Distributions of powers of the central beta matrix variates and applications. (English) Zbl 1458.62109
Summary: We consider the central Beta matrix variates of both kinds, and establish the expressions of the densities of integral powers of these variates, for all their three types of distributions encountered in the statistical literature: entries, determinant, and latent roots distributions. Applications and computation of credible intervals are presented.
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62H12 | Estimation in multivariate analysis |
| 62E15 | Exact distribution theory in statistics |
| 62G15 | Nonparametric tolerance and confidence regions |
Software:
AS 308References:
| [1] | Bekker, A.; Roux, JJ; Pham-Gia, T., The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables, Stat Pap, 50, 2, 323-338 (2009) · Zbl 1309.62029 · doi:10.1007/s00362-007-0083-2 |
| [2] | Gupta, AK; Nagar, DK, Matrix variate distributions (2000), New York: Chapman and Hall/CRC, New York · Zbl 0935.62064 |
| [3] | Lawley, D., A generalization of Fisher’s \(z\) test, Biometrika, 30, 1-2, 180-187 (1938) · Zbl 0019.12904 · doi:10.2307/2332232 |
| [4] | Mathai, AM, Extensions of Wilks’ integral equations and distributions of test statistics, Ann Inst Stat Math, 36, 2, 271-288 (1984) · Zbl 0559.62039 · doi:10.1007/BF02481970 |
| [5] | Mathai, AM, Jacobians of matrix transformations and functions of matrix arguments (1997), Singapore: World Scientific Publishing Company, Singapore · Zbl 0889.33001 |
| [6] | Mathai, AM; Saxena, R.; Haubold, H., The \(H\)-function: theory and applications (2010), New York: Springer, New York · Zbl 1181.33001 |
| [7] | McDonald, JB; Xu, YJ, A generalization of the beta distribution with applications, J Econom, 66, 1-2, 133-152 (1995) · Zbl 0813.62011 · doi:10.1016/0304-4076(94)01612-4 |
| [8] | Muirhead, RJ, Aspects of multivariate statistical theory (1982), New York: Wiley, New York · Zbl 0556.62028 |
| [9] | Nadarajah, S.; Kotz, S., A generalized beta distribution, Math Sci, 29, 36-41 (2004) · Zbl 1049.62011 |
| [10] | Pauw, J.; Bekker, A.; Roux, JJ, Densities of composite Weibullized generalized gamma variables: theory and methods, South Afr Stat J, 44, 1, 17-42 (2010) · Zbl 1397.62073 |
| [11] | Pham-Gia, T., Exact distribution of the generalized Wilks’s statistic and applications, J Multivar Anal, 99, 8, 1698-1716 (2008) · Zbl 1144.62320 · doi:10.1016/j.jmva.2008.01.021 |
| [12] | Pham-Gia, T.; Thanh, DN, Hypergeometric functions: From one scalar variable to several matrix arguments, in statistics and beyond, Open J Stat, 6, 5, 951-994 (2016) · doi:10.4236/ojs.2016.65078 |
| [13] | Pham-Gia, T.; Turkkan, N., Distributions of ratios: from random variables to random matrices, Open J Stat, 1, 2, 93-104 (2011) · doi:10.4236/ojs.2011.12011 |
| [14] | Pham-Gia, T.; Turkkan, N., Testing the equality of several covariance matrices, J Stat Comput Simul, 81, 10, 1233-1246 (2011) · Zbl 1431.62239 · doi:10.1080/00949655.2010.481790 |
| [15] | Pillai K (1954) On some distribution problems in multivariate analysis. Tech. Rep., North Carolina State University. Dept. of Statistics |
| [16] | Pillai, K., Some new test criteria in multivariate analysis, Ann Math Stat, 26, 1, 117-121 (1955) · Zbl 0064.13801 · doi:10.1214/aoms/1177728599 |
| [17] | Rencher, A.; Christensen, W., Methods of multivariate analysis (2012), New York: Wiley, New York · Zbl 1275.62011 |
| [18] | Roy, SN, On a heuristic method of test construction and its use in multivariate analysis, Ann Math Stat, 24, 2, 220-238 (1953) · Zbl 0051.36701 · doi:10.1214/aoms/1177729029 |
| [19] | Turkkan, N.; Pham-Gia, T., Computation of the highest posterior density interval in Bayesian analysis, J Stat Comput Simul, 44, 3-4, 243-250 (1993) · doi:10.1080/00949659308811461 |
| [20] | Turkkan, N.; Pham-Gia, T., Algorithm as 308: highest posterior density credible region and minimum area confidence region: the bivariate case, J R Stat Soc Ser C (Appl Stat), 46, 1, 131-140 (1997) · Zbl 0904.62040 · doi:10.1111/1467-9876.00053 |
| [21] | Wilks, S., Certain generalizations in the analysis of variance, Biometrika, 24, 3-4, 471-494 (1932) · JFM 58.1172.02 · doi:10.2307/2331979 |
| [22] | Wishart, J., The generalised product moment distribution in samples from a normal multivariate population, Biometrika, 20, 1-2, 32-52 (1928) · JFM 54.0565.02 · doi:10.1093/biomet/20A.1-2.32 |
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