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Applications of Generating Functions in Nonparametric Tests

# Nonparametric Tests

Nonparametric methods were developed to be used when the researcher knows nothing about the distribution of the variable of interest (hence the name nonparametric). Basically, there is at least one nonparametric equivalent for each parametric type of test. When you want to compare the distribution of two independent samples, usually you would use the Student -test; one of the most important alternatives for this test is the Wilcoxon rank sum test. If you have more than two independent samples to compare, usually you would use analysis of variance; the nonparametric equivalent to this method is the Kruskal-Wallis analysis of variance.

Nonparametric tests are robust (they need no assumptions about the distribution of the background population), efficient (in early days it was believed that a heavy price in loss of efficiency would have to be paid for robustness, but [2, 3] and several other authors showed clearly that the efficiency is comparable with classical tests using the assumption of normality), and easy to handle using computers.

Nevertheless, they are not widely accepted. One possible reason is that the application of a nonparametric test requires the use of (and the confidence in) a table (see, for example, [4] or [5]) of the distribution of its test statistics.

Generating such tables requires heavy algebraic manipulations and is therefore mostly beyond the scope of introductory textbooks. Published tables are sometimes inaccurate or not extensive enough (especially in the case of ties). Moreover, Mitic [6] pointed out that the entries of some published tables differ, depending on their source. Finally, if the sample size is large, it is possible to approximate most of these distributions by standard distributions, but little is known about the quality of these approximations for small sample sizes.

Thus, I believe, Mathematica users are interested in procedures that allow them to generate accurate and extensive tables of the distributions of nonparametric test statistics and plot the distribution functions of these statistics. With these procedures, Mathematica users are independent of published tables and can investigate the quality of the approximation by standard distributions.