Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality
Abstract
:1. Introduction
2. Random Fields Memory Models for Mortality Improvement Rates
2.1. Classic Mortality Models
2.2. Through a Random Field Framework
- (i)
- Mortality Surface
- (ii)
- Random Field Memory Models
- A-1
- for some ,
- A-2
- for all , where the coefficients are such that .
2.3. AR-ARCH-Type Random Fields
3. Estimation, Asymptotic Inference, and Model Selection
3.1. Quasi-Maximum Likelihood Estimator (QMLE)
- (i)
- Consistency. If the assumptions H1, H2, and H3 hold, then the QMLE estimator in Equation (11) is consistent in the sense that for each , and , we have
- (ii)
- Asymptotic normality. Under assumptions H1, H2, H3, and H4,
3.2. Optimal Model Selection
- (i)
- Likelihood ratio test statistic (LRTS)
- (ii)
- Penalized QMLE
- I-1
- , (maximal possible neighborhoods) are a finite set and , .
- I-2
- is increasing, goes to infinity as T tends to go to infinity as soon as , and tends to 0 as T tends to go to infinity for any .
- (iii)
- Bayesian Information Criterion
- (iv)
- Search Algorithm
- (v)
- Spatial Autocorrelation Function
4. Numerical and Empirical Analyses
4.1. Simulation Study
4.2. Real-World Data Application
- (i)
- Model Selection
- (ii)
- Diagnostic Checks
- (iii)
- Predictive Performance
5. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Appendix The Gradient and the Hessian of the Quasi-Likelihood Function
Appendix B. Additional Figures
Appendix B.1. Diagnostic Checks of Models’ Residuals
Appendix B.2. Out-of-Sample Analysis and Predictive Performance
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(1,1) | (2,2) | (0,1) | (1,0) | (1,1) | (2,2) | (0,1) | (1,0) | |||
1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 64.8% | |
1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 11.1% | |
1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 7.2% | |
1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 6.1% | |
1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 42.30% | |
1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 12.00% | |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 8.1% | |
1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 8.1% | |
1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 6.4% | |
1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 5.4% |
SW | AD | CVM | PC | SF | |
---|---|---|---|---|---|
USA | 0.3428 | 0.7048 | 0.7991 | 0.8523 | 0.3947 |
FRA | 0.6171 | 0.5212 | 0.5438 | 0.6279 | 0.4322 |
GBP | 0.6114 | 0.7626 | 0.8548 | 0.9309 | 0.4827 |
AR-ARCH | LC | CBD | |||
---|---|---|---|---|---|
USA | Mortality rate | RMSFE | 1.51 | 5.30 | 3.68 |
MAFE | 2.47 | 3.99 | 4.14 | ||
Life expectancy | RMSFE | 1.13 | 1.69 | 2.72 | |
MAFE | 2.75 | 3.42 | 4.34 | ||
FRA | Mortality rate | RMSFE | 1.07 | 2.30 | 5.04 |
MAFE | 2.19 | 2.84 | 4.49 | ||
Life expectancy | RMSFE | 1.66 | 2.11 | 4.01 | |
MAFE | 3.28 | 3.79 | 5.24 | ||
GBR | Mortality rate | RMSFE | 4.11 | 8.34 | 8.01 |
MAFE | 4.36 | 5.83 | 5.72 | ||
Life expectancy | RMSFE | 7.29 | 8.64 | 8.70 | |
MAFE | 7.03 | 7.87 | 7.89 |
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Doukhan, P.; Rynkiewicz, J.; Salhi, Y. Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality. Stats 2022, 5, 26-51. https://doi.org/10.3390/stats5010003
Doukhan P, Rynkiewicz J, Salhi Y. Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality. Stats. 2022; 5(1):26-51. https://doi.org/10.3390/stats5010003
Chicago/Turabian StyleDoukhan, Paul, Joseph Rynkiewicz, and Yahia Salhi. 2022. "Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality" Stats 5, no. 1: 26-51. https://doi.org/10.3390/stats5010003
APA StyleDoukhan, P., Rynkiewicz, J., & Salhi, Y. (2022). Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality. Stats, 5(1), 26-51. https://doi.org/10.3390/stats5010003