Wishart ratios with dependent structure: New members of the bimatrix beta type IV. (English) Zbl 1221.62081
Summary: In multivariate statistics under normality, the problems of interest are random covariance matrices (known as Wishart matrices) and “ratios” of Wishart matrices that arise in multivariate analysis of variance (MANOVA). The bimatrix variate beta type IV distribution (also known in the literature as bimatrix variate generalised beta; matrix variate generalization of a bivariate beta type I) arises from “ratios” of Wishart matrices. We add a further independent Wishart random variate to the “denominator” of one of the ratios; this results in deriving the exact expression for the density function of the bimatrix variate extended beta type IV distribution. The latter leads to the proposal of the bimatrix variate extended F distribution. Some interesting characteristics of these newly introduced bimatrix distributions are explored. Lastly, we focus on the bivariate extended beta type IV distribution (that is an extension of bivariate Jones’ beta) with emphasis on \(P(X_{1}<X_{2})\) where \(X_{1}\) is the random stress variate and \(X_{2}\) is the random strength variate.
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 33C90 | Applications of hypergeometric functions |
Keywords:
bimatrix variate Kummer extended beta type IV distribution; generalized Laguerre polynomial; hypergeometric function of matrix argument; invariant polynomials; Laplace transform; Meijer’s G-function; moment generating function; stress-strengthReferences:
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