On Rotational Robustness of Shapiro-Wilk Type Tests for Multivariate Normality
ABSTRACT
The Shapiro-Wilk test (SWT) for normality is well known for its competitive power against numerous one-dimensional alternatives. Several extensions of the SWT to multi-dimensions have also been proposed. This paper investigates the relative strength and rotational robustness of some SWT-based normality tests. In particular, the Royston’s H-test and the SWT-based test proposed by Villase?or-Alva and González-Estrada have R packages available for testing multivariate normality; thus they are user friendly but lack of rotational robustness compared to the test proposed by Fattorini. Numerical power comparison is provided for illustration along with some practical guidelines on the choice of these SWT-type tests in practice.
The Shapiro-Wilk test (SWT) for normality is well known for its competitive power against numerous one-dimensional alternatives. Several extensions of the SWT to multi-dimensions have also been proposed. This paper investigates the relative strength and rotational robustness of some SWT-based normality tests. In particular, the Royston’s H-test and the SWT-based test proposed by Villase?or-Alva and González-Estrada have R packages available for testing multivariate normality; thus they are user friendly but lack of rotational robustness compared to the test proposed by Fattorini. Numerical power comparison is provided for illustration along with some practical guidelines on the choice of these SWT-type tests in practice.
Cite this paper
Lee, R. , Qian, M. and Shao, Y. (2014) On Rotational Robustness of Shapiro-Wilk Type Tests for Multivariate Normality. Open Journal of Statistics, 4, 964-969. doi: 10.4236/ojs.2014.411090.
Lee, R. , Qian, M. and Shao, Y. (2014) On Rotational Robustness of Shapiro-Wilk Type Tests for Multivariate Normality. Open Journal of Statistics, 4, 964-969. doi: 10.4236/ojs.2014.411090.
References
[1] Thode Jr., H.C. (2012) Testing for Normality. Marcel Dekker, Inc., New York. [2] Shapiro, S.S. and Wilk, M.B. (1965) An Analysis of Variance Test for Normality (Complete Samples). Biometrika, 52, 591-611.
http://dx.doi.org/10.1093/biomet/52.3-4.591 [3] Royston, T.P. (1982) An Extension of Shapiro and Wilk W Test for Normality to Large Samples. Applied Statistics, 31, 115-124.
http://dx.doi.org/10.2307/2347973 [4] Royston, T.P. (1983) Some Techniques for Assessing Multivarate Normality Based on the Shapiro-Wilk W. Applied Statistics, 32, 121-133.
http://dx.doi.org/10.2307/2347291 [5] Royston, T.P. (1992) Approximating the Shapiro-Wilk W-Test for Non-Normality. Statistics and Computing, 2, 117-119.
http://dx.doi.org/10.1007/BF01891203 [6] Royston, J.P. (1995) Remark AS R94: A Remark on Algorithm AS 181: The W Test for Normality. Applied Statistics, 44, 547-551.
http://dx.doi.org/10.2307/2986146 [7] Korkmaz, S. (2013) Royston’s H Test: Multivariate Normality Test.
http://cran.r-project.org/web/packages/royston/index.html [8] Villase?or-Alva, J.A. and González-Estrada, G. (2009) A Generalization of Shapiro-Wilk’s Test for Multivariate Normality. Communications in Statistics-Theory and Methods, 38, 1870-1883.
http://dx.doi.org/10.1080/03610920802474465 [9] Gonzalez-Estrada, G. and Villase?or-Alva, J.A. (2013) Generalized Shapiro-Wilk Test for Multivariate Normality.
http://rpackages.ianhowson.com/cran/mvShapiroTest/ [10] Fattorini, L. (1986) Remarks on the Use of the Shapiro-Wilk Statistic for Testing Multivariate Normality. Statistica, 46, 209-217. [11] Henze, N. and Zirkler, B. (1990) A Class of Invariant Consistent Tests for Multivariate Normality. Communications in Statistics-Theory and Method, 19, 3595-3618.
http://dx.doi.org/10.1080/03610929008830400 [12] Malkovich, J.F. and Afifi, A.A. (1973) On Tests for Multivariate Normality. Journal of American Statistical Association, 68, 713-718.
http://dx.doi.org/10.1080/01621459.1973.10481358 [13] Mudholkar, G., Srivastava, D. and Lin, C. (1995) Some p-Variate Adaptations of the Shapiro-Wilk Test of Normality. Communications in Statistics-Theory and Method, 24, 953-985.
http://dx.doi.org/10.1080/03610929508831533 [14] Srivastava, M. and Hui, T. (1987) On Assessing Multivariate Normality Based on Shapiro-Wilk W Statistic. Statistics and Probability Letters, 5, 15-18.
http://dx.doi.org/10.1016/0167-7152(87)90019-8 [15] Shao, Y. and Zhou, M. (2010) A Characterization of Multivariate Normality through Univariate Projections. Journal of Multivariate Analysis, 101, 2637-2640.
http://dx.doi.org/10.1016/j.jmva.2010.04.015 [16] Fisher, R.A. (1936) The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7, 179-188.
http://dx.doi.org/10.1111/j.1469-1809.1936.tb02137.x [17] Looney, S.W. (1995) How to Use Tests for Univariate Normality to Assess Multivariate Normality. The American Statistician, 49, 64-70. [18] Small, N. (1980) Marginal Skewness and Kurtosis in Testing Multivariate Normality. Applied Statistics, 29, 85-87.
http://dx.doi.org/10.2307/2346414 [19] Mardia, K.V. (1974) Applications of Some Measures of Multivariate Skewness and Kurtosis in Testing Normality and Robustness Studies. Sankhyā: The Indian Journal of Statistics, Series B, 36, 115-128. [20] Srivastava, M.S. (1984) A Measure of Skewness and Kurtosis and a Graphical Method for Assessing Multivariate Normality. Statistics and Probability Letters, 2, 263-267.
http://dx.doi.org/10.1016/0167-7152(84)90062-2 [21] Zhou, M. and Shao, Y. (2014) A Powerful Test for Multivariate Normality. Journal of Applied Statistics, 41, 351-363.
http://dx.doi.org/10.1080/02664763.2013.839637
[1] Thode Jr., H.C. (2012) Testing for Normality. Marcel Dekker, Inc., New York. [2] Shapiro, S.S. and Wilk, M.B. (1965) An Analysis of Variance Test for Normality (Complete Samples). Biometrika, 52, 591-611.
http://dx.doi.org/10.1093/biomet/52.3-4.591 [3] Royston, T.P. (1982) An Extension of Shapiro and Wilk W Test for Normality to Large Samples. Applied Statistics, 31, 115-124.
http://dx.doi.org/10.2307/2347973 [4] Royston, T.P. (1983) Some Techniques for Assessing Multivarate Normality Based on the Shapiro-Wilk W. Applied Statistics, 32, 121-133.
http://dx.doi.org/10.2307/2347291 [5] Royston, T.P. (1992) Approximating the Shapiro-Wilk W-Test for Non-Normality. Statistics and Computing, 2, 117-119.
http://dx.doi.org/10.1007/BF01891203 [6] Royston, J.P. (1995) Remark AS R94: A Remark on Algorithm AS 181: The W Test for Normality. Applied Statistics, 44, 547-551.
http://dx.doi.org/10.2307/2986146 [7] Korkmaz, S. (2013) Royston’s H Test: Multivariate Normality Test.
http://cran.r-project.org/web/packages/royston/index.html [8] Villase?or-Alva, J.A. and González-Estrada, G. (2009) A Generalization of Shapiro-Wilk’s Test for Multivariate Normality. Communications in Statistics-Theory and Methods, 38, 1870-1883.
http://dx.doi.org/10.1080/03610920802474465 [9] Gonzalez-Estrada, G. and Villase?or-Alva, J.A. (2013) Generalized Shapiro-Wilk Test for Multivariate Normality.
http://rpackages.ianhowson.com/cran/mvShapiroTest/ [10] Fattorini, L. (1986) Remarks on the Use of the Shapiro-Wilk Statistic for Testing Multivariate Normality. Statistica, 46, 209-217. [11] Henze, N. and Zirkler, B. (1990) A Class of Invariant Consistent Tests for Multivariate Normality. Communications in Statistics-Theory and Method, 19, 3595-3618.
http://dx.doi.org/10.1080/03610929008830400 [12] Malkovich, J.F. and Afifi, A.A. (1973) On Tests for Multivariate Normality. Journal of American Statistical Association, 68, 713-718.
http://dx.doi.org/10.1080/01621459.1973.10481358 [13] Mudholkar, G., Srivastava, D. and Lin, C. (1995) Some p-Variate Adaptations of the Shapiro-Wilk Test of Normality. Communications in Statistics-Theory and Method, 24, 953-985.
http://dx.doi.org/10.1080/03610929508831533 [14] Srivastava, M. and Hui, T. (1987) On Assessing Multivariate Normality Based on Shapiro-Wilk W Statistic. Statistics and Probability Letters, 5, 15-18.
http://dx.doi.org/10.1016/0167-7152(87)90019-8 [15] Shao, Y. and Zhou, M. (2010) A Characterization of Multivariate Normality through Univariate Projections. Journal of Multivariate Analysis, 101, 2637-2640.
http://dx.doi.org/10.1016/j.jmva.2010.04.015 [16] Fisher, R.A. (1936) The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 7, 179-188.
http://dx.doi.org/10.1111/j.1469-1809.1936.tb02137.x [17] Looney, S.W. (1995) How to Use Tests for Univariate Normality to Assess Multivariate Normality. The American Statistician, 49, 64-70. [18] Small, N. (1980) Marginal Skewness and Kurtosis in Testing Multivariate Normality. Applied Statistics, 29, 85-87.
http://dx.doi.org/10.2307/2346414 [19] Mardia, K.V. (1974) Applications of Some Measures of Multivariate Skewness and Kurtosis in Testing Normality and Robustness Studies. Sankhyā: The Indian Journal of Statistics, Series B, 36, 115-128. [20] Srivastava, M.S. (1984) A Measure of Skewness and Kurtosis and a Graphical Method for Assessing Multivariate Normality. Statistics and Probability Letters, 2, 263-267.
http://dx.doi.org/10.1016/0167-7152(84)90062-2 [21] Zhou, M. and Shao, Y. (2014) A Powerful Test for Multivariate Normality. Journal of Applied Statistics, 41, 351-363.
http://dx.doi.org/10.1080/02664763.2013.839637