Weighted correlation tests for scale families

We show that weighted versions of recent correlation tests do not require light underlying tails. We formulate a general theorem concerning the asymptotic null distribution of weighted correlation test statistics for scale families, and demonstrate that the resulting tests may work not only for all Weibull scale families, but, with suitable choices of the weight functions, even for all Pareto scale families, in each of which the scale varies the left endpoint of the distribution.