Asymptotic power functions for the non-parametric tests of independence based on these rank statistics are obtained. They show that these tests are not unbiased in the limit (Section 9f). The ...

This paper concerns moment and tail probability inequalities and the strong law of large numbers for U-statistics with nonnegative or symmetrized kernels and their multisample and decoupled versions.

The log-likelihood is just the (natural, usually) logarithm of the ordinary likelihood function. It is numerically easier to work with, and since log is a monotonic function, equivalent.<BR><BR ...