Some Results on the Truncated Multivariate Skew-Normal Distribution

Abstract

The multivariate skew-normal distribution is useful for modeling departures from normality in data through parameters controlling skewness. Recently, several extensions of this distribution have been proposed in the statistical literature, among which the truncated multivariate skew-normal distribution is the foremost. Truncated distributions appear frequently in various theoretical and applied statistical problems. In this article, we study several properties of the truncated multivariate skew-normal distribution. We obtain distributional results through affine transformations, marginalization, and conditioning. Furthermore, the log-concavity of the joint probability density function is established.

1. Introduction

The multivariate normal distribution is one of the most important distributions in the scientific literature; however, it is used under several assumptions that are rarely satisfied in practice, such as the multivariate skewness of the data. An improvement in the multivariate normal distribution to accommodate multivariate skewness is provided by the multivariate skew-normal distribution (Azzalini and Capitanio [1,2]). This distribution allows modeling multivariate skewness through an extension of the multivariate normal distribution by introducing an additional parameter controlling the shape. The multivariate skew-normal distribution is rich in theoretical properties and is easily manipulated from a mathematical viewpoint. The mathematical tractability of this distribution has led to its application in various areas and has also motivated developments of its various extensions. Arnold and Beaver [3] defined the multivariate extended skew-normal distribution through hidden truncation. Arellano-Valle and Genton [4] studied the multivariate extended skew-t distribution, and Branco and Dey [5] considered a general class of distributions called the skew-elliptical class of distributions. Detailed work on the multivariate skew-normal distribution is contained in Genton [6] and Azzalini [7].

An extension of the multivariate skew-normal distribution is obtained by truncation, that is by deriving the conditional distribution of a d-dimensional vector having a multivariate skew-normal distribution by restricting its support to a subset of ${\mathbb{R}}^d$ (Galarza Morales et al. [8]). Truncated distributions are of great interest in statistics, since they occur naturally in theoretical and applied problems. In the multivariate setting, several studies on truncated distributions have been linked to members of the class of elliptical distributions. Some works on the class of the truncated elliptical distributions appear in Morán-Vásquez and Ferrari [9,10], Kim [11] and Arellano-Valle et al. [12]. The truncated multivariate normal and t distributions are particular cases of the truncated elliptical class of distributions. Some studies on these distributions can be found in Kan and Robotti [13], Arismendi [14], Ho et al. [15], Nadarajah [16], Horrace [17,18], Tallis [19,20,21], and Birnbaum and Meyer [22]. It is noteworthy that truncated elliptical distributions may not be appropriate to control the multivariate skewness of the data.

In the univariate case, several approaches to modeling skewed positive data are available (for example, see Ferrari and Fumes [23] and the references therein); however, modeling univariate skewed data with values restricted to an arbitrary interval are less frequent in the statistical literature. Some studies have appeared in Flecher et al. [24] and Jamalizadeh et al. [25], who considered the truncated multivariate skew-normal distribution, studied some of its properties, and presented applications to real data. In the multivariate case, there are also methodologies to model skewed multivariate positive data (for example, see Marchenko and Genton [26], Morán-Vásquez and Ferrari [9], and Morán-Vásquez et al. [27]); nevertheless, there are a few approaches to model skewed multivariate data whose values are restricted to an arbitrary subset of ${\mathbb{R}}^d$. A recent proposal appears in Galarza Morales et al. [8], where the truncated multivariate extended skew-normal distribution was defined and the moments of the doubly truncated multivariate extended skew-normal distribution were studied. A special case of this family is the truncated multivariate skew-normal distribution, which may be appropriate for modeling correlated skewed multivariate data whose values are restricted to a subset of ${\mathbb{R}}^d$.

In this article, we derive several results on the truncated multivariate skew-normal distribution. Some of these results hold for arbitrary truncation sets, while others hold only for rectangles in ${\mathbb{R}}^d$. We derive several distributional results, including marginal and conditional distributions, as well as establish the log-concavity of the joint probability density function (PDF). Our results generalize several properties already established in the literature for the multivariate skew-normal distribution and the truncated multivariate normal distribution.

This article is organized as follows. Section 2 introduces the truncated multivariate skew-normal distribution and provides some related issues. Section 3 derives several properties of the truncated multivariate skew-normal distribution. Section 4 concludes the article with final remarks.

2. The Truncated Multivariate Skew-Normal Distribution

We denote constant vectors and matrices by boldface lowercase and uppercase Greek letters, respectively. Furthermore, the components of matrices and vectors are denoted by Greek lowercase letters in normal font. For example, if $\mathbf{\alpha }\in {\mathbb{R}}^d$, then $\mathbf{\alpha }={({\alpha }_1,\dots ,{\alpha }_d)}^{\prime }$, and if $\mathbf{\Omega }(d\times h)$ is a matrix with real entries, then $\mathbf{\Omega }={\left({\omega }_{ij}\right)}_{d\times h}$. Random vectors are denoted by bold capital Roman letters. If $\mathbf{\Omega }$ is a square matrix, then $\det \left(\mathbf{\Omega }\right)$ denotes the determinant of $\mathbf{\Omega }$. Furthermore, if $\mathbf{\Omega }$ is a symmetric matrix, then $\mathbf{\Omega }>0$ means that $\mathbf{\Omega }$ is positive definite. Furthermore, ${\mathbf{\Omega }}^{1/2}$ denotes the symmetric positive definite square root of $\mathbf{\Omega }$. The rectangles in ${\mathbb{R}}^d$, denoted by the letter R, are Cartesian products of intervals (finite or infinite) $I_1,\dots ,I_d$, that is $R=I_1\times \dots \times I_d$.

The multivariate skew-normal distribution given in Definition 1 has been of great interest in theoretical and applied statistics (Azzalini and Capitanio [1], Genton [6], and Azzalini [7]).

Definition 1.

The random vector $\mathit{X}\in {\mathbb{R}}^d$ is said to have a multivariate skew-normal distribution with location parameter $\mathbf{\xi }\in {\mathbb{R}}^d$, dispersion matrix $\mathbf{\Omega }(d\times d)>0$, and shape parameter $\mathbf{\alpha }\in {\mathbb{R}}^d$, denoted by $\mathit{X}\sim {SN}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha })$, if its PDF is

$$s_{\mathit{X}}\left(\mathit{x}\right)=2{\varphi }_d(\mathit{x};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{x}-\mathbf{\xi })\right),\mathit{x}\in {\mathbb{R}}^d,$$

(1)

where $\mathbf{\omega }={(\mathbf{\Omega }\odot {\mathit{I}}_d)}^{1/2}$, with $\odot$ being the Hadamard product, ${\varphi }_d(\mathit{x};\mathbf{\xi },\mathbf{\Omega })$ the PDF of a multivariate normal vector $\mathit{X}\sim N_d(\mathbf{\xi },\mathbf{\Omega })$, and $\mathsf{\Phi }\left(z\right)$ the CDF of the standard normal random variable $Z\sim N(0,1)$.

The proof that (1) is a PDF was given by Azzalini [7] (Section 5.1). If we consider $\mathbf{\alpha }=\mathbf{0}$ in (1), then we obtain the PDF of a multivariate normal distribution. For $\mathit{x}=\mathbf{\xi }$, the PDF in (1) slides to ${\left(2\pi \right)}^{-d/2}\det {\left(\mathbf{\Omega }\right)}^{-1/2}$, for all $\mathbf{\alpha }\in {\mathbb{R}}^d$. Furthermore, if $\mathit{X}\sim {SN}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha })$, then ${(\mathit{X}-\mathbf{\xi })}^{\prime }{\mathbf{\Omega }}^{-1}(\mathit{X}-\mathbf{\xi })\sim {\chi }_d^2$. For further results and properties, one may consult Azzalini [7].

In the following definition, we present an extension of the multivariate skew-normal distribution, which can be obtained via hidden truncation (see Arnold and Beaver [3]).

Definition 2.

The random vector $\mathit{X}\in {\mathbb{R}}^d$ is said to have a multivariate extended skew-normal distribution with location parameter $\mathbf{\xi }\in {\mathbb{R}}^d$, dispersion matrix $\mathbf{\Omega }(d\times d)>0$, shape parameter $\mathbf{\alpha }\in {\mathbb{R}}^d$, and extension parameter $\tau \in \mathbb{R}$, denoted by $\mathit{X}\sim {ESN}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha },\tau )$, if its PDF is given by

$$g_{\mathit{X}}\left(\mathit{x}\right)={\displaystyle \frac{{\varphi }_d(\mathit{x};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{x}-\mathbf{\xi })+\tau )}{\mathsf{\Phi }\left({(1+{\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}\mathbf{\Omega }{\mathbf{\omega }}^{-1}\mathbf{\alpha })}^{-1/2}\tau \right)}},\mathit{x}\in {\mathbb{R}}^d,$$

(2)

where $\mathbf{\omega }={(\mathbf{\Omega }\odot {\mathit{I}}_d)}^{1/2}$, ${\varphi }_d(\mathit{x};\mathbf{\xi },\mathbf{\Omega })$ is the PDF of $\mathit{X}\sim N_d(\mathbf{\xi },\mathbf{\Omega })$, and $\mathsf{\Phi }\left(z\right)$ is the CDF of $Z\sim N(0,1)$.

The proof that (2) is a PDF can be found in Azzalini [7] (Section 5.3). Note that, when $\tau =0$ in (2), we obtain the PDF of a multivariate skew-normal distribution given in (1).

Let $\mathit{X}\sim {SN}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha })$ and B be a measurable set, $B\subseteq {\mathbb{R}}^d$. Then, the conditional distribution of $\mathit{X}$ given $\{\mathit{X}\in B\}$ is called the truncated multivariate skew-normal distribution. This distribution is defined by its PDF in Definition 3.

Definition 3.

Let $B\subseteq {\mathbb{R}}^d$ be a measurable set. The random vector $\mathit{Y}\in B$ has a truncated multivariate skew-normal distribution with support B and parameters $\mathbf{\xi }\in {\mathbb{R}}^d$, $\mathbf{\Omega }(d\times d)>0$ and $\mathbf{\alpha }\in {\mathbb{R}}^d$, denoted by $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };B)$, if its PDF is

$$f_{\mathit{Y}}\left(\mathit{y}\right)=\frac{{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })\right)}{{\int }_B{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })\right)\mathrm{d}\mathit{y}},\mathit{y}\in B.$$

(3)

The PDF of $\mathit{Y}$, $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };B)$, can be expressed in equivalent form as

$$f_{\mathit{Y}}\left(\mathit{y}\right)=\frac{s_{\mathit{X}}\left(\mathit{y}\right)}{P[\mathit{X}\in B]},\mathit{y}\in B,$$

(4)

where $s_{\mathit{X}}$ is the PDF of $\mathit{X}$, $\mathit{X}\sim {\mathrm{SN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha })$, defined in (1) and $P[\mathit{X}\in B]={\int }_B{s}_{\mathit{X}}\left(\mathit{y}\right)\mathrm{d}\mathit{y}$.

If we take $B={\mathbb{R}}^d$ in (4), then we obtain the PDF (1). That is, the multivariate skew-normal distribution is a particular case of the truncated multivariate skew-normal distribution. Furthermore, if $\mathbf{\alpha }=\mathbf{0}$ in (3), then we obtain the truncated multivariate normal distribution with support B and parameters $\mathbf{\xi }\in {\mathbb{R}}^d$ and $\mathbf{\Omega }(d\times d)>0$, denoted by $\mathit{Y}\sim {\mathrm{TN}}_d(\mathbf{\xi },\mathbf{\Omega };B)$ (see Morán-Vásquez and Ferrari [10]).

In Figure 1, we present contour plots and PDF plots of the truncated bivariate skew-normal distribution. Note that the contours are deformed ellipses in accordance with the parameter $\mathbf{\alpha }$, and they are projected onto the supports of the distributions. As $\mathbf{\alpha }$ vary, the bivariate truncated skew-normal distribution takes different forms (Figure 1a–d). In Figure 1d, where $\mathbf{\alpha }=\mathbf{0}$, the contours are truncated ellipses corresponding to the truncated bivariate normal distribution.

3. Main Results

Theorem 1 states that the truncated multivariate skew-normal distribution is closed under affine transformations.

Theorem 1.

Let $T:{\mathbb{R}}^d\to {\mathbb{R}}^d$ be the transformation $T\left(\mathit{x}\right)=\mathit{\lambda }+\mathbf{\Delta }\mathit{x}$, where $\mathit{\lambda }\in {\mathbb{R}}^d$ and $\mathbf{\Delta }(d\times d)$ is a matrix such that $\det \left(\mathbf{\Delta }\right)\ne 0$. If $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };B)$, then $T\left(\mathit{Y}\right)\sim {\mathrm{TSN}}_d(\tilde{\mathbf{\xi }},\tilde{\mathbf{\Omega }},\tilde{\mathbf{\alpha }};T\left(B\right))$ where $\tilde{\mathbf{\xi }}=\mathit{\lambda }+\mathbf{\Delta }\mathbf{\xi }$, $\tilde{\mathbf{\Omega }}=\mathbf{\Delta }\mathbf{\Omega }{\mathbf{\Delta }}^{\prime }$, and $\tilde{\mathbf{\alpha }}=\tilde{\mathbf{\omega }}{\left({\mathbf{\Delta }}^{-1}\right)}^{\prime }{\mathbf{\omega }}^{-1}\mathbf{\alpha }$, with $\tilde{\mathbf{\omega }}={(\tilde{\mathbf{\Omega }}\odot {\mathit{I}}_d)}^{1/2}$.

Proof. 

Considering the transformation $\mathit{U}=T\left(\mathit{Y}\right)=\mathit{\lambda }+\mathbf{\Delta }\mathit{Y}$ with the Jacobian $J(\mathit{y}\to \mathit{u})=\det {\left(\mathbf{\Delta }\right)}^{-1}$ in (3), we obtain

$$f_{\mathit{U}}\left(\mathit{u}\right)=\frac{{\varphi }_d({\mathbf{\Delta }}^{-1}(\mathit{u}-\mathit{\lambda });\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}{\mathbf{\Delta }}^{-1}(\mathit{u}-\tilde{\mathbf{\xi }})\right)}{{\int }_{T\left(B\right)}{\varphi }_d({\mathbf{\Delta }}^{-1}(\mathit{u}-\mathit{\lambda });\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}{\mathbf{\Delta }}^{-1}(\mathit{u}-\tilde{\mathbf{\xi }})\right)\mathrm{d}\mathit{u}},\mathit{u}\in T\left(B\right).$$

Note that ${\varphi }_d({\mathbf{\Delta }}^{-1}(\mathit{u}-\mathit{\lambda });\mathbf{\xi },\mathbf{\Omega })={\varphi }_d(\mathit{u};\tilde{\mathbf{\xi }},\tilde{\mathbf{\Omega }})$ and ${\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}{\mathbf{\Delta }}^{-1}(\mathit{u}-\tilde{\mathbf{\xi }})={\tilde{\mathbf{\alpha }}}^{\prime }{\tilde{\mathbf{\omega }}}^{-1}(\mathit{u}-\tilde{\mathbf{\xi }})$. This completes the proof. □

Corollary 1.

Let $T:{\mathbb{R}}^d\to {\mathbb{R}}^d$ be the transformation $T\left(\mathit{x}\right)=\mathit{\lambda }+\mathbf{\Delta }\mathit{x}$, where $\mathit{\lambda }\in {\mathbb{R}}^d$ and $\mathbf{\Delta }(d\times d)$ is a matrix such that $\det \left(\mathbf{\Delta }\right)\ne 0$. If $\mathit{Y}\sim {\mathrm{TN}}_d(\mathbf{\xi },\mathbf{\Omega };B)$, then $T\left(\mathit{Y}\right)\sim {\mathrm{TN}}_d(\mathit{\lambda }+\mathbf{\Delta }\mathbf{\xi },\mathbf{\Delta }\mathbf{\Omega }{\mathbf{\Delta }}^{\prime };T\left(B\right))$.

Proof. 

Substitute $\mathbf{\alpha }=\mathbf{0}$ in Theorem 1. □

The above corollary can also be obtained as a particular case of Theorem 3.3 of Morán-Vásquez and Ferrari [10].

A PDF $f:{\mathbb{R}}^d\to [0,\infty )$ is log-concave if the inequality:

$$f(\alpha \mathit{u}+(1-\alpha )\mathit{v})\ge {\left[f\left(\mathit{u}\right)\right]}^{\alpha }{\left[f\left(\mathit{v}\right)\right]}^{1-\alpha }$$

(5)

is satisfied for all $\mathit{u},\mathit{v}\in {\mathbb{R}}^d$ and for all $\alpha \in [0,1]$.

The log-concave PDFs have very good theoretical properties and play an important role in statistics. For example, if $\mathit{X}\in {\mathbb{R}}^d$ is a random vector having a log-concave PDF $f_{\mathit{X}}$, then the contours of $f_{\mathit{X}}$ are convex sets, and the marginal PDFs are log-concave. Furthermore, $f_{\mathit{X}}$ is A-unimodal, that is the set $\{\mathit{x}\in {\mathbb{R}}^d:f_{\mathit{X}}\left(\mathit{x}\right)\ge \lambda \}$ is convex for all $\lambda >0$. Moreover, the product of log-concave PDFs is also log-concave. Additionally, if $A_1,\dots ,A_m$ are subsets of ${\mathbb{R}}^d$ and ${\alpha }_1,\dots ,{\alpha }_m$ are real numbers such that ${\alpha }_i\ge 0$, $i=1,\dots ,m$, and ${\sum }_{i=1}^m{\alpha }_i=1$, then

$$P\left[\mathit{X}\in \sum _{i=1}^m{\alpha }_i{A}_i\right]\ge \prod _{i=1}^m{(P[\mathit{X}\in A_i])}^{{\alpha }_i},$$

(6)

where

$$\sum _{i=1}^m{\alpha }_i{A}_i=\left\{\mathit{z}\in {\mathbb{R}}^d:\mathit{z}=\sum _{i=1}^m{\alpha }_i{\mathit{x}}_i,{\mathit{x}}_i\in A_i,i=1,\dots ,m\right\}.$$

Inequality (6) may be useful to find the bounds for probabilities involving random vectors having log-concave PDFs. For a detailed study of the log-concavity, see Tong [28] (Section 4.2).

The PDF of the multivariate skew-normal distribution is log-concave (Azzalini [7] Proposition 5.1). In Theorem 2, we extend this result to the PDF of the truncated multivariate skew-normal distribution with support on convex sets in ${\mathbb{R}}^d$.

Theorem 2.

Let $B\subseteq {\mathbb{R}}^d$ be a measurable convex set. The PDF $f_{\mathit{Y}}$ of $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };B)$ is log-concave.

Proof. 

Let $\mathit{u},\mathit{v}\in {\mathbb{R}}^d$. Let us see that the inequality (5) is satisfied for $f_{\mathit{Y}}$ given in (4). If $\mathit{u}\notin B$ or $\mathit{v}\notin B$, then ${\left[f_{\mathit{Y}}\left(\mathit{u}\right)\right]}^{\alpha }{\left[f_{\mathit{Y}}\left(\mathit{v}\right)\right]}^{1-\alpha }=0$, for all $\alpha \in [0,1]$. Thus, (5) holds. If $\mathit{u},\mathit{v}\in B$, then $\alpha \mathit{u}+(1-\alpha )\mathit{v}\in B$, for all $\alpha \in [0,1]$, since B is a convex set. In this way, $f_{\mathit{Y}}(\alpha \mathit{u}+(1-\alpha )\mathit{v})>0$, and thus, $s_{\mathit{X}}(\alpha \mathit{u}+(1-\alpha )\mathit{v})>0$. Since $s_{\mathit{X}}$ is log-concave (Azzalini [7] Proposition 5.1), we have $s_{\mathit{X}}(\alpha \mathit{u}+(1-\alpha )\mathit{v})\ge {\left[s_{\mathit{X}}\left(\mathit{u}\right)\right]}^{\alpha }{\left[s_{\mathit{X}}\left(\mathit{v}\right)\right]}^{1-\alpha }$, for all $\alpha \in [0,1]$. Dividing each side of the previous inequality by $P[\mathit{X}\in B]$ and taking into account that $P[\mathit{X}\in B]=\left(P\right[\mathit{X}\in B\left]{)}^{\alpha }\right(P[\mathit{X}\in B]{)}^{1-\alpha }$, for all $\alpha \in [0,1]$, we obtain the result. □

Corollary 2.

The PDF of $\mathit{Y}$, $\mathit{Y}\sim {\mathrm{TN}}_d(\mathbf{\xi },\mathbf{\Omega };B)$, is log-concave.

Proof. 

Take $\mathbf{\alpha }=\mathbf{0}$ in Theorem 2. □

The result stated in the above corollary can also be obtained as a special case of Theorem 3.5 of Morán-Vásquez and Ferrari [10].

Next, in order to study marginal and conditional distributions associated with the truncated multivariate skew-normal distribution, we define the partitioned vector and partitioned matrix. We split $\mathit{Y}\in {\mathbb{R}}^d$, $\mathbf{\xi }\in {\mathbb{R}}^d$, $\mathbf{\alpha }\in {\mathbb{R}}^d$ into sub-vectors and $\mathbf{\Omega }(d\times d)>0$ into sub-matrices as

$$\mathit{Y}={({\mathit{Y}}_1^{\prime },{\mathit{Y}}_2^{\prime })}^{\prime },\mathbf{\xi }={({\mathbf{\xi }}_1^{\prime },{\mathbf{\xi }}_2^{\prime })}^{\prime },\mathbf{\Omega }=\left[\begin{array}{cc}{\mathbf{\Omega }}_{11}& {\mathbf{\Omega }}_{12}\\ {\mathbf{\Omega }}_{21}& {\mathbf{\Omega }}_{22}\end{array}\right],\mathbf{\alpha }={({\mathbf{\alpha }}_1^{\prime },{\mathbf{\alpha }}_2^{\prime })}^{\prime },$$

(7)

where ${\mathit{Y}}_1\in {\mathbb{R}}^{d_1}$, ${\mathit{Y}}_2\in {\mathbb{R}}^{d_2}$, ${\mathbf{\xi }}_1\in {\mathbb{R}}^{d_1}$, ${\mathbf{\xi }}_2\in {\mathbb{R}}^{d_2}$, ${\mathbf{\alpha }}_1\in {\mathbb{R}}^{d_1}$, ${\mathbf{\alpha }}_2\in {\mathbb{R}}^{d_2}$, ${\mathbf{\Omega }}_{11}(d_1\times d_1)>0$, ${\mathbf{\Omega }}_{22}(d_2\times d_2)>0$, and ${\mathbf{\Omega }}_{12}(d_1\times d_2)$ and ${\mathbf{\Omega }}_{21}(d_2\times d_1)$ are such that ${\mathbf{\Omega }}_{12}^{\prime }={\mathbf{\Omega }}_{21}$. We also define ${\mathbf{\omega }}_1={({\mathbf{\Omega }}_{11}\odot {\mathit{I}}_{d_1})}^{1/2}$, ${\mathbf{\omega }}_2={({\mathbf{\Omega }}_{22}\odot {\mathit{I}}_{d_2})}^{1/2}$, and the Schur complement of the block ${\mathbf{\Omega }}_{11}$ of the matrix $\mathbf{\Omega }$ as ${\mathbf{\Omega }}_{22·1}={\mathbf{\Omega }}_{22}-{\mathbf{\Omega }}_{21}{\mathbf{\Omega }}_{11}^{-1}{\mathbf{\Omega }}_{12}$. The dimension d is such that $d=d_1+d_2$. We consider as support the rectangle $R\subseteq {\mathbb{R}}^d$, which can be expressed as the Cartesian product of rectangles $R_1=I_1\times \cdots \times I_{d_1}\subseteq {\mathbb{R}}^{d_1}$ and $R_2=I_{d_1+1}\times \cdots \times I_d\subseteq {\mathbb{R}}^{d_2}$, that is,

$$R=R_1\times R_2.$$

(8)

Let $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };R)$. Further, let $\mathit{Y}\in {\mathbb{R}}^d$, $\mathbf{\xi }\in {\mathbb{R}}^d$, $\mathbf{\Omega }(d\times d)>0$, and $\mathbf{\alpha }\in {\mathbb{R}}^d$ be partitioned as in (7), and $R\subseteq {\mathbb{R}}^d$ is given by (8). Then, the marginal PDF of ${\mathit{Y}}_1$ is calculated as

$$f_{{\mathit{Y}}_1}\left({\mathit{y}}_1\right)=\frac{{\int }_{R_2}{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })\right)\mathrm{d}{\mathit{y}}_2}{{\int }_R{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })\right)\mathrm{d}\mathit{y}},{\mathit{y}}_1\in R_1.$$

(9)

The marginal PDF (9) does not necessarily have the structure of the PDF (3). Theorem 3 provides conditions on the support set R for some marginals to be truncated multivariate skew-normal distribution.

Theorem 3.

Let $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };R)$. Consider the partitions given in (7) and (8). If $R_2={\mathbb{R}}^{d_2}$, then ${\mathit{Y}}_1\sim {\mathrm{TSN}}_{d_1}({\mathbf{\xi }}_1,{\mathbf{\Omega }}_{11},{\mathbf{\alpha }}_{1\left(2\right)};R_1)$, with ${\mathbf{\alpha }}_{1\left(2\right)}={(1+{\mathbf{\alpha }}_2^{\prime }{\mathbf{\omega }}_2^{-1}{\mathbf{\Omega }}_{22·1}{\mathbf{\omega }}_2^{-1}{\mathbf{\alpha }}_2)}^{-1/2}({\mathbf{\alpha }}_1+{\mathbf{\omega }}_1{\mathbf{\Omega }}_{11}^{-1}{\mathbf{\Omega }}_{12}{\mathbf{\omega }}_2^{-1}{\mathbf{\alpha }}_2)$.

Proof. 

Note that ${\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })={\varphi }_{d_1}({\mathit{y}}_1;{\mathbf{\xi }}_1,{\mathbf{\Omega }}_{11}){\varphi }_{d_2}({\mathit{y}}_2;{\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1})$, where ${\mathbf{\xi }}_2\left({\mathit{y}}_1\right)={\mathbf{\xi }}_2+{\mathbf{\Omega }}_{21}{\mathbf{\Omega }}_{11}^{-1}({\mathit{y}}_1-{\mathbf{\xi }}_1)$, and ${\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })={\mathit{\lambda }}^{\prime }({\mathit{y}}_1-{\mathbf{\xi }}_1)+{\mathbf{\alpha }}_2^{\prime }{\mathbf{\omega }}_2^{-1}({\mathit{y}}_2-{\mathbf{\xi }}_2\left({\mathit{y}}_1\right))$, with $\mathit{\lambda }={\mathbf{\omega }}_1^{-1}{\mathbf{\alpha }}_1+{\mathbf{\Omega }}_{11}^{-1}{\mathbf{\Omega }}_{12}{\mathbf{\omega }}_2^{-1}{\mathbf{\alpha }}_2$. Therefore, in (9), we have

$$\begin{array}{cc}\hfill & f_{{\mathit{Y}}_1}\left({\mathit{y}}_1\right)=\frac{{\varphi }_{d_1}({\mathit{y}}_1;{\mathbf{\xi }}_1,{\mathbf{\Omega }}_{11}){\int }_{{\mathbb{R}}^{d_2}}{\varphi }_{d_2}({\mathit{y}}_2;{\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1})\mathsf{\Phi }({\mathit{\lambda }}^{\prime }({\mathit{y}}_1-{\mathbf{\xi }}_1)+{\mathbf{\alpha }}_2^{\prime }{\mathbf{\omega }}_2^{-1}({\mathit{y}}_2-{\mathbf{\xi }}_2\left({\mathit{y}}_1\right)))\mathrm{d}{\mathit{y}}_2}{{\int }_{R_1}{\varphi }_{d_1}({\mathit{y}}_1;{\mathbf{\xi }}_1,{\mathbf{\Omega }}_{11}){\int }_{{\mathbb{R}}^{d_2}}{\varphi }_{d_2}({\mathit{y}}_2;{\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1})\mathsf{\Phi }({\mathit{\lambda }}^{\prime }({\mathit{y}}_1-{\mathbf{\xi }}_1)+{\mathbf{\alpha }}_2^{\prime }{\mathbf{\omega }}_2^{-1}({\mathit{y}}_2-{\mathbf{\xi }}_2\left({\mathit{y}}_1\right)))\mathrm{d}{\mathit{y}}_2\mathrm{d}{\mathit{y}}_1},\hfill \end{array}$$

where ${\mathit{y}}_1\in R_1$. Now, evaluating the integral with respect to ${\mathit{y}}_2$ by using Lemma 5.3 of Azzalini [7], we have

$${\int }_{{\mathbb{R}}^{d_2}}{\varphi }_{d_2}({\mathit{y}}_2;{\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1})\mathsf{\Phi }({\mathit{\lambda }}^{\prime }({\mathit{y}}_1-{\mathbf{\xi }}_1)+{\mathbf{\alpha }}_2^{\prime }{\mathbf{\omega }}_2^{-1}({\mathit{y}}_2-{\mathbf{\xi }}_2\left({\mathit{y}}_1\right)))\mathrm{d}{\mathit{y}}_2=\mathsf{\Phi }\left({\mathbf{\alpha }}_{1\left(2\right)}^{\prime }{\mathbf{\omega }}_1^{-1}({\mathit{y}}_1-{\mathbf{\xi }}_1)\right).$$

This completes the proof. □

Corollary 3.

Let $\mathit{Y}\sim {\mathrm{TN}}_d(\mathbf{\xi },\mathbf{\Omega };R)$. Consider the partitions given in (7) and (8). If $R_2={\mathbb{R}}^{d_2}$, then ${\mathit{Y}}_1\sim {\mathrm{TN}}_{d_1}({\mathbf{\xi }}_1,{\mathbf{\Omega }}_{11};R_1)$.

Proof. 

Substitute $\mathbf{\alpha }=\mathbf{0}$ in Theorem 3. □

Horrace [17] established (see Conclusion 5) that, in general, marginal distributions of the truncated multivariate normal distribution with support as one-sided rectangle ${\mathbb{R}}_{\ge \mathit{c}}^d=[c_1,\infty )\times \dots \times [c_d,\infty )$, $\mathit{c}={(c_1,\dots ,c_d)}^{\prime }$, do not have a truncated multivariate normal distribution. However, in Corollary 3, we established a condition for these marginal distributions to belong to the same family, even more so considering rectangular supports in general.

To derive the conditional distribution of ${\mathit{Y}}_2|{\mathit{Y}}_1$ when the random vector $\mathit{Y}={({\mathit{Y}}_1^{\prime },{\mathit{Y}}_2^{\prime })}^{\prime }$ follows the truncated multivariate skew-normal distribution, we need to define the truncated multivariate extended skew-normal distribution. This distribution is constructed as the conditional distribution of $\mathit{X}\sim {\mathrm{ESN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha })$ (Definition 2) given $\{\mathit{X}\in B\}$, where $B\subseteq {\mathbb{R}}^d$ is a measurable set.

Definition 4.

Let $B\subseteq {\mathbb{R}}^d$ be a measurable set. The random vector $\mathit{Y}\in B$ has a truncated multivariate extended skew-normal distribution with support B and parameters $\mathbf{\xi }\in {\mathbb{R}}^d$, $\mathbf{\Omega }(d\times d)>0$, $\mathbf{\alpha }\in {\mathbb{R}}^d$, and $\tau \in \mathbb{R}$, denoted by $\mathit{Y}\sim {\mathrm{TESN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha },\tau ;B)$, if its PDF is

$$h_{\mathit{Y}}\left(\mathit{y}\right)=\frac{{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })+\tau )}{{\int }_B{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })+\tau )\mathrm{d}\mathit{y}},\mathit{y}\in B.$$

(10)

The PDF of $\mathit{Y}$, $\mathit{Y}\sim {\mathrm{TESN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha },\tau ;B)$, can be expressed as

$$h_{\mathit{Y}}\left(\mathit{y}\right)=\frac{g_{\mathit{X}}\left(\mathit{y}\right)}{P[\mathit{X}\in B]},\mathit{y}\in B,$$

(11)

where $g_{\mathit{X}}$ is the PDF of $\mathit{X}$, $\mathit{X}\sim {\mathrm{ESN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha },\tau )$, given in (2) and $P[\mathit{X}\in B]={\int }_B{g}_{\mathit{X}}\left(\mathit{y}\right)\mathrm{d}\mathit{y}$. The parametrization used in Definition 4 is based on the parametrization used by Azzalini and Capitanio [1] for the multivariate skew-normal distribution. Our parametrization differs slightly from the parametrization proposed by Galarza Morales et al. [8].

If we take $B={\mathbb{R}}^d$ in (11), we obtain the PDF (2). That is, the multivariate extended skew-normal distribution is a particular case of the truncated multivariate extended skew-normal distribution. Furthermore, if $\tau =0$ in (10), we obtain the PDF (3) of the truncated multivariate skew-normal distribution.

In Theorem 4, we show that if the random vector $\mathit{Y}={({\mathit{Y}}_1^{\prime },{\mathit{Y}}_2^{\prime })}^{\prime }$ has a truncated multivariate skew-normal distribution, then ${\mathit{Y}}_2|{\mathit{Y}}_1$ has a truncated multivariate extended skew-normal distribution.

Theorem 4.

Let $\mathit{Y}\sim {\mathrm{TSN}}_d(\mathbf{\xi },\mathbf{\Omega },\mathbf{\alpha };R)$. Consider the partitions given in (7) and (8). Then, ${\mathit{Y}}_2|{\mathit{Y}}_1={\mathit{y}}_1\sim {\mathrm{TESN}}_{d_2}({\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1},{\mathbf{\alpha }}_{2·1},{\tau }_{2·1};R_2)$, with ${\mathbf{\Omega }}_{22·1}$ being the Schur complement of the block ${\mathbf{\Omega }}_{11}$ of the matrix Ω, ${\mathbf{\alpha }}_{2·1}={\mathbf{\omega }}_{2·1}{\mathbf{\omega }}_2^{-1}{\mathbf{\alpha }}_2$, with ${\mathbf{\omega }}_{2·1}={({\mathbf{\Omega }}_{22·1}\odot {\mathit{I}}_{d_2})}^{1/2}$ and ${\tau }_{2·1}=({\mathbf{\alpha }}_1^{\prime }{\mathbf{\omega }}_1^{-1}+{\mathbf{\alpha }}_2^{\prime }{\mathbf{\omega }}_2^{-1}{\mathbf{\Omega }}_{21}{\mathbf{\Omega }}_{11}^{-1})({\mathit{y}}_1-{\mathbf{\xi }}_1)$.

Proof. 

From (3) and (9), the conditional PDF of ${\mathit{Y}}_2|{\mathit{Y}}_1$ is calculated as

$$f_{{\mathit{Y}}_2|{\mathit{Y}}_1}\left({\mathit{y}}_2\right)=\frac{{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })\right)}{{\int }_{R_2}{\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })\mathsf{\Phi }\left({\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })\right)\mathrm{d}{\mathit{y}}_2},{\mathit{y}}_2\in R_2.$$

Using the identities ${\varphi }_d(\mathit{y};\mathbf{\xi },\mathbf{\Omega })={\varphi }_{d_1}({\mathit{y}}_1;{\mathbf{\xi }}_1,{\mathbf{\Omega }}_{11}){\varphi }_{d_2}({\mathit{y}}_2;{\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1})$ and ${\mathbf{\alpha }}^{\prime }{\mathbf{\omega }}^{-1}(\mathit{y}-\mathbf{\xi })={\mathbf{\alpha }}_{2·1}^{\prime }{\mathbf{\omega }}_{2·1}^{-1}({\mathit{y}}_2-{\mathbf{\xi }}_2\left({\mathit{y}}_1\right))+{\tau }_{2·1}$ in the above expression and comparing the simplified expression with (10), we obtain the desired result. □

Corollary 4.

Let $\mathit{Y}\sim {\mathrm{TN}}_d(\mathbf{\xi },\mathbf{\Omega };R)$. Consider the partitions given in (7) and (8). Then, ${\mathit{Y}}_2|{\mathit{Y}}_1={\mathit{y}}_1\sim {\mathrm{TN}}_{d_2}({\mathbf{\xi }}_2\left({\mathit{y}}_1\right),{\mathbf{\Omega }}_{22·1};R_2)$.

Proof. 

Take $\mathbf{\alpha }=\mathbf{0}$ in Theorem 4. □

4. Final Remarks

In this article, we obtained several properties of the truncated multivariate skew-normal distribution, which is an extension of the multivariate skew-normal distribution and contains the truncated multivariate normal distribution as a particular case. A number of distributional results involving affine transformations, marginalization, and conditioning of random vectors having a truncated multivariate skew-normal distribution were derived. In addition, we established the log-concavity of the joint probability density function.

The main difficulty in the mathematical and computational tractability of the truncated multivariate skew-normal distribution is due to the integral in the denominator of the PDF (3). Our techniques for deriving distributional results through (3) allow the simplification of terms of the numerator and denominator. Such results are presented in Theorems 1, 3, and 4. The properties established in this paper can be studied for more general models involving truncated distributions such as the class of truncated skew-elliptical distributions, which can be built from the skew-elliptical distributions.

The implementation of maximum likelihood estimation for the truncated multivariate skew-normal distribution is an issue that we will address in a future paper, as well as the development of inferential procedures and applications to real data. Moreover, the development of computational procedures for the class of truncated skew-elliptical distributions is also interesting to address.