Testing independence when the form of the bivariate distribution is unspecified

We address the problem of testing for independence between X and Y in two situations. In the first we assume that the joint distribution of X and Y is unknown but the observations on X and Y are identifiable. In the second case we assume that the distribution of (X, Y) is exchangeable. Here we consider both when (X, Y) are identifiable and when they are not. We illustrate applications to the testing of independence in DNA databases and in twin studies.