# Dependence Modelling of Lifetimes in Egyptian Families

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## Abstract

**:**

## 1. Introduction

## 2. Data Set

## 3. Model Description

## 4. Metropolis–Hastings MCMC

- Initialise, i.e., draw ${\mathit{\theta}}_{0}$ from the prior distribution.
- For $t=1,2,\dots ,N$
- -
- Sample the proposal ${\mathit{\theta}}^{\mathbf{\prime}}\in {\mathbb{R}}^{d}$ for ${\mathit{\theta}}_{t}$ from $q\left({\mathit{\theta}}^{\prime}\right|{\mathit{\theta}}_{t-1})$.
- -
- Compute$$A=min\left(1,\frac{p\left({\mathit{\theta}}^{\mathbf{\prime}}\right|\mathit{y})q\left({\mathit{\theta}}_{t-1}\right|{\mathit{\theta}}^{\mathbf{\prime}})}{p\left({\mathit{\theta}}_{t-1}\right|\mathit{y})q\left({\mathit{\theta}}^{\mathbf{\prime}}\right|{\mathit{\theta}}_{t-1})}\right),$$
- -
- Draw $u\sim U(0,1)$. If $U<A$, accept the proposal, fixing ${\mathit{\theta}}_{t}={\mathit{\theta}}^{\mathbf{\prime}}$. Else, fix ${\mathit{\theta}}_{t}={\mathit{\theta}}_{t-1}$.

## 5. Inference Functions for Margins

## 6. Results

**Table 5.**Marginal distribution parameter estimation results for all five data sets. MCMC: estimate, acceptance rate, standard deviation (SD), integrated autocorrelation score (IAT) and standard error (SE); MLE: estimate, SE.

MCMC | MLE | |||||||
---|---|---|---|---|---|---|---|---|

Estimate | Acceptance | SD | IAT | SE | Estimate | SE | ||

(H,W) | ${m}_{h}$ | 66.79 | 0.2573 | 0.06765 | 9.303 | 0.002918 | 66.80 | 0.06923 |

${\sigma}_{h}$ | 9.076 | 0.2573 | 0.04583 | 6.609 | 0.001666 | 9.078 | 0.04436 | |

${m}_{w}$ | 86.65 | 0.2486 | 0.3638 | 25.77 | 0.02612 | 86.66 | 0.3671 | |

${\sigma}_{w}$ | 6.958 | 0.2486 | 0.1342 | 18.42 | 0.008143 | 6.955 | 0.1342 | |

(F,S) | ${m}_{f}$ | 62.61 | 0.2769 | 0.1016 | 9.590 | 0.004448 | 62.61 | 0.1004 |

${\sigma}_{f}$ | 8.973 | 0.2769 | 0.06157 | 7.239 | 0.002342 | 8.973 | 0.05980 | |

${m}_{s}$ | 75.38 | 0.2438 | 2.942 | 93.84 | 0.4030 | 74.48 | 2.583 | |

${\sigma}_{s}$ | 9.244 | 0.2438 | 0.6400 | 78.79 | 0.08033 | 9.053 | 0.5774 | |

(F,D) | ${m}_{f}$ | 65.73 | 0.2529 | 0.1091 | 10.36 | 0.004967 | 65.73 | 0.1158 |

${\sigma}_{f}$ | 10.65 | 0.2529 | 0.07023 | 7.125 | 0.002651 | 10.64 | 0.07252 | |

${m}_{d}$ | 89.91 | 0.2348 | 3.649 | 118.9 | 0.5628 | 89.29 | 3.598 | |

${\sigma}_{d}$ | 10.15 | 0.2348 | 0.7810 | 99.46 | 0.1101 | 10.03 | 0.7776 | |

(S,F) | ${m}_{s}$ | 59.69 | 0.2478 | 0.4648 | 11.26 | 0.02206 | 56.70 | 0.4417 |

${\sigma}_{s}$ | 6.263 | 0.2478 | 0.3441 | 9.471 | 0.01498 | 6.199 | 0.3225 | |

${m}_{f}$ | 91.73 | 0.2360 | 0.5434 | 8.767 | 0.02275 | 91.70 | 0.5100 | |

${\sigma}_{f}$ | 5.636 | 0.2360 | 0.3838 | 7.649 | 0.01501 | 5.554 | 0.3708 | |

(S,M) | ${m}_{s}$ | 58.67 | 0.2601 | 0.2184 | 10.94 | 0.01022 | 58.67 | 0.2144 |

${\sigma}_{s}$ | 6.640 | 0.2601 | 0.1482 | 8.864 | 0.006238 | 6.621 | 0.1488 | |

${m}_{m}$ | 94.23 | 0.2488 | 0.3679 | 9.668 | 0.01617 | 94.20 | 0.3578 | |

${\sigma}_{m}$ | 7.289 | 0.2488 | 0.2385 | 7.572 | 0.009281 | 7.248 | 0.2323 |

#### 6.1. Goodness-of-Fit

- Let B be the number of bootstrap samples. For $b=1,2,\dots ,B$
- -
- Simulate the paired remaining lifetimes $({t}_{1}^{i,b},{t}_{2}^{i,b})$, $i=1,\dots ,n$ from the estimated copula ${C}_{\widehat{\alpha}}$, with Gompertz marginals distributed as in Table 5.
- -
- Fix ${c}_{1}^{i,b}=\infty $ and ${c}_{2}^{i,b}=9$, $i=1,\dots ,n$, where 9 is the length of the observation period. If the censoring point is random, simulate ${c}_{2}^{i,b}$, $i=1,\dots ,n$ from its observed distribution.
- -
- Determine the b-th bootstrap observations ${({X}_{1}^{i,b}\left({x}_{1}^{i}\right),{X}_{2}^{i,b}\left({x}_{2}^{i}\right),{\delta}_{1}^{i,b},{\delta}_{2}^{i,b})}_{1\le i\le n}$ from the data simulated in the preceding steps.
- -
- Estimate the marginal parameters ${\mathit{\theta}}^{\mathit{b}}$ and the copula dependence parameter ${\alpha}^{b}$ of the bootstrap sample via the IFM procedure described in Section 5.
- -
- Compute the Cramér–von Mises statistic ${\widehat{\omega}}_{b}^{2}$ of the bootstrap sample using (A2).

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Copula function, generator and domain for the Clayton, Frank, Gumbel and Joe Archimedean copulas.

Copula | Generator | Domain | |
---|---|---|---|

Clayton | ${({u}^{-\alpha}+{v}^{-\alpha}-1)}^{-1/\alpha}$ | ${t}^{-\alpha}-1$ | $\alpha >0$ |

Frank | $-\frac{1}{\alpha}ln(1+\frac{({e}^{-\alpha u}-1)({e}^{-\alpha v}-1)}{({e}^{-\alpha}-1)})$ | $-ln\left(\frac{{e}^{-\alpha t}-1}{{e}^{-\alpha}-1}\right)$ | $\alpha \ne 0$ |

Gumbel | $exp\{-{[{(-ln\left(u\right))}^{\alpha}+{(-ln\left(v\right))}^{\alpha}]}^{1/\alpha}\}$ | ${(-ln\left(t\right))}^{\alpha}$ | $\alpha \ge 1$ |

Joe | $1-{[{(1-u)}^{\alpha}+{(1-v)}^{\alpha}-{(1-u)}^{\alpha}{(1-v)}^{\alpha}]}^{1/\alpha}$ | $-ln(1-{(1-t)}^{\alpha})$ | $\alpha \ge 1$ |

**Table A2.**Kendall’s tau correlation coefficient as a function of the copula dependence parameter $\alpha $, where ${D}_{1}\left(x\right)={x}^{-1}{\int}_{0}^{x}t{({e}^{t}-1)}^{-1}dt$ is the Debeye function of order 1.

Kendall’s Tau | |
---|---|

Clayton | $\frac{\alpha}{\alpha +2}$ |

Frank | $1+\frac{4}{\alpha}({D}_{1}\left(\alpha \right)-1)$ |

Gumbel | $\frac{\alpha -1}{\alpha}$ |

Joe | $1+\frac{4}{{\alpha}^{2}}{\int}_{0}^{1}tlog\left(t\right){(1-t)}^{2(1-\alpha )/\alpha}dt$ |

## Appendix B

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**Figure 2.**Comparison of Kaplan-Meier marginal distribution functions with 95% confidence intervals (black) and Gompertz marginal distribution functions with MCMC marginal parameter estimates as in Table 5 (red).

**Figure 3.**MCMC posterior density, accepted parameter ($\alpha $) traceplots and likelihood function for estimation of the Clayton, Frank, Gumbel and Joe dependence parameters. Results for (H,W) and (F,D), given in rows 1–3 and 4–6, respectively. MCMC estimates given by blue solid line, and MLE estimates by red dashed line.

Count | 10th Quantile | 25th Quantile | 50th Quantile | 75th Quantile | 90th Quantile | Mean | SD |
---|---|---|---|---|---|---|---|

20,683 | 53 | 57 | 62 | 68 | 74 | 62.9 | 8.6 |

**Table 2.**Descriptive statistics for age at entry (Entry) and age at death (Death) for each of the five samples. “Death *” gives the descriptive statistics for policyholders whose beneficiaries have also died.

Sample | Count | 10th Quantile | 25th Quantile | 50th Quantile | 75th Quantile | 90th Quantile | Mean | SD | ||
---|---|---|---|---|---|---|---|---|---|---|

(H,W) | Husband | Entry | 19,475 | 49 | 52 | 57 | 63 | 68 | 58.02 | 8.03 |

Death | 19,475 | 53 | 57 | 62 | 68 | 73 | 62.57 | 8.27 | ||

Death * | 955 | 56 | 61 | 66 | 73 | 79 | 67.02 | 8.92 | ||

Wife | Entry | 19,937 | 39 | 44 | 50 | 57 | 62 | 50.52 | 9.13 | |

Death | 955 | 53 | 59 | 65 | 71 | 77 | 64.92 | 9.46 | ||

(F,S) | Father | Entry | 13,655 | 47 | 50 | 53 | 58 | 63 | 54.07 | 7.01 |

Death | 13,655 | 50 | 53 | 57 | 62 | 67 | 57.99 | 7.19 | ||

Death * | 76 | 51.5 | 54 | 58 | 65.25 | 78.5 | 61.22 | 11.16 | ||

Son | Entry | 13,655 | 6 | 10 | 15 | 18 | 22 | 14.47 | 7.16 | |

Death | 76 | 16.5 | 19.75 | 23.5 | 35.25 | 51 | 28.34 | 16.16 | ||

(F,D) | Father | Entry | 14,274 | 47 | 50 | 55 | 61 | 67 | 56.17 | 8.55 |

Death | 14,274 | 51 | 54 | 59 | 65 | 71 | 60.17 | 8.7 | ||

Death * | 57 | 52.6 | 60 | 64 | 73 | 88 | 67.3 | 12.88 | ||

Daughter | Entry | 14,274 | 6 | 11 | 16 | 23 | 31 | 17.52 | 10.39 | |

Death | 57 | 19 | 25 | 33 | 41 | 58.4 | 36.12 | 14.66 | ||

(S,F) | Son | Entry | 218 | 43 | 48 | 51 | 55 | 58 | 49.84 | 8.06 |

Death | 218 | 45.7 | 50 | 54 | 58 | 61.3 | 53.14 | 8.23 | ||

Death * | 119 | 49 | 51 | 55 | 58 | 61 | 54.67 | 5.13 | ||

Father | Entry | 218 | 68 | 74 | 78 | 83 | 86 | 77.35 | 8.23 | |

Death | 119 | 78 | 82 | 86 | 89 | 92 | 85.71 | 5.71 | ||

(S,M) | Son | Entry | 1067 | 44 | 48 | 52 | 56 | 60 | 51.58 | 7.43 |

Death | 1067 | 47 | 51 | 56 | 60 | 64 | 55.16 | 7.56 | ||

Death * | 429 | 49 | 53 | 57 | 60 | 65 | 56.61 | 6.39 | ||

Mother | Entry | 1076 | 66 | 71 | 76 | 81 | 85 | 75.67 | 8.33 | |

Death | 429 | 76 | 80 | 85 | 89 | 93 | 84.63 | 7.16 |

Sample | Count | Pearson | Spearman | Kendall | |
---|---|---|---|---|---|

(H,W) | $0\le d<4$ | 288 | 0.946 | 0.942 | 0.819 |

$4\le d<8$ | 343 | 0.899 | 0.881 | 0.742 | |

$d\ge 8$ | 324 | 0.776 | 0.803 | 0.655 | |

Total | 955 | 0.769 | 0.771 | 0.604 | |

(F,S) | $18\le d<35$ | 28 | 0.962 | 0.923 | 0.771 |

$d\ge 35$ | 48 | 0.892 | 0.779 | 0.647 | |

Total | 76 | 0.881 | 0.781 | 0.610 | |

(F,D) | $18\le d<35$ | 31 | 0.971 | 0.924 | 0.834 |

$d\ge 35$ | 26 | 0.916 | 0.825 | 0.688 | |

Total | 57 | 0.871 | 0.771 | 0.621 | |

(S,F) | $18\le d<25$ | 34 | 0.891 | 0.871 | 0.743 |

$25\le d<35$ | 74 | 0.876 | 0.742 | 0.597 | |

$d\ge 35$ | 11 | 0.758 | 0.704 | 0.594 | |

Total | 119 | 0.544 | 0.513 | 0.385 | |

(S,M) | $18\le d<25$ | 222 | 0.820 | 0.775 | 0.618 |

$25\le d<35$ | 181 | 0.842 | 0.822 | 0.654 | |

$d\ge 35$ | 26 | 0.932 | 0.943 | 0.832 | |

Total | 429 | 0.612 | 0.575 | 0.425 |

**Table 4.**Empirical dependence measures for the husband and wife sample, split by the sex of the elder spouse, where ${X}_{h}$ represents the lifetime of the husband and ${X}_{w}$ the lifetime of the wife.

Sample | Count | Pearson | Spearman | Kendall | |
---|---|---|---|---|---|

(H,W) | ${X}_{h}$ >${X}_{w}$ | 807 | 0.819 | 0.817 | 0.659 |

${X}_{h}\le {X}_{w}$ | 148 | 0.905 | 0.911 | 0.767 |

**Table 6.**Copula dependence parameter estimation results for all five data sets. MCMC: estimate, acceptance rate, standard deviation (SD), integrated autocorrelation score (IAT) and standard error (SE). MLE: estimate, SE.

MCMC | MLE | |||||||
---|---|---|---|---|---|---|---|---|

Estimate | Acceptance | SD | IAT | SE | Estimate | SE | ||

(H,W) | Clayton | 0.1557 | 0.2523 | 0.01013 | 6.106 | 0.0003539 | 0.1553 | 0.01056 |

Frank | 2.474 | 0.2603 | 0.1390 | 6.445 | 0.004991 | 2.470 | 0.1392 | |

Gumbel | 1.322 | 0.2777 | 0.02173 | 6.472 | 0.0007817 | 1.321 | 0.02053 | |

Joe | 1.677 | 0.2448 | 0.06180 | 5.745 | 0.002095 | 1.676 | 0.05869 | |

(F,S) | Clayton | 0.06314 | 0.2703 | 0.01551 | 5.111 | 0.0004960 | 0.06225 | 0.01523 |

Frank | 1.799 | 0.2474 | 0.3997 | 5.848 | 0.01367 | 1.759 | 0.3983 | |

Gumbel | 1.179 | 0.2484 | 0.04224 | 5.819 | 0.001441 | 1.175 | 0.04116 | |

Joe | 1.412 | 0.2719 | 0.1536 | 5.052 | 0.004882 | 1.386 | 0.1456 | |

(F,D) | Clayton | 0.05751 | 0.2757 | 0.01816 | 5.360 | 0.0005944 | 0.05773 | 0.01859 |

Frank | 1.738 | 0.2410 | 0.4699 | 6.600 | 0.01707 | 1.738 | 0.4613 | |

Gumbel | 1.172 | 0.2348 | 0.05206 | 5.314 | 0.001697 | 1.174 | 0.05060 | |

Joe | 1.435 | 0.2947 | 0.1720 | 5.101 | 0.005493 | 1.427 | 0.1763 | |

(S,F) | Clayton | 0.2863 | 0.2541 | 0.06058 | 5.790 | 0.002061 | 0.2774 | 0.06242 |

Frank | 3.498 | 0.2721 | 0.5054 | 5.680 | 0.01703 | 3.457 | 0.4852 | |

Gumbel | 1.534 | 0.2444 | 0.09218 | 6.370 | 0.003290 | 1.512 | 0.09009 | |

Joe | 2.243 | 0.2835 | 0.2214 | 5.101 | 0.007071 | 2.177 | 0.2167 | |

(S,M) | Clayton | 0.3205 | 0.2480 | 0.03032 | 5.476 | 0.001003 | 0.3179 | 0.03019 |

Frank | 3.040 | 0.2817 | 0.2325 | 5.213 | 0.007506 | 3.029 | 0.2360 | |

Gumbel | 1.459 | 0.2611 | 0.04415 | 5.875 | 0.001513 | 1.454 | 0.04267 | |

Joe | 1.832 | 0.2555 | 0.09921 | 6.277 | 0.003515 | 1.814 | 0.09694 |

**Table 7.**Kendall’s tau correlation coefficient corresponding to MCMC $\alpha $ dependence parameter estimates.

Clayton | Frank | Gumbel | Joe | |
---|---|---|---|---|

Husband & wife | 0.07225 | 0.2593 | 0.2435 | 0.2738 |

Father & son | 0.03061 | 0.1937 | 0.1518 | 0.1881 |

Father & mother | 0.02795 | 0.1873 | 0.1469 | 0.1970 |

Son & father | 0.1252 | 0.3484 | 0.3480 | 0.4046 |

Son & mother | 0.1381 | 0.3100 | 0.3145 | 0.3154 |

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## Share and Cite

**MDPI and ACS Style**

Henshaw, K.; Hana, W.; Constantinescu, C.; Khalil, D.
Dependence Modelling of Lifetimes in Egyptian Families. *Risks* **2023**, *11*, 18.
https://doi.org/10.3390/risks11010018

**AMA Style**

Henshaw K, Hana W, Constantinescu C, Khalil D.
Dependence Modelling of Lifetimes in Egyptian Families. *Risks*. 2023; 11(1):18.
https://doi.org/10.3390/risks11010018

**Chicago/Turabian Style**

Henshaw, Kira, Waleed Hana, Corina Constantinescu, and Dalia Khalil.
2023. "Dependence Modelling of Lifetimes in Egyptian Families" *Risks* 11, no. 1: 18.
https://doi.org/10.3390/risks11010018