Testing bivariate independence and normality

In many statistical studies the relationship between two random variables X and Y is investigated and in particular the question whether X and Y are independent and normally distributed is of interest. Smooth tests may be used for testing this. They consist of several components, the first measuring linear (in)dependence, the next ones correlations of higher powers of X and Y. Since alternatives are not restricted to bivariate normality, not only linear (in)dependence is of interest and therefore all components come in. Moreover, normality itself is also tested by the smooth test. It is well-known that choosing the number of components in a smooth test is an important issue. Recently, data driven methods are developed for doing this. The resulting new test statistics for testing independence and normality are introduce in this paper. For a very large class of alternatives, including also independent X and Y with nonnormal marginals, consistency is proved. Monte Carlo results show that the data driven smooth test behaves very well for finite sample sizes.