Exact distributions of two non-central generalized Wilks's statistics

References

• Muirhead RJ. Aspects of multivariate statistical theory. New York (NY): Willey; 1982.
   Google Scholar
• Wilks SS. Certain generalizations in the analysis of variance. Biometrika. 1932;24(3-4):471–494. doi: 10.1093/biomet/24.3-4.471
   Google Scholar
• Hsu PL. On generalized analysis of variance (I). Biometrika. 1940;31(3-4):221–237. doi: 10.1093/biomet/31.3-4.221
   Google Scholar
• Pearson ES, Wilks SS. Methods of statistical analysis appropriate for k samples of two variables. Biometrika. 1933;25(3-4):353–378. doi: 10.1093/biomet/25.3-4.353
   Google Scholar
• Ito K. A comparison of the powers of two multivariate analysis of variance tests. Biometrika. 1962;49(3-4):455–462. doi: 10.1093/biomet/49.3-4.455
   Web of Science ®Google Scholar
• Pillai KCS, Nagarsenker BN. On the distributions of a class of statistics in multivariate analysis. J Multivar Anal. 1972;2(1):96–114. doi: 10.1016/0047-259X(72)90012-7
   Google Scholar
• Fujikoshi Y. Asymptotic formulas for the distributions of the determinant and the trace of a noncentral beta matrix. J Multivar Anal. 1972;2:208–218. doi: 10.1016/0047-259X(72)90028-0
   Google Scholar
• Mathai AM, Rathie PN. The exact distribution of Wilks' criterion. The Annals of Mathematical Statistics. 1971;42(3):1010–1019. doi: 10.1214/aoms/1177693329
   Google Scholar
• Pham-Gia T. Exact distribution of the generalized Wilks's statistic and applications. J Multivar Anal. 2008;99(8):1698–1716. doi: 10.1016/j.jmva.2008.01.021
   Web of Science ®Google Scholar
• Bekker A, Roux JJJ, Arashi M. Exact nonnull distribution of Wilks' statistic: the ratio and product of independent components. J Multivar Anal. 2011;102(3):619–628. doi: 10.1016/j.jmva.2010.11.005
   Web of Science ®Google Scholar
• Koev P, Edelman A. The efficient evaluation of the hypergeometric function of a matrix argument. Math Comput. 2006;75(254):833–846. doi: 10.1090/S0025-5718-06-01824-2
   Web of Science ®Google Scholar
• Johnstone IM, Nadler B. Roy's largest root test under rank-one alternatives. Biometrika. 2017;104(1):181–193.
   PubMed Web of Science ®Google Scholar
• Rencher AC, Christensen WF. Methods of multivariate analysis. 3rd ed. New York: Wiley; 2012.
   Google Scholar
• Roy J. Non-null distribution of the likelihood-ratio in analysis of dispersion. Sankhya: Indian J Stat. 1960;22(3-4):289–294.
   Google Scholar
• Posten HO, Bargmann RE. Power of the likelihood-ratio test of the general linear hypothesis in multivariate analysis. Biometrika. 1964;51(3-4):467–480. doi: 10.1093/biomet/51.3-4.467
   Web of Science ®Google Scholar
• Gupta AK. Noncentral distribution of Wilks' statistic in Manova. Ann Math Stat. 1971;42(4):1254–1261. doi: 10.1214/aoms/1177693238
   Google Scholar
• Hart ML, Money AH. Exact powers for the linear case of Wilks' likelihood ratio criterion. S Afr Stat J. 1975;9(1):11–26.
   Web of Science ®Google Scholar
• Hart ML, Money AH. On Wilks's multivariate generalization of the correlation ratio. Biometrika. 1976;63(1):59–67.
   Web of Science ®Google Scholar
• Pillai KCS, Jayachandran K. Power comparisons of tests of two multivariate hypotheses based on four criteria. Biometrika. 1967;54(1-2):195–210. doi: 10.1093/biomet/54.1-2.195
   PubMedGoogle Scholar
• James AT. Zonal polynomials of the real positive definite symmetric matrices. Ann Math. 1961;74(3):456–469. doi: 10.2307/1970291
   Web of Science ®Google Scholar
• Mathai AM, Saxena RK, Haubold HJ. The H-function: theory and applications. New York (NY): Springer-Verlag; 2010.
   Google Scholar
• Hua LK. Harmonic analysis of functions of several complex variables in the classical domains. Moscow: American Mathematical Society; 1963. English translation.
   Google Scholar
• James AT. The distribution of noncentral means with known covariance. The Annals of Mathematical Statistics. 1961;32:3874–882. doi: 10.1214/aoms/1177704980
   Google Scholar
• James AT. Calculation of zonal polynomial coefficients by use of the Laplace-Beltrami operator. The Annals of Mathematical Statistics. 1968;39(5):1711–1718. doi: 10.1214/aoms/1177698153
   Google Scholar
• Takemura A. Zonal Polynomials. Lecture Notes-Monograph Series. Hayward (CA): Institute of Mathematical Statistics; 1984.
   Google Scholar
• James AT. Distributions of matrix variates and latent roots derived from normal samples. Ann Math Stat. 1964;35(2):475–501. doi: 10.1214/aoms/1177703550
   Google Scholar
• Parkhurst AM, James AT. Zonal polynomial of order 1 through 12. In: H. L. Harter, D. B. Owen, editors. Selected tables in Mathematical statistics. Vol.2. The American Mathematical Society; 1974.
   Google Scholar
• McLaren ML. Algorithm AS 94: coefficients of the zonal polynomials.Journal of the Royal Statistical Society. Series C. 1976;25(1):82–87.
   Google Scholar
• Gutie'rrez R, Rodriguez J, Sa'ez AJ. Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials. Electron Trans Numer Anal. 2000;11:121–130.
   Google Scholar
• Springer M. The algebra of random variables. NewYork (NY): Wiley; 1979.
   Google Scholar
• Gupta AK, Nagar DK. Matrix variate distributions. New York (NY): Chapman and Hall/CRC; 2000.
   Google Scholar
• Nair US. The application of the moment function in the study of distribution laws in statistics. Biometrika. 1939;30(3-4):274–294. doi: 10.1093/biomet/30.3-4.274
   Google Scholar
• Muller KE, Peterson BL. Practical methods for computing power in testing the multivariate general linear hypothesis. Comput Stat Data Anal. 1984;2:143–158. doi: 10.1016/0167-9473(84)90002-1
   Web of Science ®Google Scholar
• Muller KE, LaVange LM, Ramey SL, Ramey CT. Power calculations for general linear multivariate models including repeated measures applications. J Am Stat Assoc. 1992;87(420):1209–1226. doi: 10.1080/01621459.1992.10476281
   PubMed Web of Science ®Google Scholar