A View on the Validity of Central Limit Theorem: An Empirical Study Using Random Samples from Uniform Distribution Lee, Chanmi; Kim, Seungah; Jeong, Jaesik;
We derive the exact distribution of summation for random samples from uniform distribution and then compare the exact distribution with the approximated normal distribution obtained by the central limit theorem. To check the similarity between two distributions, we consider five existing normality tests based on the difference between the target normal distribution and empirical distribution: Anderson-Darling test, Kolmogorov-Smirnov test, Cramer-von Mises test, Shapiro-Wilk test and Shaprio-Francia test. For the purpose of comparison, those normality tests are applied to the simulated data. It can sometimes be difficult to derive an exact distribution. Thus, we try two different transformations to find out which transform is easier to get the exact distribution in terms of calculation complexity. We compare two transformations and comment on the advantages and disadvantages for each transformation.
Central limit theorem;uniform distribution;normal distribution;Anderson-Darling test;Kolmogorov-Smirnov test;Cramer-von Mises test;Shapiro-Wilk test and Shapiro-Francia test;
Anderson, T. W. (1962). On the distribution of the two-sample Cramer-von Mises Criterion, The Annals of Mathematical Statistics, 33, 1148-1159.
Althouse, L. A.,Ware,W. B. and Ferron, J. M. (1998). Detecting departures from normality: A Monte Carlo simulation of a new omnibus test based on moments, Annual meeting of the American Educational Research Association.
Dufour, J. M., Farhat, A., Gardiol, L. and Khalaf, L. (1998). Simulation-based finite sample normality tests in linear regression, Econometrics Journal, 1, 154-173.
Dinov, I. D., Christou, N. and Sanchez, J. (2008). Central limit theorem: New SOCR applet and demonstration activity, Journal of Statistics Education, 16, 2.
Micheaux, P. L. and Liquet, B. (2009). Understanding convergence concepts: A visual-minded and graphical simulation-based approach, American Statistician, 63, 173-178.
Royston, J. P. (1983). A simple method for evaluating the Shapiro-Francia W' test of non-normality, The Statistician, 32, 297-300.
Smirnov, N. (1948). Table for estimating the goodness of fit of empirical distributions, The Annals of Mathematical Statistics, 19, 279-281.
Shapiro, S. S. and Wilk, M. B. (1965). An analysis of variance test for normality, Biometrika, 52, 591-611.
Shapiro, S. S. and Francia, R. S. (1972). An approximate analysis of variance test for normality, Journal of the American Statistical Association, 67, 215-216.
Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons, Journal of the American Statistical Association, 69, 730-737.
Stephens, M. A. (1976). Asymptotic results for goodness-of-fit statistics with unknown parameters, Annals of Statistics, 4, 357-369.
Stephens, M. A. (1977). Goodness of fit for the extreme value distribution, Biometrika, 64, 583-588.