Detecting positive quadrant dependence and positive function dependence

There is a lot of interest in positive dependence going beyond linear correlation. In this paper three new rank tests for testing independence against positive dependence are introduced. The first one is directed on positive quadrant dependence, the second and third one concentrate on positive function dependence. The new testing procedures are not only sensitive for positive grade linear correlation, but also for positive grade correlations of higher order. They are based on the principle of data driven tests, which consists of three steps. Firstly, parametric families are introduced spanning up the space of null hypothesis and alternatives; secondly, within the families good tests are used; thirdly, a selection rule determines the appropriate model. The new tests improve standard tests for linear correlation as Spearman's rank correlation test substantially in case some proper higher order correlations are exhibited by the data, while the loss in power under alternatives with dominating linear correlation is not very high. Monte Carlo results clearly show this behavior.