Two Wilson-Hilferty type approximations for the null distribution of the Blum, Kiefer and Rosenblatt test of bivariate independence

The Blum et al. (Ann. Math. Statist. 32 (1961) 485) test of bivariate independence, an asymptotic equivalent of Hoeffding's (Ann. Math. Statist. 19 (1948) 546) test, is consistent against all dependence alternatives. A concise tabulation of a well-considered approximation for the asymptotic percentiles of its null distribution is given in Blum et al. and a more complete selection of Monte Carlo percentiles, for samples of size 5 and larger, appears in Mudholkar and Wilding (J. Roy. Statist. Soc. 52 (2003) 1). However, neither tabulation is adequate for estimating p-values of the test. In this note we use a moment based analogue of the classical Wilson-Hilferty transformation to obtain two transformations of type T_n=(nB_n)^hn. The transformations T_n are then used to construct and compare a Gaussian and a scaled chi-square approximation for the null distribution of nB_n. Both approximations have excellent accuracy, but the Gaussian approximation is more convenient because of its portability.