Exact distributions of distribution-free test statistics (Q2711183)

Distribution-free test statistics play an important role in statistical inference when particular parametric assumptions about the population cannot be made or at least cannot be checked. In recent years another approach based on resampling techniques has been used in great extent in order to facilitate inference without parametric assumptions. However, resampling techniques are computer intensive and hence high computer recourses might be needed. On the contrary, classical distribution free test statistics can be applied rather easily and without special computational effort. A controversy against such distribution-free statistical methods is based on the fact that usually one has to use asymptotic results concerning the form of the statistics and hence loosing power.NEWLINENEWLINENEWLINEIn this thesis, the author describes algorithms that allow for exact calculation of the distribution of the statistic and, hence, allow for exact inference in distribution-free statistical tests. The algorithms are mainly based on generating functions and fast calculation of the underlying probability function through evaluation of the corresponding generating function. Symbolic computation is proposed through standard mathematical packages that offer this option.NEWLINENEWLINENEWLINEThe author discusses one-dimensional generating functions related with standard distribution-free statistics, like the Wilcoxon, Mann-Whitney and the Kolomogorov-Smirnov and two-dimensional generating functions corresponding to two-sample linear ranks statistics. A branch and bound algorithm is used to reverse the generating function. More complicated situations, including the Spearman correlation and regression rank statistics, are also discussed. A split-up algorithm, that makes use of the fact that only the calculation of the \(p\)-value is needed, and not the complete evaluation of the null distribution, is described and compared to other existing algorithms. Quadratic \(t\)-sample distribution-free statistics and multiple comparisons rank statistics are treated in this way. Finally, a description of the calculation of the non-null distribution of some of the statistics is treated. This case is much more complicated than the evaluation of the null distribution. Several, numerical examples are provided to illustrate the theory.