Abstract
Computational expressions for the exact CDF of Roy’s test statistic in MANOVA and the largest eigenvalue of a Wishart matrix are derived based upon their Pfaffian representations given in Gupta and Richards (SIAM J. Math. Anal. 16:852–858, 1985). These expressions allow computations to proceed until a prespecified degree of accuracy is achieved. For both distributions, convergence acceleration methods are used to compute CDF values which achieve reasonably fast run times for dimensions up to 50 and error degrees of freedom as large as 100. Software that implements these computations is described and has been made available on the Web.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 9th edn. Dover, New York (1970)
Aitken, A.: On Bernoulli’s numerical solution of algebraic equations. Proc. R. Soc. Edinb. 46, 289–305 (1926)
Butler, R.W.: Saddlepoint Approximations with Applications. Cambridge University Press, Cambridge (2007)
Butler, R.W., Wood, A.T.A.: Laplace approximation for hypergeometric functions of matrix argument. Ann. Stat. 30, 1155–1177 (2002)
Butler, R.W., Wood, A.T.A.: Approximation of power in multivariate analysis. Stat. Comput. 15, 281–287 (2005)
Chapra, S.C.: Applied Numerical Methods with MATLAB for Engineers and Scientists. McGraw-Hill, New York (2004)
Clark, W.D., Gray, H.L., Adams, J.E.: A note on the T-transformation of Lubkin. J. Res. Natl. Bur. Stand. B Math. Sci. 73, 25–29 (1969)
Cohen, H., Villegas, F.R., Zaigler, D.: Convergence acceleration of alternating series. Exp. Math. 9, 3–12 (2000)
Constantine, A.G.: Some noncentral distribution problems in multivariate analysis. Ann. Math. Stat. 34, 1270–1285 (1963)
Goodwin, E.T. (ed.): Modern Computing Methods, 2nd edn. Philosophical Library, New York (1961)
Gupta, R.D., Richards, D.S.P.: Hypergeometric functions of scalar matrix argument are expressible in terms of classical hypergeometric functions. SIAM J. Math. Anal. 16, 852–858 (1985)
Johnson, N.L., Kotz, S.: Continuous Multivariate Distributions. Wiley, New York (1972)
Johnstone, I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29, 295–327 (2001)
Johnstone, I.M.: Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy-Widom limits and rates of convergence. Ann. Stat. 36, 2638–2716 (2008)
Johnstone, I.M.: Approximate null distribution of the largest root in multivariate analysis. Ann. Appl. Stat. 3 (2009, to appear)
Knopp, K.: Theory and Application of Infinite Series. Dover, New York (1990)
Kres, H.: Statistical Tables for Multivariate Analysis. Springer, New York (1983)
Maple 11: Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario (2007)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Nanda, D.N.: J. Indian Soc. Agric. Stat. 3, 175–177 (1951)
Pillai, K.C.S.: On the distribution of the largest or the smallest root of a matrix in multivariate analysis. Biometrika 43, 122–127 (1956)
Pillai, K.C.S.: On the distribution of the largest characteristic root of a matrix in multivariate analysis. Biometrika 52, 405–414 (1965)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in FORTRAN 77: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Roy, S.N.: The individual sampling distribution of the maximum, the minimum, and any intermediate of the p statistics on the null hypothesis. Sankhyà 8, 133–158 (1945)
Scheid, F.: Schaum’s Outline of Theory and Problems of Numerical Analysis, 2nd edn. McGraw-Hill, New York (1989)
Sugiyama, T.: Distribution of the largest latent root and the smallest latent root of the generalized B statistic and F statistic in multivariate analysis. Ann. Math. Stat. 38, 1152–1159 (1967)
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Butler, R.W., Paige, R.L. Exact distributional computations for Roy’s statistic and the largest eigenvalue of a Wishart distribution. Stat Comput 21, 147–157 (2011). https://doi.org/10.1007/s11222-009-9154-7
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DOI: https://doi.org/10.1007/s11222-009-9154-7