Tests for multivariate normality—a critical review with emphasis on weighted L2-statistics

• Bruno Ebner ^[1] ; Norbert Henze ^[1]
  1. [1] Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, D-76133, Karlsruhe, Germany
• Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 29, Nº. 4, 2020, págs. 845-892
• Idioma: inglés
• DOI: 10.1007/s11749-020-00740-0
• Texto completo no disponible (Saber más ...)
• Resumen

  • This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted L2-statistics. Since weighted L2-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt.

• Referencias bibliográficas
  • Arcones MA (2007) Two tests for multivariate normality based on the characteristic function. Mathe Meth Stat 16:177–201
  • Axler SJ, Bourdon P, Ramey W (2001) Harmonic function theory, vol 137, 2nd edn. Graduate texts in mathematics. Springer, New York
  • Bahadur RR (1960) Stochastic comparison of tests. Ann Math Stat 31:276–295
  • Bakshaev A, Rudzkis R (2017) Goodness-of-fit tests based on the empirical characteristic function. Lith Math J 57:155–170
  • Baringhaus L, Henze N (1988) A consistent test for multivariate normality based on the empirical characteristic function. Metrika 35:339–348
  • Baringhaus L, Henze N (1991) Limit distributions for measures of multivariate skewness and kurtosis based on projections. J Multiv Anal 38:51–69
  • Baringhaus L, Henze N (1992) Limit distributions for Mardia’s measure of multivariate skewness. Ann Stat 20:1889–1902
  • Baringhaus L, Taherizadeh F (2010) Empirical Hankel transforms and their applications to goodness-of-fit tests. J Multiv Anal 101:1445–1467
  • Baringhaus L, Ebner B, Henze N (2017) The limit distribution of weighted L2-goodness-of-fit statistics under fixed alternatives, with applications....
  • Batsidis A, Martin N, Pardo L, Zografos K (2013) A necessary power divergence type family tests of multivariate normality. Commun Stat Simul...
  • Becker M, Klößner S (2017) PearsonDS: pearson Distribution System. R package version 1.1. https://CRAN.R-project.org/package=PearsonDS
  • Bosq D (2000) Linear processes in function spaces. Springer, New York
  • Bowman AW, Foster PJ (1993) Adaptive smoothing and density-based tests of multivariate normality. J Am Stat Assoc 88:529–537
  • Butsch L, Ebner B (2020) mnt: Affine invariant tests of multivariate normality. R package version 1.3. https://CRAN.R-project.org/package=mnt
  • Cardoso de Oliveira IR, Ferreira DF (2010) Multivariate extension of chi-squared univariate normality test. J Stat Comput Simul 80:513–526
  • Cox DR, Small NJH (1978) Testing multivariate normality. Biometrika 65:263–272
  • Cramér H (1936) Über eine Eigenschaft der normalen Verteilungsfunktion. Math Z 41:405–414
  • Csörgő S (1986) Testing for normality in arbitray dimension. Ann Stat 14:708–723
  • Csörgő S (1989) Consistency of some tests for multivariate normality. Metrika 36:107–116
  • Czado C, Freitag G, Munk A (2007) A nonparametric test for similarity of marginals-with applications to the assessment of bioequivalence....
  • Dette H, Munk A (2003) Some methodological aspects of validation of models in nonparametric regression. Stat Neerlandica 57:207–244
  • Dörr Ph, Ebner B, Henze N (2020) Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces. J Stat...
  • Dörr Ph, Ebner B, Henze N (2020a) A new test of multivariate normality by a double estimation in a characterizing PDE. Metrika. https://doi.org/10.1007/s00184-020-00795-x
  • Doornik JA, Hansen H (2008) An omnibus test for univariate and multivariate normality. Oxf Bull Econ Stat 70:927–939
  • Eaton ML, Perlman MD (1973) The non-singularity of generalized sample covariance matrices. Ann Stat 1:710–717
  • Ebner B (2012) Asymptotic theory for the test for multivariate normality by Cox and Small. J Multiv Anal 111:368–379
  • Enomoto R, Hanusz Z, Hara A, Seo T (2020) Multivariate normality test using normalizing transformation for Mardia’s multivariate kurtosis....
  • Epps TW, Pulley LB (1983) A test for normality based on the empirical characteristic function. Biometrika 70:723–726
  • Fang K-T, Li R-Z, Liang J-J (1998) A multivariate version of Ghosh’s T3-plot to detect non-multinormality. Comput Stat Data Anal 28:371–386
  • Farrell PJ, Salibian-Barrera M, Naczk L (2007) On tests for multivariate normality and associated simulation studies. J Stat Comput Simul...
  • Fromont M, Laurent B (2006) Adaptive goodness-of-fit tests in a density model. Ann Stat 34:680–720
  • Genz A, Bretz F (2009) Computation of multivariate normal and t probabilities, vol 195. Lecture notes in statistics. Springer, Berlin
  • Górecki T, Horváth L, Kokoszka P (2020) Tests of normality of functional data. Int Stat Rev. https://doi.org/10.1111/insr.12362
  • Groemer H (1996) Geometric applications of Fourier series and spherical harmonics. Cambridge University Press, Cambridge
  • Gürtler N (2000) Asymptotic theorems for the class of BHEP-tests for multivariate normality with fixed and variable smoothing parameter (in...
  • Hadjicosta E, Richards D (2019) Integral transform methods in goodness-of-fit testing, I: the gamma distributions. Metrika. https://doi.org/10.1007/s00184-019-00749-y
  • Hanusz Z, Tarasińska J (2008) A note on Srivastava and Hui’s test of multivariate normality. J Multiv Anal 99:2364–2367
  • Hanusz Z, Tarasińska J (2012) New tests for multivariate normality based on Small’s and Srivastava’s graphical methods. J Stat Comput Simul...
  • Hanusz Z, Enomoto R, Seo T, Koizumi K (2018) A Monte Carlo comparison of Jarque-Bera type tests and Henze-Zirkler test for multivariate normality....
  • Henze N, Zirkler B (1990) A class of invariant consistent tests for multivariate normality. Commun Stat Theory Methods 19:3595–3617
  • Henze N (1994) On Mardia’s kurtosis test for multivariate normality. Commun Stat Theory Methods 23:1031–1045
  • Henze N (1994b) The asymptotic behavior of a variant of multivariate kurtosis. Commun Stat Theory Methods 23:1047–1061
  • Henze N (1997) Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely. Stat Probab Lett 33:299–307
  • Henze N (1997b) Extreme smoothing and testing for multivariate normality. Stat Probab Lett 35:203–213
  • Henze N, Wagner Th (1997) A new approach to the class of BHEP tests for multivariate normality. J Multiv Anal 62:1–23
  • Henze N (2002) Invariant tests for multivariate normality: a critical review. Stat Pap 43:467–506
  • Henze N, Jiménez-Gamero MD (2019) A new class of tests for multinormality with i.i.d. and Garch data based on the empirical moment generating...
  • Henze N, Jiménez-Gamero MD, Meintanis SG (2019) Characterizations of multinormality and corresponding tests of fit, including for GARCH models....
  • Henze N, Visagie J (2020) Testing for nomality in any dimension based on a partial differential equation involving the moment generating function....
  • Henze N, Koch S (2020) On a test of normality based on the empirical moment generating function. Stat Papers 61:17–29
  • Holgersson HET (2006) A graphical method for assessing multivariate normality. Comput Stat 21:141–149
  • Hwu TJ, Han CP, Rogers KJ (2002) The combination test for multivariate normality. J Stat Comput Simul 72:379–390
  • Janssen A (2000) Global power function of goodness of fit tests. Ann Stat 28:239–253
  • Joenssen DW, Vogel J (2014) A power study of goodness-of-fit tests for multivariate normality implemented in R. J Stat Comput Simul 84:1055–1078
  • Jönsson K (2011) A robust test for multivariate normality. Econ Lett 113:199–201
  • Kac M, Siegert AJF (1947) An explicit representation of a stationary Gaussian process. Ann Math Stat 18:438–442
  • Kankainen A, Taskinen S, Oja H (2007) Tests of multinormality based on location vectors and scatter matrices. Stat Methods Appl 16:357–379
  • Kellner J, Celisse A (2019) A one-sample test for normality with kernel methods. Bernoulli 25:1816–1837
  • Kim N (2016) A robustified Jarque-Bera test for multivariate normality. Econ Lett 140:48–52
  • Kim I, Park S (2018) Likelihood ratio test for multivariate normality. Commun Stat Theory Methods 47:1923–1934
  • Koizumi K, Hyodo M, Pavlenko T (2014) Modified Jarque-Bera type tests for multivariate normality in a high-dimensional framework. J Stat Theory...
  • Koziol JA (1989) A note on measures of multivariate kurtosis. Biom J 31:619–624
  • Klar B (2002) A treatment of multivariate skewness, kurtosis, and related statistics. J Multiv Anal 83:141–165
  • Lee S, Ng CT (2011) Normality test for multivariate conditional heteroskedastic dynamic regression models. Econ Lett 111:75–77
  • Lee J, Lee S, Park S (2014) Maximum entropy test for GARCH models. Stat Methodol 22:8–16
  • Liang J-J, Bentler PM (1999) A t-distribution plot to detect non-multinormality. Comput Stat Data Anal 30:31–44
  • Liang J, Li R, Fang H, Fang K-T (2000) Testing multinormality based on low-dimensional projection. J Stat Plan Inference 86:129–141
  • Liang J, Pan WSY, Yang Z-H (2004) Characterization-based Q-Q-plots for testing multinormality. Stat Probab Lett 70:183–190
  • Madukaife M, Okafor FC (2018) A powerful invariant test for multivariate normality based on interpoint distances of principal components....
  • Madukaife M, Okafor FC (2019) A new large sample goodness of fit test for multivariate normality based on chi squared probability plots. Commun...
  • Majerski P, Szkutnik Z (2010) Approximations to most powerful invariant tests for multinormality against some irregular alternatives. Test...
  • Malkovich JF, Afifi AA (1973) On tests for multivariate normality. J Am Stat Assoc 68:176–179
  • Manzotti A, Quiroz AJ (2001) Spherical harmonics in quadratic forms for testing multivariate normality. Test 10:87–104
  • Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530
  • Maruyama Y (2007) On Srivastava’s multivariate sample skewness and kurtosis under non-normality. Stat Probab Lett 77:335–342
  • Mattner L (1997) Strict negative definiteness of integrals via complete monotonicity of derivatives. Trans Am Math Soc 349:3321–3342
  • Mecklin CJ, Mundfrom DJ (2004) An appraisal and Bibliography of tests for multivariate normality. Int Stat Rev 72:123–138
  • Mecklin CJ, Mundfrom DJ (2005) A Monte Carlo comparison of the type I and type II error rates of tests for multivariate normality. J Stat...
  • Meintanis SG, Hlávka Z (2010) Goodness-of-fit fests for bivariate and multivariate skew-normal distributions. Scand J Stat 37:701–714
  • Moore D, Stubblebine J (1981) Chi-square tests for multivariate normality with applications to common stock prices. Commun Stat Theory Methods...
  • Morgenstern D (2001) Proof of a conjecture by Walter Deuber concerning the distance between points of two types in Rd. Discrete Math 226:347–349
  • Móri TF, Rohatgi VK, Székely GJ (1993) On multivariate skewness and kurtosis. Theory Prob Appl 38:547–551
  • Müller C (1998) Analysis of spherical symmetries in Euclidean spaces. Springer, Berlin
  • Naito K (1998) Approximation of the power of kurtosis test for multinormality. J Multiv Anal 65:166–180
  • Nikitin YaYu (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, New York
  • Pudelko J (2005) On a new affine invariant and consistent test for multivariate normality. Probab Math Stat 25:43–54
  • R Core Team (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
  • Rizzo M, Székely G (2014) Energy: E-statistics (energy statistics). R package version 1.6.2 (2014). Available at: http://CRAN.R-project.org/package=energy
  • Ruckdeschel P, Kohl M, Stabla T, Camphausen F (2006) S4 classes for distributions. R News 6:2–6
  • Roy SN (1953) On a heuristic method of test construction and its use in multivariate analysis. Ann Math Stat 24:220–238
  • Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, London, New York
  • Srivastava MS, Hui TK (1987) On assessing multivariate normality based on Shapiro-Wilk W statistic. Stat Probab Lett 5:15–18
  • Székely G, Rizzo M (2005) A new test for multivariate normality. J Multiv Anal 93:58–80
  • Székely G, Rizzo M (2013) Energy statistics: a class of statistics based on distances. J Stat Plan Inference 143:1249–1272
  • Székely G, Rizzo M (2016) Energy distance. WIREs Comput Stat 8:27–38
  • Takeuchi K (2020) Contributions on theory of mathematical statistics. The tests for multivariate normality. Tokyo, Springer
  • Tan M, Fang H-B, Tian G-L, Wei G (2005) Testing multivariate normality in incomplete data of small sample size. J Multiv Anal 93:164–179
  • Tenreiro C (2007) On the asymptotic behaviour of location-scale invariant Bickel-Rosenblatt tests. J Stat Plan Inference 137:103–116
  • Tenreiro C (2009) On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Comput Stat Data Anal 53:1038–1053
  • Tenreiro C (2011) An affine invariant multiple test procedure for assessing multivariate normality. Comput Stat Data Anal 55:1980–1992
  • Tenreiro C (2017) A new test for multivariate normality by combining extreme and nonextreme BHEP tests. Commun Stat Simul Comput 46:1746–1759
  • Tenreiro C (2019) On the automatic selection of the tuning parameter appearing in certain families of goodness-of-fit tests. J Stat Comput...
  • Thode HC (2002) Testing for normality. Statistics: textbooks and monographs, vol 164. Marcel Dekker Inc, New York
  • Thulin M (2014) Tests for multivariate normality based on canonical correlations. Stat Methods Appl 23:189–208
  • Villasenor Alva JA, González Estrada E (2009) A generalization of Shapiro-Wilk’s test for multivariate normality. Commun Stat Theory Methods...
  • Voinov V, Pya N, Makarov R, Voinov Y (2016) New invariant and consistent chi-squared type goodness-of-fit tests for multivariate normality...
  • Volkmer H (2014) A characterization of the normal distribution. J Stat Theory Appl 13:83–85
  • Wang CC (2014) A MATLAB package for multivariate normality test. J Stat Comput Simul 85:166–188
  • Wang CC, Hwang YT (2011) A new functional statistic for multivariate normality. Stat Comput 21:501–509
  • Wellek S (2010) Testing statistical hypotheses of equivalence and noninferiority. CRC Press, Boca Raton
  • Yamada T, Himeno T (2019) Estimation of multivariate 3rd moment for high-dimensional data and its application for testing multivariate normality....
  • Yamada T, Romer M, Richards D, St P (2015) Kurtosis test for multivariate normality with monotone incomplete data. Test 24:532–557
  • Zhou M, Shao Y (2014) A powerful test for multivariate normality. J Appl Stat 41:351–363
  • Zinger AA, Kakosyan AV, Klebanov LB (1992) Characterization of distributions by mean values of statistics and certain probabilistic metrics....