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

Abstract

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 defined, consisting of water levels that persist for a fixed period of time. 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 i , 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.