# A New Generalization of the Pareto Distribution and Its Application to Insurance Data

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Distribution

#### 2.1. Probability Density Function

**Theorem**

**1.**

**Proof.**

#### 2.2. Hazard Rate Function

**Theorem**

**2.**

**Proof.**

#### 2.3. Mean Residual Life Function

**Theorem**

**3.**

**Proof.**

#### 2.4. Moments

**Theorem**

**4.**

**Proof.**

#### 2.5. Conjugate Distributions

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

#### 2.6. Stochastic Ordering

- (i)
- stochastic order (denoted by $X{\u2aaf}_{ST}Y$) if $\overline{F}\left(x\right)\le \overline{G}\left(x\right)$ for all $x,$
- (ii)
- hazard rate order (denoted by $X{\u2aaf}_{HR}Y$) if $h\left(x\right)\ge r\left(x\right)$ for all $x,$
- (iii)
- likelihood ratio order (denoted by $X{\u2aaf}_{LR}Y$) if $\frac{f\left(x\right)}{g\left(x\right)}$ decreases for all x.

**Theorem**

**7.**

**Proof.**

- (i)
- in the first-order descending stochastic dominance (denoted by $X{\u2aaf}^{1}Y$) iff ${F}_{1}^{D}\left(x\right)\le {G}_{1}^{D}\left(x\right)$ for each $x\in [a,b]$.
- (ii)
- in the second-order descending stochastic dominance (denoted by $X{\u2aaf}^{2}Y$) iff ${F}_{2}^{D}\left(x\right)\le {G}_{2}^{D}\left(x\right)$ for each $x\in [a,b]$.
- (iii)
- in the N-order descending stochastic dominance (denoted by $X{\u2aaf}^{N}Y$) iff ${F}_{N}^{D}\left(x\right)\le {G}_{N}^{D}\left(x\right)$ for each $x\in [a,b]$ and ${F}_{k}^{D}\left(a\right)\le {G}_{k}^{D}\left(a\right)$ for $2\le k\le N-1$, $N\ge 3$.

**Theorem**

**8.**

- (i)
- If ${\theta}_{1}\ge {\theta}_{2}>0$, then $X{\u2aaf}^{1}Y$.
- (ii)
- If ${\theta}_{1}\ge {\theta}_{2}>0$, then $X{\u2aaf}^{2}Y$.
- (iii)
- If ${\theta}_{1}\ge {\theta}_{2}>1$, then $X{\u2aaf}^{N}Y$ for $N\ge 3$.

**Proof.**

- (i)
- For ${\theta}_{1}\ge {\theta}_{2}>0$, we have$${F}_{1}^{D}\left(x\right)={\overline{F}}_{{\theta}_{1}}\left(x\right)=\frac{\Gamma (\lambda ,{\theta}_{1}log(x/\alpha ))}{\Gamma \left(\lambda \right)}\le \frac{\Gamma (\lambda ,{\theta}_{2}log(x/\alpha ))}{\Gamma \left(\lambda \right)}={\overline{G}}_{{\theta}_{2}}\left(x\right)={G}_{1}^{D}\left(x\right).$$Therefore, for ${\theta}_{1}\ge {\theta}_{2}>0$, $X{\u2aaf}^{1}Y$.
- (ii)
- For ${\theta}_{1}\ge {\theta}_{2}>0$, we have$${F}_{2}^{D}\left(x\right)={\int}_{x}^{\infty}{F}_{1}^{D}\left(y\right)dy\le {\int}_{x}^{\infty}{G}_{1}^{D}\left(y\right)dy={G}_{2}^{D}\left(x\right).$$Therefore, for ${\theta}_{1}\ge {\theta}_{2}>0$, $X{\u2aaf}^{2}Y$.
- (iii)
- For ${\theta}_{1}\ge {\theta}_{2}>1$, we have$${F}_{3}^{D}\left(x\right)={\int}_{x}^{\infty}{F}_{2}^{D}\left(y\right)dy\le {\int}_{x}^{\infty}{G}_{2}^{D}\left(y\right)dy={G}_{3}^{D}\left(x\right).$$Also, for ${\theta}_{1}\ge {\theta}_{2}>1$, we have$${F}_{2}^{D}\left(\alpha \right)={\int}_{\alpha}^{\infty}{\overline{F}}_{{\theta}_{1}}\left(y\right)dy={\mu}_{{\theta}_{1}}\le {\mu}_{{\theta}_{2}}={\int}_{\alpha}^{\infty}{\overline{G}}_{{\theta}_{2}}\left(y\right)dy={G}_{2}^{D}\left(\alpha \right).$$Therefore, for ${\theta}_{1}\ge {\theta}_{2}>1$, $X{\u2aaf}^{3}Y$.

## 3. Some Theoretical Financial Results

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 4. Maximum Likelihood Estimation

## 5. Numerical Application

- (1)
- Pareto distribution:$$\begin{array}{c}\hfill f\left(x\right)=\frac{\theta}{x}{\left(\frac{\alpha}{x}\right)}^{\theta},\phantom{\rule{2.em}{0ex}}x\ge \alpha ,\phantom{\rule{1.em}{0ex}}\alpha ,\theta >0.\end{array}$$
- (2)
- Shifted log-normal (SLN):$$\begin{array}{c}\hfill f\left(x\right)=\frac{\theta \sqrt{2\pi}}{x-\alpha}exp\left[-\frac{1}{2{\theta}^{2}}{(log(x-\alpha )-\lambda )}^{2}\right],\phantom{\rule{2.em}{0ex}}x\ge \alpha ,\phantom{\rule{1.em}{0ex}}\alpha ,\theta >0,\lambda \in \mathrm{I}\phantom{\rule{-2.2pt}{0ex}}\mathbb{R}\phantom{\rule{1.0pt}{0ex}}.\end{array}$$
- (3)
- Burr distribution:$$\begin{array}{c}\hfill f\left(x\right)=\frac{\theta \lambda {(x-\alpha )}^{\theta -1}}{{\left[1-{(x-\alpha )}^{\theta}\right]}^{\lambda +1}},\phantom{\rule{2.em}{0ex}}x\ge \alpha ,\phantom{\rule{1.em}{0ex}}\alpha ,\theta ,\lambda >0.\end{array}$$
- (4)
- Stoppa distribution:$$\begin{array}{c}\hfill f\left(x\right)=\frac{\lambda \theta}{x}{\left(\frac{\alpha}{x}\right)}^{\theta}{\left[1-{\left(\frac{\alpha}{x}\right)}^{\theta}\right]}^{\lambda -1},\phantom{\rule{2.em}{0ex}}x\ge \alpha ,\phantom{\rule{1.em}{0ex}}\alpha ,\theta ,\lambda >0.\end{array}$$
- (5)
- Log-gamma distribution (LG):$$\begin{array}{c}\hfill f\left(x\right)=\frac{{(1+x-\alpha )}^{-1-1/\theta}}{{\theta}^{\lambda}\Gamma \left(\lambda \right)}{log}^{\lambda -1}(1+x-\alpha ),\phantom{\rule{2.em}{0ex}}x\ge \alpha ,\phantom{\rule{1.em}{0ex}}\alpha ,\theta ,\lambda >0.\end{array}$$

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Probability density function of GTLG distribution for selected values of $\theta $ and $\lambda $ when $\alpha =1$.

**Figure 2.**Hazard rate function of GTLG distribution for selected values of $\theta $ and $\lambda $ when $\alpha =1.$

**Figure 3.**Mean residual life function of GTLG distribution for selected values of $\theta $ and $\lambda $ when $\alpha =1.$

**Figure 4.**Limited expected value function of GTLG distribution for selected values of $\theta $ and $\lambda $ when $\alpha =1.$

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

Pareto | $\theta =0.249\phantom{\rule{0.166667em}{0ex}}\left(0.057\right)$ | 77.939 | 157.878 | 158.822 | 159.822 |

SLN | $\theta =1.477\phantom{\rule{0.166667em}{0ex}}\left(0.239\right)$ | 66.080 | 136.161 | 138.05 | 140.05 |

$\lambda =1.668\phantom{\rule{0.166667em}{0ex}}\left(0.339\right)$ | |||||

Burr | $\theta =2.287\phantom{\rule{0.166667em}{0ex}}\left(0.895\right)$ | 67.352 | 138.703 | 140.592 | 142.592 |

$\lambda =0.243\phantom{\rule{0.166667em}{0ex}}\left(0.106\right)$ | |||||

Stoppa | $\theta =0.768\phantom{\rule{0.166667em}{0ex}}\left(0.159\right)$ | 66.321 | 136.643 | 138.532 | 140.532 |

$\lambda =12.013\phantom{\rule{0.166667em}{0ex}}\left(6.065\right)$ | |||||

LG | $\theta =0.802\phantom{\rule{0.166667em}{0ex}}\left(0.271\right)$ | 66.273 | 136.547 | 138.435 | 140.435 |

$\lambda =2.474\phantom{\rule{0.166667em}{0ex}}\left(0.755\right)$ | |||||

GTLG | $\theta =1.845\phantom{\rule{0.166667em}{0ex}}\left(0.606\right)$ | 65.987 | 135.974 | 137.863 | 139.863 |

$\lambda =7.401\phantom{\rule{0.166667em}{0ex}}\left(2.352\right)$ |

**Table 2.**Test statistics (p-values) of goodness-of-fit tests of the considered models when $\alpha =0.1.$

Distribution | Kolmogorov-Smirnov | Cramér-Von Misses | Anderson-Darling |
---|---|---|---|

Pareto | 0.360 (0.010) | 0.706 (0.012) | 3.574 (0.014) |

SLN | 0.116 (0.933) | 0.031 (0.970) | 0.205 (0.989) |

Burr | 0.182 (0.500) | 0.090 (0.633) | 0.483 (0.761) |

Stoppa | 0.148 (0.746) | 0.040 (0.933) | 0.242 (0.974) |

LN | 0.149 (0.731) | 0.043 (0.918) | 0.257 (0.966) |

GTLG | 0.148 (0.745) | 0.040 (0.932) | 0.242 (0.974) |

© 2018 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**

Ghitany, M.E.; Gómez-Déniz, E.; Nadarajah, S. A New Generalization of the Pareto Distribution and Its Application to Insurance Data. *J. Risk Financial Manag.* **2018**, *11*, 10.
https://doi.org/10.3390/jrfm11010010

**AMA Style**

Ghitany ME, Gómez-Déniz E, Nadarajah S. A New Generalization of the Pareto Distribution and Its Application to Insurance Data. *Journal of Risk and Financial Management*. 2018; 11(1):10.
https://doi.org/10.3390/jrfm11010010

**Chicago/Turabian Style**

Ghitany, Mohamed E., Emilio Gómez-Déniz, and Saralees Nadarajah. 2018. "A New Generalization of the Pareto Distribution and Its Application to Insurance Data" *Journal of Risk and Financial Management* 11, no. 1: 10.
https://doi.org/10.3390/jrfm11010010