Evaluating the performance of modified Anderson–Darling goodness-of-fit tests for normality

Damian Stoltmann; Piotr  Sulewski

doi:https://doi.org/10.59139/ps.2025.04.2

Damian Stoltmann Pomeranian University in Słupsk, Institute of Exact and Technical Sciences, Department of Computer Science, ul. Arciszewskiego 22a, 76–200 Słupsk, Poland ORCID:https://orcid.org/0000-0001-7053-2684 , Piotr Sulewski Pomeranian University in Słupsk, Institute of Exact and Technical Sciences, Department of Computer Science, ul. Arciszewskiego 22a, 76–200 Słupsk, Poland ORCID:https://orcid.org/0000-0002-0788-6567 Przegląd Statystyczny. Statistical Review, vol. 72, 2025, 4, pages: 25-67 Published online: 23 March 2026 DOI https://doi.org/10.59139/ps.2025.04.2 Citation: Stoltmann, D. & Sulewski, P. (2025). Evaluating the performance of modified Anderson–Darling goodness-of-fit tests for normality. Przegląd Statystyczny. Statistical Review, 72(4), 25–67. https://doi.org/10.59139/ps.2025.04.2.

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ABSTRACT

The main aim of the article is to define and practically apply the modified Anderson– Darling (MAD) goodness-of-fit tests for normality. The modifications consist in varying the formula for calculating the empirical distribution function (EDF). Additional contributions of the paper include the expansion of the EDF family with four new proposals and the creation of a family of alternative distributions, consisting of both older and newer distributions that belong to all groups of skewness and kurtosis signs thanks to their flexibility. Critical values are obtained using 106 order statistics for sample sizes of n=10,20 and at a significance level of α=0.05. Finally, the article shows the calculation of the power of the analysed tests for alternative distributions based on 10^5 values of the test statistics. Their parameters have been selected to show several similarities to the normal distribution. The effectiveness of the tests is illustrated through the analysis of real datasets.

KEYWORDS

empirical distribution function, goodness-of-fit test, Anderson–Darling test, test power

JEL

C14, C15

REFERENCES

Afeez, B. M., Maxwell, O., Otekunrin, O. A., & Happiness, O.-I. (2018). Selection and Validation of Comparative Study of Normality Test. American Journal of Mathematics and Statistics, 8(6), 190–201. https://doi.org/10.5923/j.ajms.20180806.05.

Ahmad, F., & Khan, R. A. (2015). Power Comparison of Various Normality Tests. Pakistan Journal of Statistics and Operation Research, 11(3), 331–345. https://doi.org/10.18187/pjsor.v11i3.845.

Aliaga, A. M., Martínez-González, E., Cayón, L., Argüeso, F., Sanz, J. L., & Barreiro, R. B. (2003). Goodness-of-fit tests of Gaussianity: constraints on the cumulants of the MAXIMA data. New Astronomy Reviews, 47(8–10), 821–826. https://doi.org/10.1016/j.newar.2003.07.010.

Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain ‘goodness-of-fit’ criteria based on stochastic processes. The Annals of Mathematical Statistics, 23(2), 193–212. https://doi.org/10.1214/aoms/1177729437.

Arnastauskaite, J., Ruzgas, T., & Braženas, M. (2021). An Exhaustive Power Comparison of Normality Tests. Mathematics, 9(7), 1–20. https://doi.org/10.3390/math9070788.

Bayoud, H. A. (2021). Tests of normality: new test and comparative study. Communications in Statistics – Simulation and Computation, 50(12), 4442–4463. https://doi.org/10.1080/03610918.2019.1643883.

Bloom, G. (1958). Statistical Estimates and Transformed Beta-Variables. John Wiley & Sons. https://gwern.net/doc/statistics/order/1958-blom-orderstatistics.pdf.

Bonett, D. G., & Seier, E. (2002). A test of normality with high uniform power. Computational Statistics & Data Analysis, 40(3), 435–445. https://doi.org/10.1016/S0167-9473(02)00074-9.

Bontemps, C., & Meddahi, N. (2005). Testing normality: a GMM approach. Journal of Econometrics, 124(1), 149–186. https://doi.org/10.1016/j.jeconom.2004.02.014.

Borrajo, M. I., González-Manteiga, W., & Martínez-Miranda, M. D. (2024). Goodness-of-fit test for point processes first-order intensity. Computational Statistics & Data Analysis, 194, 1–16. https://doi.org/10.1016/j.csda.2024.107929.

Brys, G., Hubert, M., & Struyf, A. (2008). Goodness-of-fit tests based on a robust measure of skewness. Computational Statistics, 23(3), 429–442. https://doi.org/10.1007/s00180-007-0083-7.

Chernobai, A., Rachev, S. T., & Fabozzi, F. J. (2005). Composite goodness-of-fit tests for lefttruncated loss samples. In C. F. Lee & J. C. Lee (Eds.), Handbook of Financial Econometrics and Statistics (pp. 575–596). Springer. https://doi.org/10.1007/978-1-4614-7750-1_20.

Coin, D. (2008). A goodness-of-fit test for normality based on polynomial regression. Computational Statistics and Data Analysis, 52(4), 2185–2198. https://doi.org/10.1016/j.csda.2007.07.012.

Cramér, H. (1928). On the Composition of Elementary Errors. Scandinavian Actuarial Journal, (1), 13–74. https://doi.org/10.1080/03461238.1928.10416862.

Desgagné, A., & Lafaye de Micheaux, P. (2018). A powerful and interpretable alternative to the Jarque-Bera test of normality based on 2nd-power skewness and kurtosis, using the Rao’s Score Test on the APD family. Journal of Applied Statistics, 45(13), 2307–2327. https://doi.org/10.1080/02664763.2017.1415311.

Desgagné, A., Lafaye de Micheaux, P., & Ouimet, F. (2023). Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on ?-th power skewness and kurtosis. Statistics. A Journal of Theoretical and Applied Statistics, 57(1), 94–122. https://doi.org/10.1080/02331888.2022.2144859.

Esteban, M. D., Castellanos, M. E., Morales, D., & Vajda, I. (2001). Monte Carlo comparison of four normality tests using different entropy estimates. Communications in Statistics – Simulation and Computation, 30(4), 761–785. https://doi.org/10.1081/SAC-100107780.

Gan, F. F., & Koehler, K. J. (1990). Goodness-of-fit tests based on P-P probability plots. Technometrics, 32(3), 289–303. https://doi.org/10.2307/1269106.

Gel, Y. R., & Gastwirth, J. L. (2008). A Robust Modification of the Jarque-Bera Test of Normality. Economics Letters, 99(1), 30–32. https://doi.org/10.1016/j.econlet.2007.05.022.

Gel, Y. R., Miao, W., & Gastwirth, J. L. (2007). Robust Directed Tests of Normality against Heavy-tailed Alternatives. Computational Statistics & Data Analysis, 51(5), 2734–2746. https://doi.org/10.1016/j.csda.2006.08.022.

Giles, D. E. (2024). New Goodness-of-Fit Tests for the Kumaraswamy Distribution. Stats, 7(2), 373–388. https://doi.org/10.3390/stats7020023.

Feuerverger, A. (2016). On Goodness of Fit for Operational Risk. International Statistical Review, 84(3), 434–455. https://doi.org/10.1111/insr.12112.

Filliben, J. J. (1975). The probability plot correlation coefficient test for normality. Technometrics, 17(1), 111–117. https://doi.org/10.1080/00401706.1975.10489279.

Harter, H. L., Khamis, H. J., & Lamb, R. E. (1984). Modified Kolmogorov-Smirnov tests of goodness of fit. Communications in Statistics – Simulation and Computation, 13(3), 293–323. https://doi.org/10.1080/03610918408812378.

Hernandez, H. (2021). Testing for Normality: What is the Best Method?. ForsChem Research Reports, 6, 101–138. https://doi.org/10.13140/RG.2.2.13926.14406.

Kellner, J., & Celisse, A. (2019). A one-sample test for normality with kernel methods. Bernoulli, 25(3), 1816–1837. https://doi.org/10.3150/18-BEJ1037.

Kendall, M. G., & Stuart, A. (1968). The advanced theory of statistics: Vol. 1. Distribution theory. Hafner Publishing Company.

Khamis, H. J. (1990). The ?-corrected Kolmogorov-Smirnov test for goodness-of-fit. Journal of Statistical Planning and Inference, 24(3), 317–335. https://doi.org/10.1016/0378-3758(90)90051-U.

Khamis, H. J. (1992). The ?-corrected Kolmogorov-Smirnov test with estimated parameters. Journal of Nonparametric Statistics, 2(1), 17–27. https://doi.org/10.1080/10485259208832539.

Khamis, H. J. (1993). A comparative study of the d-corrected Kolmogorov-Smirnov test. Journal of Applied Statistics, 20(3), 401–421. https://doi.org/10.1080/02664769300000040.

Khatun, N. (2021). Applications of normality test in statistical analysis. Open Journal of Statistics, 11(1), 113–122.

Kolmogoroff, A. (1933). Sulla determinazione empirica di una legge di distribuzione. Giornale dell’Istituto Italiano degli Attuari, (4), 83–91.

Krauczi, E. (2009). A study of the quantile correlation test of normality. TEST, 18(1), 156–165. https://doi.org/10.1007/s11749-007-0074-6.

Kuiper, N. H. (1960). Tests concerning random points on a circle. Proceedings Series A. Indagationes Mathematicae, 63, 38–47. https://doi.org/10.1016/S1385-7258(60)50006-0.

Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown. Journal of the American Statistical Association, 62(318), 399–402. https://doi.org/10.1080/01621459.1967.10482916.

Luceno, A. (2006). Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators. Computational Statistics & Data Analysis, 51(2), 904–917. https://doi.org/10.1016/j.csda.2005.09.011.

Ma, Y., Kitani, M., & Murakami, H. (2024). On modified Anderson-Darling test statistics with asymptotic properties. Communications in Statistics – Theory and Methods, 53(4), 1420–1439. https://doi.org/10.1080/03610926.2022.2101121.

Malachov, A. N. (1978). Kumulantnyj analiz słuczajnych niegaussowych processow i ich prieobrazowanij. Soviet Radio.

Marange, C. S., & Qin, Y. (2019). An Empirical Likelihood Ratio Based Comparative Study on Tests for Normality of Residuals in Linear Models. Metodološki Zvezki, 16(1), 1–16. https://doi.org/10.51936/ramh7128.

Mbah, A. K., & Paothong, A. (2015). Shapiro-Francia test compared to other normality test using expected p-value. Journal of Statistical Computation and Simulation, 85(15), 3002–3016. https://doi.org/10.1080/00949655.2014.947986.

von Mises, R. (1931). Über einige Abschätzungen von Erwartungswerten. Journal für die reine und angewandte Mathematik, 1931(165), 184–193. https://doi.org/10.1515/crll.1931.165.184.

Mishra, P., Pandey, C. M., Singh, U., Gupta, A., Sahu, C., & Keshri, A. (2019). Descriptive Statistics and Normality Tests for Statistical data. Annals of Cardiac Anaesthesia, 22(1), 67–72. https://doi.org/10.4103/aca.ACA_157_18.

Nosakhare, U. H., & Bright, A. F. (2017). Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. American Journal of Theoretical and Applied Statistics, 6(5–1), 51–61. https://doi.org/10.11648/j.ajtas.s.2017060501.18.

Noughabi, H. A., & Arghami, N. R. (2011). Monte Carlo comparison of seven normality tests. Journal of Statistical Computation and Simulation, 81(8), 965–972. https://doi.org/10.1080/00949650903580047.

Pearson, K. (1895). Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, (186) 343–414.

Razali, N. M., & Wah, Y. B. (2011). Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. Journal of Statistical Modeling and Analytics, 2(1), 21–33.

Rodriguez, J. C., Viollaz, A. J. (1995). A Cramér-von Mises type goodness of fit test with asymmetric weight function. Communications in Statistics – Theory and Methods, 24(4), 1095– 1120. https://doi.org/10.1080/03610929508831542.

Romao, X., Delgado, R., & Costa, A. (2010). An empirical power comparison of univariate goodness-of-fit tests of normality. Journal of Statistical Computation and Simulation, 80(5), 545–591. https://doi.org/10.1080/00949650902740824.

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality. Biometrika, 52(3/4), 591–611. https://doi.org/10.1093/biomet/52.3-4.591.

Sinclair, C. D., Spurr, B. D., & Ahmad, M. I. (1990). Modified Anderson darling test. Communications in Statistics – Theory and Methods, 19(10), 3677–3686. https://doi.org/10.1080/03610929008830405.

Smirnov, N. (1948). Table for estimating the goodness of fit of empirical distributions. The Annals of Mathematical Statistics, 19(2), 279–281. https://doi.org/10.1214/aoms/1177730256.

Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730–737. https://doi.org/10.1080/01621459.1974.10480196.

Stephens, M. A. (1979). The Anderson-Darling statistic. https://apps.dtic.mil/sti/tr/pdf/ADA079807.pdf.

Sulewski, P. (2019). Modification of Anderson-Darling goodness-of-fit test for normality. Afinidad, 76(588), 270–277. https://www.raco.cat/index.php/afinidad/article/view/361876.

Sulewski, P. (2021). Two component modified Lilliefors test for normality. Equilibrium. Equilibrium. Quarterly Journal of Economics and Economic Policy, 16(2), 429–455. https://doi.org/10.24136/eq.2021.016.

Sulewski, P. (2022a). Modified Lilliefors goodness-of-fit test for normality. Communications in Statistics – Simulation and Computation, 51(3), 1199–1219. https://doi.org/10.1080/03610918.2019.1664580.

Sulewski, P. (2022b). Normal distribution with plasticizing component. Communications in Statistics – Theory and Methods, 51(11), 3806–3835. https://doi.org/10.1080/03610926.2020.1837881.

Sulewski, P., & Stoltmann, D. (2026). Parameterized Kolmogorov–Smirnov Test for Normality. Applied Sciences, 16(1), 1–44. https://doi.org/10.3390/app16010366.

Tavakoli, M., Arghami, N., & Abbasnejad, M. (2019). A goodness of Fit Test For Normality Based on Balakrishnan-Sanghvi Information. Journal of The Iranian Statistical Society, 18(1), 177–190. https://doi.org/10.29252/jirss.18.1.177.

Terán-García, E., & Pérez-Fernández, R. (2024). A robust alternative to the Lilliefors test of normality. Journal of Statistical Computation and Simulation, 94(7), 1–19. https://doi.org/10.1080/00949655.2023.2291138.

Torabi, H., Montazeri, N. H., & Grané, A. (2016). A test of normality based on the empirical distribution function. SORT. Statistics and Operations Research Transactions, 40(1), 55–88. https://doi.org/10.2436/20.8080.02.35.

Uhm, T., & Yi, S. (2023). A comparison of normality testing methods by empirical power and distribution of P-values. Communications in Statistics – Simulation and Computation, 52(9), 4445–4458. https://doi.org/10.1080/03610918.2021.1963450.

Uyanto, S. S. (2022). An Extensive Comparisons of 50 Univariate Goodness-of-fit Tests for Normality. Austrian Journal of Statistics, 51(3), 45–97. https://doi.org/10.17713/ajs.v51i3.1279.

Watson, G. S. (1962). Goodness-of-fit tests on a circle. II. Biometrika, 49(1–2), 57–63. https://doi.org/10.1093/biomet/49.1-2.57.

Wijekularathna, D. K., Manage, A. B., & Scariano, S. M. (2020). Power analysis of several normality tests: A Monte Carlo simulation study. Communications in Statistics – Simulation and Computation, 51(3), 757–773. https://doi.org/10.1080/03610918.2019.1658780.

Yap, B. W., & Sim, C. H. (2011). Comparisons of various types of normality tests. Journal of Statistical Computation and Simulation, 81(12), 2141–2155. https://doi.org/10.1080/00949655.2010.520163.

Yazici, B., & Yolacan, S. (2007). Comparison of various tests of normality. Journal of Statistical Computation and Simulation, 77(2), 175–183. https://doi.org/10.1080/10629360600678310.