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Risks 2017, 5(1), 10;

Distinguishing Log-Concavity from Heavy Tails

Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
Author to whom correspondence should be addressed.
Academic Editor: Qihe Tang
Received: 14 November 2016 / Revised: 10 January 2017 / Accepted: 17 January 2017 / Published: 7 February 2017
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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 results 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. View Full-Text
Keywords: heavy-tailed; log-concave; mean excess function; principle of a single big jump heavy-tailed; log-concave; mean excess function; principle of a single big jump

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Asmussen, S.; Lehtomaa, J. Distinguishing Log-Concavity from Heavy Tails. Risks 2017, 5, 10.

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