On procedures for testing the equivalence of distribution tails

We propose a method for testing the hypothesis about the equivalence of the distribution tail of observed data and a certain distribution tail, which is the analogue of the goodness-of-fit hypothesis for statistics of extremes. The method is based on a new data transformation moving $k$ largest order statistics of a sample from the standard uniform distribution $U[0,1]$ to random variables asymptotically similar to a sample of size $k$ from $U[0,1]$. We prove that tests built by applying the proposed method are consistent on the widest alternative, specifically, on the negation of the null hypothesis.