There is interest in estimating the extremal behavior of a stationary sequence \(X_{1}, X_{2},\dots,X_{n}.\) The most commonly studied extremal feature is the distribution function, \(F_{n}(x),\) of \(M_{n}=\max(X_{1},\dots,X_{n})\) for large \(n.\) There are two aspects which influence the extremal behavior of such processes: the marginal distribution and the dependence structure of the \(X_{i}\) series. In this paper the behavior of a number of existing and new estimators for the degree of temporal dependence in extreme values of a process is studied. It is focused on the impact of the short range dependence. To clarify the effect of dependence on the behavior of \(M_{n}\) the process is taken to be independent and then dependence is introduced into the considerations whilst retaining the same marginal distribution.
A parameter \(\theta(q_{p})\) is introduced, which is a threshold based version of the extremal index \(\theta.\) The extremal index gives a measure of the short range dependence exhibited by the extremes of a process and, in particular, indicates the tendency of the extremes to occur in clusters.
This paper is mainly focused on the estimation of \(\theta\) and its threshold-based extremal index \(\theta(u).\) The behavior of a number of existing and new estimators of \(\theta\) is examined and their performance for a range of different processes is assessed. Particular emphasis is given to evaluating the benefits that can be achieved by exploiting additional knowledge about the process, such as the process being Markov.