Testing the Gumbel hypothesis via the Pot-method
The hypothesis \(H_0\) is tested that the distribution function \(F\) of an i.i.d. sample \(X_1\),\dots, \(X_n\) belongs to the extremal domain of attraction of the Gumbel distribution \(G_0(x)=\exp(-e^{-x})\). The alternative is that \(F\) belongs to the domain of attraction of the Fréchet or the Weibull distribution. The test is based on the point process \(N_{n,t_n}\) of the exceedances over a given threshold \(t_n\). Local asymptotic normality of the \(H_0\) testing problem by \(N_{n,t_n}\) with local alternatives is demonstrated in the case when \(t_n\to\infty\), but not too fast. The asymptotically optimal test sequence of fixed level \(\alpha\) for the two-sided testing problem is constructed and its asymptotic power function is evaluated. Simulation results are presented.
scientific article; zbMATH DE number 1336671