Testing the Equality and Exchangeability of Bivariate Distributions

Abstract

We consider three methods (oments, cut-points, and ranks) for testing the hypotheses of equality of two bivariate distribution functions (H 0a ) and exchangeability (H 0b ). To test H 0a , the asymptotic normality of the vector of mixed moments provides a statistic with an asymptotic chi-square distribution. With every observation, method of cut-points associates three 2 × 2 tables to record the proportions of the X, Y, and the combined samples that fall in the four regions around the observation. We measure the total squared deviations of the proportions in the combined sample from X and Y samples. The two methods are compared with the method of ranks based on the Puri and Sen (1971) multivariate two-sample rank test for location. To test H 0b we identify two bivariate distributions, one above and the other below the line of symmetry X = Y, to which a test of H 0a is applied. Under H 0b , matrix of mixed moments is symmetric and a quadratic form in differences of (r,s)-th and (s, r)-th mixed moments provides an asymptotic chi-square distribution. A permutation test is devised to apply the method of cut-points to the observations above and below the line of symmetry after they are folded. We also describe an adaption of the Puri-Sen rank test to assess H 0b . To estimate the power of the above methods under different types of alternatives and compare them to existing tests, we report on a Monte Carlo experiment that evaluates the finite-sample performance of these methods under the Plackett's family of bivariate distributions.