### A Comparison among Twenty-Seven Normality Tests

#### Abstract

This paper studies and compares the power of 27 normality tests via the Monte Carlo simulation of sample data generated from symmetric three short-tailed and three long-tailed, asymmetric three short-tailed and three long-tailed distributions under different sample sizes by using R codes. Our simulation results showed that for symmetric short-tailed Uniform (0, 1), t (10) and Beta (4, 4) distributions showed that the Agostino-Pearson K2, Jarque Bera and Kurtosis tests have better power; and Shapiro-Wilk, Cramer-Von Mises and Hegazy-Green 1 tests have lower power. For symmetric t (5), Laplace (0, 1) and Logistic (2, 1) long-tailed distributions, Robust Jarque Bera, Geary and Jarque Bera observed most powerful tests and Bonett-Seier Test, Chi-Square and Bonett-Seier tests were observed least powerful tests. Under asymmetric Weibull (2, 3), Gompertz (10, 0.002) and Gamma (1, 5) short-tailed distributions showed that the Shapiro Wilk, Kurtosis and Robust Jarque Bera test were highlighted as a more powerful test and the Cramer-Von Mises, Shapiro-Wilk and Hegazy-Green 1 tests were highlighted as a least powerful tests. Alternatively, the asymmetric Lognormal (0, 1), Exponential (1) and Chi-square (5) long-tailed distributions, the Shapiro-Wilk test was the most powerful test for all and Jarque-Bera, Hegazy-Green 1 and Bontemps-Meddahi tests were observed as the least powerful.

Keywords: Normality test, skewness, kurtosis, ROC Curve, Monte Carlo simulation, Real Life atasets

Cite this Article

Md. Siraj Ud Doulah . A Comparison

among Twenty Seven Normality Tests

Research & Reviews: Journal of Statistics .

2019; 8(3): 4 1 59 p.

#### Keywords

#### Full Text:

PDF#### References

REFERENCES

Anderson, T. W. and Darling, D. A. Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. The Annals of Mathematical Statistics, 23(2), 1952, 193-212p.

Bai J. and Ng, S. Tests for Skewness, Kurtosis, and Normality for Time Series Data. Journal of Business & Economic Statistics, 23(1), 2005, 49-60p.

Bonett D.G. and Seier, E. A test of normality with high uniform power. Computational Statistics & Data Analysis, 40, 2002, 435 – 445p.

Bontemps, C. and Meddahi, N. Testing normality: a GMM approach. J. Econom., 124(1), 2005, 149 – 186p.

Hubert, G.B. and Struyf, A. Goodness-of-fit tests based on a robust Measure of skewness. Comput. Stat. 23(3), 2007, 429 – 442p.

D’Agostino, R.B. and Pearson, E.S. Testing for departures from normality. I. Fuller empirical results for the distribution, Biometrika, 60,1973, 613–622p.

Watson, G. S. Goodness-of-fit tests on a circle. II. Biometrika, 49, 1962, 57-63p.

D'Agostino, R.B. An Omnibus Test of Normality for Moderate and Large Size Samples. Biometrika, 58(2), 1971, 341-348p.

Doornik, J.A. and Hansen, H. An Omnibus test for univariate and multivariate normality. Oxford Bulletin of Economics and Statistics, 70, 2008, 927-939p.

Farrell, P.J. and Rogers-Stewart, K. Comprehensive study of tests for normality and symmetry: Extending the Spiegelhalter test, J. Statist. Comput. Simul. 76(9), 2006, 803–816p.

Kuiper, N. H. Tests concerning random points on a circle. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen-Series A, 63, 1960, 38-47p.

Geary, R.C. The Distribution of the "Student's" Ratio for the Non-Normal Samples. Journal of the Royal Statistical Society, 3 (2), 1936, 178–184p.

Gel, Y.R. and Gastwirth, J.L. A robust modification of the Jarque–Bera test of normality. Economics Letters, 99, 2008, 30–32p.

Hegazy, Y. A. S. and Green, J. R. Some new goodness-of-fit tests using order statistics. Journal of the Royal Statistical Society- Series C (Applied Statistics), 24, 1975, 299–308p.

Hogg, R.V. and Tanis, E.A. Probability and Statistical Inference, 6th ed.; prentile, Hall USA, 2001. 57-278p.

Jarque, C. M. and Bera, A. K. A test for normality of observations and regression residuals. International Statistical Review, 55, 1987, 163–172p.

Keskin, S. Comparison of Several Univariate Normality Tests Regarding Type I Error Rate and Power of the Test in Simulation based Small Samples. Journal of Applied Science Research, 2(5), 2006, 296-300p.

Lilliefors, H.W. On the Kolmogorov-Smirnov test for normality with mean and variance unknown, Journal of the American Statistical Association. 62, 1967, 399–402p.

Oztuna, D.; Elhan, A.H. and Tuccar, E. Investigation of four different normality tests in terms of Type I error rate and power under different distributions, Turk. J. Med. Sci., 36(3), 2006, 171–176p.

Noughabi, H. A. and Arghami, N. R. Monte Carlo comparison of seven normality tests. Journal of Statistical Computation and Simulation, 81(8), 2011, 965-972p.

Shapiro. S.S. and Wilk, M.B. An analysis of variance test for normality (complete samples), Biometrika, 52, 1965, 591–611p.

Shapiro, S. S. and Francia, R.S. An approximate analysis of variance test for normality. Journal of the American Statistical Association, 67, 1972, 215–216p.

Spiegelhalter, D. J. A test for normality against symmetric alternatives. Biometrika, 64, 1977, 415–418p.

Thadewald, T. and Buning, H. Jarque–Bera test and its competitors for testing normality – a power comparison, J. Appl. Statist. 34(1), 2007, 87–105p.

Weisberg, S. and Bingham, C. An approximate analysis of variance test for non-normality suitable for machine calculation. Technometrics, 17, 1975, 133–134p.

Crawley, M.J. The R Book, 1st ed.; John Wiley & Sons Ltd, England, 2007.

Yazici, B. and Yolacan, S. A comparison of various tests of normality, J. Statist. Comput. Simul. 77(2), 2007, 175–183p.

Banik, S. and Kibria, B. M. G. Comparison of some parametric and nonparametric type one sample confidence intervals for estimating the mean of a positively skewed distribution. Communications in Statistics-Simulation and Computation, 39, 2010, 361–389p.

Mendes, M. and Pala, A. Type I Error Rate and Power of Three Normality Tests. Information Technology Journal, 2, 2003, 135-139p.

Doulah, M.S.U. Alternative Measures of Standard Deviation Coefficient of Variation and Standard Error, International Journal of Statistics and Applications, 8(6), 2018, 309-315p.

Chen, Z. and Ye, C. An alternative test for uniformity, International Journal of Reliability, Quality and Safety Engineering, 16,2009, 343-356p.

Chen, Z. Goodness-of-Fit Test Based on Arbitrarily Selected Order Statistics. Mathematics and Statistics, 2(2), 2014, 72-77p.

Goegebeur, Y. and Guillou, A. Goodness-of-fit testing for Weibull-type behavior. Journal of Statistical Planning and Inference, 140, 2010, 1417-1436p.

Keskin, S. Comparison of Several Univariate Normality Tests Regarding Type I Error Rate and Power of the Test in Simulation based Small Samples. Journal of Applied Science Research, 2(5), 2006, 296-300p.

### Refbacks

- There are currently no refbacks.