Testing extreme value conditions
A modification of the Cramér-von Mises statistics for testing the tail behaviour of i.i.d. sample CDF \(F\) is considered. Its version for nonnegative tail index \(\gamma\) is of the form \[ T_{k,n}=\int \left( {1\over \hat\gamma} (\log X_{n-[kt],n}-\log X_{n-k,n})+\log t \right)^2 t^2\, dt, \] where \(X_{i,n}\) is the \(i\)th order statistics, \(\hat\gamma\) is an estimator for \(\gamma\). It is shown that under proper conditions, \(T_{k,n}\) converges weakly to \[ T=\int_0^1 \left(B(t)+\log t \int_0^1 B(s)/s \,ds\right)^2\,dt, \] where \(B\) is a Brownian bridge, if \(F\) belongs to the domain of max-attraction of the extreme value distribution with the tail index \(\gamma\). Analogous results are obtained for testing \(\gamma\in R\). Results of simulations are presented.
scientific article; zbMATH DE number 2059060