Bootstrapping Heavy-Tailed Data and Extremes
In this chapter, we consider two topics, viz., bootstrapping heavy-tailed time series data and bootstrapping the extremes (i.e., the maxima and the minima) of stationary processes. We call a random variable heavy-tailed if its variance is infinite. For iid random variables with such heavy tails, it is well known (cf. Feller (1971b), Chapter 17) that under some regularity conditions on the tails of the underlying distribution, the normalized sample mean converges to a stable distribution. Similar results are also known for the sample mean under weak dependence. In Section 11.2, we introduce some relevant definitions and review some known results in this area. In Sections 11.3 and 11.4, we present some results on the performance of the MBB for heavy-tailed data under dependence. Like the iid case, here the MBB works if the resample size is of a smaller order than the original sample size. Consistency properties of the MBB are presented in Section 11.3, while its invalidity for a resample size equal to the sample size is considered in Section 11.4.
KeywordsExtremal Index Bootstrap Approximation Dependent Random Variable Divisible Distribution Poisson Random Measure
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