Symmetric multivariate and related distributions
Since the publication of the by now classical Johnson and Kotz Continuous Multivariate Distributions (Wiley, 1972) there have been substantial developments in multivariate distribution theory especially in the area of non-normal symmetric multivariate distributions. The book by Fang, Kotz and Ng summarizes these developments in a manner which is accessible to a reader with only limited background (advanced real-analysis calculus, linear algebra and elementary matrix calculus). Many of the results in this field are due to Kai-Tai Fang and his associates and appeared in Chinese publications only.
A thorough literature search was conducted and the book represents the latest work - as of 1988 - in this rapidly developing field of multivariate distributions. The authors are experts in statistical distribution theory.
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Spherically and elliptically symmetric distributions
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a-symmetric Anderson and Fang Assume Bentler beta distribution bution Cambanis Cauchy distribution chapter characteristic function characterizations components conditional distribution Corollary decomposition defined Definition degrees of freedom denoted density function density generator g derived Dirichlet distribution Dirichlet parameters discussion distri distribution function distribution of x(1 elliptically symmetric distributions equivalent Example exists exponential distributions following result following theorem formula gamma distribution given x(2 Gupta and Richards Hence implies independently distributed integral inverted beta Jacobian joint density Lemma marginal density marginal distributions matrix mixtures of normal multivariate normal distribution multivariate Pearson Type mutually independent n x 1 random n-dimensional version normal distribution obtain orthogonal partitioned possesses a density Problem proof Let proof of Theorem properties random variable random vector Section spherical distribution stochastic representation subclasses subvectors survival function symmetric multivariate distributions symmetric multivariate Pearson symmetric stable law transformation uniform distribution unit sphere variate