# A New Heavy Tailed Class of Distributions Which Includes the Pareto

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## Abstract

**:**

## 1. Introduction

## 2. The $\mathcal{MPLG}$ Distribution and Its Properties

**Remark**

**1.**

- Generate two random numbers ${u}_{1}$ and ${u}_{2}$ from the standard uniform distribution, $U(0,1)$.
- Generate random variates ${\tilde{a}}_{1}$ from the exponential distribution with mean $1/\theta $ and ${\tilde{a}}_{2}$ from the Erlang distribution with shape parameter 2 and rate parameter $\theta $ by using ${u}_{1}$.
- If ${u}_{2}\le p$, then set $y={\tilde{a}}_{1}$; otherwise, set $y={\tilde{a}}_{2}$.
- Generate $x={x}_{0}\phantom{\rule{0.166667em}{0ex}}exp\left\{y\right\}$.

#### 2.1. Moments and Log-Moments

#### 2.2. Unimodality

#### 2.3. Stochastic Ordering

**Definition**

**1.**

- (i)
- Stochastic order $\left({X}_{1}{\le}_{st}{X}_{2}\right)$ if ${F}_{{X}_{1}}\left(x\right)\ge {F}_{{X}_{2}}\left(x\right)$ for all x.
- (ii)
- Hazard rate order $\left({X}_{1}{\le}_{hr}{X}_{2}\right)$ if ${h}_{{X}_{1}}\left(x\right)\ge {h}_{{X}_{2}}\left(x\right)$ for all x.
- (iii)
- Mean excess order $\left({X}_{1}{\le}_{me}{X}_{2}\right)$ if ${e}_{{X}_{1}}\left(x\right)\ge {e}_{{X}_{2}}\left(x\right)$ for all x, where ${e}_{X}(\xb7)$ is mean excess function given in expression (17).

**Proposition**

**1.**

- i.
- If${x}_{01}={x}_{02}$,${\theta}_{1}\ge {\theta}_{2}$and${\lambda}_{1}\le {\lambda}_{2}$, then${X}_{1}{\le}_{lr}{X}_{2}$,${X}_{1}{\le}_{hr}{X}_{2}$and${X}_{1}{\le}_{st}{X}_{2}$.
- ii.
- If${x}_{01}\le {x}_{02}$,${\theta}_{1}={\theta}_{2}$and${\lambda}_{1}={\lambda}_{2}$, then${X}_{1}{\le}_{lr}{X}_{2}$,${X}_{1}{\le}_{hr}{X}_{2}$and${X}_{1}{\le}_{st}{X}_{2}$.

**Proof.**

#### 2.4. Integrated Tail Distribution and Equilibrium Hazard Rate

#### 2.5. Estimation

#### 2.5.1. Method of Log-Moments

#### 2.5.2. Maximum Likelihood Estimation

#### 2.6. Composite Models

#### 2.6.1. Composite Lognormal-$\mathcal{MPLG}$ Model

#### 2.6.2. Composite Weibull-$\mathcal{MPLG}$ Model

#### 2.6.3. Composite Paralogistic-$\mathcal{MPLG}$ Model

## 3. Insurance Results

#### 3.1. Mean Excess Function

#### 3.2. Excess of Loss Reinsurance

#### 3.3. Value-at-Risk and Tail Value-at-Risk

**Definition**

**2.**

_{δ}(X), is the δ-quantile of X. That is

**Proposition**

**2.**

_{δ}(X) of the$\mathcal{MPLG}(\theta ,\lambda ,{x}_{0})$distribution is

**Proof.**

_{δ}(X). As in general, the loss distributions are typically skewed, the VaR is a non-coherent risk measure due to the lack of subadditivity (see Klugman et al. 2012), for that reason, the Tail Value-at-Risk (TVaR) of X (see Acerbi and Tasche 2002) is usually considered as a more informative and more useful risk measure. The TVaR is given by

**Proposition**

**3.**

_{δ}(X) for$\mathcal{MPLG}(\theta ,\lambda ,{x}_{0})$distribution is given by

**Proof.**

#### 3.4. Limited Loss Variable for Composite Models

**Definition**

**3.**

**Definition**

**4.**

**Proposition**

**4.**

**Proof.**

## 4. Case Studies

#### 4.1. An Application to Automobile Claims Data

#### 4.2. An Application to Fire Insurance Claims

`SMPrcacticals`’ add-on package for

`R`, available from the CRAN website http://cran.r-project.org/. Parameter estimation for all the models considered has been completed by the method of maximum likelihood (which is implemented using the function ‘

`mle`’/‘

`mle2`’ in

`R`). In Table 2, parameter estimates and standard errors (S.E.) for all the composite distributions and the three measures of model validation considered in our first example are exhibited. Among the composite models, we can see that the Weibull-$\mathcal{MPLG}$ composite model gives the best fit overall for this dataset in terms of NLL, AIC and BIC values.

#### 4.3. Theil’s Income Indices

**Proposition**

**5.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Expression (1) is a genuine pdf:**

**Proof of Proposition**

**1.**

**Proof of Proposition**

**2.**

**Proof of Proposition**

**3.**

## References

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**Figure 1.**Graphs of the pdf (1) for different values of parameter $\theta $, $\lambda $ and fixed value of ${x}_{0}$.

**Figure 2.**Graphs of the hazard rate function (4) for different values of parameter $\theta $, $\lambda $ and for fixed value of ${x}_{0}$.

**Table 1.**Estimated values of different heavy-tailed models, corresponding standard errors (S.E.), negative of the log-likelihood function (NLL), Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) for automobile claims dataset (De Jong and Heller 2008).

Distribution | Parameter Estimates (S.E.) | NLL | AIC | BIC |
---|---|---|---|---|

Pareto | $\widehat{\theta}=0.661\phantom{\rule{0.166667em}{0ex}}\left(0.010\right)$ | 38,024.80 | 76,051.61 | 76,058.05 |

lognormal | $\widehat{\mu}=6.810\phantom{\rule{0.166667em}{0ex}}\left(0.017\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\sigma}=1.189\phantom{\rule{0.166667em}{0ex}}\left(0.012\right)$ | 38,852.15 | 77,708.31 | 77,721.19 |

loggamma | $\widehat{\alpha}=1.115\phantom{\rule{0.166667em}{0ex}}\left(0.021\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\beta}=5.109\phantom{\rule{0.166667em}{0ex}}\left(0.118\right)$ | 38,998.18 | 78,000.36 | 78,013.23 |

Frećhet | $\widehat{\alpha}=0.659\phantom{\rule{0.166667em}{0ex}}\left(0.019\right),\phantom{\rule{0.277778em}{0ex}}\widehat{s}=254.0\phantom{\rule{0.166667em}{0ex}}\left(11.87\right)$ | 38,408.30 | 76,822.60 | 76,841.92 |

$\widehat{b}=157.1\phantom{\rule{0.166667em}{0ex}}\left(4.279\right)$ | ||||

Weibull | $\widehat{\mu}=0.786\phantom{\rule{0.166667em}{0ex}}\left(0.008\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\sigma}=1690.8\phantom{\rule{0.166667em}{0ex}}\left(33.64\right)$ | 39,491.60 | 78,987.19 | 79,000.07 |

Lomax | $\widehat{\alpha}=2.047\phantom{\rule{0.166667em}{0ex}}\left(0.088\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\lambda}=2205.1\phantom{\rule{0.166667em}{0ex}}\left(133.1\right)$ | 39,169.85 | 78,343.70 | 78,356.58 |

PAT | $\widehat{\alpha}=0.895\phantom{\rule{0.166667em}{0ex}}\left(0.095\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\theta}=0.740\phantom{\rule{0.166667em}{0ex}}\left(0.017\right)$ | 38,006.30 | 76,016.61 | 76,029.49 |

Inverse Weibull | $\widehat{\alpha}=1.053\phantom{\rule{0.166667em}{0ex}}\left(0.012\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\sigma}=518.8\phantom{\rule{0.166667em}{0ex}}\left(7.636\right)$ | 38,595.61 | 77,195.22 | 77,208.09 |

$\mathcal{MPLG}$ | $\widehat{\theta}=0.943\phantom{\rule{0.166667em}{0ex}}\left(0.018\right),\phantom{\rule{0.277778em}{0ex}}\widehat{\lambda}=0.698\phantom{\rule{0.166667em}{0ex}}\left(0.073\right)$ | 37,965.99 | 75,935.98 | 75,948.86 |

**Table 2.**Estimated values of different heavy-tailed models, corresponding standard errors (S.E.), NLL, AIC, and BIC of different composite models for the Danish fire losses dataset.

Distribution | Parameter Estimates (S.E.) | NLL | AIC | BIC | |
---|---|---|---|---|---|

lognormal-Lomax | $\widehat{\mu}=0.104\phantom{\rule{0.166667em}{0ex}}\left(0.020\right)$ | $\widehat{\sigma}=0.182\phantom{\rule{0.166667em}{0ex}}\left(0.011\right)$ | 3860.47 | 7728.94 | 7752.223 |

$\widehat{\lambda}=0.365\phantom{\rule{0.166667em}{0ex}}\left(0.123\right)$ | $\widehat{\theta}=1.144\phantom{\rule{0.166667em}{0ex}}\left(0.029\right)$ | ||||

lognormal-Pareto | $\widehat{\mu}=0.137\phantom{\rule{0.166667em}{0ex}}\left(0.019\right)$ | $\widehat{\sigma}=0.197\phantom{\rule{0.166667em}{0ex}}\left(0.012\right)$ | 3865.86 | 7737.72 | 7755.182 |

$\widehat{\theta}=1.208\phantom{\rule{0.166667em}{0ex}}\left(0.030\right)$ | |||||

lognormal-$\mathcal{MPLG}$ | $\widehat{\mu}=0.045\phantom{\rule{0.166667em}{0ex}}\left(0.017\right)$ | $\widehat{\theta}=2.060\phantom{\rule{0.166667em}{0ex}}\left(0.039\right)$ | 3872.40 | 7752.81 | 7776.09 |

${\widehat{x}}_{0}=0.745\phantom{\rule{0.166667em}{0ex}}\left(0.070\right)$ | $\widehat{\lambda}=65.804\phantom{\rule{0.166667em}{0ex}}\left(377.425\right)$ | ||||

Weibull-Lomax | $\widehat{\tau}=15.345\phantom{\rule{0.166667em}{0ex}}\left(0.671\right)$ | $\widehat{\varphi}=0.969\phantom{\rule{0.166667em}{0ex}}\left(0.007\right)$ | 3823.70 | 7655.40 | 7678.68 |

$\widehat{\lambda}=0.561\phantom{\rule{0.166667em}{0ex}}\left(0.127\right)$ | $\widehat{\theta}=0.971\phantom{\rule{0.166667em}{0ex}}\left(0.007\right)$ | ||||

Weibull-Pareto | $\widehat{\tau}=14.048\phantom{\rule{0.166667em}{0ex}}\left(0.502\right)$ | $\widehat{\psi}=0.997\phantom{\rule{0.166667em}{0ex}}\left(0.008\right)$ | 3840.38 | 7686.76 | 7704.22 |

$\widehat{\theta}=1.003\phantom{\rule{0.166667em}{0ex}}\left(0.008\right)$ | |||||

Weibull-$\mathcal{MPLG}$ | $\widehat{\tau}=18.763\phantom{\rule{0.166667em}{0ex}}\left(0.986\right)$ | ${\widehat{x}}_{0}=0.787\phantom{\rule{0.166667em}{0ex}}\left(0.006\right)$ | 3823.30 | 7654.60 | 7677.88 |

$\widehat{\theta}=1.938\phantom{\rule{0.166667em}{0ex}}\left(0.034\right)$ | $\widehat{\lambda}=4.614\phantom{\rule{0.166667em}{0ex}}\left(0.159\right)$ | ||||

paralogistic-$\mathcal{MPLG}$ | $\widehat{\alpha}=16.719\phantom{\rule{0.166667em}{0ex}}\left(1.311\right)$ | ${\widehat{x}}_{0}=0.901\phantom{\rule{0.166667em}{0ex}}\left(0.020\right)$ | 3824.482 | 7656.96 | 7680.25 |

$\widehat{\theta}=1.989\phantom{\rule{0.166667em}{0ex}}\left(0.035\right)$ | $\widehat{\lambda}=3.379\phantom{\rule{0.166667em}{0ex}}\left(0.266\right)$ |

**Table 3.**Limited expected value for the distributions considered and different values of the policy limit u.

Composite Models | ||||||||
---|---|---|---|---|---|---|---|---|

Policy | Empirical | lognormal | lognormal | Weibull | Weibull | lognormal | Weibull | Paralogistic |

Limit u | Pareto | Lomax | Pareto | Lomax | $\mathcal{MPLG}$ | $\mathcal{MPLG}$ | $\mathcal{MPLG}$ | |

1 | 0.989 | 0.998 | 0.985 | 0.989 | 0.989 | 0.987 | 0.986 | 0.989 |

2 | 1.565 | 1.726 | 1.525 | 1.560 | 1.548 | 1.333 | 1.266 | 1.467 |

5 | 2.138 | 2.489 | 2.122 | 2.155 | 2.143 | 2.015 | 1.943 | 2.093 |

10 | 2.447 | 2.932 | 2.362 | 2.482 | 2.464 | 2.341 | 2.292 | 2.398 |

20 | 2.707 | 3.285 | 2.566 | 2.781 | 2.674 | 2.542 | 2.522 | 2.591 |

100 | 2.958 | 3.851 | 2.826 | 3.299 | 2.923 | 2.740 | 2.778 | 2.793 |

200 | 3.037 | 4.017 | 2.884 | 3.463 | 2.972 | 2.770 | 2.823 | 2.826 |

∞ | 3.063 | 4.665 | 3.005 | 4.289 | 3.060 | 2.804 | 2.885 | 2.868 |

© 2019 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**

Bhati, D.; Calderín-Ojeda, E.; Meenakshi, M. A New Heavy Tailed Class of Distributions Which Includes the Pareto. *Risks* **2019**, *7*, 99.
https://doi.org/10.3390/risks7040099

**AMA Style**

Bhati D, Calderín-Ojeda E, Meenakshi M. A New Heavy Tailed Class of Distributions Which Includes the Pareto. *Risks*. 2019; 7(4):99.
https://doi.org/10.3390/risks7040099

**Chicago/Turabian Style**

Bhati, Deepesh, Enrique Calderín-Ojeda, and Mareeswaran Meenakshi. 2019. "A New Heavy Tailed Class of Distributions Which Includes the Pareto" *Risks* 7, no. 4: 99.
https://doi.org/10.3390/risks7040099