# On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science

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## Abstract

**:**

## 1. Introduction

## 2. Lehmann’s Alternative

#### The Lehmann’s Alternative Stoppa Distribution

## 3. Extremal Properties

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

## 4. Some Candidates of Mixing Distributions

#### 4.1. Stochastic Ordering

**Definition**

**3.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 4.2. Estimation

`WinRats`), it is convenient to express this function in terms of the modified Bessel function of the first kind, ${I}_{\nu}\left(z\right)$. The relationship between both functions (see Johnson et al. 2005, p. 19) is given by

#### Conjugate Distribution

**Theorem**

**2.**

**Proof.**

#### 4.3. Novel Heavy-Tail Regression Models

#### 4.3.1. SG Case

`MAXIMIZE`in-built function in the

`WinRATS`software package by using the BFGS algorithms.

#### 4.3.2. SIG Case

## 5. Numerical Experiments

`danishuni`, which can be downloaded from the R package

`CASdatasets`and also from Extreme Value Statistics in S-plus libraries, collected at Copenhagen Reinsurance, comprising 2157 fire losses, adjusted for inflation to reflect 1985 values, over DKK 1,000,000 during the period 1980 to 1990, adjusted for inflation to reflect 1985. A detailed statistical analysis of this dataset can be found in McNeil (1988), in Albrecher et al. (2017), and also in Embrechts et al. (1997). The second dataset is

`norfire`, which is also available in the R package

`CASdatasets`, and includes 9181 fire losses over the period 1972 to 1992 from an unknown Norwegian insurer. A priority of NKR 500,000 (if this amount is exceeded, the reinsurer becomes liable to pay) was applied to derive this set of data.

`Mathematica`and has also been verified via

^{®}v.12.0`WinRATS v.7.0`. The codes are available from the authors upon request. Standard errors of the estimates were obtained by finite differentiation. The

`WinRATS v.7.0`software package also gives the option to directly calculate the maximum of the log-likelihood returning the entries of the Fisher information matrix. The parameters can also be estimated via an EM algorithm as shown in the Appendix A.

`CASdatasets`, see also Frees (2010). We consider as a response variable the claimant’s total economic loss. The empirical distribution of this variable combines losses of small, moderate, and large sizes, which makes it suitable for fitting heavy-tailed distributions. Other remarkable features of this set of data are unimodality, skewness, and a long right tail, showing a high likelihood of extremely expensive losses. Below, in Figure 5, we have plotted the histogram of this dataset. We have also superimposed the densities of the S, SG, and SIG distributions. The SIG distribution adheres closely to empirical data.

- ATTORNEY takes the value 1 if the claimant is represented by an attorney and 0 otherwise;
- CLMSEX takes the value 1 if the claimant is male and 0 otherwise;
- MARRIED takes the value 1 if the claimant is married and 0 otherwise;
- SINGLE takes the value 1 if the claimant is single and 0 otherwise;
- WIDOWED takes the value 1 if the claimant is widowed and 0 otherwise;
- CLMINSUR, whether or not the claimant’s vehicle was uninsured (=1 if yes and 0 otherwise);
- SEATBELT, whether or not the claimant was wearing the seatbelt/child restraint (=1 if yes and 0 otherwise);
- CLMAGE, claimant’s age.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Parameters Estimation for SG and SIG Distributions

#### Appendix A.1.1. SG Case

#### Appendix A.1.2. SIG Case

#### Appendix A.2. EM Algorithm for the SG and SIG Regression Models

#### Appendix A.2.1. SG Regression Model

- at the E-step, calculate the pseudo-values$$\begin{array}{ccc}\hfill {t}_{i}& =& E\left({\mathsf{\Lambda}}_{i}|{x}_{i},{\widehat{\theta}}_{i},{\widehat{\alpha}}^{\left(j\right)}\right)=\frac{{\alpha}^{\left(j\right)}+1}{{\alpha}^{\left(j\right)}-log{\psi}_{\sigma ,{\widehat{\theta}}_{i}^{\left(j\right)}}\left({x}_{i}\right)}\hfill \\ \hfill {s}_{i}& =& E\left(log{\mathsf{\Lambda}}_{i}|{x}_{i},{\widehat{\theta}}_{i},{\widehat{\alpha}}^{\left(j\right)}\right)=\Psi (1+{\widehat{\alpha}}^{\left(j\right)})-log{\widehat{\alpha}}^{\left(j\right)}\hfill \\ & & -log(1-log{\psi}_{\sigma ,{\widehat{\theta}}_{i}^{\left(j\right)}}\left({x}_{i}\right)),\hfill \end{array}$$
- at the M-step, first update the regressors ${\widehat{\mathit{\beta}}}^{\top (j+1)}$ by fitting an Stoppa regression model by including the covariates in the shape parameter as described above, by using the pseudo-values ${t}_{i}$ and ${s}_{i}$. Then update the estimate of the parameter $\alpha $ by letting$$\begin{array}{ccc}\hfill {\widehat{\alpha}}^{(j+1)}& =& exp\left[\frac{{\sum}_{i=1}^{n}{t}_{i}-{\sum}_{i=1}^{n}{s}_{i}-n}{n}+\Psi \left({\widehat{\alpha}}^{\left(j\right)}\right)\right].\hfill \end{array}$$
- If some convergence condition is satisfied then stop iterating, otherwise move back to the E-step for another iteration.

#### Appendix A.2.2. SIG Regression Model

- at the E-step, calculate numerically the pseudo-values$$\begin{array}{ccc}\hfill {m}_{i}& =& E\left(1/{\mathsf{\Lambda}}_{i}|{x}_{i},{\widehat{\theta}}_{i},{\widehat{\Gamma}}^{\left(j\right)}\right);\phantom{\rule{0.277778em}{0ex}}{t}_{i}=E\left({\mathsf{\Lambda}}_{i}|{x}_{i},{\widehat{\theta}}_{i},{\widehat{\Gamma}}^{\left(j\right)}\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\hfill \\ \hfill {s}_{i}& =& E\left(log{\mathsf{\Lambda}}_{i}|{x}_{i},{\widehat{\theta}}_{i},{\widehat{\Gamma}}^{\left(j\right)}\right).\hfill \end{array}$$
- at the M-step, first update the regressors ${\widehat{\mathit{\beta}}}^{\top (j+1)}$ by fitting an Stoppa regression model by including the covariates in the shape parameter as described above, by using the pseudo-values ${t}_{i}$ and ${s}_{i}$. Then update the estimate of the parameter $\Gamma $ by letting$$\begin{array}{ccc}\hfill {\widehat{\Gamma}}^{(j+1)}& =& \sqrt{\frac{n}{{\sum}_{i=1}^{n}{t}_{i}+{\sum}_{i=1}^{n}{m}_{i}-2n}}.\hfill \end{array}$$
- If some convergence condition is satisfied then stop iterating, otherwise move back to the E-step for another iteration.

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**Figure 1.**Different shapes of the pdf (10) for special values of the parameters.

**Figure 2.**Empirical (smoothed) distribution (histogram) and theoretical distribution model of the different distributions considered for the Danish (

**left**panel) and Norwegian (

**right**panel) datasets.

**Figure 3.**Smooth cdf of the empirical Danish (

**left**panel) and Norwegian (

**right**panel) claims data as compared to the theoretical models.

**Figure 4.**Absolute errors of the limited expected values for Danish dataset (

**left**) and Norwegian dataset (

**right**).

**Figure 5.**Histogram and densities for the S, SG, and SIG distributions for the automobile bodily injury claims dataset.

**Figure 6.**Hill’s estimator ${\theta}_{k}$ values varying k, calculated for the automobile bodily injury claims dataset.

Model | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\lambda}}$ | NLL |
---|---|---|---|

Pareto | 0.582 | – | 130.331 |

Stoppa | 1.198 | 3.861 | 119.372 |

**Table 2.**Parameter estimates and their corresponding p-values (in brackets), negative of the maximum of the log likelihood function (NLL), AIC, Kolmogorov–Smirnov and Anderson–Darling tests for the Weibull, Lognormal, Burr, and Stoppa distributions and mixture models: SG, SIG, and SGIG.

Danish Data | |||||||||

$\widehat{\alpha}$ | $\widehat{\lambda}$ | $\widehat{\beta}$ | $\widehat{\theta}$ | $\widehat{\Gamma}$ | NLL | AIC | KS | AD | |

lognormal | 1.557 | −0.290 | 3404.340 | 6812.68 | 0.2549 | 365.672 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

Weibull | 1.584 | 0.660 | 3510.610 | 7025.22 | 0.0724 | 31.752 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

Burr | 1.223 | 1.115 | 3339.440 | 6682.89 | >1 | >100 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

Stoppa | 1.163 | 1.395 | 3342.420 | 6688.84 | 0.0277 | 0.00055 | |||

(<0.001) | (<0.001) | (0.3772) | (0.4930) | ||||||

SG | 19.550 | 14.241 | 1.512 | ≈0 | 3334.788 | 6675.580 | 0.022 | 0.00031 | |

($0.002$) | ($0.008$) | (<0.001) | (0.6368) | (0.7955) | |||||

SIG | −0.5 | 5.883 | 1.517 | 11.323 | 3335.000 | 6676.000 | 0.022 | 0.000328 | |

($0.012$) | (<0.001) | (<0.001) | (0.6368) | (0.7705) | |||||

SGIG | −4.546 | 2.989 | 1.517 | 16.370 | 3335.150 | 6678.300 | 0.022 | 0.00034 | |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (0.6368) | (0.767) | ||||

Norwegian Data | |||||||||

$\widehat{\alpha}$ | $\widehat{\lambda}$ | $\widehat{\beta}$ | $\widehat{\theta}$ | $\widehat{\Gamma}$ | NLL | AIC | KS | AD | |

lognormal | 6.313 | 1.537 | 21,097.400 | 42,198.80 | 0.0682 | 23.082 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

Weibull | 0.689 | 1134.39 | 21,150.500 | 42,305.00 | 0.0765 | 37.756 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

Burr | 0.0078 | 20.295 | 23,685.620 | 47,375.20 | 0.419 | 704.395 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

Stoppa | 1.143 | 1.124 | 21,045.290 | 42,094.600 | 0.0854 | 0.00627 | |||

(<0.001) | (<0.001) | (<0.001) | (<0.001) | ||||||

SG | 6.404 | 2.686 | 1.593 | ≈0 | 20,931.033 | 41,868.100 | 0.0158 | 0.00016 | |

(<0.001) | (<0.001) | (<0.001) | (0.9011) | (0.8685) | |||||

SIG | −0.5 | 0.559 | 1.673 | 4.472 | 20,931.400 | 41,868.700 | 0.0162 | 0.00021 | |

(<0.001) | (<0.001) | (<0.001) | (0.8846) | (0.7905) | |||||

SGIG | 1.559 | 1.588 | 1.632 | 1.626 | 20,929.900 | 41,867.900 | 0.0154 | 0.00016 | |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (0.9163) | (0.8685) |

**Table 3.**Limited expected value for the different distributions considered, and different values of the fixed amount deductible x for the Danish dataset.

Deductible | Limited Expected Value | ||||
---|---|---|---|---|---|

Empirical | Stoppa | SG | SIG | SGIG | |

6.00 | 2.42371 | 2.42879 | 2.44939 | 2.43608 | 2.43616 |

6.15 | 2.43641 | 2.44278 | 2.46265 | 2.44931 | 2.44940 |

6.30 | 2.44871 | 2.45630 | 2.47544 | 2.46206 | 2.46217 |

6.45 | 2.46052 | 2.46938 | 2.48778 | 2.47437 | 2.47450 |

6.60 | 2.47209 | 2.48205 | 2.49971 | 2.48627 | 2.48640 |

6.75 | 2.48345 | 2.49433 | 2.51123 | 2.49776 | 2.49791 |

6.90 | 2.49459 | 2.50623 | 2.52238 | 2.50888 | 2.50904 |

7.05 | 2.50547 | 2.51778 | 2.53317 | 2.51964 | 2.51982 |

7.20 | 2.51607 | 2.52899 | 2.54363 | 2.53007 | 2.53025 |

7.35 | 2.52637 | 2.53989 | 2.55376 | 2.54017 | 2.54037 |

7.50 | 2.53644 | 2.55047 | 2.56359 | 2.54997 | 2.55019 |

7.65 | 2.54620 | 2.56077 | 2.57313 | 2.55949 | 2.55971 |

7.80 | 2.55563 | 2.57080 | 2.58240 | 2.56872 | 2.56896 |

7.95 | 2.56491 | 2.58056 | 2.59140 | 2.57770 | 2.57795 |

8.10 | 2.57401 | 2.59006 | 2.60015 | 2.58642 | 2.58668 |

8.25 | 2.58294 | 2.59933 | 2.60866 | 2.59491 | 2.59518 |

8.40 | 2.59165 | 2.60836 | 2.61694 | 2.60317 | 2.60345 |

8.55 | 2.60026 | 2.61717 | 2.62501 | 2.61121 | 2.61150 |

8.70 | 2.60876 | 2.62577 | 2.63286 | 2.61904 | 2.61934 |

8.85 | 2.61703 | 2.63417 | 2.64052 | 2.62667 | 2.62698 |

9.00 | 2.62514 | 2.64237 | 2.64798 | 2.63410 | 2.63443 |

9.15 | 2.63324 | 2.65039 | 2.65526 | 2.64136 | 2.64169 |

9.30 | 2.64117 | 2.65822 | 2.66236 | 2.64844 | 2.64878 |

9.45 | 2.64891 | 2.66588 | 2.66929 | 2.65534 | 2.65570 |

9.60 | 2.65653 | 2.67337 | 2.67606 | 2.66209 | 2.66245 |

9.75 | 2.66414 | 2.68070 | 2.68267 | 2.66868 | 2.66905 |

9.90 | 2.67175 | 2.68788 | 2.68913 | 2.67512 | 2.67550 |

**Table 4.**Limited expected value for the different distributions considered, and different values of the fixed amount deductible x for the Norwegian dataset.

Deductible | Limited Expected Value | ||||
---|---|---|---|---|---|

Empirical | Stoppa | SG | SIG | SGIG | |

600 | 594.306 | 592.565 | 594.307 | 594.14 | 594.189 |

1100 | 936.322 | 900.662 | 935.339 | 934.021 | 935.392 |

1600 | 1139.38 | 1084.90 | 1136.21 | 1136.55 | 1137.46 |

2100 | 1269.48 | 1214.54 | 1267.37 | 1269.27 | 1269.35 |

2600 | 1357.43 | 1313.79 | 1360.36 | 1363.11 | 1362.55 |

3100 | 1421.89 | 1393.80 | 1430.21 | 1433.22 | 1432.30 |

3600 | 1473.58 | 1460.58 | 1484.96 | 1487.80 | 1486.72 |

4100 | 1516.11 | 1517.74 | 1529.23 | 1531.64 | 1530.56 |

4600 | 1551.11 | 1567.61 | 1565.91 | 1567.72 | 1566.75 |

5100 | 1580.55 | 1611.75 | 1596.91 | 1598.02 | 1597.23 |

5600 | 1606.31 | 1651.30 | 1623.52 | 1623.88 | 1623.31 |

6100 | 1629.25 | 1687.09 | 1646.68 | 1646.24 | 1645.93 |

6600 | 1650.39 | 1719.74 | 1667.04 | 1665.8 | 1665.77 |

7100 | 1669.51 | 1749.72 | 1685.13 | 1683.08 | 1683.33 |

7600 | 1686.98 | 1777.43 | 1701.32 | 1698.47 | 1699.02 |

8100 | 1703.07 | 1803.17 | 1715.92 | 1712.29 | 1713.13 |

8600 | 1717.58 | 1827.19 | 1729.17 | 1724.77 | 1725.91 |

9100 | 1730.79 | 1849.69 | 1741.26 | 1736.11 | 1737.54 |

9600 | 1743.01 | 1870.85 | 1752.34 | 1746.46 | 1748.19 |

**Table 5.**Parameter estimates, standard errors (S.E.), and p-values for automobile bodily injury claims dataset under S, SG, and SIG regression models. NLL and AIC are included in the last two rows. The response variable is total losses.

Regression Model | |||
---|---|---|---|

Estimate (S.E.) | S | SG | SIG |

INTERCEPT | −3.704 (1.041) | 2.512 (0.019) | 17.659 (2.713) |

p-value | 0.0004 | <0.0001 | <0.0001 |

ATTORNEY | 1.651 (0.095) | 2.374 (0.244) | 20.014 (2.565) |

p-value | <0.0001 | <0.0001 | <0.0001 |

CLMSEX | 0.023 (0.087) | 2.544 (0.293) | −0.092 (0.177) |

p-value | 0.7941 | <0.0001 | 0.6022 |

MARRIED | −0.352 (0.272) | 7.009 (0.233) | −0.054 (0.582) |

p-value | 0.1951 | 0.3210 | 0.9260 |

SINGLE | −0.434 (0.281) | 2.363 (0.183) | −0.065 (0.615) |

p-value | 0.1222 | <0.0001 | 0.9158 |

WIDOWED | −1.596 (0.488) | 10.005 (14.472) | 14.631 (2.536) |

p-value | 0.0011 | 0.4802 | <0.0001 |

CLMINSUR | 0.079 (0.146) | 3.086 (0.714) | 0.271 (0.365) |

p-value | 0.5898 | <0.0001 | 0.4575 |

SEATBELT | −1.258 (0.361) | 2.508 (0.043) | −19.1757 (2.7936) |

p-value | 0.0005 | <0.0001 | <0.0001 |

CLMAGE | 0.019 (0.003) | 0.393 (0.003) | 0.0144 (0.0066) |

p-value | <0.0001 | <0.0001 | 0.0279 |

$\theta $ | 0.736 (0.015) | 1.647 (0.008) | 1.061 (0.034) |

p-value | <0.0001 | <0.0001 | <0.0001 |

$\lambda $, $\alpha $$\beta $ | 831.609 (528.599) | 1.093 (0.039) | 0.001 (0.000) |

p-value | 0.1160 | <0.0001 | 0.0236 |

–, $\beta $, $\Gamma $ | – | 0.000 (0.000) | 741.355 (104.096) |

p-value | – | <0.0001 | <0.0001 |

NLL | 2608.38 | 2566.17 | 2558.82 |

AIC | 5238.76 | 5156.34 | 5141.65 |

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Gómez-Déniz , E.; Calderín-Ojeda , E.
On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science. *Risks* **2024**, *12*, 6.
https://doi.org/10.3390/risks12010006

**AMA Style**

Gómez-Déniz E, Calderín-Ojeda E.
On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science. *Risks*. 2024; 12(1):6.
https://doi.org/10.3390/risks12010006

**Chicago/Turabian Style**

Gómez-Déniz , Emilio, and Enrique Calderín-Ojeda .
2024. "On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science" *Risks* 12, no. 1: 6.
https://doi.org/10.3390/risks12010006