A powerful test for multivariate normality. (English) Zbl 1514.62983
Summary: This paper investigates a new test for normality that is easy for biomedical researchers to understand and easy to implement in all dimensions. In terms of power comparison against a broad range of alternatives, the new test outperforms the best known competitors in the literature as demonstrated by simulation results. In addition, the proposed test is illustrated using data from real biomedical studies.
MSC:
| 62-XX | Statistics |
References:
| [1] | L. Baringhaus and N. Henze, A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1988), pp. 339-348. doi: 10.1007/BF02613322 · Zbl 0654.62046 · doi:10.1007/BF02613322 |
| [2] | K.O. Bowman and L.R. Shenton, Omnibus test contours for departures from normality based on √ \(b_1\) and \(b_2\), Biometrika 62 (1975), pp. 243-250. · Zbl 0308.62016 |
| [3] | D. Coin, A goodness-of-fit test for normality based on polynomial regression, Comput. Stat. Data Anal. 52 (2008), pp. 2185-2198. doi: 10.1016/j.csda.2007.07.012 · Zbl 1452.62307 · doi:10.1016/j.csda.2007.07.012 |
| [4] | R.B. D’Agostino, A. Belanger, and R.B. D’agostino Jr., A suggestion for using powerful and informative tests of normality, Am. Stat. 44 (1990), pp. 316-321. |
| [5] | R.B. D’Agostino and E.S. Pearson, Tests for departure from normality. Empirical results for the distribution of \(b_2\) and √ \(b_1\), Biometrika 60 (1973), pp. 613-622. · Zbl 0271.62025 |
| [6] | R.B. D’Agostino and M.A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker, Inc., New York, 1986. · Zbl 0597.62030 |
| [7] | J.A. Doornik and H. Hansen, An omnibus test for univariate and multivariate normality, Oxford Bull. Econ. Stat. 70 (2008), pp. 927-939. doi: 10.1111/j.1468-0084.2008.00537.x · doi:10.1111/j.1468-0084.2008.00537.x |
| [8] | M.L. Eaton and M.D. Perlman, The non-singularity of generalized sample covariance matrices, Ann. Stat. 1 (1973), pp. 710-717. doi: 10.1214/aos/1176342465 · Zbl 0261.62037 · doi:10.1214/aos/1176342465 |
| [9] | R.C. Elston and J.E. Grizzle, Estimation of time-response curves and their confidence bands, Biometrics 18 (1962), pp. 148-159. doi: 10.2307/2527453 · Zbl 0104.38101 · doi:10.2307/2527453 |
| [10] | T.W. Epps and L.B. Pulley, A test for normality based on the empirical characteristic function, Biometrika 70 (1983), pp. 723-726. doi: 10.1093/biomet/70.3.723 · Zbl 0523.62045 · doi:10.1093/biomet/70.3.723 |
| [11] | L. Fattorini, Remarks on the use of the Shapiro-Wilk statistic for testing multivariate normality, Statistica 46 (1986), pp. 209-217. · Zbl 0625.62037 |
| [12] | R.A. Fisher, The use of multiple measurements in taxonomic problems, Ann. Eugenics 7 (1936), pp. 179-188. doi: 10.1111/j.1469-1809.1936.tb02137.x · doi:10.1111/j.1469-1809.1936.tb02137.x |
| [13] | N. Henze and B. Zirkler, A class of invariant consistent tests for multivariate normality, Commun. Stat. Theory Methods 19 (1990), pp. 3595-3617. doi: 10.1080/03610929008830400 · Zbl 0738.62068 |
| [14] | J. Leslie, M.A. Stephens, and S. Fotopoulos, Asymptotic distribution of the Shapiro-Wilk w for testing for normality, Ann. Stat. 14 (1986), pp. 1497-1506. doi: 10.1214/aos/1176350172 · Zbl 0622.62019 · doi:10.1214/aos/1176350172 |
| [15] | S.W. Looney, How to use tests for univariate normality to assess multivariate normality, Am. Stat. 49 (1995), pp. 64-70. |
| [16] | K.V. Mardia, Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970), pp. 519-530. doi: 10.1093/biomet/57.3.519 · Zbl 0214.46302 · doi:10.1093/biomet/57.3.519 |
| [17] | E.S. Pearson, A further development of tests for normality, Biometrika 22 (1930), pp. 239-249. · JFM 56.0453.01 · doi:10.1093/biomet/22.1-2.239 |
| [18] | J. Shao, Mathematical Statistics, 2nd ed., Springer-Verlag, New York, 2003. · Zbl 1018.62001 · doi:10.1007/b97553 |
| [19] | Y. Shao and M. Zhou, A characterization of multivariate normality through univariate projections, J. Multivariate Anal. 101 (2010), pp. 2637-2640. doi: 10.1016/j.jmva.2010.04.015 · Zbl 1198.62045 · doi:10.1016/j.jmva.2010.04.015 |
| [20] | S.S. Shapiro and M.B. Wilk, An analysis of variance test for normality (complete samples), Biometrika 52 (1965), pp. 591-611. · Zbl 0134.36501 · doi:10.1093/biomet/52.3-4.591 |
| [21] | N. Small, Marginal skewness and kurtosis in testing multivariate normality, Appl. Stat. 29 (1980), pp. 85-87. doi: 10.2307/2346414 · doi:10.2307/2346414 |
| [22] | M.A. Stephens, EDF statistics for goodness of fit and some comparisons, J. Am. Stat. Assoc. 69 (1974), pp. 730-737. |
| [23] | H.C. Thode Jr., Testing for Normality, Marcel Dekker, Inc., New York, 2002. · Zbl 1032.62040 · doi:10.1201/9780203910894 |
| [24] | N.H. Timm, Applied Multivariate Analysis, Springer, New York, 2002. · Zbl 1002.62036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
