# Distinguishing Log-Concavity from Heavy Tails

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

**Lemma**

**1.**

## 3. Theoretical Results

#### 3.1. Convergence Properties

**Proposition**

**1.**

- Let X be RV$(\alpha )$ with $\alpha >1$, eventually decreasing density f. Then,$$g(d)=1-\frac{c}{d}+\mathrm{o}(1/d)\phantom{\rule{1.em}{0ex}}\mathit{where}\phantom{\rule{4.pt}{0ex}}c=\frac{2\alpha \mathbb{E}[X]}{\alpha +1}.\phantom{\rule{0.166667em}{0ex}}$$
- Let X be Weibull distributed with $\alpha <1$. Then,$$g(d)=1-\mathrm{o}({d}^{\alpha -1}).$$
- Let X be of lognormal type. Then,$$g(d)=1-\mathrm{o}({log}^{\gamma -1}x/x).$$

**Remark**

**1.**

**Proposition**

**2.**

**Theorem**

**1.**

## 4. Statistical Application: Visual Test

**Remark**

**2.**

#### 4.1. Examples and Applications

#### 4.2. Finer Diagnostics

## 5. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Theorem**

**1.**

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Graphs of $\widehat{g}(d,n)$ for $n=10,000$ for Gamma distributed random variables with shapes $0.2,1$ and 5 in figures (

**a**), (

**b**) and (

**c**), respectively. All variables are standardised to have mean 3.

**Figure 2.**Graphs of $\widehat{g}(d,n)$ for $n=10,000$ for Weibull distributed random variables with shapes $0.2,1$ and 5 in figures (

**a**), (

**b**) and (

**c**), respectively. All variables are standardised to have mean 3.

**Figure 3.**Graph of $\widehat{g}(d,n)$ from a classical set of Danish fire insurance data that can be obtained for instance from data set ‘danish’ in the R package [23]. The data is scaled to have mean 1. The sample is traditionally used to illustrate how heavy-tailed data behaves. A similar set of data was previously used in [24]. The graph supports the usual finding that the data set is heavy-tailed.

**Figure 4.**The graphs of multiple versions of $\widehat{g}(d,n)$ based on a dataset obtained from Hansjörg Albrecher (private communication) and related to occurrences of floods in a particular area. The data is scaled to have mean 1. The sample size is $n=39$. Bivariate vectors $({X}_{1},{Y}_{1}),\dots ,({X}_{19},{Y}_{19})$ were sampled several times randomly without replacement from the original data. The overall appearance of the paths points to the data being heavy- rather than light-tailed.

**Figure 5.**$d\left(1-\widehat{g}(d,n)\right)$. Pareto in the first row, lognormal in the second, and Weibull in the last. $R=5000$ (left), $R=5\times {10}^{6}$ (right).

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Asmussen, S.; Lehtomaa, J. Distinguishing Log-Concavity from Heavy Tails. *Risks* **2017**, *5*, 10.
https://doi.org/10.3390/risks5010010

**AMA Style**

Asmussen S, Lehtomaa J. Distinguishing Log-Concavity from Heavy Tails. *Risks*. 2017; 5(1):10.
https://doi.org/10.3390/risks5010010

**Chicago/Turabian Style**

Asmussen, Søren, and Jaakko Lehtomaa. 2017. "Distinguishing Log-Concavity from Heavy Tails" *Risks* 5, no. 1: 10.
https://doi.org/10.3390/risks5010010