} \right) ^\top . \end{aligned}$$

$\square$

1.5 Proof of Proposition 3

Let $\varvec{X}\sim \mathcal {MSN}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$. Then, by Proposition 1 we have that $\varvec{X}|\varvec{Y}= \varvec{y} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi }_{\varvec{y}},\varvec{\eta }_{\varvec{y}})$, with $\varvec{\Psi }_{\varvec{y}}=\varvec{\Omega }- \varvec{\eta }_{\varvec{y}} \varvec{\eta }_{\varvec{y}}^\top$ and $\varvec{\eta }_{\varvec{y}}=(1+\varvec{y}^\top \varvec{\Omega }\varvec{y})^{-1/2} \varvec{\Omega }\varvec{y}$, where $\varvec{Y}\sim {\mathcal {N}}_p(\varvec{\lambda },\varvec{\Omega }^{-1})$, with $\varvec{\lambda }=(1-\varvec{\eta }^\top \varvec{\Omega }^{-1} \varvec{\eta })^{-1/2} \varvec{\Omega }^{-1} \varvec{\eta }$ and $\varvec{\Omega }=\varvec{\Psi }+\varvec{\eta }\varvec{\eta }^\top$ as defined in Eq. (6). Hence by adapting the result of the $\mathcal{S}\mathcal{N}$-mmgf from Proposition 11 to the conditional mmgf of $\varvec{X}|\varvec{Y}=\varvec{y}\sim \mathcal{S}\mathcal{N}_p(\varvec{\xi },\varvec{\Psi }_{\varvec{y}},\varvec{\eta }_{\varvec{y}})$, we have

$$\begin{aligned} M_{\varvec{X}|\varvec{Y}=\varvec{y}}(\varvec{t})= & {} 2 \exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{t}^\top \varvec{\Omega }_{\varvec{y}} \varvec{t}\right) \Phi (\varvec{\eta }_{\varvec{y}}^\top \varvec{t})\\ \quad= & {} 2 \exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} \varvec{t}^\top \varvec{\Omega }\varvec{t}\right) \Phi \left( \frac{\varvec{y}^\top \varvec{\Omega }\varvec{t}}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\right) , \end{aligned}$$

since $\varvec{\Omega }_{\varvec{y}}=\varvec{\Psi }_{\varvec{y}}+ \varvec{\eta }_{\varvec{y}} \varvec{\eta }_{\varvec{y}}^\top =\varvec{\Omega }$. Hence, the mmgf of $\varvec{X}\sim \mathcal {MSN}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ becomes

$$\begin{aligned} M_{\varvec{X}}(\varvec{t})={\mathbb {E}}\{M_{\varvec{X}|\varvec{Y}}(\varvec{t})\}= 2 \exp \left( \varvec{t}^\top \varvec{\xi }+\frac{1}{2} t^\top \varvec{\Omega }\varvec{t}\right) {\mathbb {E}}\left\{ \Phi \left( \frac{\varvec{Y}^\top \varvec{\Omega }\varvec{t}}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\right) \right\} , \end{aligned}$$

where as before $\varvec{Y}\sim {\mathcal {N}}_p(\varvec{\lambda },\varvec{\Omega }^{-1})$, and we note that

$$\begin{aligned} {\mathbb {E}}\left\{ \Phi \left( \frac{\varvec{Y}^\top \varvec{\Omega }\varvec{t}}{\sqrt{1+ \varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\right) \right\} ={\mathbb {E}}\left\{ \Phi \left( \frac{\varvec{Z}^\top \varvec{t}}{\sqrt{1+\varvec{Z}^\top \varvec{\Omega }^{-1} \varvec{Z}}}\right) \right\} , \end{aligned}$$

where $\varvec{Z}=\varvec{\Omega }\varvec{Y} \sim {\mathcal {N}}_p(\bar{\varvec{\lambda }},\varvec{\Omega })$, with $\bar{\varvec{\lambda }}=\varvec{\Omega }\varvec{\lambda }=(1-\varvec{\eta }^\top \varvec{\Omega }^{-1} \varvec{\eta })^{-1/2} \varvec{\eta }$. $\square$

1.6 Proof of Proposition 4

From Proposition 12, the conditional mcdf of $\varvec{X}|\varvec{Y}=\varvec{y} \sim \mathcal{S}\mathcal{N}_p(\varvec{\xi }, \varvec{\Psi }_{\varvec{y}},\varvec{\eta }_{\varvec{y}})$ is given by

$$\begin{aligned} F_{\varvec{X}|\varvec{Y}=\varvec{y}}(\varvec{x})&=\int _{\varvec{u}\le \varvec{x}} 2\phi _p(\varvec{u};\varvec{\xi },\varvec{\Omega }_{\varvec{y}})\Phi \left\{ \varvec{\gamma }_{\varvec{y}}^\top \varvec{\Omega }_{\varvec{y}}^{-1/2}(\varvec{u}-\varvec{\xi })\right\} \text {d} \varvec{u}\\&=2\Phi _{p+1}(\varvec{x}^*;\varvec{\xi }^*,\varvec{\Omega }_{\varvec{y}}^*), \end{aligned}$$

where

$$\begin{aligned}{} & {} \varvec{x}^*=\left( \begin{array}{c} \varvec{x} \\ 0 \\ \end{array} \right) ,\quad \varvec{\xi }^*=\left( \begin{array}{c} \varvec{\xi }\\ 0 \\ \end{array} \right) ,\\ {}{} & {} \varvec{\Omega }_{\varvec{y}}^*=\left( \begin{array}{cc} \varvec{\Omega }_{\varvec{y}} - \varvec{\Omega }_{\varvec{y}}^{1/2} \varvec{\gamma }_{\varvec{y}} \\ - \varvec{\gamma }_{\varvec{y}}^\top \varvec{\Omega }_{\varvec{y}}^{1/2} &{} 1+ \varvec{\gamma }_{\varvec{y}}^\top \varvec{\Omega }_{\varvec{y}}^{-1} \varvec{\gamma }_{\varvec{y}} \\ \end{array} \right) =\left( \begin{array}{cc} \varvec{\Omega }&{} -\varvec{\Omega }\varvec{y} \\ -\varvec{y}^\top \varvec{\Omega }&{} 1+\varvec{y}^\top \varvec{\Omega }\varvec{y} \\ \end{array} \right) , \end{aligned}$$

since $\varvec{\Omega }_{\varvec{y}}=\varvec{\Omega }$, $\varvec{\gamma }_{\varvec{y}}=(1-\varvec{\eta }_{\varvec{y}}^\top \varvec{\Omega }_{\varvec{y}}^{-1} \varvec{\eta }_{\varvec{y}})^{-1/2} \varvec{\Omega }_{\varvec{y}}^{-1/2} \varvec{\eta }_{\varvec{y}}=\varvec{\Omega }^{1/2} \varvec{y}$ and $\varvec{\eta }_{\varvec{y}}=(1+\varvec{y}^\top \varvec{\Omega }\varvec{y})^{-1/2} \varvec{\Omega }\varvec{y}$. Therefore, the mcdf of $\varvec{X}\sim \mathcal {MSN}_p(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ becomes

$$\begin{aligned} F_{\varvec{X}}(\varvec{x})&={\mathbb {E}} \left[ \Phi _{p+1}\left\{ \left( \begin{array}{c} \varvec{x} \\ 0 \\ \end{array} \right) ;\left( \begin{array}{c} \varvec{\xi }\\ 0 \\ \end{array} \right) ,\left( \begin{array}{cc} \varvec{\Omega }&{} - \varvec{\Omega }\varvec{Y} \\ -\varvec{Y}^\top \varvec{\Omega }&{} 1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y} \\ \end{array} \right) \right\} \right] \\&={\mathbb {E}}\left[ \Phi _{p+1}\left\{ \left( \begin{array}{c} \varvec{x} \\ 0 \\ \end{array} \right) ;\left( \begin{array}{c} \varvec{\xi }\\ 0 \\ \end{array} \right) ,\left( \begin{array}{cc} \varvec{\Omega }&{} -\varvec{Z} \\ -\varvec{Z} &{} 1+\varvec{Z}^\top \varvec{\Omega }^{-1} \varvec{Z} \\ \end{array} \right) \right\} \right] , \end{aligned}$$

where as before $\varvec{Z}=\varvec{\Omega }\varvec{Y} \sim {\mathcal {N}}_p(\bar{\varvec{\lambda }},\varvec{\Omega })$, with $\bar{\varvec{\lambda }}=\varvec{\Omega }\varvec{\lambda }=(1-\varvec{\eta }^\top \varvec{\Omega }^{-1} \varvec{\eta })^{-1/2} \varvec{\eta }$. $\square$

1.7 Proof of Proposition 5

By assumption $\widetilde{\varvec{X}}=\varvec{a} + \varvec{B} \varvec{X}$, with $\varvec{X} \sim \mathcal {MSN}_p(\varvec{\xi }, \varvec{\Psi },\varvec{\eta })$, and therefore from the stochastic representation of $\varvec{X}$, given in Proposition 10, we have $\widetilde{\varvec{X}}$ has stochastic representation given by $\widetilde{\varvec{X}}\buildrel d\over = \varvec{a} + \varvec{B} \varvec{\xi }+ \dfrac{1}{\sqrt{1+\varvec{Y}^\top \varvec{\Omega }\varvec{Y}}}\,\varvec{B}\varvec{\Omega }\varvec{Y}\, T + \varvec{B}\varvec{U},$ where $T\sim \mathcal{H}\mathcal{N}(0,1)$ and $\varvec{Y}\sim {\mathcal {N}}_p(\varvec{\lambda },\varvec{\Omega }^{-1})$ are independent, and $\varvec{U}|\varvec{Y}=\varvec{y}\sim {\mathcal {N}}_p\left( \varvec{0},(\varvec{\Omega }^{-1}+\varvec{y}\varvec{y}^\top )^{-1}\right)$. Also, by conditioning $\widetilde{\varvec{X}}$ on $\varvec{Y}=\varvec{y}$ from this stochastic representation, we have that $\widetilde{\varvec{X}}|\varvec{Y}=\varvec{y}\sim \mathcal{S}\mathcal{N}_k(\varvec{a} + \varvec{B} \varvec{\xi }, \varvec{\Psi }_{\varvec{y}}, \varvec{\eta }_{\varvec{y}})$, with conditional mpdf

$$\begin{aligned} f_{\widetilde{\varvec{X}} | \varvec{Y}=\varvec{y}}(\widetilde{\varvec{x}}) = \phi _p(\widetilde{\varvec{x}};\varvec{a} + \varvec{B} \varvec{\xi },\varvec{\Psi }_{\varvec{y}}+\varvec{\eta }_{\varvec{y}}\varvec{\eta }_{\varvec{y}}^\top ) \Phi \left\{ \dfrac{\varvec{\eta }_{\varvec{y}}^\top \varvec{\Psi }_{\varvec{y}}^{-1}(\tilde{\varvec{x}}-\varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1+\varvec{\eta }_{\varvec{y}}^\top (\varvec{\Psi }_{\varvec{y}}+\varvec{\eta }_{\varvec{y}}\varvec{\eta }_{\varvec{y}}^\top )^{-1}\varvec{\eta }_{\varvec{y}}}}\right\} , \end{aligned}$$

where $\varvec{Y} \sim {\mathcal {N}}_p\left( \varvec{\lambda },\varvec{\Omega }^{-1}\right)$, $\varvec{\Psi }_{\varvec{y}}=\varvec{B}(\varvec{\Omega }^{-1}+\varvec{y}\varvec{y}^\top )^{-1}\varvec{B}^\top$ and $\varvec{\eta }_{\varvec{y}}=\dfrac{1}{\sqrt{1+\varvec{y}^\top \varvec{\Omega }\varvec{y}}}\,\varvec{B} \varvec{\Omega }\varvec{y}.$ Thus, since $f_{\widetilde{\varvec{X}}}(\widetilde{\varvec{x}}) = \int _{{\mathbb {R}}^p} f_{\widetilde{\varvec{X}} | \varvec{Y}}(\widetilde{\varvec{x}}) f_{\varvec{Y}}(\varvec{y}) d \varvec{y}$, we have, after some extensive but straightforward algebra, that the mpdf of $\widetilde{\varvec{X}}$ becomes

$$\begin{aligned} f_{\widetilde{\varvec{X}}}(\widetilde{\varvec{x}})&=2 \phi _p(\widetilde{\varvec{x}}; \varvec{a} + \varvec{B} \varvec{\xi }, \varvec{B} \varvec{\Omega }\varvec{B}^\top ) \int _{{\mathbb {R}}^p} \Phi \bigg [ \dfrac{\varvec{y}^\top \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1 + \varvec{y}^\top \{\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }\} \varvec{y} }} \bigg ] \phi _p\left( \varvec{y}; \varvec{\lambda },\varvec{\Omega }^{-1}\right) d \varvec{y}\\&=2 \phi _p(\widetilde{\varvec{x}}; \varvec{a} + \varvec{B} \varvec{\xi }, \varvec{B} \varvec{\Omega }\varvec{B}^\top ) {\mathbb {E}}\left( \Phi \bigg [ \dfrac{\varvec{Y}^\top \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1 + \varvec{Y}^\top \{\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }\} \varvec{Y} }} \bigg ]\right) , \quad \widetilde{\varvec{x}} \in {\mathbb {R}}^p, \end{aligned}$$

where $\varvec{Y} \sim {\mathcal {N}}_p(\varvec{\lambda }, \varvec{\Omega }^{-1})$.

Now, let $\varvec{Y}_B = \varvec{B} \varvec{\Omega }\varvec{Y}$ and $\varvec{Y}_C = \varvec{C} \varvec{Y}$, where $\varvec{C} = I _p - \varvec{B}^\top \varvec{(}\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }$. Note that $\varvec{B} \varvec{C}^\top = \varvec{0}$ and $\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }=\varvec{C}^\top \varvec{\Omega }\varvec{C}$ and so $\varvec{Y}^\top \{\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }\}\varvec{Y}=\varvec{Y}^\top \varvec{C}^\top \varvec{\Omega }\varvec{C} \varvec{Y}=\varvec{Y}_C^\top \varvec{\Omega }\varvec{Y}_C$. Since $\varvec{Y} \sim {\mathcal {N}}_p(\varvec{\lambda }, \varvec{\Omega }^{-1})$, it follows, from the properties of the multivariate normal distribution, that $\varvec{Y}_B \sim {\mathcal {N}}_k(\varvec{B} \varvec{\Omega }\varvec{\lambda }, \varvec{B} \varvec{\Omega }\varvec{B}^\top )$ and $\varvec{Y}_C \sim {\mathcal {N}}_k(\varvec{C} \varvec{\lambda }, \varvec{C}\varvec{\Omega }^{-1} \varvec{C}^\top )$ and they are independent since $cov (\varvec{Y}_B,\varvec{Y}_C)=\varvec{B} \varvec{C}^\top = \varvec{0}$. In turn, this means that $(\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1}\varvec{Y}_B = (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1}\varvec{B} \varvec{\Omega }\varvec{Y} \sim {\mathcal {N}}_k((\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1}\varvec{B} \varvec{\Omega }\varvec{\lambda },(\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1})$ and it is independent of $\varvec{Y}_C^\top \varvec{\Omega }\varvec{Y}_C= \varvec{Y}^\top \{\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }\}\varvec{Y}$. Thus, for the above expectation, we have, by using the same arguments as in the proof of Proposition 1, that

$$\begin{aligned}&{\mathbb {E}}\left( \Phi \bigg [ \dfrac{\varvec{Y}^\top \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1 + \varvec{Y}^\top \{\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }\} \varvec{Y} }} \bigg ]\right) \\&\quad = {\mathbb {E}}\left[ \Phi \bigg \{ \dfrac{\varvec{Y}_B^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1 + \varvec{Y}_C^\top \varvec{\Omega }\varvec{Y}_C }} \bigg \}\right] \\&\quad = {\mathbb {E}}\left[ \Phi \bigg \{ \dfrac{\varvec{\lambda }^\top \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1 + \varvec{Y}^\top \{\varvec{\Omega }- \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} \varvec{B} \varvec{\Omega }\}\varvec{Y}}\sqrt{1 + (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}} \bigg \}\right] . \end{aligned}$$

When $\varvec{B}$ is a nonsingular square matrix, this expectation, which corresponds to the skewing function of the mpdf of $\widetilde{\varvec{X}}$, reduces to $\Phi \bigg \{ \dfrac{\varvec{\lambda }^\top \varvec{\Omega }\varvec{B}^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}{\sqrt{1 + (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })^\top (\varvec{B} \varvec{\Omega }\varvec{B}^\top )^{-1} (\widetilde{\varvec{x}} - \varvec{a} - \varvec{B} \varvec{\xi })}} \bigg \},$ thus it follows that $\widetilde{\varvec{X}} \sim \mathcal {MSN}_p(\varvec{a} + \varvec{B} \varvec{\xi },\varvec{B} \varvec{\Psi }\varvec{B} ^\top , \varvec{B} \varvec{\eta })$. $\square$

1.8 Proof of Corollary 3

By considering the partitions of $\varvec{X}, \varvec{\xi },\varvec{\Psi }$, and $\varvec{\eta }$, as in Eq. (17), the mpdf of $\varvec{X}_1$ can be found using Proposition 5, putting $\varvec{a} = \varvec{0}$ and $\varvec{B} = (I _{p_1}\, \varvec{0})$ to obtain

$$\begin{aligned} f_{\varvec{X}_1}(\varvec{x}_1)&= 2 \phi _{p_1}(\varvec{x}_1; \varvec{\xi }_1, \varvec{\Omega }_{11} ) \int _{{\mathbb {R}}^p} \Phi \bigg \{ \dfrac{(\varvec{\lambda }_1 + \varvec{\Omega }_{11}^{-1} \varvec{\Omega }_{12} \varvec{\lambda }_2)^\top (\varvec{x}_1 - \varvec{\xi }_1)}{\sqrt{1 + \varvec{y}_2^\top (\varvec{\Omega }_{22}- \varvec{\Omega }_{21} \varvec{\Omega }_{11} ^{-1} \varvec{\Omega }_{12} )\varvec{y}_2 }} \bigg \}\\&\quad \times \phi _p\left( \varvec{y}; \varvec{\lambda },\varvec{\Omega }^{-1}\right) d \varvec{y}\\&= 2 \phi _{p_1}(\varvec{x}_1; \varvec{\xi }_1, \varvec{\Omega }_{11} ) \int _{{\mathbb {R}}^{p_2}} \Phi \bigg \{ \dfrac{(\varvec{\lambda }_1 + \varvec{\Omega }_{11}^{-1} \varvec{\Omega }_{12} \varvec{\lambda }_2)^\top (\varvec{x}_1 - \varvec{\xi }_1)}{\sqrt{1 + \varvec{y}_2^\top (\varvec{\Omega }_{22}- \varvec{\Omega }_{21} \varvec{\Omega }_{11} ^{-1} \varvec{\Omega }_{12} )\varvec{y}_2 }} \bigg \}\\&\quad \times \phi _{p_2}\left( \varvec{y}_2; \varvec{\lambda }_2,\varvec{\Omega }_{22\cdot 1} ^{-1}\right) d \varvec{y}_2,~\varvec{x}_1 \in {\mathbb {R}}^{p_1}, \end{aligned}$$

where $\varvec{y} = (\varvec{y}_1^\top , \varvec{y}_2^\top )^\top$, with $\varvec{y}_i \in {\mathbb {R}}^{p_i}$, and $\varvec{\lambda }= (\varvec{\lambda }_1^\top , \varvec{\lambda }_2^\top )^\top$, with $\varvec{y}_i, \varvec{\lambda }_i \in {\mathbb {R}}^{p_i}$, for $i = 1,2$, and $\varvec{\Omega }= (\varvec{\Omega }_{ij})$, with $\varvec{\Omega }_{ij} = \varvec{\Psi }_{ij} + \varvec{\eta }_i \varvec{\eta }_j ^\top$, $i,j =1,2$. Finally, by using the relations described at the beginning of the corollary we have, after some algebra, that $\dfrac{\varvec{\lambda }_1 + \varvec{\Omega }_{11}^{-1} \varvec{\Omega }_{12} \varvec{\lambda }_2}{\sqrt{1 + \varvec{y}_2^\top (\varvec{\Omega }_{22}- \varvec{\Omega }_{21} \varvec{\Omega }_{11} ^{-1} \varvec{\Omega }_{12} ) \varvec{y}_2}}=\dfrac{1}{\sqrt{1 + \varvec{y}_2^\top \varvec{\Omega }_{22\cdot 1} \varvec{y}_2 }\sqrt{1-\varvec{\eta }^\top \varvec{\Omega }^{-1} \varvec{\eta }}}\,\varvec{\Omega }_{11}^{-1}\varvec{\eta }_1=\sqrt{\dfrac{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}{1 + \varvec{y}_2^\top \varvec{\Omega }_{22\cdot 1}\varvec{y}_2}}\dfrac{1}{\sqrt{1+\varvec{\eta }_1^\top \varvec{\Psi }_{11}^{-1} \varvec{\eta }_1}}\,\varvec{\Psi }_{11}^{-1}\varvec{\eta }_1=\varvec{\lambda }_{1\cdot 2}$. $\square$

1.9 Proof of Proposition 6

To prove this, we will show that the density function of $\mathcal {MSN}_1(0,1, \eta )$ is not a log-concave function. The log-density of $\mathcal {MSN}_1(0,1, \eta )$ is

$$\begin{aligned} \log \{f(x)\} = \log \Bigg [ 2 \phi (x;0,1+\eta ^2) \Phi \Bigg \{ \dfrac{\eta x/\sqrt{1+\eta ^2}}{\sqrt{1+x^2/(1+\eta ^2)}} \Bigg \} \Bigg ]. \end{aligned}$$

The second derivative of $\log \{f(x)\}$ with respect to x is

$$\begin{aligned} \dfrac{\textrm{d}^2 \log \{f(x)\}}{\textrm{d}x^2}&= \dfrac{1}{1+\eta ^2} \Bigg [-1 - \dfrac{1}{(1+t^2)^3} \Bigg \{ \dfrac{\phi (\eta t/\sqrt{1+t^2})}{\Phi (\eta t/\sqrt{1+t^2})} \Bigg \}^2\\&\quad - \dfrac{1}{\Phi (\eta t/\sqrt{1+t^2})} \Bigg \{ \dfrac{3t \phi (\eta t/\sqrt{1+t^2}) }{(1+t^2)^{5/2}} \\&\quad + \dfrac{\eta ^2 t \phi (\eta t/\sqrt{1+t^2}) }{(1+t^2)^{7/2}} \Bigg \} \Bigg ], \end{aligned}$$

where $t = x/\sqrt{1+\eta ^2}$. It has been found numerically that the sign of $\dfrac{\textrm{d}^2 \log \{f(x)\}}{\textrm{d}x^2}$ changes for $x \in (-10,10)$ for various values of $\eta$. Thus, $\log \{f(x)\}$ is not always log-concave. $\square$

1.10 Proof of Proposition 7

Let $X \sim \mathcal {MSN}_1(\xi ,\Psi ,\eta )$. Then, the pdf of X is

$$\begin{aligned} f_X (x) = 2 \phi (x;\xi ,\Psi + \eta ^2) \Phi \bigg \{ \dfrac{\eta }{\sqrt{\Psi }} \dfrac{(x - \xi )}{\sqrt{\Psi + \eta ^2 + (x- \xi )^2}} \bigg \}, \quad x \in {\mathbb {R}}. \end{aligned}$$

For $x < \xi$ and for $\eta > 0, \dfrac{\eta }{\sqrt{\Psi }} \dfrac{(x - \xi )}{\sqrt{\Psi + \eta ^2 + (x- \xi )^2}} \rightarrow -\infty$, and for $x > \xi$, $\dfrac{\eta }{\sqrt{\Psi }} \dfrac{(x - \xi )}{\sqrt{\Psi + \eta ^2 + (x- \xi )^2}} \rightarrow \infty$, when $\Psi \rightarrow 0$. As a consequence, we have, as $\Psi \rightarrow 0$: $f_X(x) \rightarrow 0$ if $x < \xi$ and $f_X(x) \rightarrow 2 \phi (x;\xi , \eta ^2)$ if $x > \xi$, which completes the proof for $\eta >0$. The proof for $\eta < 0$ is similar. $\square$

1.11 Proof of Proposition 8

The likelihood function for $(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ based on a random sample $\varvec{x}_1,\ldots ,\varvec{x}_n$ from $\varvec{X} \sim \mathcal {MSN}_p(\varvec{\xi }, \varvec{\Psi }, \varvec{\eta })$ is

$$\begin{aligned}{} & {} L(\varvec{\xi },\varvec{\Psi },\varvec{\eta }) = \prod _{i=1}^n 2 \phi _p\left( \varvec{x}_i;\varvec{\xi },\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top \right) \\ {}{} & {} \quad \times \Phi \bigg \{ \dfrac{1}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}} \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{x}_i - \varvec{\xi })}{\sqrt{1+(\varvec{x}_i-\varvec{\xi })^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) ^{-1} (\varvec{x}_i -\varvec{\xi })}}\bigg \}, \end{aligned}$$

which becomes the profile likelihood function for the skewness parameter $\varvec{\eta }$ when $\varvec{\xi }$ and $\varvec{\Psi }$ are fixed. Similar to the $\mathcal{S}\mathcal{N}$, it can be argued exactly in the same way as in the proof of Proposition 13 that the maximum likelihood estimator of the skewness parameter is always finite for the $\mathcal {MSN}$ distribution as well. Indeed, as in the $\mathcal{S}\mathcal{N}$ case, for the skewing function of the $\mathcal {MSN}$ distribution we have

$$\begin{aligned} 0&\le \Phi \bigg \{-\sqrt{(\varvec{x}_i - \varvec{\xi })^\top \varvec{\Psi }^{-1} (\varvec{x}_i -\varvec{\xi })} \bigg \}\\ {}&\le \Phi \bigg \{ \dfrac{1}{\sqrt{1+\varvec{\eta }^\top \varvec{\Psi }^{-1} \varvec{\eta }}} \dfrac{\varvec{\eta }^\top \varvec{\Psi }^{-1} (\varvec{x}_i - \varvec{\xi })}{\sqrt{1+(\varvec{x}_i-\varvec{\xi })^\top (\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) ^{-1} (\varvec{x}_i -\varvec{\xi })}}\bigg \}\\&\le \Phi \bigg \{\sqrt{(\varvec{x}_i - \varvec{\xi })^\top \varvec{\Psi }^{-1} (\varvec{x}_i -\varvec{\xi })} \bigg \} \le 1,\quad \forall \, \varvec{\eta }\in {\mathbb {R}}^p. \end{aligned}$$

Also, as we already have established in the proof of Proposition 13, $\phi _p (\varvec{x}_i;\varvec{\xi },\varvec{\Psi }+ \varvec{\eta }\varvec{\eta }^\top ) \rightarrow 0$ for each $~i=1,\ldots ,n$ as some (or all) components of $\varvec{\eta }$ tend to $\pm \infty$. Hence, for any fixed value of $\varvec{\xi }\in {\mathbb {R}}^p$ and $\varvec{\Psi }>0$, we can say that $L(\varvec{\xi },\varvec{\Psi },\varvec{\eta }) \rightarrow 0$ whenever some (or all) components of $\varvec{\eta }$ tend to $\pm \infty$. This observation leads us to the conclusion that $L(\varvec{\xi },\varvec{\Psi },\varvec{\eta })$ is not a monotonically increasing or decreasing function of any of the components of $\varvec{\eta }$. Thus, the profile likelihood of the skewness parameter $\varvec{\eta }$ is always maximized at a finite point for the $\mathcal {MSN}$ family.

1.12 Proof of Proposition 9

The non-singularity of the matrix $\varvec{i}_{\varvec{\Psi }\varvec{\Psi }}$ is just a special case of a more general result proven in Hallin and Paindaveine (2006). Thus, the Fisher information matrix $\varvec{I}(\varvec{\xi },\varvec{\Psi }, \varvec{0})$ is nonsingular if

$$\begin{aligned} \varvec{I}_{\varvec{\xi }\varvec{\eta }} = \begin{bmatrix} \varvec{i}_{\varvec{\xi }\varvec{\xi }} &{} \varvec{i}_{\varvec{\xi }\varvec{\eta }}\\ \varvec{i}_{\varvec{\xi }\varvec{\eta }} &{} \varvec{i}_{\varvec{\eta }\varvec{\eta }} \end{bmatrix} \end{aligned}$$

is nonsingular. Let ${\mathbb {E}} \left( \dfrac{\varvec{Z} \varvec{Z}^\top }{\sqrt{1+\varvec{Z}^\top \varvec{Z}}} \right) = \varvec{U}$ and ${\mathbb {E}} \left( \dfrac{\varvec{Z} \varvec{Z}^\top }{1+\varvec{Z}^\top \varvec{Z}}\right) = \varvec{V}$. Then, $\varvec{I}_{\varvec{\xi }\varvec{\eta }}$ can be written as

$$\begin{aligned} \varvec{I}_{\varvec{\xi }\varvec{\eta }} = \varvec{\Psi }^{-1/2}\begin{bmatrix} I _p &{} \sqrt{\dfrac{2}{\pi }} \varvec{U}\\ \sqrt{\dfrac{2}{\pi }} \varvec{U} &{} \dfrac{2}{\pi } \varvec{V} \end{bmatrix} \varvec{\Psi }^{-1/2}. \end{aligned}$$

Thus, we conclude that $\varvec{I}(\varvec{\xi },\varvec{\Psi }, \varvec{0})$ is nonsingular iff the matrix $\varvec{V} - \varvec{U}^2$ is nonsingular.

Let $R=|\varvec{Z}|$, $\varvec{W}=\varvec{Z}/R$ and $\varvec{Z}_*=\varvec{Z}/\sqrt{1+R^2}.$ Since $\varvec{Z}= R \varvec{W}\sim {\mathcal {N}}_p(\varvec{0},I _p)$, R and $\varvec{W}$ are independent. Also, we know that ${\mathbb {E}}(\varvec{W})=\varvec{0}$ and ${\mathbb {V}} ar (\varvec{W})={\mathbb {E}}(\varvec{W} \varvec{W}^\top )=(1/p) I _p$. Now, note that $\varvec{Z}_*=R_* \varvec{W}$, where $R_*=R/\sqrt{1+R^2}$, and so is independent of $\varvec{W}$. Then, we have $\varvec{U}={\mathbb {C}} ov (\varvec{Z},\varvec{Z}_*)={\mathbb {C}} ov (R \varvec{W},R_* \varvec{W})={\mathbb {E}}(RR_*){\mathbb {E}}(\varvec{W} \varvec{W}^\top )=(1/p){\mathbb {E}}(R^2/\sqrt{1+R^2})I _p$ and $\varvec{V}={\mathbb {V}}ar (\varvec{Z}_*)={\mathbb {V}}ar (R_* \varvec{W})={\mathbb {E}}(R_*^2){\mathbb {E}}(\varvec{W} \varvec{W}^\top )=(1/p)E(R_*^2)I _p$. Hence, $\varvec{V}- \varvec{U}^2$ is positive definite iff $p{\mathbb {E}}(R_*^2)-\{{\mathbb {E}}(RR_*)\}^2>0$, i.e., iff $p{\mathbb {E}}\{R^2/(1+R^2)\}-\{{\mathbb {E}}(R^2/\sqrt{1+R^2})\}^2>0$.

Now, by the Cauchy–Schwartz inequality, we get

$$\begin{aligned}{} & {} \bigg \{ {\mathbb {E}} \left( \dfrac{R^2}{\sqrt{1+R^2}} \right) \bigg \}^2 \le {\mathbb {E}}(R^2) {\mathbb {E}} \left( \dfrac{R^2}{1+R^2} \right) \\ {}{} & {} \Rightarrow p{\mathbb {E}}\{R^2/(1+R^2)\}-\{{\mathbb {E}}(R^2/\sqrt{1+R^2})\}^2 \ge 0,~as ~ {\mathbb {E}}(R^2)=p. \end{aligned}$$

Since equality in the previous inequality cannot be achieved in this case, we conclude that $\varvec{V}- \varvec{U}^2$ is positive definite, and, consequently, $\varvec{I}(\varvec{\xi },\varvec{\Psi }, \varvec{0})$ is also positive definite. $\square$