Tail and dependence behavior of levels that persist for a fixed period of time

摘要

This work emerges from a study of the extremal behavior of a daily maximum sea water levels series, {X_i}, presented in Draisma (Duration of extremes at sea. In: Parametric and semi-parametric methods in E. V. T., pp. 137-143. PhD thesis, Erasmus, University, 2001). In its approach, a new series, {Y_i}, is denned, consisting of water levels that persist for a fixed period of Jime. In this paper, we study the tail behavior of {Y_i}, in case {X_i} is independent and identically distributed (i.i.d.) and in case {X_i} is a max-autoregressive sequence (we will consider two different max-autoregressive processes), whose distribution function is in the Frechet domain of attraction. We also determine Ledford and Tawn tail dependence index (Ledford and Tawn, Biometrika 83:169-187, 1996, J. R. Stat. Soc. B 59:475-499, 1997) and we analyze the asymptotic tail dependence of the random pair (Y_o, Y_(i+m)), in all considered cases. According to Drees (Bernoulli 9:617-657, 2003), we obtain the limit behavior of the tail empirical quantile function associated with a random sample (Y_1, Y_2, ...Y_n) and hence the asymptotic normality of a class of estimators of the tail index that includes Hill estimator.