# Modeling the Conditional Dependence between Discrete and Continuous Random Variables with Applications in Insurance

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## Abstract

**:**

## 1. Introduction

## 2. Motivation and Results

#### 2.1. Marginal Distributions

#### 2.2. Conditional Distributions

#### 2.3. Hypothesis Testing

## 3. Allowing for Covariates

## 4. Some Methods of Estimation

#### 4.1. Estimation of the Model without Covariates

#### The Score Vector and Fisher Information Matrix

#### 4.2. Estimation of the Model with Covariates

## 5. Empirical Analysis

#### 5.1. Example 1

`Data Descriptions`file available on the same web page. As it can be seen many explanatory variables are statistically significant at the 5% level. Only the covariates unemploy, i.e., employment status of patients and income, i.e., income compared to poverty line do not have a significant impact simultaneously on the marginal means of the variables Expenditure and Outpatient visits. In addition, the dummy variable high school degree highsch, managedcare, region and marital status, maristat are not significant for the continuous response variable. On the other hand, the explanatory variable race, i.e., race of the patient is not statistically significant at the same level for the Outpatient visits response variable.

#### 5.2. Example 2

## 6. Final Comments

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Graphs of the density for the bivariate distribution $f(x,n)$ in (14) and the corresponding contour plots for different values of the parameters. Left panel $(\gamma ,{\mu}_{1},{\mu}_{2})=(0.5,25,15)$ and right panel $(\gamma ,{\mu}_{1},{\mu}_{2})=(0.65,15,25)$.

**Figure 2.**Empirical (smoothed) distribution (histogram) and theoretical distribution model of the bivariate distribution considered.

Mean | Stand. Dev. | Minimum | Maximum | |
---|---|---|---|---|

Expenditure | 6.43 | 1.54 | 1.09 | 11.05 |

Outpatient visits | 8.39 | 14.43 | 1 | 167 |

**Table 2.**Pearson’s, Spearman’s and Kendall’s measures of correlation for the two dependent variables considered.

Measure of Correlation | Pearson’s | Spearman’s | Kendall’s |

0.5232 | 0.7663 | 0.2856 |

**Table 3.**Parameter estimates, standard errors and negative of the log-likelihood (NLL) for the model without covariates.

Parameter | Estimate | Stand. Error |
---|---|---|

$\widehat{\gamma}$ | 2.412 | 0.094 |

${\widehat{\mu}}_{1}$ | 6.428 | 0.044 |

${\widehat{\mu}}_{2}$ | 8.394 | 0.098 |

NLL | 11,014.532 | |

Observations | 1352 |

Expenditure | Outpatient Visits | |||
---|---|---|---|---|

Parameter | Estimate | $\mathit{p}$-Value | Estimate | $\mathit{p}$-Value |

age | 0.119 | 0.000 | 0.165 | 0.000 |

anylimit | 0.075 | 0.000 | 0.721 | 0.000 |

college | 0.014 | 0.092 | 0.218 | 0.000 |

highsch | −0.002 | 0.821 | 0.122 | 0.000 |

gender | 0.041 | 0.000 | 0.253 | 0.000 |

mnhpoor | 0.047 | 0.020 | 0.257 | 0.000 |

insure | 0.099 | 0.000 | 0.105 | 0.002 |

usc | 0.052 | 0.001 | 0.161 | 0.000 |

unemploy | −0.000 | 0.953 | 0.003 | 0.885 |

managedcare | 0.008 | 0.541 | −0.127 | 0.000 |

famsize | −0.010 | 0.006 | −0.048 | 0.000 |

countop | 0.070 | 0.000 | 0.199 | 0.000 |

race | 0.017 | 0.003 | 0.015 | 0.148 |

region | −0.001 | 0.687 | −0.054 | 0.000 |

education | 0.009 | 0.033 | 0.094 | 0.000 |

maristat | −0.007 | 0.219 | −0.058 | 0.000 |

income | −0.006 | 0.094 | 0.002 | 0.774 |

physical health | 0.028 | 0.000 | 0.131 | 0.000 |

constant | 1.172 | 0.000 | 0.594 | 0.000 |

$\gamma $ | 3.052 (0.000) | |||

NLL | 9163.372 | |||

Observations | 1352 |

${\mathit{n}}^{+}$ | |||||||
---|---|---|---|---|---|---|---|

$x=0.50$ | t | 0 | 1 | 2 | 3 | 4 | 5 |

0 | 1.00 | 2.00 | 3.00 | 4.00 | 5.00 | 6.00 | |

1 | 0.50 | 1.00 | 1.50 | 2.00 | 2.50 | 3.00 | |

2 | 0.33 | 0.66 | 1.00 | 1.33 | 1.66 | 2.00 | |

3 | 0.25 | 0.50 | 0.75 | 1.00 | 1.25 | 1.50 | |

4 | 0.20 | 0.40 | 0.60 | 0.80 | 1.00 | 1.20 | |

5 | 0.16 | 0.33 | 0.50 | 0.66 | 0.83 | 1.00 | |

${n}^{+}$ | |||||||

$x=1.00$ | t | 0 | 1 | 2 | 3 | 4 | 5 |

0 | 2.00 | 4.00 | 6.00 | 8.00 | 10.00 | 12.00 | |

1 | 0.66 | 1.33 | 2.00 | 2.66 | 3.33 | 4.00 | |

2 | 0.40 | 0.80 | 1.20 | 1.60 | 2.00 | 2.40 | |

3 | 0.28 | 0.57 | 0.85 | 1.14 | 1.42 | 1.71 | |

4 | 0.22 | 0.44 | 0.66 | 0.88 | 1.11 | 1.33 | |

5 | 0.18 | 0.36 | 0.54 | 0.72 | 0.90 | 1.09 | |

${n}^{+}$ | |||||||

$x=1.50$ | t | 0 | 1 | 2 | 3 | 4 | 5 |

0 | 3.00 | 6.00 | 9.00 | 12.00 | 15.00 | 18.00 | |

1 | 0.75 | 1.50 | 2.25 | 3.00 | 3.75 | 4.50 | |

2 | 0.42 | 0.85 | 1.28 | 1.71 | 2.14 | 2.57 | |

3 | 0.30 | 0.60 | 0.90 | 1.20 | 1.50 | 1.80 | |

4 | 0.23 | 0.46 | 0.69 | 0.92 | 1.15 | 1.38 | |

5 | 0.18 | 0.37 | 0.56 | 0.75 | 0.93 | 1.12 |

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**MDPI and ACS Style**

Gómez-Déniz, E.; Calderín-Ojeda, E. Modeling the Conditional Dependence between Discrete and Continuous Random Variables with Applications in Insurance. *Mathematics* **2021**, *9*, 45.
https://doi.org/10.3390/math9010045

**AMA Style**

Gómez-Déniz E, Calderín-Ojeda E. Modeling the Conditional Dependence between Discrete and Continuous Random Variables with Applications in Insurance. *Mathematics*. 2021; 9(1):45.
https://doi.org/10.3390/math9010045

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, and Enrique Calderín-Ojeda. 2021. "Modeling the Conditional Dependence between Discrete and Continuous Random Variables with Applications in Insurance" *Mathematics* 9, no. 1: 45.
https://doi.org/10.3390/math9010045