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Year 2021, Volume: 50 Issue: 3, 872 - 894, 07.06.2021

Orhan Kesemen Buğra Kaan Tiryaki Özge Tezel Eda Özkul

https://doi.org/10.15672/hujms.644516

Abstract

References

  • [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.

A new goodness of fit test for multivariate normality

Year 2021, Volume: 50 Issue: 3, 872 - 894, 07.06.2021

Orhan Kesemen Buğra Kaan Tiryaki Özge Tezel Eda Özkul

https://doi.org/10.15672/hujms.644516

Abstract

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.

Keywords

multivariate empirical distribution function goodness of fit test elliptical distributions

References

  • [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.
There are 49 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Orhan Kesemen 0000-0002-5160-1178

Buğra Kaan Tiryaki 0000-0003-0995-7389

Özge Tezel 0000-0003-2815-686X

Eda Özkul 0000-0002-9840-8818

Publication Date June 7, 2021
Published in Issue Year 2021 Volume: 50 Issue: 3

Cite

APA Kesemen, O., Tiryaki, B. K., Tezel, Ö., Özkul, E. (2021). A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics, 50(3), 872-894. https://doi.org/10.15672/hujms.644516
AMA Kesemen O, Tiryaki BK, Tezel Ö, Özkul E. A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics. June 2021;50(3):872-894. doi:10.15672/hujms.644516
Chicago Kesemen, Orhan, Buğra Kaan Tiryaki, Özge Tezel, and Eda Özkul. “A New Goodness of Fit Test for Multivariate Normality”. Hacettepe Journal of Mathematics and Statistics 50, no. 3 (June 2021): 872-94. https://doi.org/10.15672/hujms.644516.
EndNote Kesemen O, Tiryaki BK, Tezel Ö, Özkul E (June 1, 2021) A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics 50 3 872–894.
IEEE O. Kesemen, B. K. 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, 2021, doi: 10.15672/hujms.644516.
ISNAD Kesemen, Orhan et al. “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.
JAMA Kesemen O, Tiryaki BK, Tezel Ö, Özkul E. A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics. 2021;50:872–894.
MLA Kesemen, Orhan et al. “A New Goodness of Fit Test for Multivariate Normality”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, 2021, pp. 872-94, doi:10.15672/hujms.644516.
Vancouver Kesemen O, Tiryaki BK, Tezel Ö, Özkul E. A new goodness of fit test for multivariate normality. Hacettepe Journal of Mathematics and Statistics. 2021;50(3):872-94.

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