A new affine invariant test for multivariate normality based on beta probability plots

A new technique for assessing multivariate normality (MVN) is proposed in this work based on a beta transform of the multivariate normal data set. The statistic is the sum of interpoint squared distances between an ordered set of the transformed observations and the set of the beta population pth quantiles. We showed that the statistic is affine invariant. The critical values of the test were evaluated for different sample sizes and different random vector dimensions through extensive simulations. For some selected sample sizes and random vector dimensions, the empirical type-I-error rates and powers of the proposed test were compared with those of others already in use tests for MVN. The results showed that the test is a good and competitive tool for testing MVN.

Ahn, S.K. (1992). F-probability plot and its application to multivariate normality.Communications in Statistics-Theory and Methods, 21, 997-1023.

Baringhaus, L. and Henze, N. (1988). A consistent test for multivariate normality based on the empirical characteristic function. Metrika, 35, 339-348.

Blom G. (1958). Statistical estimates and transformed beta-variables. Wiley. New York.

Cardoso de Oliveira, I.R.C. and Ferreira, D.F. (2010). Multivariate extension of chi-squared univariate normality test. Journal of Statistical Computation and Simulation, 80, 513-525.

Cox, D.R. and Wermuth, N. (1994). Tests of linearity, multivariate normality and the adequacy of linear scores. Applied Statistics, 43, 347-355.

Fan, Y. (1997). Goodness of fit tests for a multivariate distribution by the empirical characteristic function. Journal of Multivariate Analysis, 62, 36-63.

Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annual Eugenics, 7, Part II, 179-188.

Gnanadesikan, R. and Kettenring, J. R. (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 28, 81-124.

Hanusz, Z. and Tarasinska, J. (2012). New tests for multivariate normality based on Smalls and Srivastavas graphical methods. Journal of Statistical Computation and Simulation,80, 513-526.

Hawkins, D. N. (1981). A new test for multivariate normality and homoscedasticity.Technometrics, 23, 105-110.

Healy M. J. R. (1968). Multivariate normal plotting. Applied Statistics, 17, 157-161.

Henze, N. (2002). Invariant tests for multivariate normality: A critical review. Statistical Papers, 43, 467-506.

Henze, N. and Zirkler, B. (1990). A class of invariant consistent tests for multivariate normality. Communications in Statistics-Theory and Methods, 19, 3595-3618.

Hwu, T., Han, C., and Rogers, K. J. (2002). The combination test for multivariate normality. Journal of Statistical Computation and Simulation, 72, 379-390.

Liang, J., Fang, M. L. and Chang, P. S. (2009). A generalized Shapiro-Wilk W statistic for testing high dimensional normality. Computational Statistics and Data Analysis, 53, 3883-3891.

Madukaife, M. S. and Okafor, F. C. (2017). A powerful affine invariant test for multivariate normality based on interpoint distances of principal components. Communications in Statistics-Simulation and Computation, DOI: 10.1080/03610918.2017.1309667.

Madukaife, M. S. and Okafor, F. C. (2018). A new large sample goodness of fit test for multivariate normality based on chi squared probability plots. Communications in Statistics-Simulation and Computation, DOI: 10.1080/03610 918.2017.1422749.

Malkovich, J. F. and Afifi, A. A. (1973). On tests for multivariate normality. Journal of the American Statistical Association, 68, 176-179.

Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications.Biometrika, 573, 519-530.

Mecklin, C. J. and Mundfrom, D. J. (2004). An appraisal and bibliography of tests for multivariate normality. International Statistical Review, 72, 123-138.

Mecklin, C. J. and Mundfrom, D. J. (2005). A Monte Carlo comparison of the type I and type II error rates of tests of multivariate normality. Journal of Statistical Computation and Simulation, 75, 93-107.

Muirhead, R. J. (2005). Aspects of multivariate statistical theory. Wiley. New Jersey.

Nwagbata, A. C. (2016). A discriminant analysis of features of babies at birth. An unpublished project report, Department of Statistics, University of Nigeria, Nsukka.

Royston, J. P. (1983). Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Applied Statistics, 32, 121-133.

Scrucca, L. (2000). Assessing multivariate normality through interactive dynamic graphics.Quaderni di statistica, 2, 221-240.

Singh, A. (1993). Omnibus robust procedures for assessment of multivariate normality and detection of multivariate outliers, in Multivariate Environmental Statistics, eds. G.P. Patil and C.R. Rao. North Holland. Amsterdam.

Small, N. J. H. (1978). Plotting squared radii. Biometrika, 65, 657-658.

Weiss, I. (1958). A test of fit for multivariate distributions. Annals of Mathematical Statistics, 29, 595-599.

Wilks S. S. (1962). Mathematical statistics. Wiley. New York.