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# Tests of fit to a given distribution

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# Tests of fit to a given distribution

Chi-square — The chi-square test is a very simple method for testing the fit of a particular distribution, such as the Poisson, to a sample set of data. It compares the observed frequency distribution in the sample, O, with the expected frequency distribution, E, in a selected distribution for which parameters have been pre-specified or have been estimated from the sample

Anderson-Darling — The Anderson-Darling (AD) statistic is a goodness-of-fit test that is primarily used for deciding whether a sample of size n is drawn from a specified distribution, most commonly whether the sample data is drawn from a Normal distribution. In this context it is widely believed to be one of the best statistics of this type available, even with relatively small sample sizes

Shapiro-Wilk — The Shapiro-Wilk (SW) test for Normality uses the observation that a Normal probability plot that examines the fit of a sample dataset to the Normal is rather like linear regression — the diagonal line of the graph is the line of perfect fit, with divergence from this line being similar to the residuals in regression. By analyzing the scale of this variation (analysis of variance) the quality of the fit can be examined. The authors recommended the use of their statistic use with smaller samples (e.g. <20) and using empirical tests of its power and sensitivity against a range of other tests and a variety of non-Normal distributions showed that it is indeed an effective and sensitive measure

Kolmogorov-Smirnov — Tests if a sample comes from a continuous distribution with specified parameters, against the alternative that it does not come from that distribution. The two-sample Kolmogorov-Smirnov variant tests if two samples come from the same continuous distribution, against the alternative that they do not come from the same distribution. This is a distance-based test which relies (strictly speaking) on knowledge of the target distribution parameters in the one-sample case. The Lilliefors test may be more appropriate for tests of fit to the Normal distribution

Jarque-Bera — Tests if a sample comes from a Normal distribution with unknown mean and variance, against the alternative that it does not come from a Normal distribution.

Lilliefors — Tests if a sample comes from a distribution in the Normal family, against the alternative that it does not come from a Normal distribution

Wilcoxon rank sum/Mann-Witney U test — Tests if two independent samples come from identical continuous distributions with equal medians, against the alternative that they do not have equal medians. A nonparametric alternative to the 2-sample t-test

Sign rank test — one-sample or paired-sample Wilcoxon signed rank test. Tests if a sample comes from a continuous distribution symmetric about a specified median, against the alternative that it does not have that median

Sign test — One-sample or paired-sample sign test. Tests if a sample comes from an arbitrary continuous distribution with a specified median, against the alternative that it does not have that median. A nonparametric alternative to 1-sample z-tests and 1-sample t-tests