# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 1 Introduction  
The multivariate skew-normal distribution ([PERSON] and [PERSON] [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 ([PERSON] and [PERSON] [2], [PERSON] and [PERSON] [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, [PERSON] and [PERSON] [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 [PERSON] and [PERSON] [4] and [PERSON] and [PERSON] [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 [PERSON] 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 ([PERSON] and [PERSON] [7, 8]) and the truncated multivariate skew-normal distribution ([PERSON] et al. [9], [PERSON] 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 [PERSON] and [PERSON] [7, 8],[PERSON] [11], [PERSON] et al. [12], [PERSON] and [PERSON] [13], [PERSON] [14], [PERSON] et al. [15], [PERSON] [16], [PERSON] [17; 18], [PERSON] [19; 20; 21], and [PERSON] and [PERSON] [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 [PERSON] and [PERSON] [23], [PERSON] and [PERSON] [7], and [PERSON] 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 ([PERSON] 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 [PERSON] et al. [9]. Additionally, applications of our results establish some new properties on the truncated multivariate \(t\) distribution. The paper is organized as follows.