An affine invariant multiple test procedure for assessing multivariate normality. (English) Zbl 1328.62357

Tenreiro, Carlos

Comput. Stat. Data Anal. 55, No. 5, 1980-1992 (2011).

Summary: A multiple test procedure for assessing multivariate normality (MVN) is proposed. The new test combines a finite set of affine invariant test statistics for MVN through an improved Bonferroni method. The usefulness of such an approach is illustrated by a multiple test including the Mardia and BHEP (Baringhaus-Henze-Epps-Pulley) tests that are among the most recommended procedures for testing MVN. A simulation study carried out for a wide range of alternative distributions, in order to analyze the finite sample power behavior of the proposed multiple test procedure, indicates that the new test demonstrates a good overall performance against other highly recommended MVN tests.

MSC:

62H15	Hypothesis testing in multivariate analysis
62G20	Asymptotic properties of nonparametric inference

Keywords:

multivariate normality tests; affine invariance; multiple testing; mardia tests; BHEP tests

Software:

R

References:

[1]	Arcones, M.A., Two tests for multivariate normality based on the characteristic function, Math. methods statist., 16, 177-201, (2007) · Zbl 1231.62102
[2]	Baringhaus, L.; Henze, N., A consistent test for multivariate normality based on the empirical characteristic function, Metrika, 35, 339-348, (1988) · Zbl 0654.62046
[3]	Baringhaus, L.; Henze, N., Limit distributions for mardia’s measure of multivariate skewness, Ann. statist., 20, 1889-1902, (1992) · Zbl 0767.62041
[4]	Bowman, A.W.; Foster, P.J., Adaptive smoothing and density-based tests of multivariate normality, J. amer. statist. assoc., 88, 529-537, (1993) · Zbl 0775.62086
[5]	Coin, D., A goodness-of-fit test for normality based on polynomial regression, Comput. statist. data anal., 52, 2185-2198, (2008) · Zbl 1452.62307
[6]	Chiu, S.N.; Liu, K.I., Generalized cramérvon Mises goodness-of-fit tests for multivariate distributions, Comput. statist. data anal., 53, 3817-3834, (2009) · Zbl 1453.62070
[7]	Csörgő, S., Testing for normality in arbitrary dimension, Ann. statist., 14, 708-723, (1986) · Zbl 0615.62060
[8]	Csörgő, S., Consistency of some tests for multivariate normality, Metrika, 36, 107-116, (1989) · Zbl 0705.62053
[9]	Doornik, J.A., Hansen, H., 1994. An omnibus test for univariate and multivariate normality. Working Paper, Nuffield College, Oxford.
[10]	Dykstra, R.L., Establishing the positive definiteness of the sample covariance matrix, Ann. math. statist., 41, 2153-2154, (1970) · Zbl 0212.22202
[11]	Epps, T.W.; Pulley, L.B., A test for normality based on the empirical characteristic function, Biometrika, 70, 723-726, (1983) · Zbl 0523.62045
[12]	Fan, Y., Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters, Econometric theory, 14, 604-621, (1998)
[13]	Farrel, P.J.; Salibian-Barrera, M.; Naczk, K., On tests for multivariate normality and associated simulation studies, J. stat. comput. simul., 77, 1065-1080, (2007) · Zbl 1255.62163
[14]	Filliben, J.J., The probability plot correlation coefficient test for normality, Technometrics, 17, 111-117, (1975) · Zbl 0295.62049
[15]	Fromont, M.; Laurent, B., Adaptive goodness-of-fit tests in a density model, Ann. statist., 34, 680-720, (2006) · Zbl 1096.62040
[16]	Henze, N.; Zirkler, B., A class of invariante consistent tests for multivariate normality, Comm. stat. theory methods, 19, 3595-3617, (1990) · Zbl 0738.62068
[17]	Henze, N., On mardia’s kurtosis test for multivariate normality, Comm. statist. theory methods, 23, 1047-1061, (1994) · Zbl 0825.62141
[18]	Henze, N., Extreme smoothing and testing for multivariate normality, Statist. probab. lett., 35, 203-213, (1997) · Zbl 0955.62060
[19]	Henze, N., Invariant tests for multivariate normality: a critical review, Statist. papers, 43, 467-506, (2002) · Zbl 1008.62060
[20]	Henze, N.; Wagner, T., A new approach to the BHEP tests for multivariate normality, J. multivariate anal., 62, 1-23, (1997) · Zbl 0874.62043
[21]	Horswell, R.L.; Looney, S.W., A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis, J. stat. comput. simul., 42, 21-38, (1992)
[22]	Jiménez-Gamero, M.D.; Alba-Fernández, V.; Muñoz-García, J.; Chalco-Cano, Y., Goodness-of-fit tests based on empirical characteristic functions, Comput. statist. data anal., 53, 3957-3971, (2009) · Zbl 1453.62119
[23]	Johnson, M.E., Multivariate statistical simulation, (1987), Wiley New York · Zbl 0604.62056
[24]	Kotz, S.; Kozubowski, T.; Podgorski, K., The Laplace distribution and generalizations, (2001), Birkhauser Boston
[25]	Liang, J.; Pan, W.S.Y.; Yang, Z.-H., Characterization-based Q-Q plots for testing multinormality, Statist. probab. lett., 70, 183-190, (2005) · Zbl 1056.62069
[26]	Liang, J.; Tang, M.-L.; Chan, P.S., A generalized shapiro – wilk W statistic for testing high-dimensional normality, Comput. statist. data anal., 53, 3883-3891, (2009) · Zbl 1454.62026
[27]	Mardia, K.V., Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519-530, (1970) · Zbl 0214.46302
[28]	Mardia, K.V.; Foster, K., Omnibus tests of multinormality based on skewness and kurtosis, Comm. statist. theory methods, 12, 207-221, (1983)
[29]	Mardia, K.V.; Kent, J.T., Rao score tests for goodness of fit and independence, Biometrika, 78, 355-363, (1991) · Zbl 0735.62056
[30]	Mecklin, C.J., Mundfrom, D.J., 2000. Comparing of the power of classical and newer tests of multivariate normality. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, April 24-28, 2000.
[31]	Mecklin, C.J.; Mundfrom, D.J., An appraisal and bibliography of tests for multivariate normality, Int. stat. rev., 72, 123-138, (2004) · Zbl 1211.62095
[32]	Mecklin, C.J.; Mundfrom, D.J., A Monte Carlo comparison of type I and type II error rates of tests of multivariate normality, J. stat. comput. simul., 75, 93-107, (2005) · Zbl 1096.62058
[33]	R Development Core Team, 2009. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org.
[34]	Romeu, J.L.; Ozturk, A., A comparative study of goodness-of-fit tests for multivariate normality, J. multivariate anal., 46, 309-334, (1993) · Zbl 0778.62051
[35]	Royston, J.P., Some techniques for assessing multivarate normality based on the shapiro – wilk W, J. roy. statist. soc. ser., C 32, 121-133, (1983) · Zbl 0536.62043
[36]	Royston, J.P., Approximating the shapiro – wilk W-test for non-normality, Stat. comput., 2, 117-119, (1992)
[37]	Shapiro, S.S; Wilks, M.B., An analysis of variance test for normality (complete samples), Biometrika, 52, 591-611, (1965) · Zbl 0134.36501
[38]	Shorack, G.R.; Wellner, J.A., Empirical processes with applications to statistics, (1986), Wiley New York · Zbl 1170.62365
[39]	Sürücü, B., Goodness-of-fit tests for multivariate distributions, Comm. statist. theory methods, 35, 1319-1331, (2006) · Zbl 1106.62062
[40]	Székely, G.J.; Rizzo, M.L., A new test for multivariate normality, J. multivariate anal., 93, 58-80, (2005) · Zbl 1087.62070
[41]	Tenreiro, C., On the choice of the smoothing parameter for the BHEP goodness-of-fit test, Comput. statist. data anal., 53, 1038-1053, (2009) · Zbl 1452.62322