Tests of Bivariate Exchangeability

We discuss the hypothesis of bivariate exchangeability and show that testing bivariate exchangeability is related to the two-sample testing of equality of distribution functions. We consider three test statistics based on the ordering of the Euclidean interpoint distances. The runs test of exchangeability counts the runs among the observations and their mirror images on the minimal spanning tree. The nearest neighbour test of exchangeability is based on the number of nearest neighbour type coincidences among the observations and their folded images on the plane. The rank test of exchangeability compares the within and between ranks of the interpoint distances. We also consider the sign test of exchangeability, which uses the signs of the observations in specific regions, and a bootstrap test of exchangeability based on the maximum distance between the mirror images. We compare the power of these methods in a Monte Carlo study which shows different power orderings of the methods, depending on the alternative hypothesis.