On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Stupfler, Gilles

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 esti-mators 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.

Citation

Stupfler, G. (2019). On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails. Extremes, 22(4), 749–769. https://doi.org/10.1007/s10687-019-00351-5

• Journal Article Type: Article
• Acceptance Date: May 17, 2019
• Online Publication Date: May 28, 2019
• Publication Date: 2019-12
• Deposit Date: May 20, 2019
• Publicly Available Date: Jul 9, 2019
• Journal: Extremes
• Print ISSN: 1386-1999
• Electronic ISSN: 1572-915X
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 22
• Issue: 4
• Pages: 749–769
• 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
• Public URL: https://nottingham-repository.worktribe.com/output/2068366
• Publisher URL: https://link.springer.com/article/10.1007%2Fs10687-019-00351-5