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- W1978648601 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." @default.
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- W1978648601 date "1968-08-01" @default.
- W1978648601 modified "2023-10-18" @default.
- W1978648601 title "Multivariate Exponential-type Distributions" @default.
- W1978648601 doi "https://doi.org/10.1214/aoms/1177698257" @default.
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