Abstract
Threshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time.
Highlights
Extreme value analysis of heavy-tailed distributions is important for various applications
Let X1, . . . , Xn be independent and identically distributed (i.i.d.) random variables with a distribution function F, where F is in the domain of attraction (DoA) of an extreme value distribution Gγ with extreme value index γ > 0
We suggest a smooth estimator for the asymptotic mean squared error (AMSE) of the Hill estimator, called SAMSEE
Summary
Extreme value analysis of heavy-tailed distributions is important for various applications. For n1, a data-driven, but computationally expensive selection method is provided, where the whole bootstrap procedure is repeated for various possible values of n1 Another estimator for kopt is given in Beirlant et al (2002), which employs least squares estimates from an exponential regression approach. This example illustrates how the new methods enable the selection of a threshold that depends on covariates.
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