A nonparametric test of independence dealing with tied observations

Abstract

Based on a population model we develop probability theoretical methods from basic definition over the weak law of large numbers to Sluzky's theorem. These tools provide an extension of Lindeberg's central limit theorem. Due to this extension, the independence of the random variables can be replaced by a weak dependence, such as the dependence between ranks in a sample. The presented nonparametric test of independence for paired data is a robust method based on linear rank statistics. From a sample of paired, at least ordinal data, we attempt to ascertain whether one component has a significant influence on the other one or vice verse. Therefore, we test the hypothesis of independence against an association between the components. The test statistic compares the relative magnitude of one component to the other. Thus, it is similar to Spearman's rho.<br />The limitation of the classical approach arises from the appearance of ties. The appearance of ties is related with the underlying probability distribution of the random variables. The distributions of the corresponding ranks are no longer given. To provide a null distribution for the test we assume that the appearance of the ties is given by a certain sample. So, we derive a conditional distribution free test given the appearance of the ties.<br />This thesis shows that asymptotic normality of the test statistic can be established under minor assumptions. Thus, we assume that one component is governed by a distribution which is the mixture between a continuous distribution and a discrete distribution which has just a finite number of jumps. Therefore, we get some regularity assumptions on the nature of ties. This implication on the level of ties, some properties of order statistics and the extension of the Lindeberg's central limit theorem are sufficient to show asymptotic normality of the test statistic.<br />