Abstract
This paper gives a synopsis on affine invariant tests of the hypothesis that the unknown distribution of a d-dimensional random vector X is some nondegenerate d-variate normal distribution, on the basis of i.i.d. copies X 1,...,X n of X. Particular emphasis is given to progress that has been achieved during the last decade. Furthermore, we stress the typical diagnostic pitfall connected with purportedly ‘directed’ procedures, such as tests based on measures of multivariate skewness.
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References
Ahn, S.K. (1992). F-probability plot and its application to multivariate normality. Commun. Statist. — Th. Meth., 21, 997–1023.
Anderson, H.H., Hall, P. and Titterington, D.M. (1994). Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates. Corny. Statist. Data Anal., 30, 119–142.
Baringhaus, L., and Henze, N. (1988). A consistent test for multivariate normality based on the empirical characteristic function. Metrika, 35, 339–348.
Baringhaus, L., and Henze, N. (1991). Limit distributions for measures of multivariate skewness and kurtosis based on projections. J. Multiv. Anal., 38, 51–69.
Baringhaus, L., and Henze, N. (1992). Limit distributions for Mardia’s measure of multivariate skewness. Ann. Statist., 20, 1889–1902.
Baxter, M.J., and Gale, N.H. (1998). Testing for multivariate normality via univariate tests: A case study using lead isotope ratio data. J. Appl. Statist., 25, 671–683.
Beirlant, J., Mason, D.M. and Vynckier, C. (1999). Goodness-of-fit analysis for multivariate normality based on generalized quantiles. Comp. Statist. Data Anal., 30, 119–142.
Bera, A., and John, S. (1983). Tests for multivariate normality with Pearson alternatives. Comm. Statist. — Th. Meth., 12, 103–117.
Bogdan, M. (1999). Data driven smooth tests for bivariate normality. J. Multiv. Anal., 68, 26–53.
Bowman, A.W., and Foster, P.J. (1993). Adaptive smoothing and density-based tests of multivariate normality. J. Amer. Statist. Ass., 88, 529–537.
Cox, D.R., and Small, N.J.H. (1978). Testing multivariate normality. Biometrika, 65, 263–272.
Cox, D.R., and Wermuth, N. (1994). Tests of linearity, multivariate normality and the adequacy of linear scores. Appl. Statist., 43, 347–355.
Csorgo, S. (1986). Testing for normality in arbitrary dimension. Ann. Statist., 14, 708–723.
Csorgo, S. (1989). Consistency of some tests for multivariate normality. Metrika, 36, 107–116.
D’Agostino, R.B., and Stephens, M.A. (eds) (1986). Goodness-of-fit techniques. Marcel Dekker, New York.
De Wet, T., and Randies, R.H. (1987). On the effect of substituting parameter estimators in limiting χ2, U and V statistics. Ann. Statist., 15, 398–412.
De Wet, T., Venter, J.H., and Van Wyk, J.W.J. (1979). The null distributions of some test criteria of multivariate normality. S. Afr. Statist. J., 13, 153–177.
Dunn, Ch.L. (1995). Critical values and powers for tests of uniformity of directions under multivariate normality. Comm. Statist. — Th. Meth., 24, 2541–2560.
Eaton, M.L.(1989). Group invariance applications in statistics. Regional Conf. Ser. in Probab. and Statist. Vol. 1. Inst, of Mathem. Statistics.
Eaton, M.L., and Perlman, M.D. (1973). The non-singularity of generalized sample covariance matrices. Ann. Statist., 1, 710–717.
Epps, T.W. (1999). Limiting behavior of the ICF test for multivariate normality under Gram-Charlier alternatives. Statist & Prob. Lett., 42, 175–184.
Epps, T.W., and Pulley, L.B. (1983). A test for normality based on the empirical characteristic function. Biometrika, 70, 723–726.
Fang, K.T., Li, R.Z. and Liang, J.J. (1998). A multivariate version of Ghosh’s T 3-plot to detect non-multinormality. Comp. Statist. Data Anal., 28, 371–386.
Fang, K.T., and Wang, Y. (1993). Number-theoretic methods in ststistics. Monographs on statistics and applied probability. Chapman and Hall, London.
Fattorini, L. (1986). Remarks on the use of the Shapiro-Wilk statistic for testing multivariate normality. Statistica, 46, 209–217.
Fattorini, L. (1988). On a large class of tests for multivariate normality. Quaderni dell’ Istituto di Statistica Universita’ degli Studi di Siena, Facolta’ di Scienze Economiche e Bancarie, Vol. 66, Siena, Italy.
Fattorini, L. (2001). On the assessment of multivariate normality. Atti della XL Riunoine della Societa Italiana die Statistica, Firenze, 26–28 Aprile 2001, 313–324, Italy.
Fattorini, L., and Pisani, C. (2000). Assessing multivariate normality on the “worst” sample configuration. Metron, 58, 23–38.
Geary, R.C. (1947). Testing for normality. Biometrika, 34, 209–242.
Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations. Wiley, New York.
Gnanadesikan, R., and Kettenring, J.R. (1972). Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 28, 81–124.
Gürtler, N. (2000). Asymptotic theorems for the class of BHEP-tests for multivariate normality with fixed and variable smoothing parameter (in German). Doctoral dissertation, University of Karlsruhe, Germany.
Gutjahr, S., Henze, N., and Folkers, M. (1999). Shortcomings of generalized affine invariant skewness measures. J. Mult. Anal., 71, 1–23.
Hall, P. (1984). Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multiv. Anal., 14, 1–15.
Hasofer, A.M., and Stein, G.Z. (1990). Testing for multivariate normality after coordinate transformation. Comm. Statist. — Th. Meth., 19, 1403–1418.
Henze, N. (1990). An approximation to the limit distribution of the Epps-Pulley test statistic for normality. Metrika, 37, 7–18.
Henze, N. (1994a). On Mardia’s kurtosis test for multivariate normality. Comm. Statist. — Th. Meth., 23, 1031–1045.
Henze, N. (1994b). The asymptotic behavior of a variant of multivariate kurtosis. Comm. Statist. — Th. Meth., 23, 1047–1061.
Henze, N. (1994c). Tests for normality (in German). Allg. Statistisches Archiv, 78, 293–317.
Henze, N. (1997a). Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Szekely. Statist. & Prob. Lett., 33, 299–307.
Henze, N. (1997b). Do components of smooth tests of fit have diagnostic properties? Metrika, 45, 121–130.
Henze, N. (1997c). Extreme smoothing and testing for multivariate normality. Statist. & Prob. Lett., 35, 203–213.
Henze, N., and Klar, B. (1996). Properly rescaled components of smooth tests of fit are diagnostic. Austral. J. Statist., 38, 61–74.
Henze, N., and Wagner, Th. (1997). A new approach to the BHEP tests for multivariate normality. J. Multiv. Anal., 62(1), 1–23.
Henze, N., and Zirkler, B. (1990). A class of invariant and consistent tests for multivariate normality. Coram. Statist. — Th. Meth., 19, 3595–3617.
Horswell, R.L., and Looney, St.W. (1992a). Diagnostic limitations of skewness coefficients in assessing departures from univariate and multivariate normality. Comm. Statist. — Simul., 22, 437–439.
Horswell, R.L., and Looney, St.W. (1992b). A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis. J. Statist. Comp. Simul., 42, 21–38.
Isogai, T. (1982). On a measure of multivariate skewness and a test for multivariate normality. Ann. Inst. Statist. Math., 34, 531–541.
Isogai, T. (1983). On measures of multivariate skewness and kurtosis. Math. Japonica, 28, 251–261.
Isogai, T. (1989). On using influence functions for testing multivariate normality. Ann. Inst. Statist. Math., 41, 169–186.
Janssen, A. (2000). Global power functions of goodness of fit tests. Ann. Statist., 28, 239–253.
Kallenberg, W.C.M., Ledwina, T., and Rafajlowicz, E. (1997). Testing bivariate independence and normality. Sankhya Ser. A, 59, 42–59.
Kariya, T., and George, E.I. (1994). Locally best invariant tests for multivariate normality in curved families with μ known. IMS Lecture Notes — Monograph Series, 24, 311–322.
Kariya, T., and George, E.I. (1995). Locally best invariant tests for multivariate normality in curved families and Mardias’s test. Sankhya Ser. A, 57, 440–451.
Kariya, T., Tsai, R.S., Terui, N., and Li, H. (1999). Tests for multinormality with applications to time series. Comm. Statist. — Th. Meth., 28, 519–536.
Klar, B. (1998). Classical and new goodness-of-fit tests (in German). Doctoral dissertation, University of Karlsruhe, Germany.
Koziol, J.A. (1982). A class of invariant procedures for assessing multivariate normality. Biometrika, 69, 423–427.
Koziol, J.A. (1983). On assessing multivariate normality. J. Roy. Statist. Soc. Ser. B, 45, 358–361.
Koziol, J.A. (1986a). A note on the asymptotic distribution of Mardia’s measure of multivariate kurtosis. Comrn. Statist. — Th. Meth., 15, 1507–1513.
Koziol, J.A. (1986b). Assessing multivariate normality: A compendium. Comm. Statist. — Th. Meth., 15, 2763–2783.
Koziol, J.A. (1987). An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives. Metrika, 34, 17–24.
Koziol, J.A. (1989). A note on measures of multivariate kurtosis. Biom. J., 31, 619–624.
Koziol, J.A. (1993). Probability plots for assessing multivariate normality. The Statistician, 42, 161–173.
Kuwana, Y., and Kariya, T. (1991). LBI tests for multivariate normality in exponential power distributions. J. Multiv. Anal., 39, 117–134.
Liang, J., and Bentler, P.M. (1999). A t-distribution plot to detect non-multinormality. Comput. Statist. Data Anal., 30, 31–44.
Liang, J., Li, R., Fang, H., and Fang, K.T. (2000). Testing multinormality based on lowdimensional projection. J. Statist. Plann. Infer., 86, 129–141.
Liu, Y., and Cheng, P. (1997). The limit distribution of PP Cramer-von Mises normality test statistics for high dimensions and large sample size. Northeast. Math. J., 13, 213–228.
Machado, S.G. (1983). Two statistics for testing for multivariate normality. Biometrika, 70, 713–718.
Malkovich, J.F., and Afifi, A.A. (1973). On tests for multivariate normality. J. Amer. Statist. Ass., 68, 176–179.
Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530.
Mardia, K.V. (1974). Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya Ser. B, 36, 115–128.
Mardia, K.V. (1975). Assessment of multinormality and the robustness of Hotelling’s T 2-test. Appl. Statist., 24, 163–171.
Mardia, K.V. (1980). Tests of univariate and multivariate normality. In: Handbook of Statistics 1. Analysis of Variance (P.R. Krishnaiah, ed., North-Holland, Amsterdam., 279–290.
Mardia, K.V., and Kanazawa, M. (1983). The null distribution of multivariate kurtosis. Comm. Statist. — Simul., 12, 569–576.
Mardia, K.V., and Kent, J.T. (1991). Rao score tests for goodness of fit and independence. Biometrika, 78, 355–363.
Moore, D., and Stubblebine, J. (1981). Chi-square tests for multivariate normality with applications to common stock prices. Comm. Statist. — Th. Meth., 10, 713–738.
Móri, T.F., Rohatgi, V.K., and Székely, G.J. (1993). On multivariate skewness and kurtosis. Th. Prob. Appl., 38, 547–551.
Mudholkar, G.S., McDermott, M., and Srivastava, D.K. (1992). A test of p-variate normality. Biometrika, 79, 850–854.
Mudholkar, G.S., Srivastava, D.K., and Lin, C.Th. (1995). Some p-variate adaptions of the Shapiro-Wilk test of normality. Comm. Statist. — Th. Meth., 24, 953–985.
Naito, K. (1996). On weighting the studentized empirical characteristic function for testing normality. Comm. Statist. — Simul., 25, 201–213.
Naito, K. (1998). Approximation of the power of kurtosis test for multinormality. J. Multiv. Anal., 65, 201–213.
Oja, H. (1983). Descriptive statistics for multivariate distributions. Statist. Prob. Lett., 1, 327–332.
Ozturk, A., and Romeu, J.L. (1992). A new method for assessing multivariate normality with graphical applications. Comm. Statist. — Simula., 21, 15–34.
Park, C. (1999). A note on the chi-square test for multivariate normality based on the sample Mahalanobis distances. J. Korean Statist. Soc., 28, 479–488.
Paulson, A.S., Roohan, P., and Sullo, P. (1987). Some empirical distribution function tests for multivariate normality. J. Statist. Comput. Simul., 28, 15–30.
Prentice, M.J. (1978). On invariant tests of uniformity for directions and orientations. Ann. Statist., 6, 169–176.
Quiroz, A.J., and Dudley, R.M. (1991). Some new tests for multivariate normality. Prob. Th. Rel. Fields, 87, 521–546.
Quiroz, A.J., and Trabucco, J.C. (1998). Kolmogorov-Smirnov type statistics for testing multivariate normality. Publ. Mat. Urug., 7, 67–82.
Rao, C.R., and Ali, H. (1998). An overall test for multivariate normality. Student, 2, 317–324.
Rayleigh, L (1919). On the problems of random vibrations and of random flights in one, two or three dimensions. Phil. Mag., 57, 321–347.
Rayner, J.C.W., and Best, D.J. (1989). Smooth tests of goodness of fit. Oxford University Press, New York.
Rayner, J.C.W., Best, D.J., and Matthews, K.L. (1995). Interpreting the skewness coefficient. Coram. Statist. — Th. Meth., 24, 593–600.
Romeu, J.L. (1992). A comparative visualization of a graphical multivariate normality GOF test. Computing Science and Statistics. Proc. 24rd Symp. Interface, 305–309.
Romeu, J.L., and Ozturk, A. (1993). A comparative study of goodness-of-fit tests for multivariate normality. J. Multiv. Anal., 46, 309–334.
Romeu, J.L., and Ozturk, A. (1996). A new graphical test for multivariate normality. Amer. J. Manage. Sci., 16, 5–48.
Roy, S.N. (1953). On a heuristic method of test construction and its use in multivariate analysis. Ann. Math. Statist., 24, 220–238.
Ruymgaart, F. (1998). A note on weak convergence of density estimators in Hilbert spaces. Statistics, 30, 331–343.
Schwager, S.J., and Margolin, B.H. (1982). Detection of multivariate normal outliers. Ann. Statist., 10, 943–954.
Singh, A. (1993). Omnibus robust procedures for assessment of multivariate normality and detection of multivariate outliers. In: Multivariate Environmental Statistics (G.P. Patil and C.R. Rao, eds.) North-Holland, Amsterdam, 445–488.
Srivastava, M.S. (1984). A measure of skewness and kurtosis and a graphical method for assessing multivariate normality. Statist. Prob. Lett., 2, 263–267.
Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Prob., 21, 34–71.
Szkutnik, Z. (1988). Most powerful invariant tests for binormality. Ann. Statist., 16, 292–301.
Tsai, K.T., and Koziol, J.A (1988). A correlation type procedure for assessing multivariate normality. Comm. Statist. — Simula., 17, 637–651.
Vasicek, O. (1976). A test for normality based on sample entropy. J. Roy. Statist. Soc. A, 139, 54–59.
Versluis, C. (1996). Comparison of tests for bivariate normality with unknown parameters by transformation to a univariate statistic. Comm. Statist. — Th. Meth., 25, 647–665.
Viollaz, A.J. (1995). Convergence in weigthed supremum norms of the Skorokhod representation of the estimated empirical process. Comm. Statist. — Th. Meth., 24, 2829–2839.
Wagner, Th. (1995). Asymptotic results on a class of tests for multivariate normality (in German). Diploma thesis, University of Karlsruhe, Germany.
Zhu, L., Fang, K.T., and Bhatti, M.I. (1997). On estimated projection pursuit Cramer-von Mises statistics. J. Multiv. Anal., 63, 1–14.
Zhu, L., Wong, H.L., and Fang, K.T. (1995). A test for multivariate normality based on sample entropy and projection pursuit. J. Statist. Plann. Infer., 45, 373–385.
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Henze, N. Invariant tests for multivariate normality: a critical review. Statistical Papers 43, 467–506 (2002). https://doi.org/10.1007/s00362-002-0119-6
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DOI: https://doi.org/10.1007/s00362-002-0119-6
Keywords and Phrases
- Tests for multivariate normality
- affine invariance
- consistency
- multivariate skewness
- multivariate kurtosis
- Roy’s union-intersection principle
- empirical characteristic function
- angles and radii
- projection pursuit
- locally best invariant test