# Small Area Estimation Using a Semiparametric Spatial Model with Application in Insurance

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Semiparametric Mixed-Effects Model

**Proposition**

**1.**

## 3. Life Insurance Data Analysis

## 4. Simulation Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Proposition 1.**

## References

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**Figure 1.**Scatterplot of the provincial per capita number of life insurance contracts (response) versus the work experience of insurance sales branches in Iran.

**Figure 2.**Boxplot of the EMSPE of ${\widehat{\mathit{\eta}}}_{i}^{EB}$ of the number of life insurance contracts in IRAN under semiparametric model (1) and parametric model (2) under the semiparametric spatial GLMM.

**Figure 3.**Boxplots of the EMSPE of ${\widehat{\mathit{\eta}}}_{i}^{EB}$ for the semiparametric and parametric spatial Poisson models.

**Figure 4.**Heat map of the MSPE of ${\widehat{\mathit{\eta}}}_{i}^{EB}$ per provinces in the proposed model (

**left**) and parametric model (

**right**).

**Figure 5.**Predicted values of the nonparametric component for the two semiparametric (Spline) and parametric (Linear) models in the simulated data.

Parameter | Estimate | Standard Error |
---|---|---|

$\mathit{\beta}$ | −0.1498 | 0.02699 |

${B}_{0}$ | −0.1915 | 0.06451 |

${B}_{1}$ | 0.06252 | 0.01494 |

${B}_{2}$ | −0.01838 | 0.06013 |

${B}_{3}$ | −0.0603 | 0.0588 |

${B}_{4}$ | −0.01853 | 0.07475 |

${B}_{5}$ | 0.02873 | 0.04812 |

${\lambda}_{u}$ | 0.2397 | 0.01419 |

**Table 2.**Model parameter estimates and corresponding standard errors using the MLE approach for the semiparametric spatial Poisson model.

Parameter | Estimate | Standard Error |
---|---|---|

$\mathit{\beta}$ | −76.0308 | 22.10532 |

${B}_{0}$ | 69.5480 | 22.31195 |

${B}_{1}$ | 75.4873 | 22.03408 |

${B}_{2}$ | 73.3040 | 22.15233 |

${B}_{3}$ | 72.5719 | 22.08202 |

${B}_{4}$ | 77.8309 | 22.12026 |

${B}_{5}$ | 78.4698 | 22.10168 |

${\lambda}_{u}$ | 1.2003 | 0.17648 |

**Table 3.**Model parameter estimates and corresponding standard errors using the MLE approach for the parametric spatial Poisson model.

Parameter | Estimate | Standard Error |
---|---|---|

${\mathit{\beta}}_{0}$ | −2.4955 | 0.05152 |

${\mathit{\beta}}_{1}$ | 0.8918 | 0.03687 |

${\lambda}_{u}$ | 0.1378 | 0.01313 |

**Table 4.**Value of the EMSPE of ${\widehat{\mathit{\eta}}}_{i}^{EB}$ and ISQR for the semiparametric and parametric spatial Poisson models.

Semiparametric | Parametric | |
---|---|---|

$\mathrm{MSPE}$ | $0.009085778$ | $0.02691221$ |

$\mathrm{ISQR}$ | $0.009277414$ | $0.02369608$ |

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**MDPI and ACS Style**

Hosseini, S.E.; Shahsavani, D.; Rabiei, M.R.; Arashi, M.; Baghishani, H.
Small Area Estimation Using a Semiparametric Spatial Model with Application in Insurance. *Symmetry* **2022**, *14*, 2194.
https://doi.org/10.3390/sym14102194

**AMA Style**

Hosseini SE, Shahsavani D, Rabiei MR, Arashi M, Baghishani H.
Small Area Estimation Using a Semiparametric Spatial Model with Application in Insurance. *Symmetry*. 2022; 14(10):2194.
https://doi.org/10.3390/sym14102194

**Chicago/Turabian Style**

Hosseini, Seyede Elahe, Davood Shahsavani, Mohammad Reza Rabiei, Mohammad Arashi, and Hossein Baghishani.
2022. "Small Area Estimation Using a Semiparametric Spatial Model with Application in Insurance" *Symmetry* 14, no. 10: 2194.
https://doi.org/10.3390/sym14102194