Multivariate Exponential-type Distributions

Abstract

Let x and θ denote s-dimensional column vectors. The components x1, x2,⋯ xs of x are random variables jointly following an s-variate distribution and components θ1, θ2,⋯, θs of θ are real numbers. The random vector x is said to follow an s-variate Exponential-type distribution with the parameter vector (pv) θ, 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*} x ε Rs and $\mathbf{\theta} \varepsilon (\mathbf{a}, \mathbf{b}) \subset R_s. R_s$ denotes the s-dimensional Euclidean space. The s-dimensional open interval (a, b) may or may not be finite. h(x) is a function of x, independent of θ, and q(θ) is a bounded analytic function of θ1, θ2,⋯ θs, independent of x. We note that f(x, θ), 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 x1, x2,⋯, xs 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. φ(t), m(t), k(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 x. The components ti of the s-dimensional column vector t are all real.