On Some Tests of the Covariance Matrix Under General Conditions

Anderson T.W. (1984). An Introduction to Multivariate Analysis (2nd ed). John Wiley & Sons, New York Bartlett M.S. (1938). Further aspects of the theory of multiple regression. Mathematical Proceedings of the Cambridge Philosophical Society 34:33–40 Bartlett M.S. (1947). Multivariate analysis. Journal of the Royal Statistical Society (Suppl). 9:176–190 Bhattacharya, R. N., & Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Annals of Statistics, 6, 434–451; Corrigendum, ibid. 8 (1980). David A.W. (1977). A differential equation approach to linear combinations of independent chi-squares. Journal of the American Statistical Association 72:212–214 Fang K., Zhang Y. (1990). Generalized Multivariate Analysis. Springer-Verlag, New York Fujikoshi Y. (1976). Asymptotic expansions for the distributions of some multivariate tests. In: Krishinaiah P.R. (eds). Multivariate Analysis, Vol 4. North Holland, Amsterdam, pp. 55–71 Fujikoshi Y. (1997). An asymptotic expansion of the distribution of Hotelling’s T 2-statistic. Journal of Multivariate Analysis 61:187–193 Fujikoshi Y. (2002a). Asymptotic expansions for the distributions for multivariate basic statistics and one-way MANOVA tests under nonnormality. Journal of Statistical Planning and Inference 108:263–282 Fujikoshi Y. (2002b). Some recent results on asymptotic expansions of multivariate test statistics for mean vectors under non-normality. Calcutta Statistical Association Bulletin, 52 (Special 4th Triennial Proceedings Volume), Nos. 205–208, 1–46 Gupta A.K. (1977). On the distribution of sphericity test criterion in the multivariate Gaussian distribution. Australian & New Zealand Journal of Statistics 19:202–205 Gupta A.K., Chattopadhyay A.K., Krishnaiah P.R. (1975). Asymptotic distributions of the determinants of some random matrices. Communications in Statistics 4:33–47 Gupta A.K., Nagar D.K. (1988). Asymptotic expansion of the nonnull distribution of likelihood ratio statistic for testing multisample sphericity. Communications in Statistics. Theory and Methods 17:3145–3155 Gupta A.K., Tang J. (1984). Distribution of likelihood ratio statistic for testing equality of covariance matrices of multivariate Gaussian models. Biometrika 71:555–559 Gupta, A. K., Xu, J., & Fujikoshi, Y. (2005). An asymptotic expansion of the distribution of Rao’s U-statistic under a general condition. Journal of Multivariate Analysis (to appear) Hall P. (1992). The Bootstrap and Edgeworth Expansions. Springer-Verlag, New York Ito K. (1968). On the effect of heteroscedasticity and nonnormality upon some multivariate test procedures. In: Krishinaiah P.R. (eds). Multivariate Analysis, Vol 2. Academic Press, New York, pp. 87–120 Jensen D.R., Solomon H. (1972). A Gaussian approximation to the distribution of a definite quadratic form. Journal of the American Statistical Association 67:898–902 Johnson N.I., Kotz S. (1970). Continuous Univariate Distributions, Vol 2. John Wiley & Sons, New York Jones M.C., & Faddy, M. J. (2003). A skew extension of the t-distribution, with applications. Journal of the Royal Statistical Society. Series B, 65, Part 1, 159–174 Kano Y. (1995). An asymptotic expansion of the distribution of Hotellings T 2-statistic under general condition. American Journal of Mathematical and Management Sciences 15:317–341 Kariya, T., & Kim, P. (1997). Finite sample robustness of tests: an overview. In G. S. Maddala & C. R. Rao (Eds). Handbook of Statistics, Vol. 15, Robust Inference (pp. 645–660). Konishi S., Niki N., Gupta A.K. (1988). Asymptotic expansions for the distribution of quadratic forms in normal variables. Annals of the Institute of Statistical Mathematics 40:279–296 Kuonen D. (1999). Saddlepoint approximations for distributions of quadratic forms in normal variables. Biometrika 86: 929–935 Lawley D.N. (1959). Tests of significance in canonical analysis. Biometrika 46:59–66 Lu Z. -H., King M. (2002). Improving the numerical technique for computing the accumulated distribution of a quadratic form in normal variables. Econometric Reviews 21:149–165 Mardia K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530 Mathai A.M., Provost S.B. (1992). Quadratic forms in random variables. Marcel Dekker, New York Muirhead R.J., Waternaux C.M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika 67:31–43 Muirhead R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley & Sons, New York Muirhead R.J. (1985). Estimating a particular function of the multiple correlation coefficient. Journal of the American Statistical Association 80:923–925 Rao C.R. (1973). Linear Statistical Inference and Its Applications (2nd Ed). Wiley, New York Siotani M., Hayakawa T., Fujikoshi Y. (1985). Modern Multivariate Statistical Analysis: A Graduate Course and Handbook. American Sciences Press, Columbus, OH Solomon H., Stephens M.A. (1977). Distribution of a sum of weighted chi-square variables. Journal of the American Statistical Association 72:881–885 Stuart A., Ord J.K. (1987). Kendall’s Advanced Theory of Statistics, Vol 1 (5th Ed). Oxford University Press, New York Sugiura N. (1972). Locally best invariant test for sphericity and the limiting distribution. Annals of Mathematical Statistics 43:1312–1316 Tang J., Gupta A.K. (1990). Testing independence of several groups of variables. Journal of Statistical Computation and Simulation 37:27–35 Xu J., Gupta A.K. (2005). Confidence intervals for the mean value of response function in generalized linear models. Statistica Sinica 15:1081–1096 Yanagihara H., Matsumoto C., Tonda T. (2004). Asymptotic expansion of the null distribution of the modified normal likelihood ratio criterion for testing Σ=Σ0 under nonnormality. Hiroshima Mathematical Journal 34:81–100