# Log-Normal or Over-Dispersed Poisson?

## Abstract

**:**

## 1. Introduction

## 2. Empirical Illustration of the Problem

## 3. Overview of the Rival Models

#### 3.1. Data

#### 3.2. Identification

`apc`(Nielsen 2015), as well as in the Python package of the same name (Harnau 2017).

#### 3.3. Over-Dispersed Poisson Model

#### 3.3.1. Assumptions

#### 3.3.2. Estimation

#### 3.3.3. Sampling Scheme

#### 3.3.4. Asymptotic Theory

**Corollary**

**1.**

#### 3.4. Generalized Log-Normal Model

#### 3.4.1. Assumptions

#### 3.4.2. Estimation

#### 3.4.3. Sampling Scheme

#### 3.4.4. Asymptotic Theory

## 4. Encompassing Tests

#### 4.1. Identifiable Differences

#### 4.2. Null Model: Over-Dispersed Poisson

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Theorem**

**1.**

#### 4.3. Null Model: Generalized Log-Normal

**Lemma**

**4.**

**Theorem**

**2.**

#### 4.4. Distribution of Ratios of Quadratic Forms

**Lemma**

**5.**

#### 4.5. Power

**Theorem**

**3.**

## 5. Simulations

`quad_form_ratio`(Harnau 2018b) and

`apc`(Harnau 2017). The package was inspired by the R (R Core Team 2017) package

`apc`(Nielsen 2015) with similar functionality.

#### 5.1. Quality of Saddle Point Approximations

`quad_form_ratio`. Then, for each Monte Carlo quantile ${q}_{\alpha}$, we compute the difference ${\widehat{P}}^{SP}(\mathrm{R}\le {q}_{\alpha})-\alpha $. Taking the Monte Carlo cdf as the truth, we refer to this as the saddle point approximation error.

#### 5.2. Finite Sample Approximations under the Null

#### 5.3. Power

#### 5.3.1. Finite Sample Approximations Under the Alternative

#### 5.3.2. Increasing Mean Dispersion in Limiting Distributions

## 6. Empirical Applications

#### 6.1. Empirical Illustration Revisited

#### 6.2. Sensitivity to Invalid Model Reductions

#### 6.3. A General to Specific Testing Procedure

## 7. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Corollary 1

#### Appendix A.2. Proof of Lemma 1

#### Appendix A.3. Proof of Lemma 2

#### Appendix A.4. Proof of Lemma 3

#### Appendix A.5. Proof of Theorem 1

#### Appendix A.6. Proof of Lemma 4

#### Appendix A.7. Proof of Theorem 2

#### Appendix A.8. Proof of Lemma 5

#### Appendix A.9. Proof of Theorem 3

**Figure A1.**Bar chart of maximum absolute errors for the considered combinations of R and $\widehat{\mathrm{R}}$. Ordered by the sum of errors within combination across data generating processes and parameterizations increasing from top to bottom. Sum of maximum absolute errors across parameterizations indicated by “+”. $VNJ$, $TA$, and $BZ$ is short for parameters set to their estimates from the Verrall et al. (2010), Taylor and Ashe (1983) and Barnett and Zehnwirth (2000) data, respectively. Based on ${10}^{5}$ repetitions for each parametrization. $s=1$.

**Table A1.**Insurance run-off triangle taken from Barnett and Zehnwirth (2000, Table 3.5) as used in the empirical application in Section 6.2 and the simulations in Section 5.

$\mathit{i},\mathit{j}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

01 | 153,638 | 188,412 | 134,534 | 87,456 | 60,348 | 42,404 | 31,238 | 21,252 | 16,622 | 14,440 | 12,200 |

02 | 178,536 | 226,412 | 158,894 | 104,686 | 71,448 | 47,990 | 35,576 | 24,818 | 22,662 | 18,000 | - |

03 | 210,172 | 259,168 | 188,388 | 123,074 | 83,380 | 56,086 | 38,496 | 33,768 | 27,400 | - | - |

04 | 211,448 | 253,482 | 183,370 | 131,040 | 78,994 | 60,232 | 45,568 | 38,000 | - | - | - |

05 | 219,810 | 266,304 | 194,650 | 120,098 | 87,582 | 62,750 | 51,000 | - | - | - | - |

06 | 205,654 | 252,746 | 177,506 | 129,522 | 96,786 | 82,400 | - | - | - | - | - |

07 | 197,716 | 255,408 | 194,648 | 142,328 | 105,600 | - | - | - | - | - | - |

08 | 239,784 | 329,242 | 264,802 | 190,400 | - | - | - | - | - | - | - |

09 | 326,304 | 471,744 | 375,400 | - | - | - | - | - | - | - | - |

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

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

**Table A2.**Insurance run-off triangle taken from Taylor and Ashe (1983) as used in the empirical application in Section 6.3 and the simulations in Section 5.

$\mathit{i},\mathit{j}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

01 | 357,848 | 766,940 | 610,542 | 482,940 | 527,326 | 574,398 | 146,342 | 139,950 | 227,229 | 67,948 |

02 | 352,118 | 884,021 | 933,894 | 1,183,289 | 445,745 | 320,996 | 527,804 | 266,172 | 425,046 | - |

03 | 290,507 | 1,001,799 | 926,219 | 1,016,654 | 750,816 | 146,923 | 495,992 | 280,405 | - | - |

04 | 310,608 | 1,108,250 | 776,189 | 1,562,400 | 272,482 | 352,053 | 206,286 | - | - | - |

05 | 443,160 | 693,190 | 991,983 | 769,488 | 504,851 | 470,639 | - | - | - | - |

06 | 396,132 | 937,085 | 847,498 | 805,037 | 705,960 | - | - | - | - | - |

07 | 440,832 | 847,631 | 1,131,398 | 1,063,269 | - | - | - | - | - | - |

08 | 359,480 | 1,061,648 | 1,443,370 | - | - | - | - | - | - | - |

09 | 376,686 | 986,608 | - | - | - | - | - | - | - | - |

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

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**Figure 1.**Approximation error of the first order saddle point approximation to ${\mathrm{R}}_{GLN}$, shown in (

**a**), and ${\mathrm{R}}_{ODP}$, displayed in (

**b**). Monte Carlo simulation with ${10}^{7}$ draws taken as the truth. One and two Monte Carlo standard errors shaded in blue and green, respectively.

**Figure 2.**Bar chart of the area under the curve of absolute approximation errors (also roughly mean absolute error) for the considered combinations of R and $\widehat{\mathrm{R}}$. Ordered by the sum of errors within the combination across the data generating processes and parameterizations increasing from top to bottom. Sum of maximum absolute errors across parameterizations indicated by “+”. $VNJ$, $TA$ and $BZ$ are short for parameters set to their estimates from the Verrall et al. (2010) data in Table 1, the Taylor and Ashe (1983) data in Table A2 and the Barnett and Zehnwirth (2000) data in Table A1, respectively. Based on ${10}^{5}$ repetitions for each parametrization. $s=1$.

**Figure 3.**Box plots of size error at $5\%$ critical values over parameterizations ($VNJ$, $TA$ and $BZ$) and data generating processes (generalized log-normal and over-dispersed Poisson). Results for $s=1$ shown in (

**a**) and for $s=2$ in (

**b**). Medians indicated by blue lines inside the boxes. The boxes show the interquartile range. Whiskers represent the full range.

**Figure 4.**Power as t increases from zero to ${t}^{max}$. Values for ${t}^{max}$ are $1.083$ for $VNJ$, $1.396$ for $TA$ and $1.103$ for $BZ$. (

**a**) shows power when the null model is generalized log-normal, and (

**b**) shows the difference in power between the two models.

**Figure 5.**Critical values as t increases. (

**a**) shows critical values of ${\mathrm{R}}_{GLN}^{(t)}$ and (

**b**) the ratio of critical values of ${\mathrm{R}}_{ODP}^{(t)}$ to ${\mathrm{R}}_{GLN}^{(t)}$.

**Table 1.**Run-off triangle taken from Verrall et al. (2010) with an indication for splitting into sub-samples corresponding to the first and last five accident years. Accident years i in the rows; development years j in the columns.

$\mathit{i},\mathit{j}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

01 | 451,288 | 339,519 | 333,371 | 144,988 | 093,243 | 045,511 | 25,217 | 20,406 | 31,482 | 1729 |

02 | 448,627 | 512,882 | 168,467 | 130,674 | 056,044 | 033,397 | 56,071 | 26,522 | 14,346 | - |

03 | 693,574 | 497,737 | 202,272 | 120,753 | 125,046 | 037,154 | 27,608 | 17,864 | - | - |

04 | 652,043 | 546,406 | 244,474 | 200,896 | 106,802 | 106,753 | 63,688 | - | - | - |

05 | 566,082 | 503,970 | 217,838 | 145,181 | 165,519 | 091,313 | - | - | - | - |

06 | 606,606 | 562,543 | 227,374 | 153,551 | 132,743 | - | - | - | - | - |

07 | 536,976 | 472,525 | 154,205 | 150,564 | - | - | - | - | - | - |

08 | 554,833 | 590,880 | 300,964 | - | - | - | - | - | - | - |

09 | 537,238 | 701,111 | - | - | - | - | - | - | - | - |

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

**Table 2.**Power in % at $5\%$ critical values for large $\tau $ (over-dispersed Poisson DGP) and small ${\omega}^{2}$ (generalized log-normal DGP) along with the power gap in pp for the top four performers from Table 3. DGP is short for data generating process. Based on ${10}^{5}$ repetitions. $s=1$.

$\mathit{P}({\mathbf{R}}_{\mathbf{GLN}}\le {\mathit{c}}_{\mathbf{ODP}})$ | $\mathit{P}(\mathit{R}\le {\mathit{c}}_{\mathbf{ODP}}^{\widehat{\mathbf{R}}})-\mathit{P}({\mathbf{R}}_{\mathbf{GLN}}\le {\mathit{c}}_{\mathbf{ODP}})$ | ||||||

${H}_{0}$ | DGP | $\mathsf{\Pi}$ | $({R}_{ls}^{*},{\widehat{\mathrm{R}}}_{ls}^{*})$ | $({R}_{ls}^{*},{\widehat{\mathrm{R}}}_{ls})$ | $({R}_{ls},{\widehat{\mathrm{R}}}_{ql})$ | $({R}_{ls}^{*},{\widehat{\mathrm{R}}}_{ql}^{*})$ | |

$GLN$ | $ODP$ | $VNJ$ | 99.02 | 00.25 | 00.14 | 0.18 | 00.22 |

$BZ$ | 94.61 | −0.94 | −0.96 | −0.97 | −0.94 | ||

$TA$ | 65.39 | 04.18 | 03.41 | 5.28 | 03.15 | ||

$\mathit{P}({\mathbf{R}}_{\mathit{ODP}}>{\mathit{c}}_{\mathit{GLN}})$ | $\mathit{P}(\mathit{R}>{\mathit{c}}_{\mathit{ODP}}^{\widehat{\mathbf{R}}})$-$\mathbf{P}({\mathbf{R}}_{\mathit{ODP}}>{\mathit{c}}_{\mathit{GLN}})$ | ||||||

$({R}_{ls}^{*},{\widehat{\mathrm{R}}}_{ls}^{*})$ | $({R}_{ls}^{*},{\widehat{\mathrm{R}}}_{ls})$ | $({R}_{ls},{\widehat{\mathrm{R}}}_{ql})$ | $({R}_{ls}^{*},{\widehat{\mathrm{R}}}_{ql}^{*})$ | ||||

$ODP$ | $GLN$ | $VNJ$ | 99.23 | −0.30 | −0.25 | −0.34 | −0.20 |

$BZ$ | 94.67 | −1.14 | −1.15 | −1.12 | −1.12 | ||

$TA$ | 64.73 | 00.81 | 00.15 | 04.68 | 02.49 |

${\mathit{H}}_{0}$: Generalized Log-Normal | ${\mathit{H}}_{0}$: Over-Dispersed Poisson | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{\mathit{ls}}$ | ${\mathit{R}}_{\mathit{ql}}$ |