Geodesic
Computing the distribution of a linear combination of inverted gamma variables
Kybernetika, Tome 37 (2001) no. 1, pp. 79-90 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A formula for evaluation of the distribution of a linear combination of independent inverted gamma random variables by one-dimensional numerical integration is presented. The formula is direct application of the inversion formula given by Gil–Pelaez [gil-pelaez]. This method is applied to computation of the generalized $p$-values used for exact significance testing and interval estimation of the parameter of interest in the Behrens–Fisher problem and for variance components in balanced mixed linear model.
A formula for evaluation of the distribution of a linear combination of independent inverted gamma random variables by one-dimensional numerical integration is presented. The formula is direct application of the inversion formula given by Gil–Pelaez [gil-pelaez]. This method is applied to computation of the generalized $p$-values used for exact significance testing and interval estimation of the parameter of interest in the Behrens–Fisher problem and for variance components in balanced mixed linear model.
Classification : 62E15, 62J05, 65C60, 65D30
Keywords: generalized $p$-value; inversion formula; Behrens-Fisher problem 
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     title = {Computing the distribution of a linear combination of inverted gamma variables},
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     zbl = {1263.62022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2001_37_1_a5/}
}
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Witkovský, Viktor. Computing the distribution of a linear combination of inverted gamma variables. Kybernetika, Tome 37 (2001) no. 1, pp. 79-90. http://geodesic.mathdoc.fr/item/KYB_2001_37_1_a5/

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