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generalized $p$-value; inversion formula; Behrens-Fisher problem
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.
[1] Abramowitz M., Stegun I. A.: Handbook of Mathematical Functions. Dover, New York 1965 Zbl 0643.33002
[2] Amos D. E.: A portable package for bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software 12 (1986), 265–273 DOI 10.1145/7921.214331 | MR 0889069 | Zbl 0613.65013
[3] Davies R. B.: Numerical inversion of a characteristic function. Biometrika 60 (1973), 415–417 DOI 10.1093/biomet/60.2.415 | MR 0321152 | Zbl 0263.65115
[4] Gil–Pelaez J.: Note on the inversion theorem. Biometrika 38 (1951), 481–482 DOI 10.1093/biomet/38.3-4.481 | MR 0045992 | Zbl 0045.07204
[5] Imhof J. P.: Computing the distribution of quadratic forms in normal variables. Biometrika 48 (1961), 419–426 DOI 10.1093/biomet/48.3-4.419 | MR 0137199 | Zbl 0136.41103
[6] Khuri A. I., Mathew, T., Sinha B. K.: Statistical Tests for Mixed Linear Models. Wiley, New York 1998 MR 1601351 | Zbl 0893.62009
[7] Prudnikov A. P., Brychkov Y. A., Marichev O. I.: Integraly i Rjady. (Integrals and Series). Nauka, Moscow 1981 MR 0635931
[8] Tsui K. W., Weerahandi S.: Generalized $p$ values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 (1989), 602–607 MR 1010352
[9] Waller L. A., Turnbull B. W., Hardin J. M.: Obtaining distribution functions by numerical inversion of characteristic functions with applications. Amer. Statist. 49 (1995), 4, 346–350
[10] Weerahandi S.: Testing variance components in mixed models with generalized $p$ values. J. Amer. Statist. Assoc. 86 (1991), 151–153
[11] Weerahandi S.: Exact Statistical Methods for Data Analysis. Springer–Verlag, New York 1995 MR 1316663 | Zbl 1050.62003
[12] Witkovský V.: On the exact computation of the density and of the quantiles of linear combinations of $t$ and $F$ random variables. J. Statist. Plann. Inference 94 (2001), 1, 1–13 DOI 10.1016/S0378-3758(00)00208-1 | MR 1820167 | Zbl 0971.62012
[13] Zhou L., Mathew T.: Some tests for variance components using generalized $P$-values. Technometrics 36 (1994), 394–402 MR 1304900 | Zbl 0825.62603
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