# A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory

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## Abstract

**:**

## 1. Introduction

**Proposition**

**1.**

**Proof.**

## 2. Bimodal Extension of the Exponential Distribution

#### Reliability, Hazard Rate Function and Moments

## 3. Results in Risk Theory

**Proposition**

**2.**

**Proof.**

## 4. Methods of Estimation and Simulation

`FindMaximum`built-in function in

`Mathematica`software package. These methods include the Newton–Raphson and the Broyden–Fletcher–Goldfarb-Shanno (BGGS) algorithms. The same results were achieved under these two optimization functions. Although a more general structure, such as kernel regression or neural network, could provide accurate estimates, the approach used in this paper does not require training data. It can also work well, even if the fit to data is not perfect. Additionally, this method is easier to understand and interpret results, i.e., a parametric test for the significance of the parameter estimates can lead to a rejection of the null hypothesis rather than the non-parametric counterpart. Finally, from the actuarial perspective, the practitioner may be interested in the parametric approach since it provides appealing closed-form expressions, as is the case of this BE representation.

#### Simulation Experiment

## 5. A suitable Regression Model

## 6. Empirical Results

#### 6.1. Dataset 1

#### 6.2. Dataset 2

## 7. Conclusions and Extensions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The graphs of the pdf of the bimodal exponential distribution for different values of the parameters $\alpha $ and $\theta $.

**Figure 2.**The hazard rate function of the $BE$ distribution for different values of the parameters $\alpha $ and $\theta $.

**Figure 4.**Histogram of the empirical data and density of the $BE$, $GM$, and exponential distributions superimposed for the US Term Life Insurance dataset.

**Figure 5.**Histogram of the empirical data and density of the $BE$, $GM$, and exponential distributions superimposed for the Swedish motor insurance.

**Figure 6.**Empirical and fitted limited expected values for the Swedish motor insurance dataset. Exponential (dashed line), BE (thin line), and empirical (thick line).

**Table 1.**Means, standard deviations (SD) and percentage of coverage probabilities (C) for different values of $\alpha $ and $\theta $.

$\mathit{n}=50$ | $\mathit{n}=100$ | ||||
---|---|---|---|---|---|

$\mathbf{\alpha}$ | $\mathbf{\theta}$ | $\widehat{\mathbf{\alpha}}$(SD)(C) | $\widehat{\mathbf{\theta}}$(SD)(C) | $\widehat{\mathbf{\alpha}}$(SD)(C) | $\widehat{\mathbf{\theta}}$(SD)(C) |

1.0 | 1.0 | 1.0415(0.1793)(92.9) | 0.9997(0.2508)(95.3) | 1.0128(0.1042)(94.1) | 0.990155(0.1579)(94.6) |

2.0 | 1.0133(0.1021)(93.5 | 2.0564(0.4298)(94.7) | 1.0040(0.0701)(95.0) | 2.0488(0.2953)(96.0) | |

3.0 | 1.0063(0.0856)(94.3) | 3.1707(0.8710)(94.0) | 1.0051(0.0600)(95.2) | 3.0881(0.5573)(94.9) | |

2.0 | 1.0 | 2.0615(0.3737)(93.8) | 0.9813(0.2605)(94.7) | 2.0262(0.2094)(93.4) | 0.9971(0.1579)(93.9) |

2.0 | 2.0133(0.2013)(93.9) | 2.0671(0.4328)(94.6) | 2.0083(0.1414)(95.4) | 2.0266(0.2916)(95.3) | |

3.0 | 2.0147(0.1718)(95.5) | 3.2368(0.9209)(92.9) | 2.0101(0.1200)(94.0) | 3.0804(0.5570)(94.5) | |

3.0 | 1.0 | 3.1000(0.5494)(93.1) | 1.0011(0.2582)(96.3) | 3.0544(0.3125)(94.2) | 1.0085(0.1573)(95.4) |

2.0 | 3.0277(0.3052)(93.9) | 2.0567(0.4303)(92.8) | 3.0265(0.2130)(95.1) | 2.0296(0.2919)(95.9) | |

3.0 | 3.0013(0.2545)(95.8) | 3.2297(0.8992)(94.6) | 3.0076(0.1795)(95.2) | 3.1114(0.5715)(94.4) | |

$\mathbf{n}=\mathbf{150}$ | $\mathbf{n}=\mathbf{200}$ | ||||

$\mathbf{\alpha}$ | $\mathbf{\theta}$ | $\widehat{\mathbf{\alpha}}$(SD)(C) | $\widehat{\mathbf{\theta}}$(SD)(C) | $\widehat{\mathbf{\alpha}}$(SD)(C) | $\widehat{\mathbf{\theta}}$(SD)(C) |

1.0 | 1.0 | 1.0139(0.0843)(93.3) | 1.0097(0.1275)(95.3) | 1.0092(0.0731)(95.8) | 1.0014(0.1098)(92.8) |

2.0 | 1.0023(0.0574)(94.4) | 2.0158(0.2348)(95.4) | 1.0029(0.0495)(95.2) | 2.0255(0.2039)(95.2) | |

3.0 | 1.0010(0.0485)(95.4) | 3.0794(0.4495)(95.3) | 1.0030(0.0422)(93.5) | 3.0395(0.3785)(95.3) | |

2.0 | 1.0 | 2.0184(0.1696)(95.1) | 0.9979(0.1271)(94.2) | 2.0096(0.1457)(95.3) | 1.0011(0.1098)(95.7) |

2.0 | 2.0068(0.1147)(95.6) | 2.0116(0.2336)(95.9) | 2.0063(0.0992)(93.2) | 2.0133(0.2020)(95.3) | |

3.0 | 2.0016(0.0972)(95.2) | 3.0647(0.4466)(94.7) | 2.0059(0.0844)(95.3) | 3.0312(0.3763)(94.8) | |

3.0 | 1.0 | 3.0231(0.2533)(94.5) | 0.9943(0.1265)(95.2) | 3.0227(0.2193)(95.9) | 1.0037(0.1100)(95.5) |

2.0 | 3.0051(0.1719)(93.9) | 2.0268(0.2365)(96.0) | 3.0016(0.1486)(95.2) | 2.0109(0.2017)(95.1) | |

3.0 | 3.0030(0.1458)(94.5) | 3.0851(0.4518)(95.8) | 2.9982(0.1261)(95.9) | 3.0397(0.3784)(93.6) |

Variable | Description |
---|---|

Gender | Gender of the survey respondent |

Age | Age of the survey respondent |

Marstat | Marital status of the survey respondent |

(=1 if married, =2 if living with partner, and =0 otherwise) | |

Education | Number of years of education of the survey respondent |

Ethnicity | Ethnicity |

Smarstat | Marital status of the respondent’s spouse |

Sgender | Gender of the respondent’s spouse |

Sage | Age of the respondent’s spouse |

Seducation | Education of the respondent’s spouse |

Numhh | Number of household members |

Income | Annual income of the family |

Totincome | Total income |

Charity | Charitable contributions |

**Table 3.**Estimates and standard error (in brackets) for $BE$, $GM$, and exponential distributions for the US Term Life Insurance and Swedish motor insurance datasets without covariates.

Dataset 1 | Dataset 2 | |||||
---|---|---|---|---|---|---|

Exponential | $\mathit{GM}$ | $\mathit{BE}$ | Exponential | $\mathit{GM}$ | $\mathit{BE}$ | |

$\widehat{\alpha}$ | 0.243 | 1.534 | 0.285 | 0.114 | 1.353 | 0.129 |

(0.011) | (0.053) | (0.012) | (0.009) | (0.076) | (0.014) | |

$\widehat{\theta}$ | - | 7.475 | 1.429 | - | 10.893 | 1.172 |

- | (0.691) | (0.091) | - | (1.412) | (0.169) | |

$\widehat{p}$ | - | 0.45 | - | 0.198 | ||

- | (0.022) | - | (0.034) | |||

${\ell}_{max}$ | −1206.92 | −1073.63 | −959.524 | −430.685 | −419.870 | −393.317 |

AIC | 2415.84 | 2153.26 | 1923.05 | 863.369 | 845.741 | 790.635 |

CAIC | 2421.05 | 2168.90 | 1933.48 | 867.282 | 857.479 | 798.460 |

**Table 4.**Parameter estimates, standard errors, t-Wald statistics and p-values for the $BE$ and exponential (in brackets) regression models for the US Term Life Insurance dataset.

Variable | Estimate | S.E. | $\left|\mathit{t}\right|$-Statistic | $\mathbf{Pr}>\left|\mathit{t}\right|$ |
---|---|---|---|---|

gender | 1.688 (1.023) | 0.190 (0.192) | 8.879 (5.308) | 0.00 (0.00) |

age | −0.023 (0.018) | 0.007 (0.007) | 3.371 (2.618) | 0.00 (0.00) |

marstat | −1.639 (−1.310) | 0.158 (0.189) | 10.344 (6.906) | 0.00 (0.01) |

education | 0.206 (0.221) | 0.019 (0.019) | 10.622 (11.462) | 0.00 (0.00) |

ethnicity | 0.002 (−0.128) | 0.032 (0.035) | 0.078 (3.622) | 0.93 (0.00) |

smarstat | 0.201 (0.613) | 0.112 (0.111) | 1.789 (5.487) | 0.07 (0.00) |

sgender | −1.132 (0.031) | 0.268 (0.296) | 4.223 (0.104) | 0.00 (0.91) |

sage | 0.048 (0.010) | 0.007 (0.008) | 6.093 (1.287) | 0.00 (0.19) |

seducation | 0.134 (0.044) | 0.020 (0.024) | 6.492 (1.821) | 0.00 (0.07) |

numhh | −0.126 (0.145) | 0.034 (0.044) | 3.679 (3.282) | 0.00 (0.00) |

income | 0.261 (0.369) | 0.030 (0.030) | 8.730 (12.053) | 0.00 (0.00) |

totincome | 0.125 (0.097) | 0.037 (0.037) | 3.348 (2.616) | 0.00 (0.01) |

charity | −0.612 (−0.630) | 0.106 (0.122) | 5.762 (5.170) | 0.00 (0.00) |

$\theta $ | 1.495 | 0.102 | 14.646 | 0.00 |

constant | −5.018 (−8.959) | 0.585 (0.519) | 8.570 (17.246) | 0.00 (0.00) |

${\ell}_{max}=-628.150\phantom{\rule{0.166667em}{0ex}}(-759.280)$ | ||||

$\mathrm{AIC}=1286.300\phantom{\rule{0.166667em}{0ex}}\left(1546.560\right)$ | ||||

$\mathrm{CAIC}=1364.520\phantom{\rule{0.166667em}{0ex}}\left(1619.560\right)$ |

Variable | Description |
---|---|

km | Distance driven by a vehicle, grouped into five categories |

zone | Graphic zone of a vehicle, grouped into seven categories |

bonus | Driver claim experience, grouped into seven categories |

make | Type of a vehicle |

claims | Number of claims |

**Table 6.**Parameter estimates, standard errors, t-Wald statistics and p-values for the $BE$ and exponential (in brackets) regression models for the Swedish motor insurance dataset.

Variable | Estimate | S.E. | $\left|\mathit{t}\right|$-Statistic | $\mathbf{Pr}>\left|\mathit{t}\right|$ |
---|---|---|---|---|

km | −0.303 (−0.290) | 0.070 (0.076) | 4.303 (3.824) | 0.00 (0.00) |

zone | −0.335 (−0.255) | 0.049 (0.052) | 6.707 (4.887) | 0.00 (0.00) |

make | −0.030 (−0.206) | 0.056 (0.079) | 0.535 (2.588) | 0.59 (0.01) |

claims | 0.034 (0.027) | 0.004 (0.004) | 8.400 (6.759) | 0.00 (0.00) |

$\theta $ | 1.799 | 0.226 | 7.932 | 0.00 |

constant | 2.752 (3.044) | 0.479 (0.481) | 5.735 (6.317) | 0.00 (0.00) |

${\ell}_{max}=-297.607\phantom{\rule{0.166667em}{0ex}}(-299.280)$ | ||||

$\mathrm{AIC}=607.213\phantom{\rule{0.166667em}{0ex}}\left(608.560\right)$ | ||||

$\mathrm{CAIC}=630.689\phantom{\rule{0.166667em}{0ex}}\left(628.123\right)$ |

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**MDPI and ACS Style**

Reyes, J.; Gómez-Déniz, E.; Gómez, H.W.; Calderín-Ojeda, E. A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory. *Symmetry* **2021**, *13*, 679.
https://doi.org/10.3390/sym13040679

**AMA Style**

Reyes J, Gómez-Déniz E, Gómez HW, Calderín-Ojeda E. A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory. *Symmetry*. 2021; 13(4):679.
https://doi.org/10.3390/sym13040679

**Chicago/Turabian Style**

Reyes, Jimmy, Emilio Gómez-Déniz, Héctor W. Gómez, and Enrique Calderín-Ojeda. 2021. "A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory" *Symmetry* 13, no. 4: 679.
https://doi.org/10.3390/sym13040679