On testing the extreme value index via the pot-method

Consider an iid sample $Y_1, \dots, Y_n$ of random variables with common distribution function F, whose upper tail belongs to a neighborhood of the upper tail of a generalized Pareto distribution $H_{\beta}, \beta \epsilon \mathbb{R}$. We investigate the testing problem $\beta = \beta_0$ against a sequence $\beta = \beta_n$ of contiguous 0 n alternatives, based on the point processes $N_n$ of the exceedances among $Y_i$ over a sequence of thresholds $t_n$. It turns out that the (random) number of exceedances $\tau (n)$ over $t_n$ is the central sequence for the log-likelihood ratio $d \mathsf{L}_{\beta_n} (N_n)/ d \mathsf{L}_{\beta_0} (N_n)$, yielding its local asymptotic normality (LAN). This result implies in particular that $\tau (n)$ carries asymptotically all the information about the underlying parameter $\beta$, which is contained in $N_n$. We establish sharp bounds for the rate at which $\tau (n)$ becomes asymptotically sufficient, which show, however, that this is quite a poor rate. These results remain true if we add an unknown scale parameter.