Drezner, Zvi and Turel, Ofir and Zerom, Dawit (2008): A modified Kolmogorov-Smirnov test for normality.
Download (309kB) | Preview
In this paper we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, a sample is compared with a normal distribution where the sample mean and the sample variance are used as parameters of the distribution. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data. This is like shifting and stretching the reference normal distribution so that it fits the data in the best possible way. If this shifting and stretching does not lead to an acceptable fit, the data is probably not normal. We also introduce a fast easily implementable algorithm for the proposed test. A study of the power of the proposed test indicates that the test is able to discriminate between the normal distribution and distributions such as uniform, bi-modal, beta, exponential and log-normal that are different in shape, but has a relatively lower power against the student t-distribution that is similar in shape to the normal distribution. In model settings, the former distinction is typically more important to make than the latter distinction. We demonstrate the practical significance of the proposed test with several simulated examples.
|Item Type:||MPRA Paper|
|Original Title:||A modified Kolmogorov-Smirnov test for normality|
|Keywords:||Closest fit; Kolmogorov-Smirnov; Normal distribution|
|Subjects:||C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics|
|Depositing User:||Dawit Zerom|
|Date Deposited:||01. Apr 2009 04:38|
|Last Modified:||07. Jan 2014 19:14|
Dallal G. E. and L. Wilkinson (1986) \An analytic approximation to the distribution of Lilliefors's test statistic for normality," The American Statistician, 40, 294-296.
Jarque, C.M. and A.K. Bera (1980) \E±cient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals," Economics Letters, 6(3), 255-259.
Lilliefors H. W. (1967) \On the Kolmogorov-Smirnov test for normality with mean and vari- ance unknown," Journal of the American Statistical Association, 62, 399-402.
Massey F. J. (1951) \The Kolmogorov-Smirnov test for goodness of ¯t," Journal of the Amer- ican Statistical Association, 46, 68-78.
Megiddo N. (1983) \Linear-time algorithms for linear programming inR3 and related prob- lems, SIAM Journal on Computing, 12, 759-776.
Royston, J. P. (1983) \A Simple Method for Evaluating the Shapiro-Francia W' Test of Non-Normality," Statistician, 32(3) (September), 297-300.
Shapiro, S. S. and R. S. Francia (1972) \An Approximate Analysis of Variance Test for Normality," Journal of the American Statistical Association, 67, 215-216.
Shapiro, S. S. and M. B. Wilk (1965) \An Analysis of Variance Test for Normality (Complete Samples)," Biometrika, 52(3/4) (December), 591-611.
Sobieszczanski-Sobieski, J., Laba, K. and R. Kincaid (1998) \Bell-curve evolutionary opti- mization algorithm," Proceedings of the 7th AIAA Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MO, 2-4 September, AIAA paper 98-4971.
Stephens, M.A. (1974) \EDF statistics for goodness of ¯t and some comparisons, " Journal of the American Statistical Association, 69, 730-737.
Weber, M., Leemis, L. and R. Kincaid (2006) \Minimum Kolmogorov-Smirnov test statistic parameter estimates," Journal of Statistical Computation and Simulation, 76, 3, 195-206.