Abstract
Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.
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Acknowledgments
The author acknowledges an anonymous Associate Editor and two anonymous reviewers for their helpful comments that led to a substantially improved version of this paper, and in particular to the writing of Theorem 3 and Corollary 3. The author also gratefully acknowledges Laurens de Haan for a couple of very instructive conversations about tail empirical processes which led him to write this paper.
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Stupfler, G. On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails. Extremes 22, 749–769 (2019). https://doi.org/10.1007/s10687-019-00351-5
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DOI: https://doi.org/10.1007/s10687-019-00351-5
Keywords
- Average excess
- Conditional tail moment
- Heavy-tailed distribution
- Hill estimator
- Joint convergence
- Random right-truncation
- Tail homogeneity
- Tail index
- Threshold