Semiparametric Lower Bounds for Tail Index Estimation

We consider semiparametric estimation of the tail index parameter from i.i.d. observations in Pareto and Weibull type models, using a local and asymptotic approach. The slowly varying function describing the non-tail behavior of the distribution is considered as infinite dimensional nuisance parameter. Without further regularity conditions, we derive a Local Asymptotic Normality (LAN) result that describes essentially the least favorable submodel for the tail index parameter. From this result, we immediately obtain the optimal rate of convergence of tail index parameter estimators for more specific models previously studied. On top of the optimal rate of convergence, our LAN result also gives the minimal limiting variance of (regular) semiparametric estimators through the convolution theorem. We show that the Hill estimator is also semiparametrically efficient in the Pareto case in this much stronger sense. We also discuss the Weibull model in this respect. Running title: Semiparametric tail index estimatio