# EM Estimation for the Bivariate Mixed Exponential Regression Model

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## Abstract

**:**

## 1. Introduction

- finite mixture models: see, for example, (Fung et al. 2021; Lee and Lin 2010; Miljkovic and Grün 2016; Tzougas et al. 2014, 2018, 2019);
- composite or splicing models: see for instance, (Bakar et al. 2015; Calderín-Ojeda and Kwok 2016; Cooray and Ananda 2005; Grün and Miljkovic 2019; Nadarajah and Bakar 2014; Parodi 2020; Pigeon and Denuit 2011; Scollnik 2007; Scollnik and Sun 2012).
- combinations of finite mixtures and composite models: see Reynkens et al. (2017).

## 2. The Bivariate Mixed Exponential Regression Model

#### 2.1. Bivariate Pareto Regression Model

#### 2.2. Bivariate Exponential-Inverse Gaussian Regression Model

## 3. The EM Algorithm for the Bivariate Mixed Exponential Regression Model

**E-step:**The Q-function, $Q(\Theta ;{\Theta}^{\left(r\right)})$, which is the conditional posterior expectation of Equation (11), is given by$$Q(\Theta ;{\Theta}^{\left(r\right)})=\sum _{j=1}^{n}\sum _{i=1}^{2}\left(-log\left({\mu}_{i,j}^{\left(r\right)}\right)-\frac{{y}_{i,j}}{{\mu}_{i,t}}{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]\right)+\sum _{j=1}^{n}{\mathbb{E}}_{z,j}^{\left(r\right)}[log\left({g}_{\varphi}\left(z\right)\right)],$$$${\mathbb{E}}_{z,j}^{\left(r\right)}\left[h\left(z\right)\right]=\mathbb{E}\left[h\left(z\right)\right|{\Theta}^{\left(r\right)},{\mathbf{y}}_{j},{\mathbf{x}}_{1,j},{\mathbf{x}}_{2,j}]={\int}_{0}^{\infty}h\left(z\right)\pi \left(z\right|{\Theta}^{\left(r\right)},{\mathbf{y}}_{t},{\mathbf{x}}_{1,j},{\mathbf{x}}_{2,j})dz,$$$$\pi \left(z\right|{\Theta}^{\left(r\right)},{\mathbf{y}}_{j},{\mathbf{x}}_{1,j},{\mathbf{x}}_{2,j})=\frac{\frac{1}{{\mu}_{1,j}{\mu}_{2,j}{z}^{2}}exp\left\{-\frac{{y}_{1,j}}{{\mu}_{1,j}z}-\frac{{y}_{2,j}}{{\mu}_{2,j}z}\right\}{g}_{\varphi}\left(z\right)}{{f}_{\mathbf{Y}}\left({\mathbf{y}}_{t}\right)}.$$**M-step:**After calculating the Q-function, we find its maximum global point, ${\Theta}^{(j+1)}$, i.e., we update the parameters by computing the gradient function, $g(.)$, and the Hessian matrix, $H(.)$, of the Q-function. In particular, the Newton–Raphson algorithm is used for maximizing the Q-function and the parameters ${\beta}_{1},{\beta}_{2}$ for the Exponential part and the parameter $\varphi $ for the randnom effect part are updated separately as shown below.- -
- For the Exponential part,$$\begin{array}{cc}\hfill {\mathit{\beta}}_{i}^{(r+1)}& ={\mathit{\beta}}_{i}^{\left(r\right)}-{H}^{-1}\left({\mathit{\beta}}_{i}^{\left(r\right)}\right)g\left({\mathit{\beta}}_{i}^{\left(r\right)}\right),\phantom{\rule{1.em}{0ex}}i=1,2\hfill \\ \hfill g\left({\mathit{\beta}}_{i}^{\left(r\right)}\right)& ={\mathbf{X}}_{i}^{T}{\mathbf{V}}_{i}\phantom{\rule{1.em}{0ex}}H\left({\mathit{\beta}}_{i}^{\left(r\right)}\right)={\mathbf{X}}_{i}^{T}{\mathbf{D}}_{i}{\mathbf{X}}_{i}\hfill \\ \hfill {\mathbf{V}}_{i}& =\left({\left\{\frac{{y}_{i,j}}{{\mu}_{i,j}^{\left(r\right)}}{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]-1\right\}}_{j=1,\cdots ,n}\right)\hfill \\ \hfill {\mathbf{D}}_{i}& =\mathrm{diag}\left({\left\{-\frac{{y}_{i,j}}{{\mu}_{i,j}^{\left(r\right)}}{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]\right\}}_{j=1,\cdots ,n}\right),\hfill \end{array}$$
- -
- For the random effect part, we derive the first and second order derivatives of $log{g}_{\mathit{\varphi}}\left(\mathit{\theta}\right)$ and then we take the posterior expectations to construct its gradient functions and the Hessian matrix. In what follows, we derive the derivatives for the IGA and IG densities which were defined in the previous section. Finally, we update $\varphi $ using the one-step ahead Newton iteration$${\varphi}^{(r+1)}={\varphi}^{\left(r\right)}-\frac{g\left({\varphi}^{\left(r\right)}\right)}{h\left({\varphi}^{\left(r\right)}\right)}.$$In what follows, we will show how Equation (11) can be modified in the case of the IGA and IG mixing densities.

**Inverse Gamma mixing density**The first and second derivatives of the term ${\sum}_{j=1}^{n}log{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{g}_{\varphi}\left(z\right)\right]$ are given by$$\begin{array}{cc}\hfill g\left({\varphi}^{\left(r\right)}\right)& =n\left(1+\frac{1}{{\varphi}^{\left(r\right)}}+log\left({\varphi}^{\left(r\right)}\right)-\mathsf{\Psi}({\varphi}^{\left(r\right)}+1)\right)-\sum _{j=1}^{n}({\mathbb{E}}_{z,j}^{\left(r\right)}[logz]+{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right])\hfill \\ \hfill \mathrm{and}\\ \hfill h\left({\varphi}^{\left(r\right)}\right)& =n\left(-\frac{1}{{\left({\varphi}^{\left(r\right)}\right)}^{2}}+\frac{1}{{\varphi}^{\left(r\right)}}-{\mathsf{\Psi}}^{\prime}({\varphi}^{\left(r\right)}+1)\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]& =\frac{{\varphi}^{\left(r\right)}+3}{{\varphi}^{\left(r\right)}+\frac{{y}_{1,j}}{{\mu}_{1,j}^{\left(r\right)}}+\frac{{y}_{2,j}}{{\mu}_{2,j}^{\left(r\right)}}}\hfill \\ \hfill \mathrm{and}\\ \hfill {\mathbb{E}}_{z,j}^{\left(r\right)}[logz]& =log\left({\varphi}^{\left(r\right)}+\frac{{y}_{1,j}}{{\mu}_{1,j}^{\left(r\right)}}+\frac{{y}_{2,j}}{{\mu}_{2,j}^{\left(r\right)}}\right)-\mathsf{\Psi}({\varphi}^{\left(r\right)}+3).\hfill \end{array}$$**Inverse Gaussian density**In the case of the IG mixing density, the first and second derivatives of the term ${\sum}_{j=1}^{n}log{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{g}_{\varphi}\left(z\right)\right]$ are given by$$\begin{array}{cc}\hfill g\left({\varphi}^{\left(r\right)}\right)& =n\left(\frac{1}{{\varphi}^{\left(r\right)}}+2{\varphi}^{\left(r\right)}\right)-{\varphi}^{\left(r\right)}\sum _{j=1}^{n}\left({\mathbb{E}}_{z,j}^{\left(r\right)}\left[z\right]+{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]\right)\hfill \\ \hfill \mathrm{and}\\ \hfill h\left({\varphi}^{\left(r\right)}\right)& =n\left(-\frac{1}{{\left({\varphi}^{\left(r\right)}\right)}^{2}}+2\right)-\sum _{j=1}^{n}\left({\mathbb{E}}_{z,j}^{\left(r\right)}\left[z\right]+{\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]\right).\hfill \end{array}$$Furthermore, one can easily see that at iteration r, the posterior density in this case is a GIG$\left({\left({\varphi}^{\left(r\right)}\right)}^{2},{\left({\varphi}^{\left(r\right)}\right)}^{2}+\frac{2{y}_{1,j}}{{\mu}_{1,j}^{\left(r\right)}}+\frac{{y}_{2,j}}{{\mu}_{2,j}^{\left(r\right)}},-\frac{5}{2}\right)$. Therefore, the expectations involved in the E-step of the algorithm are given by$$\begin{array}{cc}\hfill {\mathbb{E}}_{z,j}^{\left(r\right)}\left[z\right]& =\sqrt{\frac{{\left({\varphi}^{\left(r\right)}\right)}^{2}+\frac{2{y}_{1,j}}{{\mu}_{1,j}^{\left(r\right)}}+\frac{{y}_{2,j}}{{\mu}_{2,j}^{\left(r\right)}}}{{\left({\varphi}^{\left(r\right)}\right)}^{2}}}\frac{\eta +1}{\eta +\frac{3}{\eta}+3}\hfill \\ \hfill \mathrm{and}\\ \hfill {\mathbb{E}}_{z,j}^{\left(r\right)}\left[{z}^{-1}\right]& =\sqrt{\frac{{\left({\varphi}^{\left(r\right)}\right)}^{2}}{{\left({\varphi}^{\left(r\right)}\right)}^{2}+\frac{2{y}_{1,j}}{{\mu}_{1,j}^{\left(r\right)}}+\frac{{y}_{2,j}}{{\mu}_{2,j}^{\left(r\right)}}}}\frac{\eta +1}{\eta +\frac{3}{\eta}+3}+\frac{5}{{\left({\varphi}^{\left(r\right)}\right)}^{2}+\frac{2{y}_{1,j}}{{\mu}_{1,j}^{\left(r\right)}}+\frac{{y}_{2,j}}{{\mu}_{2,j}^{\left(r\right)}}},\hfill \end{array}$$

## 4. Empirical Analysis

- The variable Driver’s age. Policyholders aged 18 to 90 years old.
- The variable Vehicle’s age. Vehicles aged 0 to 60 years old.
- The variable Car cubism, ’CC’, consists of four categories. Vehicles with horse power ’0–1299 cc’ (C1), ’1300–1399 cc’ (C2), ’1400–1599 cc’ and ’greater or equal 1600 cc’ (C3).
- The variable ’PT’ consisted of three types of policy, ’Economic type which includes only MTPL coverage’ (C1) , ’Middle type which includes apart from MTPL coverage other types of coverage’ (C2), and ’Expensive type—Own coverage’ (C3).
- The variable ’Region’ consisted of three board regions, ’Capital city’ (C1), ’province cities of the mainland’ (C2), and ’province cities of the island area’ (C3).

**R**software. The vector of parameters $\Theta =\{{\beta}_{1},{\beta}_{2},\varphi \}$ was estimated using the EM algorithm which was presented in Section 3 and their standard deviations were obtained through expressions that were directly produced by the EM algorithm for the observed information matrix of each model. The fit of the competing models was compared by employing the classic hypothesis/specification tests Akaike information criterion (AIC) and Bayesian information criterion (BIC). The results are presented in Table 3. We see that the values of the estimated regression coefficients of the variables Driver’s Age, Vehicle’s Age and Region have a a similar effect (positive and/or negative) and are almost identical for both response variables in the case of the bivariate claim size models. Furthermore, we observe that the best fitting performances are provided by the BEIG regression models since according to a very well known rule of thumb, two models can be considered to be significantly different if the difference in their respective AIC and SBC values is greater than ten and five, respectively, see Anderson and Burnham (2004) and Raftery (1995), respectively.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EM | Expectation—Maximization |

IGA | Inverse Gamma |

IG | Inverse Gaussian |

GIG | Generalized Inverse Gaussian |

BPA | Bivariate Pareto |

BEIG | Bivariate Exponential-Inverse Gaussian |

## Note

1 | Please note that the EM algorithms which are used for fitting the BPA and BEIG regression models are direct extensions from the univariate to the multivariate case of the EM algorithms which were developed by Tzougas and Karlis (2020). |

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**Table 1.**Summary statistics of two types of d claim amount. The correlation test is an one-sided test, where the alternative hypothesis is ’true correlation is greater than 0’.

Aggregated Claim | Min | Median | Mean | Max | Standard Deviation | Correlation | p-Value |
---|---|---|---|---|---|---|---|

${Y}_{1}$ | 0.9 | 2413.4 | 11,017.3 | 251,958.2 | 27,128.85 | 0.1095 | 0.000 |

${Y}_{2}$ | 6.2 | 1012.4 | 1871.2 | 14,818.2 | 2217.138 |

Horse Power (CC) | Policy Type (PT) | Region | |
---|---|---|---|

C1 | 2036 | 1144 | 4220 |

C2 | 2417 | 1940 | 2333 |

C3 | 1833 | 4179 | 710 |

C4 | 977 | - | - |

**Table 3.**Estimated parameters and standard errors in parentheses for the BPA and BEIG regressions model. AIC: Akaike information criterion; BIC: Bayesian information criterion.

BPA | BEIG | |||
---|---|---|---|---|

Response | ${\mathit{Y}}_{\mathbf{1}}$ | ${\mathit{Y}}_{\mathbf{2}}$ | ${\mathit{Y}}_{\mathbf{1}}$ | ${\mathit{Y}}_{\mathbf{2}}$ |

$\varphi $ | 0.5258 | 0.7905 | ||

(0.0394) | (0.016) | |||

Intercept | 8.6756 | 8.0076 | 8.5108 | 7.8137 |

(0.0979) | (0.0905) | (0.0905) | (0.0861) | |

Driver’s Age | 0.0010 | 0.0028 | 0.0007 | 0.0028 |

(0.0014) | (0.0012) | (0.0014) | (0.0013) | |

CC: C2 | −0.0854 | 0.0761 | −0.0523 | 0.0918 |

(0.0486) | (0.0431) | (0.0481) | (0.0455) | |

CC: C3 | 0.0517 | 0.0661 | 0.0498 | 0.0615 |

(0.0517) | (0.0463) | (0.0517) | (0.0489) | |

CC: C4 | −0.0064 | 0.1104 | 0.0238 | 0.1183 |

(0.0633) | (0.0564) | (0.0625) | (0.0595) | |

PT: C2 | 0.4555 | −0.0352 | 0.3859 | −0.0684 |

(0.0614) | (0.0535) | (0.0599) | (0.0564) | |

PT: C3 | 0.4057 | −0.0764 | 0.3622 | −0.0989 |

(0.0559) | (0.0482) | (0.0540) | (0.0506) | |

Vehcle’s Age | 0.0155 | −0.0015 | 0.0139 | −0.0021 |

(0.0035) | (0.0031) | (0.0035) | (0.0033) | |

Region: C2 | −0.1552 | 0.0502 | −0.1125 | 0.0736 |

(0.0417) | (0.0369) | (0.0411) | (0.0389) | |

Region: C3 | 0.2422 | −0.0306 | 0.2493 | −0.0189 |

(0.0644) | (0.0577) | (0.0643) | (0.0609) | |

AIC | 267,937.7 | 267,843.1 | ||

BIC | 268,082.4 | 267,987.8 |

**Table 4.**Model comparison between Gaussian copula with two PA marginals and BPA regression model. Data are generated from Gaussian copula with two PA marginals. The AIC and BIC values are for the bivariate model.

${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ||||||
---|---|---|---|---|---|---|---|

True | Copula with PA | BPA | True | Copula with PA | BPA | ||

$\varphi $ | 2 | 2.0713 | 2.6455 | $\varphi $ | 3 | 3.0353 | 2.6455 |

Intercept | −1 | −1.0445 | −1.0447 | Intercept | −1.5 | −1.5936 | −1.5675 |

${\nu}_{1,1}$ | 0.0003 | 0.0012 | 0.0008 | ${\nu}_{1,2}$ | 0.003 | 0.0065 | 0.0070 |

${\nu}_{2,1}C2$ | −0.4 | −0.3527 | −0.3603 | ${\nu}_{2,2}C2$ | −0.3 | −0.3182 | −0.3152 |

${\nu}_{3,1}C2$ | −0.05 | 0.0125 | −0.0089 | ${\nu}_{2,2}C3$ | −0.2 | −0.2113 | −0.2205 |

${\nu}_{3,1}C3$ | 0.1 | 0.0385 | 0.0190 | ${\nu}_{3,2}C2$ | −0.05 | −0.0329 | −0.0227 |

${\nu}_{4,1}C2$ | 0.2 | 0.1422 | 0.1651 | ${\nu}_{3,2}C3$ | 0.15 | 0.1675 | 0.1628 |

${\nu}_{4,1}C3$ | 0.3 | 0.2438 | 0.2571 | ${\nu}_{4,2}C2$ | 0.25 | 0.2746 | 0.2700 |

${\nu}_{4,1}C4$ | 0.4 | 0.3157 | 0.3216 | ${\nu}_{4,2}C3$ | 0.35 | 0.4538 | 0.4366 |

${\nu}_{4,2}C4$ | 0.45 | 0.5197 | 0.5063 | ||||

$\rho $ | 0.2 | 0.1941 | |||||

AIC | −4486.499 | −4418.983 | BIC | −4356.155 | −4301.673 |

**Table 5.**Model comparison between Gaussian copula model with two EIG marginals and BEIG regression model. Data are generated from Gaussian copula with two EIG marginals. The AIC and BIC values are for the bivariate model.

${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ||||||
---|---|---|---|---|---|---|---|

True | Copula with EIG | BEIG | True | Copula with EIG | BEIG | ||

$\varphi $ | 2 | 2.0168 | 2.1322 | $\varphi $ | 3 | 2.7243 | 2.1322 |

Intercept | −1 | −1.0676 | −1.0694 | Intercept | −1.5 | −1.4777 | −1.4709 |

${\nu}_{1,1}$ | 0.0003 | 0 | −0.0002 | ${\nu}_{1,2}$ | 0.003 | −0.0020 | -0.0008 |

${\nu}_{2,1}C2$ | −0.4 | −0.3679 | −0.0002 | ${\nu}_{2,2}C2$ | −0.3 | −0.2583 | −0.2654 |

${\nu}_{3,1}C2$ | −0.05 | −0.0411 | −0.0348 | ${\nu}_{2,2}C3$ | −0.2 | −0.1480 | −0.1601 |

${\nu}_{3,1}C3$ | 0.1 | 0.1598 | 0.1676 | ${\nu}_{3,2}C2$ | −0.05 | −0.0230 | −0.0279 |

${\nu}_{4,1}C2$ | 0.2 | 0.2485 | 0.2507 | ${\nu}_{3,2}C3$ | 0.15 | 0.1724 | 0.1661 |

${\nu}_{4,1}C3$ | 0.3 | 0.3525 | 0.3589 | ${\nu}_{4,2}C2$ | 0.25 | 0.2030 | 0.2083 |

${\nu}_{4,1}C4$ | 0.4 | 0.4232 | 0.4334 | ${\nu}_{4,2}C3$ | 0.35 | 0.3321 | 0.3378 |

${\nu}_{4,2}C4$ | 0.45 | 0.4537 | 0.4450 | ||||

$\rho $ | 0.2 | 0.1920 | |||||

AIC | −3310.483 | −3260.53 | BIC | −3180.139 | −3143.22 |

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## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Dassios, A.; Tzougas, G.
EM Estimation for the Bivariate Mixed Exponential Regression Model. *Risks* **2022**, *10*, 105.
https://doi.org/10.3390/risks10050105

**AMA Style**

Chen Z, Dassios A, Tzougas G.
EM Estimation for the Bivariate Mixed Exponential Regression Model. *Risks*. 2022; 10(5):105.
https://doi.org/10.3390/risks10050105

**Chicago/Turabian Style**

Chen, Zezhun, Angelos Dassios, and George Tzougas.
2022. "EM Estimation for the Bivariate Mixed Exponential Regression Model" *Risks* 10, no. 5: 105.
https://doi.org/10.3390/risks10050105