On the conventional wisdom regarding two consistent tests of bivariate independence

Hoeffding's test of bivariate independence and its asymptotic equivalent due to Blum, Kiefer and Rosenblatt are well known to be consistent against all dependence alternatives. However, the two tests, which are often treated as interchangeable, are rarely used in data analysis mainly because their finite sample null distributions are unavailable, and little is known about their operating characteristics. In this paper the conventional wisdom regarding the equivalence of these tests and their distributions is examined by first tabulating their null distributions for sample sizes n = 5, 6, . . ., 25, 30, . . ., 50, 60, . . ., 100, and then studying their power functions empirically. The power functions are compared with those of the commonly used methods based on the product moment correlation, the rank correlation and Kendall's τ, for bivariate normal and log-normal populations, as well as a variety of dependence models such as the well-known copulas due to Morgenstern, Gumbel, Plackett, Marshall and Olkin, Raftery, Clayton and Frank. It is seen that the Blum, Kiefer and Rosenblatt test is generally preferable in terms of power against positive dependence alternatives and that the conventional wisdom deserves a revision.