Jarque-Bera test and its competitors for testing normality: A power comparison

For testing normality we investigate the power of several tests, first of all, the well known test of Jarque and Bera (1980) and furthermore the tests of Kuiper (1960) and Shapiro and Wilk (1965) as well as tests of Kolmogorov-Smirnov and Cramer-von Mises type. The tests on normality are based, first, on independent random variables (model I) and, second, on the residuals in the classical linear regression (model II). We investigate the exact critical values of the Jarque-Bera test and the Kolmogorov-Smirnov and Cramer-von Mises tests, in the latter case for the original and standardized observations where the unknown parameters u and o have to be estimated. The power comparison is carried out via Monte Carlo simulation assuming the model of contaminated normal distributions with varying parameters u and o and different proportions of contamination. It turns out that for the Jarque-Bera test the approximation of critical values by the chi-square distribution does not work very well. The test is superior in power to its competitors for symmetric distributions with medium up to long tails and for slightly skewed distributions with long tails. The power of the Jarque-Bera test is poor for distributions with short tails, especially if the shape is bimodal, sometimes the test is even biased. In this case a modification of the Cramer-von Mises test or the Shapiro-Wilk test may be recommended. --goodness-of-fit tests, tests of Kolmogorov-Smirnov and Cramervon Mises type, Shapiro-Wilk test, Kuiper test, skewness, kurtosis, contaminated normal distribution, Monte-Carlo simulation, critical values, power comparison

Robustness and power of modified Lepage, Kolmogorov-Smirnov and Crame´r-von Mises two-sample tests

For the two-sample problem with location and/or scale alternatives, as well as different shapes, several statistical tests are presented, such as of Kolmogorov-Smirnov and Cramér-von Mises type for the general alternative, and such as of Lepage type for location and scale alternatives. We compare these tests with the t-test and other location tests, such as the Welch test, and also the Levene test for scale. It turns out that there is, of course, no clear winner among the tests but, for symmetric distributions with the same shape, tests of Lepage type are the best ones whereas, for different shapes, Cramér-von Mises type tests are preferred. For extremely right-skewed distributions, a modification of the Kolmogorov-Smirnov test should be applied.

APL and the teaching of statistics

Naeve P. APL and the teaching of statistics. In: Büning H, ed. Computational statistics: Wolfgang Wetzel zur Vollendung seines 60. Lebensjahres. Berlin [u.a.]: de Gruyter; 1981: 215-230

Jarque-Bera Test and its Competitors for Testing Normality - A Power Comparison

For testing normality we investigate the power of several tests, first of all, the well-known test of Jarque & Bera (1980) and furthermore the tests of Kuiper (1960) and Shapiro & Wilk (1965) as well as tests of Kolmogorov-Smirnov and Cramer-von Mises type. The tests on normality are based, first, on independent random variables (model I) and, second, on the residuals in the classical linear regression (model II). We investigate the exact critical values of the Jarque-Bera test and the Kolmogorov-Smirnov and Cramer-von Mises tests, in the latter case for the original and standardized observations where the unknown parameters μ and σ have to be estimated. The power comparison is carried out via Monte Carlo simulation assuming the model of contaminated normal distributions with varying parameters μ and σ and different proportions of contamination. It turns out that for the Jarque-Bera test the approximation of critical values by the chi-square distribution does not work very well. The test is superior in power to its competitors for symmetric distributions with medium up to long tails and for slightly skewed distributions with long tails. The power of the Jarque-Bera test is poor for distributions with short tails, especially if the shape is bimodal - sometimes the test is even biased. In this case a modification of the Cramer-von Mises test or the Shapiro-Wilk test may be recommended.Goodness-of-fit tests, tests of Kolmogorov-Smirnov and Cramer-von Mises type, Shapiro-Wilk test, Kuiper test, skewness, kurtosis, contaminated normal distribution, Monte Carlo simulation, critical values, power comparison,