# Misspecification Tests for Log-Normal and Over-Dispersed Poisson Chain-Ladder Models

## Abstract

**:**

## 1. Introduction

## 2. Data and Sub-Samples

## 3. Log-Normal Model

#### 3.1. Model and Hypotheses

#### 3.2. Estimation

#### 3.2.1. Estimation in Unrestricted Model ${M}^{LN}$

#### 3.2.2. Estimation with Common Variances in ${M}_{{\sigma}^{2}}^{LN}$

#### 3.2.3. Estimation with Common Variances and Linear Predictors in ${M}_{\mu ,{\sigma}^{2}}^{LN}$

#### 3.2.4. Remarks

#### 3.3. Testing for Common Variances

#### 3.4. Testing for Common Linear Predictors

**Theorem**

**1.**

## 4. Over-Dispersed Poisson

#### 4.1. Model and Hypotheses

#### 4.2. Estimation

#### 4.2.1. Estimation in Unrestricted Model ${M}^{ODP}$

#### 4.2.2. Estimation with Common Variances in ${M}_{{\sigma}^{2}}^{ODP}$

#### 4.2.3. Estimation with Common Variances and Linear Predictors in ${M}_{\mu ,{\sigma}^{2}}^{ODP}$

#### 4.3. Sampling Scheme

#### 4.4. Asymptotic Testing for Common Over-Dispersion

**Theorem**

**2.**

#### 4.5. Asymptotic Testing for Common Linear Predictors

**Lemma**

**1.**

## 5. Empirical Applications

`apc`(Nielsen 2015) for the empirical applications and simulations below.

#### 5.1. Log-Normal Chain-Ladder

#### 5.2. Over-Dispersed Poisson Chain-Ladder

#### 5.3. Log-Normal (Extended) Chain-Ladder

## 6. Simulations

#### 6.1. Performance of Bartlett test ${\chi}^{2}$ Approximation

#### 6.2. Rejection Frequencies of Tests for Common Variance in Log-Normal Model

#### 6.3. Performance of Over-Dispersed Poisson Model Asymptotics

#### 6.3.1. Rejection Frequencies of Tests for Common Over-Dispersion

#### 6.3.2. Independence of Test for Common Linear Predictors

^{6}triangles per scenario.

#### 6.4. Remark

## 7. Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Theorem 1

#### Appendix A.2. Proof of Theorem 2

#### Appendix A.3. Proof of Lemma 1

**Table A1.**Insurance run-off triangle taken from Verrall et al. (2010, Table 1) as used in the empirical application in Section 5.1 and the simulations in Section 6.

i, j | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 451,288 | 339,519 | 333,371 | 144,988 | 093,243 | 045,511 | 25,217 | 20,406 | 31,482 | 1729 |

2 | 448,627 | 512,882 | 168,467 | 130,674 | 056,044 | 033,397 | 56,071 | 26,522 | 14,346 | - |

3 | 693,574 | 497,737 | 202,272 | 120,753 | 125,046 | 037,154 | 27,608 | 17,864 | - | - |

4 | 652,043 | 546,406 | 244,474 | 200,896 | 106,802 | 106,753 | 63,688 | - | - | - |

5 | 566,082 | 503,970 | 217,838 | 145,181 | 165,519 | 091,313 | - | - | - | - |

6 | 606,606 | 562,543 | 227,374 | 153,551 | 132,743 | - | - | - | - | - |

7 | 536,976 | 472,525 | 154,205 | 150,564 | - | - | - | - | - | - |

8 | 554,833 | 590,880 | 300,964 | - | - | - | - | - | - | - |

9 | 537,238 | 701,111 | - | - | - | - | - | - | - | - |

10 | 684,944 | - | - | - | - | - | - | - | - | - |

**Table A2.**Insurance run-off triangle taken from Barnett and Zehnwirth (2000, Table 3.5) as used in the empirical application in Section 5.1 and the simulations in Section 6.

i, j | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 153,638 | 188,412 | 134,534 | 087,456 | 060,348 | 42,404 | 31,238 | 21,252 | 16,622 | 14,440 | 12,200 |

2 | 178,536 | 226,412 | 158,894 | 104,686 | 071,448 | 47,990 | 35,576 | 24,818 | 22,662 | 18,000 | - |

3 | 210,172 | 259,168 | 188,388 | 123,074 | 083,380 | 56,086 | 38,496 | 33,768 | 27,400 | - | - |

4 | 211,448 | 253,482 | 183,370 | 131,040 | 078,994 | 60,232 | 45,568 | 38,000 | - | - | - |

5 | 219,810 | 266,304 | 194,650 | 120,098 | 087,582 | 62,750 | 51,000 | - | - | - | - |

6 | 205,654 | 252,746 | 177,506 | 129,522 | 096,786 | 82,400 | - | - | - | - | - |

7 | 197,716 | 255,408 | 194,648 | 142,328 | 105,600 | - | - | - | - | - | - |

8 | 239,784 | 329,242 | 264,802 | 190,400 | - | - | - | - | - | - | - |

9 | 326,304 | 471,744 | 375,400 | - | - | - | - | - | - | - | - |

10 | 420,778 | 590,400 | - | - | - | - | - | - | - | - | - |

11 | 496,200 | - | - | - | - | - | - | - | - | - | - |

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**Figure 1.**Examples for splits of run-off triangles into two (

**a**), three (

**b**) and four (

**c**) sub-samples. Sub-samples are denoted by ${\mathcal{I}}_{\ell}$. Accident years i are in the rows, development years j in the columns.

**Figure 2.**Log-normal chain-ladder model for Verrall et al. (2010) data. Sub-sample structure shown in (

**a**), estimation and test results in (

**b**).

**Figure 3.**Over-dispersed Poisson chain-ladder model for Taylor and Ashe (1983) data. Sub-sample structure shown in (

**a**), estimation and test results in (

**b**).

**Figure 4.**Log-normal chain-ladder ($LN$) and extended chain-ladder ($LNe$) model for Barnett and Zehnwirth (2000) data. Sub-sample structure shown in (

**a**), estimation and test results in (

**b**).

**Figure 5.**pp-plots for the adjusted Bartlett distribution $\mathrm{Ba}/C$ against ${\chi}^{2}$ for varying degrees of freedom. (

**a**) and (

**b**) show results for degrees of freedom corresponding to the empirical applications and half those degrees of freedom, respectively.

**Figure 6.**Power curves for log-normal dispersion tests based on sub-sample structures from empirical applications. Empirical maximum to minimum ratios indicated by horizontal lines. BZ is short for Barnett and Zehnwirth (2000), VNJ for Verrall et al. (2010), and TA for Taylor and Ashe (1983).

**Figure 7.**Power gap for log-normal dispersion tests based on sub-sample structures from empirical applications. Empirical maximum to minimum ratios indicated by horizontal lines. Rejection frequencies shown in (

**a**), gap to asymptotic rejection frequencies in (

**b**).

**Table 1.**Insurance run-off triangle taken from Taylor and Ashe (1983) as an example for a generalized trapezoid. Entries are aggregate incremental paid amounts for claims of accident year i and development year j. Calendar years $k=i+j-1$ are on the diagonals increasing from the top left.

i, j | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 357,848 | 0766,940 | 0610,542 | 0482,940 | 527,326 | 574,398 | 146,342 | 139,950 | 227,229 | 67,948 |

2 | 352,118 | 0884,021 | 0933,894 | 1,183,289 | 445,745 | 320,996 | 527,804 | 266,172 | 425,046 | - |

3 | 290,507 | 1,001,799 | 0926,219 | 1,016,654 | 750,816 | 146,923 | 495,992 | 280,405 | - | - |

4 | 310,608 | 1,108,250 | 0776,189 | 1,562,400 | 272,482 | 352,053 | 206,286 | - | - | - |

5 | 443,160 | 0693,190 | 0991,983 | 0769,488 | 504,851 | 470,639 | - | - | - | - |

6 | 396,132 | 0937,085 | 0847,498 | 0805,037 | 705,960 | - | - | - | - | - |

7 | 440,832 | 0847,631 | 1,131,398 | 1,063,269 | - | - | - | - | - | - |

8 | 359,480 | 1,061,648 | 1,443,370 | - | - | - | - | - | - | - |

9 | 376,686 | 0986,608 | - | - | - | - | - | - | - | - |

10 | 344,014 | - | - | - | - | - | - | - | - | - |

**Table 2.**$P(\mathrm{Ba}/C>{c}_{\alpha})$ where ${c}_{\alpha}$ is the ${\chi}^{2}$ $\alpha $ critical value. Results are in %. Degrees of freedom shown as $(d{f}_{1},\cdots ,d{f}_{m})$.

(13, 3) | (1, 2, 4) | (3, 3, 4) | (3, 1, 3, 3) | (26, 6) | (3, 5, 8) | (6, 6, 9) | (6, 3, 6, 6) | |
---|---|---|---|---|---|---|---|---|

$\alpha $ = 10% | 9.94 | 9.05 | 9.86 | 9.20 | 9.98 | 9.93 | 9.97 | 9.92 |

$\alpha $ = 5% | 4.93 | 4.22 | 4.85 | 4.37 | 4.98 | 4.92 | 4.97 | 4.92 |

$\alpha $ = 1% | 0.95 | 0.69 | 0.92 | 0.76 | 0.99 | 0.95 | 0.98 | 0.96 |

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Harnau, J.
Misspecification Tests for Log-Normal and Over-Dispersed Poisson Chain-Ladder Models. *Risks* **2018**, *6*, 25.
https://doi.org/10.3390/risks6020025

**AMA Style**

Harnau J.
Misspecification Tests for Log-Normal and Over-Dispersed Poisson Chain-Ladder Models. *Risks*. 2018; 6(2):25.
https://doi.org/10.3390/risks6020025

**Chicago/Turabian Style**

Harnau, Jonas.
2018. "Misspecification Tests for Log-Normal and Over-Dispersed Poisson Chain-Ladder Models" *Risks* 6, no. 2: 25.
https://doi.org/10.3390/risks6020025