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Linear combination and product of inverted Dirichlet components. (English) Zbl 1149.62304

Summary: It is well known that \(X+Y\) has the \(F\) distribution and \(X/(X+Y)\) has the beta distribution when \(X\) and \(Y\) follow the inverted Dirichlet distribution. We derive the exact distributions of \(S= \alpha X+\beta Y\) and \(P= XY\) (involving the Gauss hypergeometric function) and the corresponding moment properties. We also propose approximations for \(S\) and \(P\) and discuss evidence of their robustness based on the powerful Kolmogorov-Smirnov test.

MSC:

62E15 Exact distribution theory in statistics
62G10 Nonparametric hypothesis testing
33C90 Applications of hypergeometric functions
62G35 Nonparametric robustness
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