  We suggest novel correlation coefficients which equal the maximum correlation
for a class of bivariate Lancaster distributions while being only slightly
smaller than maximum correlation for a variety of further bivariate
distributions. In contrast to maximum correlation, however, our correlation
coefficients allow for rank and moment-based estimators which are simple to
compute and have tractable asymptotic distributions. Confidence intervals
resulting from these asymptotic approximations and the covariance bootstrap
show good finite-sample coverage. In a simulation, the power of asymptotic as
well as permutation tests for independence based on our correlation measures
compares favorably with competing methods based on distance correlation or rank
coefficients for functional dependence, among others. Moreover, for the
bivariate normal distribution, our correlation coefficients equal the absolute
value of the Pearson correlation, an attractive feature for practitioners which
is not shared by various competitors. We illustrate the practical usefulness of
our methods in applications to two real data sets.
