# A Note on Parameter Estimation in the Composite Weibull–Pareto Distribution

## Abstract

**:**

## 1. Introduction

## 2. Parameter Estimation and Model Selection

## 3. A Simulation Analysis

- If $u\le r$, then$$x=\lambda \phantom{\rule{0.166667em}{0ex}}{\left\{-log\left[1-\frac{u}{r}\left(1-exp\left[-{\left(\frac{\theta}{\lambda}\right)}^{\alpha}\right]\right)\right]\right\}}^{1/\alpha}$$
- If $u>r$, then$$x=(\sigma +\theta ){\left(\frac{1-u}{1-r}\right)}^{-1/\beta}-\sigma $$

- Average bias of the simulated estimates:$$A{B}_{n}(\widehat{\mathsf{\Theta}})=\frac{1}{N}\phantom{\rule{0.166667em}{0ex}}\sum _{i=1}^{N}({\widehat{\mathsf{\Theta}}}_{i}-\mathsf{\Theta})\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\mathsf{\Theta}=(\widehat{\alpha},\widehat{\lambda},\widehat{\sigma},\widehat{\theta})$$
- Average root-mean-square errors:$$RMS{E}_{n}(\widehat{\mathsf{\Theta}})=\sqrt{\frac{1}{N}\phantom{\rule{0.166667em}{0ex}}\sum _{i=1}^{N}{({\widehat{\mathsf{\Theta}}}_{i}-\mathsf{\Theta})}^{2}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\mathsf{\Theta}=(\widehat{\alpha},\widehat{\lambda},\widehat{\sigma},\widehat{\theta})$$
- Coverage probability: percentage of confidence intervals containing the true value of $\mathsf{\Theta}$ at the 95% confidence level.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationship between claim rank and claim size in log–log scale and graphs of the inverse of the survival function of Weibull (W, dashed), Lomax (L, dotted) and composite Weibull–Pareto (CWL, solid) distributions.

**Figure 2.**Relationship between claim rank and claim size in log–log scale and graphs of the inverse of the survival function of composite Weibull–Pareto model in Bakar et al. (2015) (CWL2, solid).

**Table 1.**R optimization function, parameter estimates, standard errors (S.E.), Akaike’s information criterion (AIC) and Schwarz’s Bayesian criterion (SBC) under composite Weibull–Pareto (CWL) distribution model for allocated loss adjustment expenses (ALAE) dataset.

Model | R Function | Estimate (S.E.) | NLL | AIC | SBC |
---|---|---|---|---|---|

Weibull Lomax | mle | $\widehat{\alpha}=1.0375\phantom{\rule{0.166667em}{0ex}}(0.0386)$ | 5047.110 | 10,102.220 | 10,123.473 |

$\widehat{\lambda}=6.3207\phantom{\rule{0.166667em}{0ex}}(0.4898)$ | |||||

$\widehat{\sigma}=9.5937\phantom{\rule{0.166667em}{0ex}}(2.2659)$ | |||||

$\widehat{\theta}=8.0407\phantom{\rule{0.166667em}{0ex}}(2.0436)$ | |||||

Weibull Lomax | mle2 | $\widehat{\alpha}=1.0375\phantom{\rule{0.166667em}{0ex}}(0.0386)$ | 5047.110 | 10,102.220 | 10,123.473 |

$\widehat{\lambda}=6.3207\phantom{\rule{0.166667em}{0ex}}(0.4892)$ | |||||

$\widehat{\sigma}=9.5937\phantom{\rule{0.166667em}{0ex}}(2.2646)$ | |||||

$\widehat{\theta}=8.0407\phantom{\rule{0.166667em}{0ex}}(2.0407)$ |

**Table 2.**Average bias (AB) for the composite Weibull–Pareto (CWL) distribution model involving 1000 simulations of samples of size n for the mle (second column) and mle2 (third column) functions.

R Function | ||
---|---|---|

Sample Size $\mathit{n}$ | mle | mle2 |

$n=1000$ | $AB(\widehat{\alpha})=0.0194$ | $AB(\widehat{\alpha})=0.0318$ |

$AB(\widehat{\lambda})=-0.2747$ | $AB(\widehat{\lambda})=-0.4641$ | |

$AB(\widehat{\sigma})=-0.1697$ | $AB(\widehat{\sigma})=-0.3226$ | |

$AB(\widehat{\theta})=0.0174$ | $AB(\widehat{\theta})=0.0532$ | |

$n=1500$ | $AB(\widehat{\alpha})=0.0105$ | $AB(\widehat{\alpha})=0.0127$ |

$AB(\widehat{\lambda})=-0.1800$ | $AB(\widehat{\lambda})=-0.1710$ | |

$AB(\widehat{\sigma})=0.0374$ | $AB(\widehat{\sigma})=-0.2408$ | |

$AB(\widehat{\theta})=-0.0653$ | $AB(\widehat{\theta})=0.1194$ | |

$n=2000$ | $AB(\widehat{\alpha})=0.0078$ | $AB(\widehat{\alpha})=0.0059$ |

$AB(\widehat{\lambda})=-0.1177$ | $AB(\widehat{\lambda})=-0.1070$ | |

$AB(\widehat{\sigma})=-0.0909$ | $AB(\widehat{\sigma})=-0.2437$ | |

$AB(\widehat{\theta})=0.0350$ | $AB(\widehat{\theta})=0.1371$ |

**Table 3.**Average root-mean-square errors (RMSE) for the composite Weibull–Pareto (CWL) model involving 1000 simulations of samples of size n for the mle function (second column) and mle2 function (third column).

R Function | ||
---|---|---|

Sample Size $\mathit{n}$ | mle | mle2 |

$n=1000$ | $RMSE(\widehat{\alpha})=0.0763$ | $RMSE(\widehat{\alpha})=0.0957$ |

$RMSE(\widehat{\lambda})=1.1908$ | $RMSE(\widehat{\lambda})=1.5083$ | |

$RMSE(\widehat{\sigma})=3.5354$ | $RMSE(\widehat{\sigma})=4.4062$ | |

$RMSE(\widehat{\theta})=3.8563$ | $RMSE(\widehat{\theta})=3.7280$ | |

$n=1500$ | $RMSE(\widehat{\alpha})=0.0557$ | $RMSE(\widehat{\alpha})=0.0588$ |

$RMSE(\widehat{\lambda})=0.9361$ | $RMSE(\widehat{\lambda})=0.9129$ | |

$RMSE(\widehat{\sigma})=2.7904$ | $RMSE(\widehat{\sigma})=2.8334$ | |

$RMSE(\widehat{\theta})=3.1370$ | $RMSE(\widehat{\theta})=3.2292$ | |

$n=2000$ | $RMSE(\widehat{\alpha})=0.0459$ | $RMSE(\widehat{\alpha})=0.0437$ |

$RMSE(\widehat{\lambda})=0.7717$ | $RMSE(\widehat{\lambda})=0.7283$ | |

$RMSE(\widehat{\sigma})=2.3913$ | $RMSE(\widehat{\sigma})=2.4617$ | |

$RMSE(\widehat{\theta})=2.7425$ | $RMSE(\widehat{\theta})=2.7723$ |

**Table 4.**Average coverage probability (CP) at the 95% confidence level for the composite Weibull–Pareto (CWL) model involving 1000 simulations of samples of size n for the mle (second column) and mle2 (third column) functions.

R Function | ||
---|---|---|

Sample Size $\mathit{n}$ | mle | mle2 |

$n=1000$ | $CP(\widehat{\alpha})=0.9220$ | $CP(\widehat{\alpha})=0.9170$ |

$CP(\widehat{\lambda})=0.8720$ | $CP(\widehat{\lambda})=0.8690$ | |

$CP(\widehat{\sigma})=0.9050$ | $CP(\widehat{\sigma})=0.9050$ | |

$CP(\widehat{\theta})=0.8270$ | $CP(\widehat{\theta})=0.8330$ | |

$n=1500$ | $CP(\widehat{\alpha})=0.9420$ | $CP(\widehat{\alpha})=0.9150$ |

$CP(\widehat{\lambda})=0.8980$ | $CP(\widehat{\lambda})=0.8880$ | |

$CP(\widehat{\sigma})=0.9100$ | $CP(\widehat{\sigma})=0.9220$ | |

$CP(\widehat{\theta})=0.8580$ | $CP(\widehat{\theta})=0.8590$ | |

$n=2000$ | $CP(\widehat{\alpha})=0.9310$ | $CP(\widehat{\alpha})=0.9310$ |

$CP(\widehat{\lambda})=0.8970$ | $CP(\widehat{\lambda})=0.9020$ | |

$CP(\widehat{\sigma})=0.9190$ | $CP(\widehat{\sigma})=0.9210$ | |

$CP(\widehat{\theta})=0.8830$ | $CP(\widehat{\theta})=0.8860$ |

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Calderín-Ojeda, E.
A Note on Parameter Estimation in the Composite Weibull–Pareto Distribution. *Risks* **2018**, *6*, 11.
https://doi.org/10.3390/risks6010011

**AMA Style**

Calderín-Ojeda E.
A Note on Parameter Estimation in the Composite Weibull–Pareto Distribution. *Risks*. 2018; 6(1):11.
https://doi.org/10.3390/risks6010011

**Chicago/Turabian Style**

Calderín-Ojeda, Enrique.
2018. "A Note on Parameter Estimation in the Composite Weibull–Pareto Distribution" *Risks* 6, no. 1: 11.
https://doi.org/10.3390/risks6010011