Year 2021, Volume 50 , Issue 3, Pages 872 - 894 2021-06-07

This paper presents a multivariate Kolmogorov-Smirnov (MVKS) goodness of fit test for multivariate normality. The proposed test is based on the difference between the empirical distribution function and the theoretical distribution function. While calculating them in multivariate case, the problem is that the variables cannot be distribution-free as in the univariate case. Firstly, the variables are made independent to solve this problem and the Rosenblatt transform is applied for independence of variates. Then the theoretical and empirical distribution values are calculated and the MVKS test statistic is computed. It provides an easy calculation for d-dimensional data by using the same algorithm and critical table values. This paper demonstrates the effectiveness of the MVKS for different dimensions with a simulation study which also includes the comparison of the MVKS critical tables with univariate Kolmogorov-Smirnov (KS) critical table and the power comparisons of the MVKS (bivariate case) against with the existing bivariate normality tests. Lastly, the MVKS is applied to two different multivariate data sets to confirm that it achieves consistent, accurate and correct results.
multivariate empirical distribution function, goodness of fit test, elliptical distributions
  • [1] K. Alam, R. Abernathy and C.L. Williams, Multivariate goodness-of-fit tests based on statistically equivalent blocks, Comm. Statist. Theory Methods 22 (6), 1515-1533, 1993.
  • [2] K. Alam and C.L. Williams, A multivariate goodness-of-fit test for stochastically ordered distributions, Biom. J. 37 (8), 945-956, 1995.
  • [3] F.R. Bach and M.I. Jordan, Kernel independent component analysis, J. Mach. Learn. Res. 3 (Jul), 1-48, 2002.
  • [4] L. Baringhaus and N. Henze, A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1), 339-348, 1988.
  • [5] H. Bozdogan and D.E. Ramirez, Testing for model fit: Assessing the Box–Cox transformations of multivariate data to nearnormality, Comput. Stat. Q. 3, 127-150, 1986.
  • [6] A. Cabaña and E.M. Cabaña, Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions, Ann. Statist. 25 (6), 2388- 2409, 1997.
  • [7] R. Chicheportiche and J.P. Bouchaud, Goodness-of-fit tests with dependent observations, J. Stat. Mech. Theory Exp. 9, P09003, 2011.
  • [8] S.N. Chiu and K.I. Liu, Generalized Cramér-von Mises goodness-of-fit tests for multivariate distributions, Comput. Statist. Data Anal. 53 (11), 3817-3834, 2009.
  • [9] W.G. Cochran, The χ2 test of goodness of fit, Ann. Math. Statist. 23 (3), 315-345, 1952.
  • [10] H. Cramér, On the composition of elementary errors: First paper: Mathematical deductions, Scand. Actuar. J. 1928 (1), 13-74, 1928.
  • [11] S. Csorgo, Testing for normality in arbitrary dimension, Ann. Statist. 14 (2), 708-723, 1986.
  • [12] S. Facchinetti, A procedure to find exact critical values of Kolmogorov-Smirnov test, Statistica Applicata – Italian Journal of Applied Statistics 21 (34), 337–359, 2009.
  • [13] Y. Fan, Goodness-of-fit tests for a multivariate distribution by the empirical characteristic function, J. Multivariate Anal. 62 (1), 36-63, 1997.
  • [14] G. Fasano and A. Franceschini, A multidimensional version of the Kolmogorov- Smirnov test, Mon. Notices Royal Astron. Soc. 225 (1), 155-170, 1987.
  • [15] S. Ghosh and F.H. Ruymgaart, Applications of empirical characteristic functions in some multivariate problems, Canad. J. Statist. 20 (4), 429-440, 1992.
  • [16] A. Hyvärinen, J. Karhunen and E. Oja, Independent Component Analysis, John Wiley & Sons, 2004.
  • [17] R.A. Johnson and D.W. Wichern, Applied Multivariate Statistical Analysis, 5th ed., Prentice Hall, New Jersey, 2001.
  • [18] A. Justel, D. Peña and R. Zamar, A multivariate Kolmogorov-Smirnov test of goodness of fit, Statist. Probab. Lett. 35 (3), 251-259, 1997.
  • [19] J.F. Kenneys, Mathematics of Statistics, D. Van Nostrand Company Inc, 2013.
  • [20] A. Kolmogorov, Sulla determinazione empirica di una lgge di distribuzione (in Italian), G. Inst. Ital. Attuari 4 (1933), 83-91, 1933.
  • [21] J.A. Koziol, Assessing multivariate normality: A compendium, Comm. Statist. Theory Methods 15 (9), 2763-2783, 1986.
  • [22] N.H. Kuiper, Tests concerning random points on a circle, Nederl. Akad. Wetensch. Proc. Ser. A 63 (1), 38-47, 1960.
  • [23] H.W. Lilliefors, On the Kolmogorov-Smirnov test for normality with mean and variance unknown, J. Amer. Statist. Assoc. 62 (318), 399-402, 1967.
  • [24] S.W. Looney, How to use tests for univariate normality to assess multivariate normality, Amer. Statist. 49 (1), 64-70, 1995.
  • [25] R. Lopes, I. Reid and P. Hobson, The two-dimensional Kolmogorov-Smirnov test, Proceedings of XI International Workshop on Advanced Computing and Analysis Techniques in Physics Research, Amsterdam, The Netherlands, 2007.
  • [26] J.F. Malkovich and A.A. Afifi, On tests for multivariate normality, J. Amer. Statist. Assoc. 68 (341), 176-179, 1973.
  • [27] K.V. Mardia, Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (3), 519-530, 1970.
  • [28] K.V Mardia and J.T. Kent, Multivariate Analysis, Academic Press, London, 1979.
  • [29] K.V. Mardia and J.T. Kent, Rao score tests for goodness of fit and independence, Biometrika 78 (2), 355-363, 1991.
  • [30] J.R. Massey and J. Frank, The Kolmogorov-Smirnov test for goodness of fit, J. Amer. Statist. Assoc. 46 (253), 68-78, 1951.
  • [31] M.P. McAssey, An empirical goodness-of-fit test for multivariate distributions, J. Appl. Stat. 40 (5), 1120-1131, 2013.
  • [32] M.S. Mecibah, T.E. Boukelia, R. Tahtah and K. Gairaa, Introducing the best model for estimation the monthly mean daily global solar radiation on a horizontal surface (Case study: Algeria), Renew. Sust. Energ. Rev. 36, 194-202, 2014.
  • [33] D.S. Moore and J.B. Stubblebine, Chi-square tests for multivariate normality with application to common stock prices, Comm. Statist. Theory Methods 10 (8), 713-738, 1981.
  • [34] G.S. Mudholkar, M. McDermott and D.K. Srivastava, A test of p-variate normality, Biometrika 79 (4), 850-854, 1992.
  • [35] A. Novobilski and F. Kamangar, Absolute Percent Error Based Fitness Functions for Evolving Forecast Models, Proceedings of FLAIRS Conference, FL, USA, 591-595, 2001.
  • [36] J.A. Peacock, Two-dimensional goodness-of-fit testing in astronomy, Mon. Notices Royal Astron. Soc. 202 (3), 615-627, 1983.
  • [37] N.M. Razali and Y.B. Wah, Power comparisons of Shapiro-Wilk, Kolmogorov- Smirnov, Lilliefors and Anderson-Darling tests, Journal of Statistical Modeling and Analytics 2 (1), 21-33, 2011.
  • [38] J.L. Romeu and A. Ozturk, A comparative study of goodness-of-fit tests for multivariate normality, J. Multivariate Anal. 46 (2), 309-334, 1993.
  • [39] M. Rosenblatt, Remarks on a multivariate transformation, Ann. Math. Statist. 23 (3), 470-472, 1952.
  • [40] J.P. Royston, Some techniques for assessing multivarate normality based on the Shapiro-Wilk W, J. R. Stat. Soc. Ser. C. Appl. Stat. 32 (2), 121-133, 1983.
  • [41] N.J. Small, Marginal skewness and kurtosis in testing multivariate normality, J. R. Stat. Soc. Ser. C. Appl. Stat. 29, 85-87, 1980.
  • [42] N.V. Smirnov, On the estimation of the discrepancy between empirical curves of distribution for two independent samples, Bull. Math. Univ. Moscou 2 (2), 3-14, 1939.
  • [43] M.S. Srivastava, A measure of skewness and kurtosis and a graphical method for assessing multivariate normality, Statist. Probab. Lett. 2 (5), 263-267, 1984.
  • [44] M.S. Srivastava and T.K. Hui, On assessing multivariate normality based on Shapiro- Wilk W statistic, Statist. Probab. Lett. 5 (1), 15-18, 1987.
  • [45] B. Sürücü, Goodness-of-fit tests for multivariate distributions, Comm. Statist. Theory Methods 35 (7), 1319-1331, 2006.
  • [46] G.J. Székely and M.L. Rizzo, A new test for multivariate normality, J. Multivariate Anal. 93 (1), 58-80, 2005.
  • [47] R. Von Mises, Wahrscheinlichkeitsrechnung und ihre anwendung in der statistik und theorestischen physik, Leipzig, Deutsche, 1931.
  • [48] C. Zhang, Y. Xiang and X. Shen, Some multivariate goodness-of-fit tests based on data depth, J. Appl. Stat. 39 (2), 385-397, 2012.
  • [49] M. Zhou and Y. Shao, A powerful test for multivariate normality, J. Appl. Stat. 41 (2), 351-363, 2014.
Primary Language en
Subjects Statistics and Probability
Journal Section Statistics
Authors

Orcid: 0000-0002-5160-1178
Author: Orhan KESEMEN
Institution: Karadeniz Technical University
Country: Turkey


Orcid: 0000-0003-0995-7389
Author: Buğra Kaan TİRYAKİ (Primary Author)
Institution: Karadeniz Technical University
Country: Turkey


Orcid: 0000-0003-2815-686X
Author: Özge TEZEL
Institution: Karadeniz Technical University
Country: Turkey


Orcid: 0000-0002-9840-8818
Author: Eda ÖZKUL
Institution: Karadeniz Technical University
Country: Turkey


Dates

Publication Date : June 7, 2021

Bibtex @research article { hujms644516, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2021}, volume = {50}, pages = {872 - 894}, doi = {10.15672/hujms.644516}, title = {A new goodness of fit test for multivariate normality}, key = {cite}, author = {Kesemen, Orhan and Tiryaki, Buğra Kaan and Tezel, Özge and Özkul, Eda} }
APA Kesemen, O , Tiryaki, B , Tezel, Ö , Özkul, E . (2021). A new goodness of fit test for multivariate normality . Hacettepe Journal of Mathematics and Statistics , 50 (3) , 872-894 . DOI: 10.15672/hujms.644516
MLA Kesemen, O , Tiryaki, B , Tezel, Ö , Özkul, E . "A new goodness of fit test for multivariate normality" . Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 872-894 <https://dergipark.org.tr/en/pub/hujms/issue/62731/644516>
Chicago Kesemen, O , Tiryaki, B , Tezel, Ö , Özkul, E . "A new goodness of fit test for multivariate normality". Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 872-894
RIS TY - JOUR T1 - A new goodness of fit test for multivariate normality AU - Orhan Kesemen , Buğra Kaan Tiryaki , Özge Tezel , Eda Özkul Y1 - 2021 PY - 2021 N1 - doi: 10.15672/hujms.644516 DO - 10.15672/hujms.644516 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 872 EP - 894 VL - 50 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.644516 UR - https://doi.org/10.15672/hujms.644516 Y2 - 2021 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics A new goodness of fit test for multivariate normality %A Orhan Kesemen , Buğra Kaan Tiryaki , Özge Tezel , Eda Özkul %T A new goodness of fit test for multivariate normality %D 2021 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 50 %N 3 %R doi: 10.15672/hujms.644516 %U 10.15672/hujms.644516
ISNAD Kesemen, Orhan , Tiryaki, Buğra Kaan , Tezel, Özge , Özkul, Eda . "A new goodness of fit test for multivariate normality". Hacettepe Journal of Mathematics and Statistics 50 / 3 (June 2021): 872-894 . https://doi.org/10.15672/hujms.644516
AMA Kesemen O , Tiryaki B , Tezel Ö , Özkul E . A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 872-894.
Vancouver Kesemen O , Tiryaki B , Tezel Ö , Özkul E . A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 872-894.
IEEE O. Kesemen , B. Tiryaki , Ö. Tezel and E. Özkul , "A new goodness of fit test for multivariate normality", Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 872-894, Jun. 2021, doi:10.15672/hujms.644516