Testing the Existence of Clustering in the Extreme Values

This paper introduces an estimator for the extremal index as the ratio of the number of elements of two point processes defined by threshold sequences {un}, {vn} and a partition of the sequence in different blocks of the same size. The first point process is defined by the sequence of the block maxima that exceed {un}. This paper introduces a thinning of this point process, defined by a threshold {vn} with {vn}> {un}, and with the appealing property that under some mild conditions the ratio of the number of elements of both point processes is a consistent estimator of the extremal index. The method supports a hypothesis test for the extremal index, and hence for testing the existence of clustering in the extreme values. Other advantages are that it allows some freedom to choose {un}, and it is not very sensitive to the choice of the partition. Finally, the stylized facts found in financial returns (clustering, skewness, heavy tails) are tested via the extremal index, in this case for the DaX returns