| Title:
|
Computing the distribution of a linear combination of inverted gamma variables (English) |
| Author:
|
Witkovský, Viktor |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
37 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
[79]-90 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
generalized $p$-value |
| Keyword:
|
inversion formula |
| Keyword:
|
Behrens-Fisher problem |
| MSC:
|
62E15 |
| MSC:
|
62J05 |
| MSC:
|
65C60 |
| MSC:
|
65D30 |
| idZBL:
|
Zbl 1263.62022 |
| idMR:
|
MR1825758 |
| . |
| Date available:
|
2009-09-24T19:37:18Z |
| Last updated:
|
2015-03-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135390 |
| . |
| Reference:
|
[1] Abramowitz M., Stegun I. A.: Handbook of Mathematical Functions.Dover, New York 1965 Zbl 0643.33002 |
| Reference:
|
[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 Zbl 0613.65013, MR 0889069, 10.1145/7921.214331 |
| Reference:
|
[3] Davies R. B.: Numerical inversion of a characteristic function.Biometrika 60 (1973), 415–417 Zbl 0263.65115, MR 0321152, 10.1093/biomet/60.2.415 |
| Reference:
|
[4] Gil–Pelaez J.: Note on the inversion theorem.Biometrika 38 (1951), 481–482 Zbl 0045.07204, MR 0045992, 10.1093/biomet/38.3-4.481 |
| Reference:
|
[5] Imhof J. P.: Computing the distribution of quadratic forms in normal variables.Biometrika 48 (1961), 419–426 Zbl 0136.41103, MR 0137199, 10.1093/biomet/48.3-4.419 |
| Reference:
|
[6] Khuri A. I., Mathew, T., Sinha B. K.: Statistical Tests for Mixed Linear Models.Wiley, New York 1998 Zbl 0893.62009, MR 1601351 |
| Reference:
|
[7] Prudnikov A. P., Brychkov Y. A., Marichev O. I.: Integraly i Rjady.(Integrals and Series). Nauka, Moscow 1981 MR 0635931 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[10] Weerahandi S.: Testing variance components in mixed models with generalized $p$ values.J. Amer. Statist. Assoc. 86 (1991), 151–153 |
| Reference:
|
[11] Weerahandi S.: Exact Statistical Methods for Data Analysis.Springer–Verlag, New York 1995 Zbl 1050.62003, MR 1316663 |
| Reference:
|
[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 Zbl 0971.62012, MR 1820167, 10.1016/S0378-3758(00)00208-1 |
| Reference:
|
[13] Zhou L., Mathew T.: Some tests for variance components using generalized $P$-values.Technometrics 36 (1994), 394–402 Zbl 0825.62603, MR 1304900 |
| . |
