Some Statistical Aspects of the Truncated Multivariate Skew-t Distribution

Abstract

The multivariate skew-t distribution plays an important role in statistics since it combines skewness with heavy tails, a very common feature in real-world data. A generalization of this distribution is the truncated multivariate skew-t distribution which contains the truncated multivariate t distribution and the truncated multivariate skew-normal distribution as special cases. In this article, we study several distributional properties of the truncated multivariate skew-t distribution involving affine transformations, marginalization, and conditioning. The generation of random samples from this distribution is described.

1. Introduction

The multivariate skew-normal distribution (Azzalini and Capitanio [1]) and its extensions have received growing attention in recent years. Although this distribution controls the skewness of the data, its sensitivity to the presence of outliers is one of its limitations. The multivariate skew-t distribution (Azzalini and Capitanio [2], Arellano-Valle and Genton [3]) is an alternative for dealing with skewness and outliers in the data since it has heavier tails than that of skew-normal. Several generalizations of this distribution have been studied in the statistical literature. For example, Arellano-Valle and Genton [3] introduced and studied the multivariate extended skew-t distribution. These authors have also defined the multivariate unified skew-t distribution. In a more general setup, the class of multivariate unified skew-elliptical distributions has been studied by Arellano-Valle and Azzalini [4] and Arellano-Valle and Genton [5].

Another extension of the multivariate skew-t distribution is obtained as the conditional distribution resulting from restricting its domain. Such distribution is called the truncated multivariate skew-t distribution and is defined in Galarza Morales et al. [6]. These authors studied the moments of the doubly truncated selection elliptical distributions from which the moments of the doubly truncated multivariate skew-t distribution can be obtained as particular cases. The truncated multivariate skew-t distribution is the focus of the present paper. This distribution has special cases such as the truncated multivariate t distribution (Morán-Vásquez and Ferrari [7,8]) and the truncated multivariate skew-normal distribution (Morán-Vásquez et al. [9], Galarza Morales et al. [10]). Many statistical models in Bayesian statistics, regression analysis and survival analysis involve truncated multivariate distributions. These distributions have received considerable attention in the scientific literature since they often arise in a wide variety of applied sciences such as physics, economics, biology, medicine, engineering, among others. Several studies relating truncation have been developed, mainly in the elliptical class and in two of its members, the multivariate normal and t distributions. For example, see Morán-Vásquez and Ferrari [7,8], Kim [11], Arellano-Valle et al. [12], Kan and Robotti [13], Arismendi [14], Ho et al. [15], Nadarajah [16], Horrace [17,18], Tallis [19,20,21], and Birnbaum and Meyer [22].

The modeling of skewed multivariate data with values restricted to a subset of ${\mathbb{R}}^p$ is of statistical interest, especially in the presence of outliers. Examples of this situation appear in Marchenko and Genton [23], Morán-Vásquez and Ferrari [7], and Morán-Vásquez et al. [24]. These works present parametric methodologies to model correlated multivariate positive data which are skewed and heavy-tailed. However, the statistical literature when this type of data is restricted to an arbitrary subset of ${\mathbb{R}}^p$ is less frequent. The truncated multivariate skew-t distribution may be appropriate for modeling correlated skewed and heavy-tailed multivariate data whose values are restricted to a subset of ${\mathbb{R}}^p$. This type of data occurs, for example, in environmental studies where variables such as pH, grades, and humidity have upper and lower physical bounds, and their densities are not necessarily symmetric within these bounds (Flecher et al. [25]).

In this paper, we establish several distributional properties on the truncated multivariate skew-t distribution involving affine transformations, marginalization, and conditioning. In addition, we describe a procedure based on rejection sampling to simulate random samples from this distribution. Our results generalize some properties on the truncated multivariate skew-normal distribution studied by Morán-Vásquez et al. [9]. Additionally, applications of our results establish some new properties on the truncated multivariate t distribution.

The paper is organized as follows. Section 2 defines the truncated multivariate skew-t distribution and presents other related families. Section 3 and Section 4 focus on distributional properties and random samples generation of the truncated multivariate skew-t distribution, respectively. Section 5 closes the paper with conclusions and future work.

2. The Truncated Multivariate Skew-t Distribution

In what follows, if $\mathsf{\Omega }$ is a square matrix, then $det\left(\mathsf{\Omega }\right)$ denotes the determinant of $\mathsf{\Omega }$. If $\mathsf{\Omega }$ is a symmetric matrix, then $\mathsf{\Omega }>0$ means that $\mathsf{\Omega }$ is positive definite. Furthermore, ${\mathsf{\Omega }}^{1/2}$ denotes the unique symmetric positive definite square root of $\mathsf{\Omega }>0$.

It is well-known that a random vector $\mathit{W}\in {\mathbb{R}}^p$ has a multivariate t distribution with location vector $\mathsf{\xi }\in {\mathbb{R}}^p$, dispersion matrix $\mathsf{\Omega }(p\times p)>0$, and degrees of freedom $\nu >0$, if its probability density function (PDF) is

$$t_p(\mathit{w};\mathsf{\xi },\mathsf{\Omega },\nu )=\frac{\mathsf{\Gamma }\left[\right(\nu +p)/2]}{det{\left(\mathsf{\Omega }\right)}^{1/2}{\left(\pi \nu \right)}^{p/2}\mathsf{\Gamma }(\nu /2)}{\left(1+\frac{{\delta }_{\mathsf{\Omega }}(\mathit{w},\mathsf{\xi })}{\nu }\right)}^{-(\nu +p)/2},\mathit{w}\in {\mathbb{R}}^p,$$

(1)

where ${\delta }_{\mathsf{\Omega }}(\mathit{w},\mathsf{\xi })={(\mathit{w}-\mathsf{\xi })}^{\prime }{\mathsf{\Omega }}^{-1}(\mathit{w}-\mathsf{\xi })$ is the square of the Mahalanobis distance between $\mathit{w}$ and $\mathsf{\xi }$ with respect to $\mathsf{\Omega }$.

When $\nu \to \infty$ in (1), we obtain the PDF of the multivariate normal vector $\mathit{W}\sim {\mathrm{N}}_p(\mathsf{\xi },\mathsf{\Omega })$, which is given by ${\varphi }_p(\mathit{w};\mathsf{\xi },\mathsf{\Omega })={\left(2\pi \right)}^{-p/2}det{\left(\mathsf{\Omega }\right)}^{-1/2}exp[-{\delta }_{\mathsf{\Omega }}(\mathit{w},\mathsf{\xi })/2]$. In addition, if $\nu =1$ in (1), we retrieve the PDF of the multivariate Cauchy distribution.

A detailed study on the multivariate t distribution appear in Kotz and Nadarajah [26].

The PDF of the multivariate skew-t distribution can be expressed in terms of the PDF (1) as in Definition 1 (Arellano-Valle and Genton [3]).

Definition 1.

The random vector $\mathit{X}\in {\mathbb{R}}^p$ is said to have a multivariate skew-t distribution with location vector $\mathsf{\xi }\in {\mathbb{R}}^p$, dispersion matrix $\mathsf{\Omega }(p\times p)>0$, shape parameter $\mathsf{\alpha }\in {\mathbb{R}}^p$ and $\nu >0$ degrees of freedom, denoted by $\mathit{X}\sim {\mathrm{S}\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )$, if its PDF is

$${\mathrm{S}\mathrm{T}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )=2t_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{x},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{x}-\mathsf{\xi });\nu +p\right),$$

(2)

where $\mathit{x}\in {\mathbb{R}}^p$, $\omega ={(\mathsf{\Omega }\odot {\mathit{I}}_p)}^{1/2}$, with ⊙ being the Hadamard product, and $T(z;\nu )$ is the CDF of a standard t random variable with $\nu >0$ degrees of freedom.

The multivariate skew-normal distribution is obtained as a limiting case of the multivariate skew-t distribution when $\nu \to \infty$, with the PDF given by

$$\mathrm{S}{\mathrm{N}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha })=2{\varphi }_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega })\mathsf{\Phi }\left({\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{x}-\mathsf{\xi })\right),\mathit{x}\in {\mathbb{R}}^p,$$

(3)

where $\mathsf{\Phi }\left(z\right)$ is the CDF of a standard normal random variable. We write $\mathit{X}\sim \mathrm{S}{\mathrm{N}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha })$ if the PDF of $\mathit{X}$ is given by (3). In addition, the multivariate t distribution is obtained from the multivariate skew-t distribution when $\mathsf{\alpha }=\mathbf{0}={(0,\dots ,0)}^{\prime }$.

A stochastic representation for $\mathit{X}\sim {\mathrm{S}\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )$ is $\mathit{X}|U=u\sim \mathrm{S}{\mathrm{N}}_p(\mathsf{\xi },u^{-1}\mathsf{\Omega },\mathsf{\alpha })$, $U\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\nu /2,\nu /2)$; see Arellano-Valle and Genton [3]. From this representation we have

$${\mathrm{S}\mathrm{T}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )={\int }_0^{\infty }\mathrm{S}{\mathrm{N}}_p(\mathit{x};\mathsf{\xi },u^{-1}\mathsf{\Omega },\mathsf{\alpha })h(u;\nu )\mathrm{d}u,\mathit{x}\in {\mathbb{R}}^p,$$

(4)

where $h(u;\nu )$ is the PDF of U, $U\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\nu /2,\nu /2)$, given by

$$h(u;\nu )=\frac{{(\nu /2)}^{\nu /2}}{\mathsf{\Gamma }(\nu /2)}{u}^{\nu /2-1}exp(-\nu u/2),\nu >0,u>0.$$

(5)

Next, we define an extension of the multivariate skew-t distribution. This definition can be found in Arellano-Valle and Genton [3].

Definition 2.

The random vector $\mathit{X}\in {\mathbb{R}}^p$ is said to have a multivariate extended skew-t distribution with location parameter $\mathsf{\xi }\in {\mathbb{R}}^p$, dispersion matrix $\mathsf{\Omega }(p\times p)>0$, shape parameter $\mathsf{\alpha }\in {\mathbb{R}}^p$, extension parameter $\tau \in \mathbb{R}$ and $\nu >0$ degrees of freedom, denoted by $\mathit{X}\sim \mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )$, if its PDF is given by

$$\mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )=\frac{t_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{x},\mathsf{\xi })}}\right]}^{1/2}({\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{x}-\mathsf{\xi })+\tau );\nu +p\right)}{T({[1+{\mathsf{\alpha }}^{\prime }{\omega }^{-1}\mathsf{\Omega }{\omega }^{-1}\mathsf{\alpha }]}^{-1/2}\tau ;\nu )},$$

(6)

where $\mathit{x}\in {\mathbb{R}}^p$.

For $\tau =0$ in (6), we obtain the PDF of a multivariate skew-t distribution given in (2). If $\mathsf{\alpha }=\mathbf{0}$ and $\tau =0$ in (6), we obtain the PDF of a multivariate t distribution given in (1). The same occurs when $\tau \to \infty$. If $\tau \to -\infty$, the density degenerates to zero. The behavior of the multivariate extended skew-t distribution according to the values of the extension parameter $\tau$ is studied in detail in Arellano-Valle and Genton [3] where several properties about this distribution and an application have also been studied.

The conditional distribution of $\mathit{X}$, $\mathit{X}\sim {\mathrm{S}\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )$, given $\{\mathit{X}\in A\}$, with $A\subseteq {\mathbb{R}}^p$ being a measurable set, is called the truncated multivariate skew-t distribution. This distribution is defined via its PDF in Definition 3.

Definition 3.

Let $A\subseteq {\mathbb{R}}^p$ be a measurable set. The random vector $\mathit{Y}\in A$ has a truncated multivariate skew-t distribution with support A and parameters $\mathsf{\xi }\in {\mathbb{R}}^p$, $\mathsf{\Omega }(p\times p)>0$, $\mathsf{\alpha }\in {\mathbb{R}}^p$ and $\nu >0$ degrees of freedom, denoted by $\mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu ;A)$, if its PDF is given by

$$f_{\mathit{Y}}\left(\mathit{y}\right)=\frac{t_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right)}{{\displaystyle {\int }_A{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right)\mathrm{d}\mathit{y}}},\mathit{y}\in A.$$

(7)

The PDF of $\mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu ;A)$ can be expressed in equivalent form as

$$f_{\mathit{Y}}\left(\mathit{y}\right)=\frac{{\mathrm{S}\mathrm{T}}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )}{{\mathrm{S}\mathrm{T}}_p(A;\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )},\mathit{y}\in A,$$

(8)

where ${\mathrm{S}\mathrm{T}}_p(A;\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )={\int }_A{\mathrm{S}\mathrm{T}}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )\mathrm{d}\mathit{y}$, being ${\mathrm{S}\mathrm{T}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )$ the PDF of $\mathit{X}$, $\mathit{X}\sim {\mathrm{S}\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu )$, given in (2).

Note that if $A={\mathbb{R}}^p$ in (8), we obtain the multivariate skew-t distribution (Definition 1) as a particular case of the truncated multivariate skew-t distribution. If we take $\mathsf{\alpha }=\mathbf{0}$ in (7), then we obtain the PDF of a random vector $\mathit{Y}\in A$ with truncated multivariate t distribution with support A and parameters $\mathsf{\xi }\in {\mathbb{R}}^p$, $\mathsf{\Omega }(p\times p)>0$ and $\nu >0$ degrees of freedom, which we denote by $\mathit{Y}\sim \mathrm{T}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\nu ;A)$ (Morán-Vásquez and Ferrari [8]). As expected, the truncated multivariate skew-normal distribution (Morán-Vásquez et al. [9]) is obtained as a limiting case of the truncated multivariate skew-t distribution when $\nu \to \infty$. Additional special cases of the truncated multivariate skew-t distribution are the multivariate t distribution ($A={\mathbb{R}}^p$ and $\mathsf{\alpha }=\mathbf{0}$ in (8)) and the multivariate normal distribution ($A={\mathbb{R}}^p$, $\mathsf{\alpha }=\mathbf{0}$ and $\nu \to \infty$ in (8)).

Figure 1 exhibit different shapes of the truncated bivariate skew-t distribution with rectangular support according to its parameters. We consider different parameter settings and plot the surface of the PDF (7) and its contours for different levels, namely $f_{\mathit{Y}}\left(\mathit{y}\right)=k$, where $k=0.33,0.25,0.14,0.06,0.02,0.006,0.002$. The legend indicates the values of all the parameters considered in the first plot and the values of the parameters that are changed from a plot to the subsequent one (in alphabetical order). In Figure 1a, where $\mathsf{\alpha }=\mathbf{0}$, the contours are truncated ellipses. When $\mathsf{\alpha }\ne \mathbf{0}$, these ellipses are deformed (Figure 1b). Figure 1c shows the effect of parameter ${\omega }_{12}$. As the degrees of freedom parameter grows, the contours of the truncated bivariate skew-t distribution tend to the contours of truncated bivariate skew-normal distribution (Figure 1c–d). The tails of the truncated bivariate skew-t distribution are heavier for smaller values of $\nu$.

The truncated multivariate skew-t distribution is appropriate for modeling multivariate data whose values are restricted to a subset of ${\mathbb{R}}^p$, and they are possibly skewed and heavy-tailed. This type of data occurs in a variety of situations. For instance, the dataset studied in Flecher et al. [25] contains measurements of the relative humidity (in %) of an air–water mixture recorded by the INRA (National Institute of Agricultural Research) weather station located in Toulouse, South of France, between 1972 and 1999. These data are restricted to the interval $[0,100]$, and the skewness is apparent (Flecher et al. [25] Figure 1). In addition, the dataset presented and discussed in Morán-Vásquez and Ferrari [7] refers to observations on vitamins B2 (in mg), B3 (in mg), B12 (in mcg), and D (in mcg) intakes based on the first 24 h dietary recall interview. These data are positively correlated, their bivariate distributions are skewed, and outliers are present (Morán-Vásquez and Ferrari [7] Figure 2). The models proposed by these authors for analyzing this dataset are associated with truncated distributions. Thus, the datasets in Flecher et al. [25] and Morán-Vásquez and Ferrari [7] are practical examples where the truncated multivariate skew-t distribution can be used.

3. Distributional Properties

In Theorem 1, we state and prove the closure property of the truncated multivariate skew-t distribution under affine transformations.

Theorem 1.

Let $T:{\mathbb{R}}^p\to {\mathbb{R}}^p$ be the transformation $T\left(\mathit{x}\right)=\lambda +\Delta \mathit{x}$, being $\lambda \in {\mathbb{R}}^p$ a constant vector and $\Delta (p\times p)$ a non-singular constant matrix. If $\mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu ;A)$, then $T\left(\mathit{Y}\right)=\lambda +\Delta \mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\tilde{\mathsf{\xi }},\tilde{\mathsf{\Omega }},\tilde{\mathsf{\alpha }},\nu ;T\left(A\right))$, where $\tilde{\mathsf{\xi }}=\lambda +\Delta \mathsf{\xi }$, $\tilde{\mathsf{\Omega }}=\Delta \mathsf{\Omega }{\Delta }^{\prime }$, and $\tilde{\mathsf{\alpha }}=\tilde{\omega }{\left({\Delta }^{-1}\right)}^{\prime }{\omega }^{-1}\mathsf{\alpha }$, with $\tilde{\omega }={(\tilde{\mathsf{\Omega }}\odot {\mathit{I}}_p)}^{1/2}$.

Proof. 

Applying the transformation $\mathit{W}=T\left(\mathit{Y}\right)=\lambda +\Delta \mathit{Y}$ in (7) with the Jacobian $J(\mathit{y}\to \mathit{w})=det{(\Delta )}^{-1}$, we have

$$f_{\mathit{W}}\left(\mathit{w}\right)=\frac{t_p({\Delta }^{-1}(\mathit{w}-\lambda );\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}({\Delta }^{-1}(\mathit{w}-\lambda ),\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}({\Delta }^{-1}(\mathit{w}-\lambda )-\mathsf{\xi });\nu +p\right)}{{\displaystyle {\int }_{T\left(A\right)}{t}_p({\Delta }^{-1}(\mathit{w}-\lambda );\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}({\Delta }^{-1}(\mathit{w}-\lambda ),\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}({\Delta }^{-1}(\mathit{w}-\lambda )-\mathsf{\xi });\nu +p\right)\mathrm{d}\mathit{w}}},$$

where $\mathit{w}\in T\left(A\right)$. By noting ${\delta }_{\mathsf{\Omega }}({\Delta }^{-1}(\mathit{w}-\lambda ),\mathsf{\xi })={\delta }_{\tilde{\mathsf{\Omega }}}(\mathit{w},\tilde{\mathsf{\xi }})$, $t_p({\Delta }^{-1}(\mathit{w}-\lambda );\mathsf{\xi },\mathsf{\Omega },\nu )=det(\Delta )t_p(\mathit{w};\tilde{\mathsf{\xi }},\tilde{\mathsf{\Omega }},\nu )$ and ${\mathsf{\alpha }}^{\prime }{\omega }^{-1}({\Delta }^{-1}(\mathit{w}-\lambda )-\mathsf{\xi })={\tilde{\mathsf{\alpha }}}^{\prime }{\tilde{\omega }}^{-1}(\mathit{w}-\tilde{\mathsf{\xi }})$, we obtain that

$$f_{\mathit{W}}\left(\mathit{w}\right)=\frac{t_p(\mathit{w};\tilde{\mathsf{\xi }},\tilde{\mathsf{\Omega }},\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\tilde{\mathsf{\Omega }}}(\mathit{w},\tilde{\mathsf{\xi }})}}\right]}^{1/2}{\tilde{\mathsf{\alpha }}}^{\prime }{\tilde{\omega }}^{-1}(\mathit{w}-\tilde{\mathsf{\xi }});\nu +p\right)}{{\displaystyle {\int }_{T\left(A\right)}{t}_p(\mathit{w};\tilde{\mathsf{\xi }},\tilde{\mathsf{\Omega }},\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\tilde{\mathsf{\Omega }}}(\mathit{w},\tilde{\mathsf{\xi }})}}\right]}^{1/2}{\tilde{\mathsf{\alpha }}}^{\prime }{\tilde{\omega }}^{-1}(\mathit{w}-\tilde{\mathsf{\xi }});\nu +p\right)\mathrm{d}\mathit{w}}},$$

which shows that $T\left(\mathit{Y}\right)\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\tilde{\mathsf{\xi }},\tilde{\mathsf{\Omega }},\tilde{\mathsf{\alpha }},\nu ;T\left(A\right))$. □

Corollary 1.

Let $T:{\mathbb{R}}^p\to {\mathbb{R}}^p$ be the transformation $T\left(\mathit{x}\right)=\lambda +\Delta \mathit{x}$, being $\lambda \in {\mathbb{R}}^p$ a constant vector and $\Delta (p\times p)$ a non-singular constant matrix. If $\mathit{Y}\sim \mathrm{T}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\nu ;A)$, then $T\left(\mathit{Y}\right)\sim \mathrm{T}{\mathrm{T}}_p(\lambda +\Delta \mathsf{\xi },\Delta \mathsf{\Omega }{\Delta }^{\prime },\nu ;T\left(A\right))$.

Proof. 

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

The result stated in the above corollary can also be obtained as a particular case of Theorem 3.3 of Morán-Vásquez and Ferrari [8].

Next, we present results on marginal and conditional distributions involving sub-vectors of the random vector having truncated multivariate skew-t distribution. For this, we consider partitions of $\mathit{Y}\in {\mathbb{R}}^p$, $\mathsf{\xi }\in {\mathbb{R}}^p$, $\mathsf{\alpha }\in {\mathbb{R}}^p$, and $\mathsf{\Omega }(p\times p)>0$ as follows

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

(9)

where ${\mathit{Y}}_1\in {\mathbb{R}}^{p_1}$, ${\mathit{Y}}_2\in {\mathbb{R}}^{p_2}$, ${\mathsf{\xi }}_1\in {\mathbb{R}}^{p_1}$, ${\mathsf{\xi }}_2\in {\mathbb{R}}^{p_2}$, ${\mathsf{\alpha }}_1\in {\mathbb{R}}^{p_1}$, ${\mathsf{\alpha }}_2\in {\mathbb{R}}^{p_2}$, ${\mathsf{\Omega }}_{11}(p_1\times p_1)>0$, ${\mathsf{\Omega }}_{22}(p_2\times p_2)>0$, and ${\mathsf{\Omega }}_{12}(p_1\times p_2)$ and ${\mathsf{\Omega }}_{21}(p_2\times p_1)$ are such that ${\mathsf{\Omega }}_{12}^{\prime }={\mathsf{\Omega }}_{21}$. In addition, we define ${\omega }_1={({\mathsf{\Omega }}_{11}\odot {\mathit{I}}_{p_1})}^{1/2}$ and ${\omega }_2={({\mathsf{\Omega }}_{22}\odot {\mathit{I}}_{p_2})}^{1/2}$. The Schur complement of the block ${\mathsf{\Omega }}_{11}$ of the matrix $\mathsf{\Omega }$ is given by ${\mathsf{\Omega }}_{22·1}={\mathsf{\Omega }}_{22}-{\mathsf{\Omega }}_{21}{\mathsf{\Omega }}_{11}^{-1}{\mathsf{\Omega }}_{12}$. In addition, we define ${\mathsf{\xi }}_{2·1}={\mathsf{\xi }}_2+{\mathsf{\Omega }}_{21}{\mathsf{\Omega }}_{11}^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)$. The dimension p is such that $p=p_1+p_2$. We assume that the support set of the truncated multivariate skew-t distribution is a rectangle $R\subseteq {\mathbb{R}}^p$, which can be written as $R=I_1\times \cdots \times I_p$, where $I_1,\dots ,I_p$ are finite or infinite intervals. Furthermore, R can be expressed as

$$R=R_1\times R_2,$$

(10)

where $R_1=I_1\times \cdots \times I_{p_1}\subseteq {\mathbb{R}}^{p_1}$ and $R_2=I_{p_1+1}\times \cdots \times I_p\subseteq {\mathbb{R}}^{p_2}$.

Now, consider the partitions given in (9) and (10) for $\mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu ;R)$. Integrating (7) with respect to ${\mathit{y}}_2$, we find the marginal PDF of ${\mathit{Y}}_1$ as

$$f_{{\mathit{Y}}_1}\left({\mathit{y}}_1\right)=\frac{{\displaystyle {\int }_{R_2}{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right)\mathrm{d}{\mathit{y}}_2}}{{\displaystyle {\int }_R{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right)\mathrm{d}\mathit{y}}},$$

(11)

where ${\mathit{y}}_1\in R_1$. It is noteworthy that the above PDF does not necessarily have the structure of the PDF (7). In Theorem 2, we establish conditions on the support set R for some marginals to preserve the truncated multivariate skew-t distribution. To proceed, we need the following preliminary results.

Lemma 1.

Let $h(u;\nu )$ be the PDF given in (5). Then,

$${\int }_0^{\infty }{\varphi }_p(\mathit{x};\mathsf{\xi },u^{-1}\mathsf{\Omega })\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{x}-\mathsf{\xi })\right)h(u;\nu )\mathrm{d}u=t_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{x},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{x}-\mathsf{\xi });\nu +p\right).$$

(12)

Proof. 

The result follows from the representation (4). □

Lemma 2.

If $\mathit{G}(p\times p)>0$, $\mathbf{b},\mathbf{c}\in {\mathbb{R}}^p$, and k is a scalar, then

$${\left(2\pi \right)}^{-p/2}det{\left(\mathit{G}\right)}^{-1/2}{\int }_{{\mathbb{R}}^p}exp\left(-\frac{{\mathit{x}}^{\prime }{\mathit{G}}^{-1}\mathit{x}-2{\mathbf{b}}^{\prime }\mathit{x}}{2}\right)\mathsf{\Phi }(k+{\mathbf{c}}^{\prime }\mathit{x})\mathrm{d}\mathit{x}=exp\left(\frac{{\mathit{b}}^{\prime }\mathit{G}\mathit{b}}{2}\right)\mathsf{\Phi }\left(\frac{k+{\mathit{c}}^{\prime }\mathit{G}\mathit{b}}{\sqrt{1+{\mathit{c}}^{\prime }\mathit{G}\mathit{c}}}\right).$$

Proof. 

See Azzalini [27] Lemma 5.3. □

Theorem 2.

Let $\mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu ;R)$. Consider the partitions given in (9) and (10). If $R_2={\mathbb{R}}^{p_2}$, then ${\mathit{Y}}_1\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_{p_1}({\mathsf{\xi }}_1,{\mathsf{\Omega }}_{11},{\mathsf{\alpha }}_{1\left(2\right)},\nu ;R_1)$, with ${\mathsf{\alpha }}_{1\left(2\right)}={(1+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}{\mathsf{\Omega }}_{22·1}{\omega }_2^{-1}{\mathsf{\alpha }}_2)}^{-1/2}({\mathsf{\alpha }}_1+{\omega }_1{\mathsf{\Omega }}_{11}^{-1}{\mathsf{\Omega }}_{12}{\omega }_2^{-1}{\mathsf{\alpha }}_2)$.

Proof. 

From (11) and (12), we have

$$f_{{\mathit{Y}}_1}\left({\mathit{y}}_1\right)=\frac{{\displaystyle {\int }_{{\mathbb{R}}^{p_2}}{\int }_0^{\infty }{\varphi }_p(\mathit{y};\mathsf{\xi },u^{-1}\mathsf{\Omega })\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })\right)h(u;\nu )\mathrm{d}u\mathrm{d}{\mathit{y}}_2}}{{\displaystyle {\int }_R{\int }_0^{\infty }{\varphi }_p(\mathit{y};\mathsf{\xi },u^{-1}\mathsf{\Omega })\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })\right)h(u;\nu )\mathrm{d}u\mathrm{d}\mathit{y}}},{\mathit{y}}_1\in R_1.$$

By noting ${\varphi }_p(\mathit{Y};\mathsf{\xi },u^{-1}\mathsf{\Omega })={\varphi }_{p_1}({\mathit{Y}}_1;{\mathsf{\xi }}_1,u^{-1}{\mathsf{\Omega }}_{11}){\varphi }_{p_2}({\mathit{Y}}_2;{\mathsf{\xi }}_{2·1},u^{-1}{\mathsf{\Omega }}_{22·1})$ and by applying Fubini’s theorem, we have

$$f_{{\mathit{Y}}_1}\left({\mathit{y}}_1\right)=\frac{{\displaystyle {\int }_0^{\infty }{\varphi }_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,u^{-1}{\mathsf{\Omega }}_{11})\left[{\int }_{{\mathbb{R}}^{p_2}}{\varphi }_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},u^{-1}{\mathsf{\Omega }}_{22·1})\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })\right)\mathrm{d}{\mathit{y}}_2\right]h(u;\nu )\mathrm{d}u}}{{\displaystyle {\int }_{R_1}{\int }_0^{\infty }{\varphi }_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,u^{-1}{\mathsf{\Omega }}_{11})\left[{\int }_{{\mathbb{R}}^{p_2}}{\varphi }_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},u^{-1}{\mathsf{\Omega }}_{22·1})\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })\right)\mathrm{d}{\mathit{y}}_2\right]h(u;\nu )\mathrm{d}u\mathrm{d}{\mathit{y}}_1}}.$$

(13)

To evaluate the integral with respect to ${\mathit{y}}_2$, we use the identity ${\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })={(1+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}{\mathsf{\Omega }}_{22·1}{\omega }_2^{-1}{\mathsf{\alpha }}_2)}^{1/2}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}({\mathit{y}}_2-{\mathsf{\xi }}_{2·1})$ and subsequently change the variable $\mathit{x}={\mathit{y}}_2-{\mathsf{\xi }}_{2·1}$. Thus we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^{p_2}}{\varphi }_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},u^{-1}{\mathsf{\Omega }}_{22·1})\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })\right)\mathrm{d}{\mathit{y}}_2={\left(2\pi \right)}^{-p_2/2}det{\left(u^{-1}{\mathsf{\Omega }}_{22·1}\right)}^{-1/2}\hfill \\ \hfill & \times {\int }_{{\mathbb{R}}^{p_2}}exp\left(-\frac{u{\mathit{x}}^{\prime }{\mathsf{\Omega }}_{22·1}^{-1}\mathit{x}}{2}\right)\mathsf{\Phi }\left(\sqrt{u}[{(1+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}{\mathsf{\Omega }}_{22·1}{\omega }_2^{-1}{\mathsf{\alpha }}_2)}^{1/2}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}\mathit{x}]\right)\mathrm{d}\mathit{x}\hfill \\ \hfill & =\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)\right),\hfill \end{array}$$

where the last line is derived by using Lemma 2 with $\mathit{G}=u^{-1}{\mathsf{\Omega }}_{22·1}$, $\mathit{b}=\mathbf{0}$, $\mathit{c}=\sqrt{u}{\omega }_2^{-1}{\mathsf{\alpha }}_2$ and $k=\sqrt{u}{(1+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}{\mathsf{\Omega }}_{22·1}{\omega }_2^{-1}{\mathsf{\alpha }}_2)}^{1/2}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)$.

Therefore, in (13) we have

$$\begin{array}{cc}\hfill f_{{\mathit{Y}}_1}\left({\mathit{y}}_1\right)& =\frac{{\displaystyle {\int }_0^{\infty }{\varphi }_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,u^{-1}{\mathsf{\Omega }}_{11})\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)\right)h(u;\nu )\mathrm{d}u}}{{\displaystyle {\int }_{R_1}{\int }_0^{\infty }{\varphi }_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,u^{-1}{\mathsf{\Omega }}_{11})\mathsf{\Phi }\left(\sqrt{u}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1)\right)h(u;\nu )\mathrm{d}u\mathrm{d}{\mathit{y}}_1}}\hfill \\ \hfill & =\frac{t_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,{\mathsf{\Omega }}_{11},\nu )T\left({\left[{\displaystyle \frac{\nu +p_1}{\nu +{\delta }_{{\mathsf{\Omega }}_{11}}({\mathit{y}}_1,{\mathsf{\xi }}_1)}}\right]}^{1/2}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1);\nu +p_1\right)}{{\displaystyle {\int }_{R_1}{t}_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,{\mathsf{\Omega }}_{11},\nu )T\left({\left[{\displaystyle \frac{\nu +p_1}{\nu +{\delta }_{{\mathsf{\Omega }}_{11}}({\mathit{y}}_1,{\mathsf{\xi }}_1)}}\right]}^{1/2}{\mathsf{\alpha }}_{1\left(2\right)}^{\prime }{\omega }_1^{-1}({\mathit{y}}_1-{\mathsf{\xi }}_1);\nu +p_1\right)\mathrm{d}{\mathit{y}}_1}},\hfill \end{array}$$

where the last line is obtained by using (12). This completes the proof. □

Corollary 2.

Let $\mathit{Y}\sim \mathrm{T}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\nu ;R)$. Consider the partitions given in (9) and (10). If $R_2={\mathbb{R}}^{p_2}$, then ${\mathit{Y}}_1\sim \mathrm{T}{\mathrm{T}}_{p_1}({\mathsf{\xi }}_1,{\mathsf{\Omega }}_{11},\nu ;R_1)$.

Proof. 

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

From Theorem 3.7 of Morán-Vásquez and Ferrari [8], we can establish that if ${\mathsf{\Omega }}_{12}=\mathbf{0}$, then the marginals of the truncated multivariate t distribution belong to the class of truncated elliptical distributions (but do not necessarily belong to the truncated multivariate t family). Corollary 2 allows us to establish closedness of the truncated multivariate t distribution for some marginals by restricting the support set without constraining the values of $\mathsf{\Omega }$.

In Definition 4, we present the truncated multivariate extended skew-t distribution, which is needed to derive the conditional distribution of ${\mathit{Y}}_2|{\mathit{Y}}_1$, where $\mathit{Y}={({\mathit{Y}}_1^{\prime },{\mathit{Y}}_2^{\prime })}^{\prime }$ has a truncated multivariate skew-t distribution. The truncated multivariate extended skew-t distribution is obtained by conditioning $\mathit{X}\sim \mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )$ (Definition 2) on $\{\mathit{X}\in A\}$, where $A\subseteq {\mathbb{R}}^p$ is a measurable set.

Definition 4.

Let $A\subseteq {\mathbb{R}}^p$ be a measurable set. The random vector $\mathit{Y}\in A$ has a truncated multivariate extended skew-t distribution with support A and parameters $\mathsf{\xi }\in {\mathbb{R}}^p$, $\mathsf{\Omega }(p\times p)>0$, $\mathsf{\alpha }\in {\mathbb{R}}^p$, extension parameter $\tau \in \mathbb{R}$ and $\nu >0$ degrees of freedom, denoted by $\mathit{Y}\sim \mathrm{T}\mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu ;A)$, if its PDF is

$$h_{\mathit{Y}}\left(\mathit{y}\right)=\frac{t_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}({\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })+\tau );\nu +p\right)}{{\displaystyle {\int }_A{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}({\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })+\tau );\nu +p\right)\mathrm{d}\mathit{y}}},$$

(14)

where $\mathit{y}\in A$.

The PDF of $\mathit{Y}\sim T\mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu ;A)$ can be expressed as

$$f_{\mathit{Y}}\left(\mathit{y}\right)=\frac{{\mathrm{EST}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )}{{\mathrm{EST}}_p(A;\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )},\mathit{y}\in A,$$

(15)

where $\mathrm{E}\mathrm{S}{\mathrm{T}}_p(A;\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )={\int }_A\mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )\mathrm{d}\mathit{y}$, with $\mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathit{x};\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )$ being the PDF of $\mathit{X}$, $\mathit{X}\sim \mathrm{E}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\tau ,\nu )$ (Definition 2).

For $A={\mathbb{R}}^p$ in (15), we have the PDF of a multivariate extended skew-t distribution given in (6). If $\tau =0$ in (14), we obtain the PDF of a truncated multivariate skew-t distribution given in (7). The truncated multivariate t distribution is a limiting case of the truncated multivariate extended skew-t distribution when $\tau \to \infty$. The truncated multivariate extended skew-normal distribution (Morán-Vásquez et al. [9]) is obtained from (14) when $\nu \to \infty$. If we take $\mathsf{\alpha }=\mathbf{0}$ in (14), we obtain a new truncated elliptical distribution with the PDF

$$h_{\mathit{Y}}\left(\mathit{y}\right)=\frac{t_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}\tau ;\nu +p\right)}{{\displaystyle {\int }_A{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}\tau ;\nu +p\right)\mathrm{d}\mathit{y}}},\mathit{y}\in A,$$

(16)

which is also an extension of the distribution presented in Equation (2) of Arellano-Valle and Genton [3]. Furthermore, the PDF of the truncated multivariate t distribution can also be obtained from (14) when $\mathsf{\alpha }=\mathbf{0}$ and $\tau =0$.

In Theorem 3, we derive the conditional distribution of ${\mathit{Y}}_2|{\mathit{Y}}_1$, when $\mathit{Y}={({\mathit{Y}}_1^{\prime },{\mathit{Y}}_2^{\prime })}^{\prime }$ has a truncated multivariate skew-t distribution.

Theorem 3.

Let $\mathit{Y}\sim \mathrm{T}\mathrm{S}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\mathsf{\alpha },\nu ;R)$. Consider the partitions given in (9) and (10). Then, ${\mathit{Y}}_2|{\mathit{Y}}_1={\mathit{y}}_1\sim \mathrm{T}\mathrm{E}\mathrm{S}{\mathrm{T}}_{p_2}({\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\mathsf{\alpha }}_{2·1},{\tau }_{2·1}^{*},{\nu }_{2·1};R_2)$, with ${\mathsf{\Omega }}_{22·1}^{*}={\lambda }_{{\delta }_1}{\mathsf{\Omega }}_{22·1}$, ${\lambda }_{{\delta }_1}=(\nu +{\delta }_{{\mathsf{\Omega }}_{11}}({\mathit{y}}_1,{\mathsf{\xi }}_1))/(\nu +p_1)$, ${\mathsf{\alpha }}_{2·1}={\omega }_{2·1}{\omega }_2^{-1}{\mathsf{\alpha }}_2$, with ${\omega }_{2·1}={({\mathsf{\Omega }}_{22·1}\odot {\mathit{I}}_{p_2})}^{1/2}$, ${\nu }_{2·1}=\nu +p_1$, and ${\tau }_{2·1}^{*}={\lambda }_{{\delta }_1}^{-1/2}{\tau }_{2·1}$, with ${\tau }_{2·1}=({\mathsf{\alpha }}_1^{\prime }{\omega }_1^{-1}+{\mathsf{\alpha }}_2^{\prime }{\omega }_2^{-1}{\mathsf{\Omega }}_{21}{\mathsf{\Omega }}_{11}^{-1})({\mathit{y}}_1-{\mathsf{\xi }}_1)$.

Proof. 

From (7) and (11), the conditional PDF of ${\mathit{Y}}_2|{\mathit{Y}}_1$ is calculated as

$$f_{{\mathit{Y}}_2|{\mathit{Y}}_1}\left({\mathit{y}}_2\right)=\frac{t_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right)}{{\displaystyle {\int }_{R_2}{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right)\mathrm{d}{\mathit{y}}_2}},{\mathit{y}}_2\in R_2.$$

Using the identities (Arellano-Valle and Genton [3] Equation (24))

$$t_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )=t_{p_1}({\mathit{y}}_1;{\mathsf{\xi }}_1,{\mathsf{\Omega }}_{11},\nu )t_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\nu }_{2·1})$$

and ${\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi })={\mathsf{\alpha }}_{2·1}^{\prime }{\omega }_{2·1}^{-1}({\mathit{y}}_2-{\mathsf{\xi }}_{2·1})+{\tau }_{2·1}$, ${\omega }_{2·1}={({\mathsf{\Omega }}_{22·1}\odot {\mathit{I}}_{p_2})}^{1/2}$, in the above expression we have

$$f_{{\mathit{Y}}_2|{\mathit{Y}}_1}\left({\mathit{y}}_2\right)=\frac{t_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\nu }_{2·1})T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}({\mathsf{\alpha }}_{2·1}^{\prime }{\omega }_{2·1}^{-1}({\mathit{y}}_2-{\mathsf{\xi }}_{2·1})+{\tau }_{2·1});\nu +p\right)}{{\displaystyle {\int }_{R_2}{t}_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\nu }_{2·1})T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}({\mathsf{\alpha }}_{2·1}^{\prime }{\omega }_{2·1}^{-1}({\mathit{y}}_2-{\mathsf{\xi }}_{2·1})+{\tau }_{2·1});\nu +p\right)\mathrm{d}{\mathit{y}}_2}},$$

where ${\mathit{y}}_2\in R_2$. Taking into account the equalities $\nu +p={\nu }_{2·1}+p_2$ and

$${\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}={\lambda }_{{\delta }_1}^{-1}\frac{{\nu }_{2·1}+p_2}{{\nu }_{2·1}+{\delta }_{{\mathsf{\Omega }}_{22·1}^{*}}({\mathit{y}}_2,{\mathsf{\xi }}_{2·1})},$$

in the above expression, we arrive at

$$f_{{\mathit{Y}}_2|{\mathit{Y}}_1}\left({\mathit{y}}_2\right)=\frac{t_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\nu }_{2·1})T\left({\left[{\displaystyle \frac{{\nu }_{2·1}+p_2}{{\nu }_{2·1}+{\delta }_{{\mathsf{\Omega }}_{22·1}^{*}}({\mathit{y}}_2,{\mathsf{\xi }}_{2·1})}}\right]}^{1/2}({\mathsf{\alpha }}_{2·1}^{\prime }{\omega }_{2·1}^{*-1}({\mathit{y}}_2-{\mathsf{\xi }}_{2·1})+{\tau }_{2·1}^{*});{\nu }_{2·1}+p_2\right)}{{\displaystyle {\int }_{R_2}{t}_{p_2}({\mathit{y}}_2;{\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\nu }_{2·1})T\left({\left[{\displaystyle \frac{{\nu }_{2·1}+p_2}{{\nu }_{2·1}+{\delta }_{{\mathsf{\Omega }}_{22·1}^{*}}({\mathit{y}}_2,{\mathsf{\xi }}_{2·1})}}\right]}^{1/2}({\mathsf{\alpha }}_{2·1}^{\prime }{\omega }_{2·1}^{*-1}({\mathit{y}}_2-{\mathsf{\xi }}_{2·1})+{\tau }_{2·1}^{*});{\nu }_{2·1}+p_2\right)\mathrm{d}{\mathit{y}}_2}},$$

where ${\mathit{y}}_2\in R_2$ and ${\omega }_{2·1}^{*}={\lambda }_{{\delta }_1}^{1/2}{\omega }_{2·1}={({\mathsf{\Omega }}_{22·1}^{*}\odot {\mathit{I}}_{p_2})}^{1/2}$. This completes the proof. □

Note that for ${\mathsf{\alpha }}_2=\mathbf{0}$ in Theorem 3, the conditional distribution of ${\mathit{Y}}_2|{\mathit{Y}}_1$ has the structure of the distribution with PDF given in (16).

Corollary 3.

Let $\mathit{Y}\sim \mathrm{T}{\mathrm{T}}_p(\mathsf{\xi },\mathsf{\Omega },\nu ;R)$. Consider the partitions given in (9) and (10). Then, ${\mathit{Y}}_2|{\mathit{Y}}_1={\mathit{Y}}_1\sim \mathrm{T}{\mathrm{T}}_{p_2}({\mathsf{\xi }}_{2·1},{\mathsf{\Omega }}_{22·1}^{*},{\nu }_{2·1};R_2)$.

Proof. 

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

4. Random Vector Generation

Simulations of truncated multivariate distributions have been widely used in various fields of statistics. This section deals with the random vector generation from the truncated multivariate skew-t distribution. Our proposal is based on the rejection sampling method, which can be extended to unnormalized PDFs avoiding the computation of normalizing constants (Maatouk and Bay [28] Prop. 1).

Let $A\subseteq {\mathbb{R}}^p$ be a measurable set. Define the functions

$$\tilde{f}\left(\mathit{y}\right)=t_p(\mathit{Y};\mathsf{\xi },\mathsf{\Omega },\nu )T\left({\left[{\displaystyle \frac{\nu +p}{\nu +{\delta }_{\mathsf{\Omega }}(\mathit{y},\mathsf{\xi })}}\right]}^{1/2}{\mathsf{\alpha }}^{\prime }{\omega }^{-1}(\mathit{y}-\mathsf{\xi });\nu +p\right),\mathit{y}\in A,$$

(17)

and

$$\tilde{g}\left(\mathit{y}\right)=\frac{t_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )}{{\displaystyle {\int }_A{t}_p(\mathit{y};\mathsf{\xi },\mathsf{\Omega },\nu )\mathrm{d}\mathit{y}}},\mathit{y}\in A.$$

(18)

Note that (17) corresponds to the kernel of the PDF of the truncated multivariate skew-t distribution. The expression (18) is the PDF of the truncated multivariate t distribution with support A, which can be obtained from (7) when $\mathsf{\alpha }=\mathbf{0}$. It is straightforward to show that

$$\tilde{f}\left(\mathit{y}\right)\le \tilde{g}\left(\mathit{y}\right),\mathit{y}\in A.$$

A study on the generation of random samples from the truncated multivariate t distribution appear in Geweke [29]. In a more general approach, Morán-Vásquez and Ferrari [7] proposed an algorithm based on Gibbs sampling for the generation of random vectors of the truncated elliptical distributions with rectangular support.

The generation of a random vector from the truncated multivariate skew-t distribution is carried out through the following steps:

(1)

Generate $\mathit{y}$ from a truncated multivariate t distribution with PDF $\tilde{g}$ given in (18).

(2)

Generate u from a uniform distribution on $[0,1]$. If $u\tilde{g}\left(\mathit{y}\right)\le \tilde{f}\left(\mathit{y}\right)$, accept $\mathit{y}$; otherwise, go back to step 1.

Figure 2 displays scatter plots of random samples of size 2000 generated from truncated bivariate skew-t distributions with rectangular supports. These scatter plots are overlaid with contours of the PDF (7) for different levels, namely $f_{\mathit{Y}}\left(\mathit{y}\right)=k$, where $k=0.18,0.1,0.03,0.008,0.002$. The random sample of Figure 2a corresponds to a truncated bivariate t distribution with heavy tails ($\mathsf{\alpha }=\mathbf{0}$ and small $\nu$), and that of Figure 2b corresponds to a truncated bivariate normal distribution ($\mathsf{\alpha }=\mathbf{0}$ and large $\nu$). Figure 2c–e show skewed and heavy-tailed random samples generated from a truncated bivariate skew-t distribution with $\mathsf{\alpha }\ne \mathbf{0}$ and small $\nu$. The random sample showed in Figure 2f was generated with the same parameter values as in Figure 2e, except $\nu$, which is large; so we have a random sample from a truncated bivariate skew-normal distribution. It is noteworthy that the random samples generated from the truncated bivariate skew-t distributions with small $\nu$ exhibit outliers in relation to those generated with large $\nu$ (Figure 2a,b,e,f). Multivariate skewness is also evident in the random samples generated from the truncated bivariate skew-t distributions with $\mathsf{\alpha }\ne \mathbf{0}$ (Figure 2c–f). The plots suggest that the methodology to generate random vectors of the truncated multivariate skew-t distribution seems to be adequate.

Figure 3 presents the histograms of the random samples from the marginal $Y_1$ (Figure 3a) and the conditional $Y_2|Y_1=-4.7$ (Figure 3b) obtained from the bivariate random sample displayed in Figure 2e. These histograms are overlaid with the marginal PDFs given in (11) and the one established according to Theorem 3, respectively. Since the support of the bivariate skew-t random sample shown in Figure 2e is $R=(-8,-2)\times (-2,4)$, Theorem 2 is not applicable, and therefore, the random sample of $Y_1$ does not come from a truncated univariate skew-t distribution, but it does come from a distribution with PDF given by (11). On the other hand, the random sample of $Y_2$ given that $Y_1=-4.7$ has size 70 and comes from the truncated univariate extended skew-t distribution, $Y_2|Y_1=-4.7\sim \mathrm{T}\mathrm{E}\mathrm{S}{\mathrm{T}}_1(-1.07,0.87,2.84,0.24,5;(-2,4))$. The plots presented in Figure 3 suggest a good performance of the random samples obtained from $Y_1$ and $Y_2|Y_1=-4.7$.

5. Conclusions and Future work

In this paper, we showed the property of closedness under affine transformations of random vectors having a truncated multivariate skew-t distribution. We also provided conditions for the truncated multivariate skew-t family to be preserved under marginalization. Additionally, we derived the conditional distribution of subvectors of a random vector having truncated multivariate skew-t distribution. On the other hand, we described a procedure based on the rejection sampling method to generate random vectors from the truncated multivariate skew-t distribution. This procedure can be useful for performing simulation studies on statistical models based on this distribution.

The implementation of maximum likelihood estimation for the truncated multivariate skew-t distribution may be challenging since an efficient computation of the integral involved in (7) is required. This topic will be addressed in a future paper as well as alternative methods for the generation of random vectors, simulation studies, and applications to real data. Another interesting issue is to test the hypothesis that a dataset has a truncated multivariate skew-t distribution, which may be achieved by following the ideas of Avdović and Jevremović [30] and Opheim and Roy [31]. In addition, it is interesting to study the properties derived in this paper in more general truncated distributions, such as the truncated skew-elliptical distributions, which can be defined through the skew-elliptical distributions (Azzalini [27] Ch. 6).