Second order conditions provide a natural framework for establishing asymptotic results about estimators for tail related quantities. Such conditions are typically tailored to the estimation principle at hand, and may be vastly different for estimators based on the block maxima (BM) method or the peak-over-threshold (POT) approach. In this paper we provide details on the relationship between typical second order conditions for BM and POT methods in the multivariate case. We show that the two conditions typically imply each other, but with a possibly different second order parameter. The latter implies that, depending on the data generating process, one of the two methods can attain faster convergence rates than the other. The class of multivariate Archimax copulas is examined in detail; we find that this class contains models for which the second order parameter is smaller for the BM method and vice versa. The theory is illustrated by a small simulation study.
One of the most widely applied unit root test, Phillips-Perron test, enjoys in general high powers, but suffers from size distortions when moving average noise exists. As a remedy, this paper proposes a nonparametric bootstrap unit root test that specifically targets moving average noise. Via a bootstrap functional central limit theorem, the consistency of this bootstrap approach is established under general assumptions which allows a large family of non-linear time series. In simulation, this bootstrap test alleviates the size distortions of the Phillips-Perron test while preserving its high powers. *
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