Weak convergence of a bootstrap geometric-type estimator with applications to risk theory

Based on least square considerations, Brito and Moreira Freitas [Brito, M., Moreira Freitas, A.C., 2003. Limiting behaviour of a geometric-type estimator for tail indices. Insurance: Math. Econ. 33, 211–226] proposed a geometric-type estimator for estimating an exponential tail coefficient. We consider here the tail bootstrap method introduced by Bacro and Brito [Bacro, J.N., Brito, M., 1998. A tail bootstrap procedure for estimating the tail Pareto index. J. Stat. Plan. Infer. 71, 245–260] and show that this procedure works for this estimator. Moreover, we extend the application given in Brito and Moreira Freitas [Brito, M., Moreira Freitas, A.C., 2003. Limiting behaviour of a geometric-type estimator for tail indices. Insurance: Math. Econ. 33, 211–226], by showing that the results obtained may be applied to the related problem of estimating the adjustment coefficient in the Sparre Andersen model, under the standard conditions.