# Claims Modelling with Three-Component Composite Models

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## Abstract

**:**

## 1. Introduction

#### 1.1. Current Literature

#### 1.2. Proposed Composite Models

## 2. Weibull-Lognormal-Pareto Model

## 3. Weibull-Lognormal-GPD Model

## 4. Weibull-Lognormal-Burr Model

## 5. Application to Two Data Sets

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Density ratios of Weibull (3.8511, 0.7717) to Lognormal (1, 1) (left graph) and Weibull (0.7071, 2) to Lognormal (−0.5881, 0.4915) (right graph).

**Figure A2.**History plot, posterior distribution function, posterior density function, and autocorrelation plot of Weibull-lognormal-Pareto model parameters for Danish fire insurance claims data. (The blue and purple lines represent two separate chains of simulations.)

## Notes

1 | The AIC is defined as $-2l+2{n}_{p}$, and the BIC as $-2l+{n}_{p}\mathrm{ln}{n}_{d}$, where $l$ is the computed maximum log-likelihood value, ${n}_{p}$ is the effective number of parameters estimated, and ${n}_{d}$ is the number of observations. The KS test statistic is calculated as $\mathrm{max}|{F}_{n}(x)-F(x)|$, that is, the maximum distance between the empirical and fitted distribution functions. The DIC is computed as the posterior mean of the deviance plus the effective number of parameters under the Bayesian framework (Spiegelhalter et al. 2003). |

2 | The link functions are $\mathrm{l}\mathrm{n}\varnothing ={\rho}_{\mathrm{1,0}}+{\rho}_{\mathrm{1,1}}{x}_{1}+{\rho}_{\mathrm{1,2}}{x}_{2}+{\rho}_{\mathrm{1,3}}{x}_{3}+{\rho}_{\mathrm{1,4}}{x}_{4}$, $\mu ={\rho}_{\mathrm{2,0}}+{\rho}_{\mathrm{2,1}}{x}_{1}+{\rho}_{\mathrm{2,2}}{x}_{2}+{\rho}_{\mathrm{2,3}}{x}_{3}+{\rho}_{\mathrm{2,4}}{x}_{4}$, and $\mathrm{l}\mathrm{n}\beta ={\rho}_{\mathrm{3,0}}+{\rho}_{\mathrm{3,1}}{x}_{1}+{\rho}_{\mathrm{3,2}}{x}_{2}+{\rho}_{\mathrm{3,3}}{x}_{3}+{\rho}_{\mathrm{3,4}}{x}_{4}$, where $\rho $’s are the regression coefficients and ${x}_{1}$, ${x}_{2}$, ${x}_{3}$, ${x}_{4}$ are the four covariates. We have checked the covariates in the data, and there is no multicollinearity issue. |

## References

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**Figure 2.**Examples of density functions of three-component composite models with weights of 20%, 60%, and 20%, respectively.

**Figure 4.**3D map of 14 models’ 99% VaR estimates against BIC and KS values for Danish fire insurance claims data (left) and vehicle insurance claims data (right). The three major categories are noted as traditional models (triangles), two-component composite models (empty circles), and new three-component composite models (solid circles).

Model | NLL | AIC | BIC | KS | DIC |
---|---|---|---|---|---|

Weibull | 5270.47 (14) | 10,544.94 (14) | 10,556.58 (14) | 0.2555 (13) | 33,495 (14) |

Lognormal | 4433.89 (12) | 8871.78 (12) | 8883.42 (12) | 0.1271 (12) | 31,822 (12) |

Pareto | 5051.91 (13) | 10107.81 (13) | 10119.45 (13) | 0.2901 (14) | 33,058 (13) |

Burr | 3835.12 (7) | 7676.24 (6) | 7693.70 (6) | 0.0383 (9) | 30,625 (6) |

GB2 | 3834.77 (6) | 7677.53 (7) | 7700.82 (7) | 0.0602 (11) | 30,626 (7) |

Lognormal-Pareto | 3865.86 (11) | 7737.73 (11) | 7755.19 (11) | 0.0323 (8) | 30,687 (11) |

Lognormal-GPD | 3860.47 (10) | 7728.94 (10) | 7752.23 (9) | 0.0196 (6) | 30,677 (10) |

Lognormal-Burr | 3857.83 (9) | 7725.65 (9) | 7754.76 (10) | 0.0193 (5) | 30,673 (9) |

Weibull-Pareto | 3840.38 (8) | 7686.75 (8) | 7704.21 (8) | 0.0516 (10) | 30,636 (8) |

Weibull-GPD | 3823.70 (5) | 7655.40 (5) | 7678.68 (3) | 0.0255 (7) | 30,604 (5) |

Weibull-Burr | 3817.57 (4) | 7645.14 (3) | 7674.24 (2) | 0.0147 (4) | 30,593 (4) |

Weibull-Lognormal-Pareto | 3815.89 (2) | 7641.77 (1) | 7670.88 (1) | 0.0114 (2) | 30,589 (1) |

Weibull-Lognormal-GPD | 3815.88 (1) | 7643.76 (2) | 7678.69 (4) | 0.0113 (1) | 30,590 (3) |

Weibull-Lognormal-Burr | 3815.89 (3) | 7645.77 (4) | 7686.52 (5) | 0.0114 (3) | 30,590 (2) |

Quantile | Empirical | Weibull-Lognormal-Pareto | Weibull-Lognormal-GPD | Weibull-Lognormal-Burr |
---|---|---|---|---|

1% | 0.845 | 0.811 | 0.811 | 0.811 |

5% | 0.905 | 0.905 | 0.905 | 0.905 |

10% | 0.964 | 0.967 | 0.967 | 0.967 |

25% | 1.157 | 1.164 | 1.164 | 1.164 |

50% | 1.634 | 1.620 | 1.619 | 1.620 |

75% | 2.645 | 2.654 | 2.651 | 2.654 |

90% | 5.080 | 5.081 | 5.080 | 5.081 |

95% | 8.406 | 8.303 | 8.317 | 8.303 |

99% | 24.614 | 25.971 | 26.172 | 25.971 |

**Table 3.**Parameter estimates of fitting three-component composite models to Danish fire insurance claims data.

Model | Maximum Likelihood | Bayesian MCMC (Posterior Distribution) | |||
---|---|---|---|---|---|

Estimate | Standard Error | Mean | Median | Standard Deviation | |

Weibull-Lognormal-Pareto | τ = 16.253 | 1.290 | 16.127 | 16.073 | 1.351 |

σ = 0.649 | 0.089 | 0.716 | 0.705 | 0.110 | |

α = 1.411 | 0.040 | 1.416 | 1.415 | 0.042 | |

θ_{1} = 0.947 | 0.011 | 0.952 | 0.951 | 0.013 | |

θ_{2} = 1.976 | 0.189 | 2.113 | 2.078 | 0.254 | |

Weibull-Lognormal-GPD | τ = 16.252 | 1.289 | 16.165 | 16.101 | 1.373 |

σ = 0.648 | 0.088 | 0.728 | 0.719 | 0.113 | |

α = 1.402 | 0.097 | 1.440 | 1.432 | 0.096 | |

λ = −0.018 | 0.174 | 0.041 | 0.034 | 0.178 | |

θ_{1} = 0.947 | 0.011 | 0.952 | 0.951 | 0.013 | |

θ_{2} = 1.988 | 0.218 | 2.106 | 2.070 | 0.291 | |

Weibull-Lognormal-Burr | τ = 16.253 | 1.290 | 16.14 | 16.106 | 1.376 |

σ = 0.649 | 0.089 | 0.725 | 0.718 | 0.113 | |

α = 0.449 | 1.575 | 0.526 | 0.477 | 0.193 | |

γ = 3.143 | 1.015 | 3.069 | 2.994 | 1.010 | |

β = 0.001 | 0.039 | 0.391 | 0.358 | 0.260 | |

θ_{1} = 0.947 | 0.011 | 0.952 | 0.951 | 0.013 | |

θ_{2} = 1.976 | 0.189 | 2.045 | 2.015 | 0.273 |

Model | NLL | AIC | BIC | KS | DIC |
---|---|---|---|---|---|

Weibull | 7132.74 (14) | 14,269.47 (14) | 14,282.02 (14) | 0.1414 (13) | 50,289 (14) |

Lognormal | 6567.94 (12) | 13,139.87 (12) | 13,152.42 (12) | 0.0816 (5) | 49,160 (12) |

Pareto | 6906.02 (13) | 13,816.03 (13) | 13,828.57 (13) | 0.1471 (14) | 49,836 (13) |

Burr | 6292.07 (10) | 12,590.15 (10) | 12,608.96 (10) | 0.0911 (10) | 48,609 (10) |

GB2 | 6300.41 (11) | 12,608.82 (11) | 12,633.90 (11) | 0.0783 (4) | 48,627 (11) |

Lognormal-Pareto | 6281.18 (9) | 12,568.36 (9) | 12,587.17 (9) | 0.0934 (12) | 48,587 (9) |

Lognormal-GPD | 6153.72 (7) | 12,315.43 (7) | 12,340.52 (7) | 0.0853 (8) | 48,333 (7) |

Lognormal-Burr | 6076.13 (4) | 12,162.27 (4) | 12,193.62 (4) | 0.0766 (3) | 48,178 (4) |

Weibull-Pareto | 6249.84 (8) | 12,505.67 (8) | 12,524.49 (8) | 0.0933 (11) | 48,524 (8) |

Weibull-GPD | 6144.36 (6) | 12,296.72 (6) | 12,321.81 (6) | 0.0891 (9) | 48,314 (6) |

Weibull-Burr | 6062.21 (3) | 12,134.43 (3) | 12,165.78 (3) | 0.0827 (7) | 48,150 (3) |

Weibull-Lognormal-Pareto | 6088.95 (5) | 12,187.91 (5) | 12,219.27 (5) | 0.0822 (6) | 48,204 (5) |

Weibull-Lognormal-GPD | 5971.78 (1) | 11,955.56 (1) | 11,993.19 (1) | 0.0764 (2) | 48,090 (2) |

Weibull-Lognormal-Burr | 6025.74 (2) | 12,065.48 (2) | 12,109.38 (2) | 0.0743 (1) | 46,355 (1) |

Quantile | Empirical | Weibull-Lognormal-Pareto | Weibull-Lognormal-GPD | Weibull-Lognormal-Burr |
---|---|---|---|---|

1% | 0.234 | 0.250 | 0.250 | 0.252 |

5% | 0.338 | 0.318 | 0.314 | 0.317 |

10% | 0.354 | 0.361 | 0.353 | 0.358 |

25% | 0.440 | 0.510 | 0.493 | 0.496 |

50% | 1.045 | 0.964 | 0.961 | 0.968 |

75% | 2.560 | 2.257 | 2.346 | 2.473 |

90% | 5.813 | 5.464 | 5.762 | 5.711 |

95% | 8.993 | 9.600 | 8.852 | 8.887 |

99% | 18.845 | 28.889 | 18.167 | 18.842 |

**Table 6.**Parameter estimates of fitting three-component composite models to vehicle insurance claims data.

Model | Maximum Likelihood | Bayesian MCMC (Posterior Distribution) | |||
---|---|---|---|---|---|

Estimate | Standard Error | Mean | Median | Standard Deviation | |

Weibull-Lognormal-Pareto | τ = 7.373 | 0.331 | 7.383 | 7.376 | 0.326 |

σ = 1.789 | 0.047 | 1.797 | 1.795 | 0.063 | |

α = 2.632 | 0.267 | 2.492 | 2.521 | 0.254 | |

θ_{1} = 0.365 | 0.004 | 0.365 | 0.365 | 0.004 | |

θ_{2} = 1312 | 1077 | 1054 | 1057 | 544 | |

Weibull-Lognormal-GPD | τ = 7.707 | 0.304 | 7.856 | 7.851 | 0.244 |

σ = 16.917 | 0.053 | 17.454 | 17.451 | 0.386 | |

α = 4.483 | 0.016 | 4.428 | 4.428 | 0.039 | |

λ = 12.717 | 0.054 | 12.444 | 12.443 | 0.122 | |

θ_{1} = 0.366 | 0.003 | 0.357 | 0.357 | 0.003 | |

θ_{2} = 4.626 | 0.033 | 4.699 | 4.699 | 0.093 | |

Weibull-Lognormal-Burr | τ = 7.647 | 0.341 | 7.784 | 7.943 | 0.258 |

σ = 12.401 | 0.210 | 12.392 | 12.288 | 0.231 | |

α = 9.034 | 0.110 | 9.164 | 9.232 | 0.218 | |

γ = 0.724 | 0.020 | 0.667 | 0.607 | 0.067 | |

β = 35.198 | 0.371 | 35.297 | 35.595 | 0.514 | |

θ_{1} = 0.367 | 0.004 | 0.366 | 0.366 | 0.003 | |

θ_{2} = 3.538 | 0.092 | 3.683 | 3.848 | 0.255 |

**Table 7.**Parameter estimates and standard errors of fitting Weibull-lognormal-GPD regression model to vehicle insurance claims data with covariates.

Model Component | Covariate | Estimate | Standard Error | t-Ratio | p-Value |
---|---|---|---|---|---|

Weibull Component (small claims) | Intercept | 0.850 | 0.644 | 1.32 | 0.19 |

Exposure | −0.085 | 0.055 | −1.54 | 0.12 | |

Vehicle Age | 0.233 | 0.253 | 0.92 | 0.36 | |

Driver Age | 1.921 | 0.013 | 143.55 | 0.00 | |

Gender | −0.012 | 0.009 | −1.29 | 0.20 | |

Lognormal Component (medium claims) | Intercept | −57.411 | 26.128 | −2.20 | 0.03 |

Exposure | 8.179 | 26.821 | 0.30 | 0.76 | |

Vehicle Age | 7.670 | 5.425 | 1.41 | 0.16 | |

Driver Age | −5.023 | 4.670 | −1.08 | 0.28 | |

Gender | −1.221 | 11.118 | −0.11 | 0.91 | |

GPD Component (large claims) | Intercept | 2.269 | 0.192 | 11.80 | 0.00 |

Exposure | −1.028 | 0.186 | −5.52 | 0.00 | |

Vehicle Age | −0.116 | 0.043 | −2.71 | 0.01 | |

Driver Age | −0.049 | 0.023 | −2.08 | 0.04 | |

Gender | 0.275 | 0.074 | 3.72 | 0.00 |

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**MDPI and ACS Style**

Li, J.; Liu, J.
Claims Modelling with Three-Component Composite Models. *Risks* **2023**, *11*, 196.
https://doi.org/10.3390/risks11110196

**AMA Style**

Li J, Liu J.
Claims Modelling with Three-Component Composite Models. *Risks*. 2023; 11(11):196.
https://doi.org/10.3390/risks11110196

**Chicago/Turabian Style**

Li, Jackie, and Jia Liu.
2023. "Claims Modelling with Three-Component Composite Models" *Risks* 11, no. 11: 196.
https://doi.org/10.3390/risks11110196