Multivariate Exponential-type Distributions

Abstract

Let $\mathbf{x}$ and $\mathbf{\theta}$ denote $s$-dimensional column vectors. The components $x_1, x_2,\cdots x_s$ of $\mathbf{x}$ are random variables jointly following an $s$-variate distribution and components $\theta_1, \theta_2,\cdots, \theta_s$ of $\mathbf{\theta}$ are real numbers. The random vector $\mathbf{x}$ is said to follow an $s$-variate Exponential-type distribution with the parameter vector (pv) $\mathbf{\theta}$, if its probability function (pf) is given by \begin{equation*}\tag{1.1} f(\mathbf{x}, \mathbf{\theta}) = h(\mathbf{x}) \exp \{\mathbf{x'\theta} - q(\mathbf{\theta})\},\end{equation*} $\mathbf{x} \varepsilon R_s$ and $\mathbf{\theta} \varepsilon (\mathbf{a}, \mathbf{b}) \subset R_s. R_s$ denotes the $s$-dimensional Euclidean space. The $s$-dimensional open interval $(\mathbf{a}, \mathbf{b})$ may or may not be finite. $h(\mathbf{x})$ is a function of $\mathbf{x}$, independent of $\mathbf{\theta}$, and $q(\mathbf{\theta})$ is a bounded analytic function of $\theta_1, \theta_2,\cdots \theta_s$, independent of $\mathbf{x}$. We note that $f(\mathbf{x}, \mathbf{\theta})$, given by (1.1), defines the class of multivariate exponential-type distributions which includes distributions like multivariate normal, multinomial, multivariate negative binomial, multivariate logarithmic series, etc. This paper presents a theoretical study of the structural properties of the class of multivariate exponential-type distributions. For example, different distributions connected with a multivariate exponential-type distribution are derived. Statistical independence of the components $x_1, x_2,\cdots, x_s$ is discussed. The problem of characterization of different distributions in the class is studied under suitable restrictions on the cumulants. A canonical representation of the characteristic function of an infinitely divisible (id), purely discrete random vector, whose moments of second order are all finite, is also obtained. $\varphi(\mathbf{t}), m(\mathbf{t}), k(\mathbf{t})$ denote, throughout this paper, the characteristic function (ch. f.), the moment generating function (mgf), and the cumulant generating function (cgf), respectively, of a random vector $\mathbf{x}$. The components $t_i$ of the $s$-dimensional column vector $\mathbf{t}$ are all real.