# Sebastian Neblung: Sliding and disjoint blocks estimators for the extremal index

**Time: **
Mon 2020-03-02 15.15 - 16.15

**Lecturer: **
Sebastian Neblung, Hamburg

**Location: **
KTH, F11

### Abstract

Extreme value estimators based on the observations of a time series are often constructed from blockwise defined statistics. These blocks can be specified as disjoint blocks \((X_t)_{(i-1)s+1\leq t\leq is}\), \(1\leq i\leq \lfloor \frac{n}{s}\rfloor\), of length \(s\), or alternatively as sliding blocks \((X_t)_{i\leq t\leq i+s-1}\), \(1\leq i\leq n-s+1\). For specific statistics, it has been observed that the sliding block statistic leads to a smaller asymptotic variance. Yet another approach are offered by runs estimators, which can be interpret as a special type of sliding blocks estimators. All three types of estimators can be analysed asymptotically in an unified framework.

We will discuss this unified, peak-over-threshold framework by the example of the extremal index, which is the reciprocal of the mean cluster length of extremes. The asymptotics of all three type of estimators will be established under similar conditions. In addition, we will show that the asymptotic variance of the sliding block estimator is less than or equal to the asymptotic variance of the runs estimator and the disjoint block estimator.