Performance Analysis of Powers of Skewness and Kurtosis Based Multivariate Normality Tests and Use of Extended Monte Carlo Simulation for Proposed Novelty Algorithm
American Journal of Theoretical and Applied Statistics
Volume 4, Issue 2-1, March 2015, Pages: 11-18
Received: Dec. 12, 2014; Accepted: Dec. 13, 2014; Published: Mar. 11, 2015
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Authors
Vishwa Nath Maurya, Department of Pure & Applied Mathematics and Statistics, School of Science & Technology, The University of Fiji, Lautoka, Fiji Islands
Ram Bilas Misra, Division of Applied Mathematics, State University of New York, Incheon, Republic of Korea & Ex-Vice Chancellor, Dr. R. M. L. Avadh University, Faizabad, UP, India
Chandra K. Jaggi, Department of Operations Research, University of Delhi, New Delhi, India
Avadhesh Kumar Maurya, Department of Electronics & Communication Engineering, Lucknow Institute of Technology, U. P. Technical University, Lucknow, India
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Abstract
An ample study of the comparative powers of a number of omnibus multivariate normality tests is main object in this paper. Since testing for multivariate normality tests is considerably more challenging process than for testing of univariate one and therefore, study of testing for multivariate normality tests has its increasing demand. Through this paper, we have explored several techniques for assessing multivariate normality (MVN) and as well as comparative analysis for their competence have also been demonstrated. The results of extensive Monte Carlo simulation study of the size corrected power of various tests of multivariate normality for drawn samples from contaminated normal distributions have been explored as well. Moreover, a novel algorithm has been proposed in order to evaluate the size corrected powers for testing multivariate normality. The algorithm proposed herein is a fast easily implementable algorithm and it can be applied for both types of univariate and multivariate normality tests. Using Different omnibus tests for sample size 50 and 200, graphs for empirical powers of multivariate normal data with lower and upper contamination have been presented. Finally, some significant conclusions of our present study have been drawn.
Keywords
Multivariate Normality Tests, Goodness-of-Fit Tests, Correlation Coefficient, Skewness, Kurtosis, Monte Carlo Simulation Technique
To cite this article
Vishwa Nath Maurya, Ram Bilas Misra, Chandra K. Jaggi, Avadhesh Kumar Maurya, Performance Analysis of Powers of Skewness and Kurtosis Based Multivariate Normality Tests and Use of Extended Monte Carlo Simulation for Proposed Novelty Algorithm, American Journal of Theoretical and Applied Statistics. Special Issue: Scope of Statistical Modeling and Optimization Techniques in Management Decision Making Process. Vol. 4, No. 2-1, 2015, pp. 11-18. doi: 10.11648/j.ajtas.s.2015040201.12
References
[1]
Anscombe F.J. and Glynn W.J., Distribution of the kurtosis statistic for normal samples, Biometrika, Vol. 70, pp.227–234, 1983.
[2]
Bera A.K., A new test for normality, Economics Letter, Vol. 9, pp.263-268, 1982.
[3]
Bowman K.O. and Shenton L.R., Omnibus test contours for departures from normality based on and , Biometrika, Vol. 62, pp.243–250, 1975.
[4]
Cox D.R. and Small N.J.H., Testing multivariate normality, Biometrika, Vol. 65, pp.267-372, 1978, doi:10.1093/biomet/65.2.263.
[5]
D’Agostino R. and Pearson E.S., Tests for departure from normality: Empirical results for the distributions of and , Biometrika, Vol. 60, pp.613–622, 1973.
[6]
D'Agostino R.B., Belanger A. and Jr D'Agostino R.B., A Suggestion for Using Powerful and Informative Tests of Normality, The American Statistician, Vol. 44, pp.316-321, 1990.
[7]
D'Agostino R.B., Transformation to Normality of the Null Distribution of , Biometrika, Vol. 57, pp.679-681, 1970.
[8]
Doornik J.A. and Hansen H., An Omnibus Test for Univariate and Multivariate Normality, Oxford Bulletin of Economics and Statistics, Vol.70, pp. 0305-9049, 2008.
[9]
Enomoto R., Okamoto N., and Seo T., On the distribution of test statistic using Srivastava’s skewness and kurtosis, Technical Report No. 10-07, Statistical Research Group, Hiroshima University, 2010.
[10]
Friedman J.H. and Rafsky L.C., Multivariate Generalizations of the Wald-Wolfowitz and Smirnov Two-Sample Tests, The Annals of Statistics Vol. 7(4), pp. 697, 1979. doi:10.1214/aos/1176344722.
[11]
Grianadesikan R., Methods for statistical data analysis of multivariate observations, Wiley, New York, USA, 1977.
[12]
Inhof J.P., Computing the distribution of quadratic forms in normal variables, Biometrika, Vol.48, pp.419-426, 1961.
[13]
Isogai T., On a measure of multivariate skewness and a test for multivariate normality, Ann. Inst. Statist. Math, 34, pp.531-541, 1982.
[14]
Isogai T., On using influence functions for testing multivariate normality, Ann. Inst. Statist. Math, Vol. 41, pp.169-180, 1989.
[15]
Jarque C.M. and Bera A.K., A test for normality of observations and regression residuals, Internat Stat Rev, Vol. 55, pp.163–172, 1987.
[16]
Koizumi K., Okamoto N. and Seo T., On Jarque–Bera tests for assessing multivariate normality, J Stat Adv. Theory Appl, Vol.1, pp.207–220, 2009.
[17]
Koziol J.A., Assessing multivariate normality: a compendium, Communications in Statistics-Theory and Methods, Vol. 15, pp. 2763-2783, 1986.
[18]
Looney S.W., How to use tests for univariate normality to assess multivariate normality, Amer. Statist, Vol. 39, pp.75-79, 1995.
[19]
Malkovich J.F. and Afifi A.A., On tests for multivariate normality, Journal of the American Statistical Association, Vol. 68, pp.176–179, 1973.
[20]
Mardia K.V. and Foster K., Omnibus tests of multinormality based on skewness and kurtosis, Commun. Statist, Vol.12, pp.207-221, 1983.
[21]
Mardia K.V., Applications of some measures of multivariate skewness and kurtosis for testing normality and robustness studies, Sankhya, Vol. 36, pp.115-128, 1974.
[22]
Mardia K.V., Measures of multivariate skewness and kurtosis with applications, Biometrika, Vol. 57, pp. 519-530, 1970.
[23]
Maurya A.K., Singh R.K., Maurya V.N. and Misra R.B., Analysis and simulation of harmonics current in power electronics equipment generated by nonlinear loads: hysteresis current control approach, American Journal of Engineering Technology, Academic and Scientific Publishing, USA, Vol.2(1), pp. 1-13, 2014
[24]
Maurya Avadhesh Kumar and Maurya V.N., A novel algorithm for optimum balancing energy consumption LEACH protocol using numerical simulation technique, International Journal of Electronics Communication and Electrical Engineering, Algeria, Vol. 3(4), pp. 1-19, 2013, ISSN: 2277-7040
[25]
Maurya V.N. and Maurya A.K., Polynomial simulation and refutation of complex formulas of resolution over linear equations in propositional proof system, American Journal of Modeling and Optimization, Science & Education Publishing, USA, Vol.2(2), pp. 34-38, 2014, ISSN (Print) 2333-1143, ISSN (Online) 2333-1267
[26]
Maurya V.N., Arora Diwinder Kaur and Maurya Avadhesh Kumar, A survey report of parameter and structure learning in Bayesian network inference, International Journal of Information Technology & Operations Management, Academic and Scientific Publishing, New York, USA, Vol. 1(2), pp. 11-28, 2013, ISSN: 2328-8531
[27]
Maurya V.N., Arora Diwinder Kaur, Maurya Avadhesh Kumar and Gautam Ram Asrey, Numerical simulation and design parameters in solar photovoltaic water pumping systems, American Journal of Engineering Technology, Academic & Scientific Publishing, New York, USA, Vol.1(1), pp. 1-09, 2013
[28]
Maurya V.N., Arora Diwinder Kaur, Maurya Avadhesh Kumar and Gautam R.A., Exact modeling of annual maximum rainfall with Gumbel and Frechet distributions using parameter estimation techniques, World of Sciences Journal, Engineers Press Publishing, Vienna, Austria, Vol. 1(2), pp.11-26, 2013, ISSN: 2307-3071
[29]
Maurya V.N., Maurya A.K. and Kaur D., A survey report on nonparametric hypothesis testing including Kruskal-Wallis ANOVA and Kolmogorov–Smirnov goodness-fit-test, International Journal of Information Technology & Operations Management, Academic and Scientific Publishing, New York, USA, Vol.1(2), pp. 29-40, 2013, ISSN: 2328-8531
[30]
Maurya V.N., Misra R.B., Jaggi C.K., Arneja C.S., Maurya A.K. and and Maharaj Yogesh, Comparative analysis for the maximum precision using systematic and stratified random sampling techniques, Edited Book on Dynamics of Business through Management, Engineering, Science & Technology, Mohit Publications, New Delhi, India, 2014
[31]
Maurya V.N., Misra R.B., Jaggi Chadra K., Maurya A.K. and Arora D.K., Design and estimate of the optimal parameters of adaptive control chart model using Markov chains technique, Special Issue: Scope of Statistical Modeling and Optimization Techniques in Management and Decision Making Process, American Journal of Theoretical and Applied Statistics, Science Publishing Group, USA, 2014
[32]
Maurya V.N., Numerical simulation for nutrients propagation and microbial growth using finite difference approximation technique, International Journal of Mathematical Modeling and Applied Computing, Academic & Scientific Publishing, New York, USA, Vol. 1(7), pp. 64-76, 2013, ISSN: 2332-3744
[33]
Maurya V.N., Singh Bijay, Singh V.V., Maurya A.K. and Arora D.K., Statistical modeling and parameter estimates of dbh-crown diameter prediction using sampling technique: A case study, Edited Book on Dynamics of Business through Management, Engineering, Science & Technology, Mohit Publications, New Delhi, India, 2014
[34]
Maurya V.N., Singh V.V., and Yusuf Madaki Umar, Statistical analysis on the rate of kidney (renal) failure, Application and Future Scope of Fundamental Mathematical and Computational Sciences in Engineering & Technology, American Journal of Applied Mathematics and Statistics, Science & Education Publishing, USA, Vol. 2, No. 6A, pp. 6-12, 2014
[35]
Nakagawa S., Hashiguchi H. and Niki N., Improved omnibus test statistic for normality, Computational Statistics, DOI 10.1007/s00180-011-0258-0, 2011.
[36]
Okamoto N. and Seo T., On the distribution of multivariate sample skewness for assessing multivariate normality, Technical Report No. 08-01, Statistical Research Group, Hiroshima University, Hiroshima, 2008.
[37]
Ozturk A. and Romeu J.L., A new method for assessing multivariate normality with graphical applications, Communications in Statistics-Simulation and Computation, Vol. 21, pp.15-34, 1992.
[38]
Pearson E.S., D'Agostino R.B. and Bowman K.O., Test for departure from normality: Comparison of powers, Biometrika, Vol. 64, pp.231-246, 1977.
[39]
Rencher A.C., Methods of Multivariate Analysis, New York: Wiley, 2002.
[40]
Romeu J.L. and Ozturk A., A comparative study of goodness-of-fit tests for multivariate normality, Journal of Multivariate Analysis, Vol. 46, pp. 309-334, 1993.
[41]
Roy S.N., On a heuristic method of test construction and its use in multivariate analysis, Ann. Math. Statist., Vol. 24, pp.220-238, 1953.
[42]
Shenton L.R. and Bowman K.O., A bivariate model for the distribution of and , Journal of the American Statistical Association, Vol. 72, pp. 206–211, 1977.
[43]
Small N.J.H., Marginal skewness and kurtosis in testing multivariate normality, Appl. Statist., Vol. 29, pp.85-87, 1980.
[44]
Smith S.P. and Jain A.K., A test to determine the multivariate normality of a data set. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 10(5), pp. 757, 1988, doi:10.1109/34.6789.
[45]
Srivastava M.S., A measure of skewness and kurtosis and a graphical method for assessing multivariate normality, Statist. Probab. Lett, Vol.2, pp.263-267, 1984.
[46]
Thode H., Testing for Normality, Marcel Dekker, New York, 2002.
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