EXACT STATISTICAL INFERENCE: THE PERMUTATION PARADIGM

ABSTRACT
In recent years, the use of software for the calculation of statistical tests has
become widespread. For many nonparametric tests, a number of statistical programs calculate significance levels based on algorithms appropriate for large samples only.
In scientific experiments, small samples are common. This requires the use of the exact statistical test. The aim of this research is to formulate a simple but logical method of obtaining unconditional exact permutation distribution for test statistics, especially when small samples are involved. This research is also aimed at providing exact critical values for several test statistics in order to ensure that the probability of a type I error is exactly . The methodology for obtaining distinct exhaustive permutation of paired and unpaired observations or ranks in multisample experiments starts by choosing the test statistic and the acceptable significance level  . For rank order statistics, rank the observations of the experiment as required by the test statistic and compute the observed value of the test statistic. Obtain a distinct permutation of the observations or ranks and then compute the test statistic for the permutation. This process of obtaining distinct permutation and computing the test statistic is repeated until all the distinct permutations are exhausted. An empirical cumulative distributionof the test statistic is constructed for all the distinct values of the test statistic; this gives all the critical values of the test statistic. This research work produces the exact permutation distributions of several test statistics. Exact permutation distributions and exact critical values are produced for test statistics such as t (paired and unpaired observations), One-way ANOVA, Kruskal-Wallis, Mood, Bivariate correlation, Savage, Wilcoxon rank sum and Ansari-Bradley.