Testing hypotheses on distribution tails

We consider main approaches to asymptotic testing hypotheses on distributions when only k maximal order statistics of a sample are observed. We assume that $k\to\infty$ and $k=o(n)$ where $n$ is the sample size. A general criterion for a distribution tail belonging to one of two classes is suggested provided all tails from one of the classes are lighter than the tails from another one. Further, we consider a problem of distinguish a simple hypothesis on a distribution tail and a close (contigual) alternative, and evaluate the limit distribution of the corresponding likelihood ratio.