Distinguishing log-concavity from heavy tails

By Søren Asmussen and Jaakko Lehtomaa
Thiele Research Reports
No. 06, December 2016
Abstract:

Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum $$X_1+X_2$$ occur as result of a single big jump with heavy tails whereas $$X_1,X_2$$ are of equal order of magnitude in the light-tailed case. The method is based on the ratio $$|X_1-X_2|/(X_1+X_2)$$, for which sharp asymptotic result are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot.

Keywords: Heavy-tailed; log-concave; Mean excess function; Principle of a single big jump Please add some keywords

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