Scale and shape mixtures of matrix variate extended skew normal distributions

Abstract

In this paper, we propose a matrix extension of the scale and shape mixtures of multivariate skew normal distributions and present some particular cases of this new class. We also present several formal properties of this class, such as the marginal distributions, the moment generating function, the distribution of linear and quadratic forms, and the selection and stochastic representations. In addition, we introduce the matrix variate tail conditional expectation measure and derive this risk measure for the scale and shape mixtures of matrix variate extended skew normal distributions. We present an efficient EM-type algorithm for the computation of maximum likelihood estimates of parameters in some special cases of the proposed class. Finally, we conduct a small simulation study and fit various special cases of the new class to a real dataset.

Introduction

Matrix variate distributions have a central role in multivariate analysis, and have been useful in many fields such as psychology, physics and economics. This type of distributions have proven useful for describing and modeling repeated measurements in multivariate variables. In this context, the observations are not necessarily independent. As an example, consider $n$ blood tests for $k$ individuals where in each test, $p$ variables are measured. In this situation, a random matrix $\mathit{X}$ of size $p\times n$ is then observed for each individual. In most cases, however, the focus has been focused on matrix variate elliptical distributions, particularly on the matrix variate normal distribution. In the books of Fang and Zhang [11], Gupta and Nagar [15] and Gupta et al. [17] we can find a comprehensive collection of the most important results on matrix variate elliptical distributions.

The matrix variate normal distribution is one of the most well-known matrix variate elliptical distribution. For example, its quadratic forms are used in the analysis of the multivariate linear models. A random matrix $\mathit{X}$ of size $p\times n$ is said to have a matrix variate normal distribution with parameters $\mathit{M}$, $\mathbf{\Sigma }$ and $\mathbf{\Psi }$ and is denoted by $\mathit{X}\sim {\mathcal{N}}_{p\times n}(\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma })$, if its probability density function (pdf) is given by

$${\varphi }_{p\times n}(\mathit{X};\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma })={\left(2\pi \right)}^{-\frac{np}{2}}{\left|\mathbf{\Psi }\right|}^{-\frac{p}{2}}{\left|\mathbf{\Sigma }\right|}^{-\frac{n}{2}}\mathrm{etr}\left\{-\frac{1}{2}{\mathbf{\Psi }}^{-1}{(\mathit{X}-\mathit{M})}^{\top }{\mathbf{\Sigma }}^{-1}(\mathit{X}-\mathit{M})\right\},$$

where $\mathrm{etr}\left\{\mathit{A}\right\}=exp\left\{\mathrm{tr}\left(\mathit{A}\right)\right\}$. Here, $\mathit{M}$ is a matrix of size $p\times n$ corresponding to the mean of $\mathit{X}$, while $\mathbf{\Sigma }$ and ${\mathbf{\Psi }}_{n\times n}$ are positive definite matrices of size $p\times p$ and $n\times n$, respectively, such that $\mathbf{\Psi }\otimes \mathbf{\Sigma }$ is the covariance matrix of vec$\left(\mathit{X}\right)$, where vec$\left(\cdot \right)$ denotes the vectorization operator.

Matrix variate elliptical distributions, however, cannot take into account the skewness present in the data, which can lead to incorrect conclusions in our analysis. For this reason, in recent years there has been a growing interest in building more flexible matrix variate distributions so that they can adjust the skewness of the data. For instance, a first skew version of the matrix variate normal distribution is introduced by Chen and Gupta [6]. They proposed properties of matrix variate skew normal distribution. A few years later, Domínguez-Molina et al. [10] introduced the matrix variate closed skew normal distribution and gave two constructions for it; and Harrar and Gupta [18] generalized the results of Chen and Gupta [6] and provided a stochastic representation for the matrix variate skew normal distribution. More recently, Zheng et al. [24] studied the inverse problem of matrix variate skew normal distributions; Zheng et al. [25] obtained moments and quadratic forms of matrix variate skew normal distributions by using the moment generating function of the matrix variate closed skew normal distribution. In the same way, Ning and Gupta [21] introduced the matrix variate extended skew normal (ESN) distribution based on the result of Harrar and Gupta [18]. According to these authors, a random matrix $\mathit{X}$ of size $p\times n$ is said to have a ESN distribution, denoted by $\mathit{X}\sim {\mathcal{ESN}}_{p\times n}(\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma },\mathbf{\Omega },\mathit{\lambda },\mathit{\delta })$, if its pdf is given by

$$f_{ESN}(\mathit{X};\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma },\mathbf{\Omega },\mathit{\lambda },\mathit{\delta })=\frac{1}{{\Phi }_n(\mathit{\delta };\mathbf{\Omega }+{\mathit{\lambda }}^{\top }\mathit{\lambda }\mathbf{\Psi })}{\varphi }_{p\times n}(\mathit{X};\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma }){\Phi }_n\left(\mathit{\delta }+{(\mathit{X}-\mathit{M})}^{\top }{\mathbf{\Sigma }}^{-\frac{1}{2}}\mathit{\lambda };\mathbf{\Omega }\right),$$

where $\mathit{M}$ is a $p\times n$ location matrix, $\mathbf{\Sigma }$ is a $p\times p$ positive definite matrix, $\mathbf{\Omega }$ and $\mathbf{\Psi }$ are $n\times n$ positive definite matrices, ${\varphi }_{p\times n}(.;\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma })$ is the pdf of ${\mathcal{N}}_{p\times n}(\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma })$ distribution, $\mathit{\lambda }\in {\mathbb{R}}^p$, $\mathit{\delta }\in {\mathbb{R}}^n$ and ${\Phi }_n(.;\mathbf{\Omega })$ is the cumulative distribution function (cdf) of the $n$-variate normal distribution with mean vector and covariance matrix $\mathbf{0}$ and $\mathbf{\Omega }$, respectively. The moment generating function (mgf) of ${\mathcal{ESN}}_{p\times n}(\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma },\mathbf{\Omega },\mathit{\lambda },\mathit{\delta })$ is

$${\mathcal{M}}_{ESN}(\mathit{T};\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma },\mathbf{\Omega },\mathit{\lambda },\mathit{\delta })=\frac{\mathrm{etr}\left\{{\mathit{T}}^{\top }\mathit{M}+\frac{1}{2}\mathbf{\Sigma }\mathit{T}\mathbf{\Psi }{\mathit{T}}^{\top }\right\}}{{\Phi }_n(\mathit{\delta };\mathbf{\Omega }+{\mathit{\lambda }}^{\top }\mathit{\lambda }\mathbf{\Psi })}{\Phi }_n\left(\mathit{\delta }+\mathbf{\Psi }{\mathit{T}}^{\top }{\mathbf{\Sigma }}^{\frac{1}{2}}\mathit{\lambda };\mathbf{\Omega }+{\mathit{\lambda }}^{\top }\mathit{\lambda }\mathbf{\Psi }\right),$$

where $\mathit{T}\in {\mathbb{R}}^{p\times n}$.

On the other hand, there is relatively little work for mixtures of matrix variate distributions. For example, Díaz-García and Gutiérrez-Jáimez [8] proposed the class of scale mixture of matrix variate Kotz-type distributions; Doğru et al. [9] considered finite mixtures of matrix variate $t$ distributions; Gallaugher and McNicholas [13] used some finite mixtures of skewed matrix variate distributions for clustering; Gallaugher and McNicholas [12], [14] developed a total of four skewed matrix variate distributions based on a matrix normal variance-mean mixture. In same way, Gupta and Varga [16] proposed the scale mixture of matrix variate normal (SMN) distributions and provided its pdf as follows:

$$g\left(\mathit{X}\right)={\int }_{S_J}{\varphi }_{p\times n}\left(\mathit{X};\mathit{M},\mathbf{\Psi }\otimes L\left(s\right)\mathbf{\Sigma }\right)dJ\left(s\right),$$

where ${\varphi }_{p\times n}\left(\cdot ;\mathit{M},\mathbf{\Psi }\otimes L\left(s\right)\mathbf{\Sigma }\right)$ is the pdf of ${\mathcal{N}}_{p\times n}\left(\mathit{M},\mathbf{\Psi }\otimes L\left(s\right)\mathbf{\Sigma }\right)$, $L$ is a weight function and $J$ is a distribution function on $(0,+\infty ]$. We denote it by ${\mathcal{SMN}}_{p\times n}\left(\mathit{M},\mathbf{\Psi }\otimes \mathbf{\Sigma };L,J\right)$.

The main purpose of this paper is to develop the theory of matrix variate flexible distributions by defining the scale and shape mixtures of ESN distributions (hereinafter SSMESN), presenting some particular cases and studying main properties of them. The remainder of this paper is laid out as follows. In Section 2, we give the definition of the SSMESN distributions and propose some subclasses such as scale mixtures of matrix variate ESN (hereinafter SMESN) distributions. Some basic properties are studied in Section 3. In Section 4, we discuss the different representations of the SSMESN distributions. The matrix variate tail conditional expectation (TCE) measure is defined and, also, it is derived for the SSMESN family in Section 5. In Section 6, we obtain maximum likelihood estimator of some special cases of the SSMESN distributions by using the EM algorithm. In Section 7, a small simulation study is presented for evaluating the performance of the maximum likelihood estimators. Finally, some matrix variate distributions in the SSMESN family are fitted to a real dataset in Section 8.