# Credible Regression Approaches to Forecast Mortality for Populations with Limited Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mortality Modelling: A Review of Methods

#### 2.1. The Lee–Carter Model

#### 2.2. The Cairns–Blake–Dowd Model

#### 2.3. The Random Coefficients Regression Model

## 3. Credible Regression Mortality Models

#### 3.1. A Credibility Regression Approach with Randomly Varying Coefficients

- (i)
- The pairs $({A}_{{x}_{0}},{\mathit{Y}}_{{x}_{0}})$, $({A}_{{x}_{1}},{\mathit{Y}}_{{x}_{1}})$, $\cdots$, $({A}_{{x}_{k-1}},{\mathit{Y}}_{{x}_{k-1}})$ are independent and $A}_{{x}_{0}},\cdots ,\phantom{\rule{4pt}{0ex}}{A}_{{x}_{k-1}$ are independent and identically distributed.
- (ii)
- $E({\mathit{Y}}_{x}|{A}_{x})={\mathit{Z}}_{x}\mathbf{\beta}({A}_{x})$, where $\mathit{Z}}_{x$ is a fixed $n\times p$ design matrix of full rank $p\phantom{\rule{4pt}{0ex}}(<n)$ and $\mathbf{\beta}({A}_{x})$ is an unknown regression vector of length $p$.
- (iii)
- $\mathrm{Cov}({\mathit{Y}}_{x}|{A}_{x})=\mathrm{diag}\left[{d}_{{t}_{0}{t}_{0}}({A}_{x}),\cdots ,{d}_{{t}_{n-1}{t}_{n-1}}({A}_{x})\right]$,$\mathrm{where}\phantom{\rule{4pt}{0ex}}{d}_{tt}({A}_{x})={\sigma}_{1}^{2}({A}_{x})+\sum _{k=2}^{p}{\sigma}_{k}^{2}({A}_{x})\phantom{\rule{4pt}{0ex}}{Z}_{kt,x}^{2},\phantom{\rule{4pt}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}{\sigma}_{1}^{2}({A}_{x})={\sigma}_{01}^{2}({A}_{x})+{\sigma}_{11}^{2}({A}_{x}),$or in matrix formulation:$\mathrm{Cov}({\mathit{Y}}_{x}|{A}_{x})$ = $\left(\begin{array}{cc}{\sigma}_{1}^{2}({A}_{x})+{\displaystyle \sum _{k=2}^{p}}{\sigma}_{k}^{2}({A}_{x})\phantom{\rule{4pt}{0ex}}{Z}_{k{t}_{0},x}^{2}& 0\\ \hfill \ddots \\ 0& {\sigma}_{1}^{2}({A}_{x})+{\displaystyle \sum _{k=2}^{p}}{\sigma}_{k}^{2}({A}_{x})\phantom{\rule{4pt}{0ex}}{Z}_{k{t}_{n-1},x}^{2}\end{array}\right)$.

**Proposition**

**1.**

**Proof.**

#### 3.2. Estimation of Structural Parameters

**Remark**

**2.**

#### 3.3. Credibility Regression with Fixed Coefficients and Weights: A Special Case

## 4. Extrapolation Methods for Estimating Future Mortality Rates

#### 4.1. Standard Extrapolation Method (SEM)

#### 4.2. Other Extrapolation Methods

**Remark**

**3.**

## 5. Empirical Illustration

#### 5.1. Forecasting Results

#### Credibility Effects on Mortality Modelling

#### 5.2. Applying the Bühlmann Credibility Approach

#### 5.3. Application in Insurance-Related Products

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bozikas, Apostolos, and Georgios Pitselis. 2018. An Empirical Study on Stochastic Mortality Modelling under the Age-Period-Cohort Framework: The Case of Greece with Applications to Insurance Pricing. Risks 6: 44. [Google Scholar] [CrossRef]
- Brouhns, Natacha, Michel Denuit, and Jeroen K. Vermunt. 2002. A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics 31: 373–93. [Google Scholar] [CrossRef]
- Bühlmann, Hans. 1967. Experience Rating and Credibility. ASTIN Bulletin 4: 199–207. [Google Scholar] [CrossRef] [Green Version]
- Bühlmann, Hans, and Alois Gisler. 2005. A Course in Credibility Theory and Its Applications. Berlin and Heidelberg: Springer. [Google Scholar]
- Cairns, Andrew J. G., David Blake, and Kevin Dowd. 2006. A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance 73: 687–718. [Google Scholar] [CrossRef]
- Cairns, Andrew J. G., David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, Alen Ong, and Igor Balevich. 2009. A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal 13: 1–35. [Google Scholar] [CrossRef]
- De Vylder, F. Etienne. 1978. Parameter estimation in credibility theory. ASTIN Bulletin 10: 99–112. [Google Scholar] [CrossRef]
- Gong, Maxwell, Zhuangdi Li, Maria Milazzo, Kristen Moore, and Matthew Provencher. 2018. Credibility Methods for Individual Life Insurance. Risks 6: 144. [Google Scholar] [CrossRef]
- Goovaerts, Marc. J., Rob Kaas, A. E. Van Heerwaarden, and T. Bauwelinckx. 1990. Effective Actuarial Methods. Amsterdam: North-Holland. [Google Scholar]
- Greene, William H. 2012. Econometric Analysis, International ed. London: Pearson Education Limited. [Google Scholar]
- Hachemeister, Charles. 1975. Credibility for Regression Models with Application to Trend (Reprint). In Credibility: Theory and Applications. Edited by P. Kahn. New York: Academic Press, Inc., pp. 307–48. [Google Scholar]
- Hansen, Hendrik. 2013. The forecasting performance of mortality models. AStA Advances in Statistical Analysis 97: 11–31. [Google Scholar] [CrossRef]
- Hardy, M. R., and H. H. Panjer. 1998. A credibility approach to mortality risk. Astin Bulletin 28: 269–83. [Google Scholar] [CrossRef]
- Hildreth, Clifford, and James P. Houck. 1968. Some Estimators for a Linear Model with Random Coefficients. Journal of the American Statistical Association 63: 584–95. [Google Scholar]
- Hsiao, Cheng. 1986. Analysis of Panel Data. In Econometric Society Monographs. New York: Cambridge University Press. [Google Scholar]
- Huang, Fei, and Bridget Browne. 2017. Mortality forecasting using a modified Continuous Mortality Investigation Mortality Projections Model for China I: Methodology and country-level results. Annals of Actuarial Science 11: 20–45. [Google Scholar] [CrossRef]
- Human Mortality Database. 2017. University of California, Berkeley (USA) and Max Planck Institute for Demographic Research (Germany). Available online: www.mortality.org (accessed on 20 April 2018).
- Klugman, Stuart A., Harry H. Panjer, and Gordon E. Willmot. 2012. Loss Models: From Data to Decisions, 4th ed. New York: John Wiley & Sons. [Google Scholar]
- Lee, Ronald D., and Lawrence R. Carter. 1992. Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association 87: 659–71. [Google Scholar] [CrossRef]
- Ledolter, Johannes, Stuart Klugman, and Chang-Soo Lee. 1991. Credibility models with time-varying trend components. Astin Bulletin 21: 73–91. [Google Scholar] [CrossRef]
- Li, Hong, and Yang Lu. 2018. A Bayesian non-parametric model for small population mortality. Scandinavian Actuarial Journal 2018: 605–28. [Google Scholar] [CrossRef]
- Li, Nan, Ronald Lee, and Shripad Tuljapurkar. 2004. Using the Lee–Carter Method to Forecast Mortality for Populations with Limited Data. International Statistical Review 72: 19–36. [Google Scholar] [CrossRef]
- Luan, Xiang. 2015. A Pseudo Non-Parametric Buhlmann Credibility Approach to Modeling Mortality Rates. Master’s thesis, Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC, Canada. [Google Scholar]
- Norberg, Ragnar. 1980. Empirical bayes credibility. Scandinavian Actuarial Journal 1980: 177–94. [Google Scholar] [CrossRef]
- Pitselis, Georgios. 2004. Credibility models with cross-section effect and with both cross-section and time effects. Blätter der DGVFM 26: 643–63. [Google Scholar] [CrossRef]
- Plat, Richard. 2009. On stochastic mortality modeling. Insurance: Mathematics and Economics 45: 393–404. [Google Scholar]
- R Core Team. 2017. A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing, Available online: https://www.r-project.org/ (accessed on 10 February 2018).
- Rao, C. Radhakrishna. 1973. Linear Statistical Inference and Its Applications, 2nd ed. New York: Wiley. [Google Scholar]
- Renshaw, Arthur E., and Steven Haberman. 2006. A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics 38: 556–70. [Google Scholar] [CrossRef]
- Salhi, Yahia, Pierre-E. Thérond, and Julien Tomas. 2016. A credibility approach of the Makeham mortality law. European Actuarial Journal 6: 61–96. [Google Scholar] [CrossRef]
- Salhi, Yahia, and Pierre-E. Thérond. 2018. Age-Specific Adjustment of Graduated Mortality. ASTIN Bulletin 48: 543–69. [Google Scholar] [CrossRef]
- Schinzinger, Edo, Michel M. Denuit, and Marcus C. Christiansen. 2016. A multivariate evolutionary credibility model for mortality improvement rates. Insurance: Mathematics and Economics 69: 70–81. [Google Scholar] [CrossRef]
- Shang, Han Lin, Heather Booth, and Rob J. Hyndman. 2011. Point and interval forecasts of mortality rates and life expectancy: A comparison of ten principal component methods. Demographic Research 25: 173–214. [Google Scholar] [CrossRef] [Green Version]
- Tsai, Cary Chi-Liang, and Tzuling Lin. 2017a. A Bühlmann credibility approach to modeling mortality rates. North American Actuarial Journal 21: 204–27. [Google Scholar] [CrossRef]
- Tsai, Cary Chi-Liang, and Tzuling Lin. 2017b. Incorporating the Bühlmann credibility into mortality models to improve forecasting performances. Scandinavian Actuarial Journal 2017: 419–40. [Google Scholar] [CrossRef]
- Tsai, Cary Chi-Liang, and Shuai Yang. 2015. A Linear Regression Approach to Modeling Mortality Rates of Different Forms. North American Actuarial Journal 19: 1–23. [Google Scholar] [CrossRef]
- Van Berkum, Frank, Katrien Antonio, and Michel Vellekoop. 2016. The impact of multiple structural changes on mortality predictions. Scandinavian Actuarial Journal 2016: 581–603. [Google Scholar] [CrossRef]
- Zhao, Bojuan Barbara. 2012. A modified Lee-Carter model for analysing short-base-period data. Population Studies 66: 39–52. [Google Scholar] [CrossRef] [PubMed]

1 | Medical, biological, environmental or other factors that affect mortality evolution of each corresponding age over consecutive years are treated as unknown or exogenous due to the lack of specific data. |

2 | The software, which is not part of CRAN, is available from http://www.macs.hw.ac.uk/~andrewc/lifemetrics/. |

3 | For instance, use of MAFE is demonstrated in the modelling comparison study of Shang et al. (2011), while RMSFE in Hansen (2013) and Van Berkum et al. (2016). |

4 | To distinguish one from the other, MAFE and RMSFE averaged error values are rounded to four decimal points, while, for MAPFE values, two decimal points are enough. |

5 | According to the World Bank database (https://data.worldbank.org/indicator/SP.POP.TOTL), total population counts for 2016 were 11.35 million for Belgium, 5.50 for Finland, 5.23 for Norway, 4.77 for Ireland, 5.43 for Slovakia and 4.69 for New Zealand. |

**Figure 1.**Observed $logm(t,x)$ of the period 1981–2010 in Greece, for males (

**left**) and females (

**middle**) at the age of 40, 60 and 80. Average male and female $logm(t,x)$ values over ages 15–84 are illustrated in (

**right**), where straight lines show the corresponding trends in mortality decline.

**Figure 2.**Observed $logitq(t,x)$ of the period 1981–2010 in Greece, for males (

**left**) and females (

**middle**) at the age of 40, 60 and 80. Average male and female $logitq(t,x)$ values over ages 15–84 are illustrated in (

**right**), where straight lines show the corresponding trends in mortality decline.

**Figure 3.**Linear trend of the observed $logitq(t,x)$ of the period 1981–2010 in Greece, for males (

**left**) and females (

**right**) at the age of 55, 65 and 75.

**Figure 4.**AFE values against age of $logitq(2000+h,x)$, $h=1,\cdots ,10$ between the actual rates and the rates produced from the best performing models with and without credibility for males (

**left**) and females (

**right**) over $[2001,2010]$, fitted to pension ages $[65,84]$ for years $[1981,2000]$.

**Figure 5.**Intercept and slope estimates of $logitq(2000+h,x)$ for $h=1,\cdots ,10$ and ages $x=66$ for males and $x=67$ for females, with credibility (dot-dashed lines for FC-MEM and RC-MEM) and without credibility (dashed lines for LC and dot lines for CBD). Solid lines show the actual mortality and its trend.

**Figure 6.**AFE values against age of life insurance and annuity products for the top LC, CBD and credibility regression models for males (left panels) and females (right panels): (

**a**) life insurance AFEs for males; (

**b**) life insurance AFEs for females; (

**c**) pure endowment AFEs for males; (

**d**) pure endowment AFEs for females; (

**e**) life annuity AFEs for males; and (

**f**) life annuity AFEs for females.

Fitting Ages | Fitting Period | Forecasting Period |
---|---|---|

$[{x}_{0},{x}_{k-1}]$ | $[{t}_{0},{t}_{n-1}]$ | $[{t}_{n-1}+1,{t}_{n-1}+H]$ |

$[15,84]$ | $[1981,2000]$ | $[2001,2010]$ |

$[15,84]$ | $[1986,2000]$ | $[2001,2010]$ |

$[15,84]$ | $[1991,2000]$ | $[2001,2010]$ |

$[55,84]$ | $[1981,2000]$ | $[2001,2010]$ |

$[55,84]$ | $[1986,2000]$ | $[2001,2010]$ |

$[55,84]$ | $[1991,2000]$ | $[2001,2010]$ |

(a) MAFE Values | |||||||||

$\mathit{MAFE}}_{[\mathbf{15},\mathbf{84}]$ | Lee–Carter | Random Coefficients (RC) | Fixed Coefficients (FC) | ||||||

Fitting Period | Gender | LC | LC-Poisson | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1981,2000]$ | Male | 0.1513 | 0.1569 | 0.1338 | 0.1205 | 0.1322 | 0.1352 | 0.1256 | 0.1361 |

Female | 0.0831 | 0.0861 | 0.0702 | 0.0740 | 0.0711 | 0.0691 | 0.0657 | 0.0690 | |

$[1986,2000]$ | Male | 0.1684 | 0.1514 | 0.1175 | 0.1196 | 0.1158 | 0.1203 | 0.1221 | 0.1206 |

Female | 0.0625 | 0.0799 | 0.0650 | 0.0696 | 0.0758 | 0.0608 | 0.0651 | 0.0613 | |

$[1991,2000]$ | Male | 0.1468 | 0.1681 | 0.1275 | 0.1257 | 0.1280 | 0.1288 | 0.1289 | 0.1289 |

Female | 0.0763 | 0.0959 | 0.0705 | 0.0678 | 0.0750 | 0.0622 | 0.0663 | 0.0669 | |

Average | 0.1147(7) | 0.1231(8) | 0.0974(5) | 0.0962(3) | 0.0997(6) | 0.0961(2) | 0.0956(1) | 0.0971(4) | |

(b) RMSFE Values | |||||||||

$\mathit{RMSFE}}_{[\mathbf{15},\mathbf{84}]$ | Lee–Carter | Random Coefficients (RC) | Fixed Coefficients (FC) | ||||||

Fitting Period | Gender | LC | LC-Poisson | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1981,2000]$ | Male | 0.3165 | 0.3220 | 0.2661 | 0.2349 | 0.2629 | 0.2716 | 0.2511 | 0.2745 |

Female | 0.1791 | 0.1825 | 0.1398 | 0.1594 | 0.1457 | 0.1376 | 0.1365 | 0.1374 | |

$[1986,2000]$ | Male | 0.3543 | 0.3200 | 0.2257 | 0.2265 | 0.2204 | 0.2362 | 0.2364 | 0.2375 |

Female | 0.1307 | 0.1742 | 0.1410 | 0.1509 | 0.1700 | 0.1264 | 0.1385 | 0.1288 | |

$[1991,2000]$ | Male | 0.3180 | 0.4010 | 0.2478 | 0.2457 | 0.2470 | 0.2570 | 0.2551 | 0.2516 |

Female | 0.1694 | 0.2415 | 0.1580 | 0.1511 | 0.1707 | 0.1302 | 0.1438 | 0.1476 | |

Average | 0.2447(7) | 0.2735(8) | 0.1964(4) | 0.1948(3) | 0.2028(6) | 0.1932(1) | 0.1936(2) | 0.1962(5) |

(a) MAFE Values | |||||||||||

$\mathit{MAFE}}_{[\mathbf{55},\mathbf{84}]$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | ||||||||

Fitting Period | Gender | LC | LC-Poisson | CBD | CBD-Poisson | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1981,2000]$ | Male | 0.3191 | 0.3322 | 0.2924 | 0.3247 | 0.2885 | 0.2642 | 0.2846 | 0.2871 | 0.2673 | 0.2870 |

Female | 0.1884 | 0.1933 | 0.1694 | 0.1884 | 0.1624 | 0.1458 | 0.1611 | 0.1629 | 0.1448 | 0.1627 | |

$[1986,2000]$ | Male | 0.2928 | 0.3186 | 0.2682 | 0.2988 | 0.2506 | 0.2547 | 0.2494 | 0.2544 | 0.2581 | 0.2541 |

Female | 0.1577 | 0.1769 | 0.1618 | 0.1708 | 0.1287 | 0.1377 | 0.1344 | 0.1289 | 0.1351 | 0.1288 | |

$[1991,2000]$ | Male | 0.3091 | 0.3622 | 0.2790 | 0.3348 | 0.2483 | 0.2461 | 0.2464 | 0.2538 | 0.2493 | 0.2525 |

Female | 0.1723 | 0.2126 | 0.1659 | 0.1868 | 0.1324 | 0.1350 | 0.1363 | 0.1363 | 0.1382 | 0.1361 | |

Average | 0.2399(8) | 0.2660(10) | 0.2228(7) | 0.2507(9) | 0.2018(3) | 0.1973(1) | 0.2020(4) | 0.2039(6) | 0.1988(2) | 0.2035(5) | |

(b)RMSFEValues | |||||||||||

$\mathit{RMSFE}}_{[\mathbf{55},\mathbf{84}]$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | ||||||||

Fitting Period | Gender | LC | LC-Poisson | CBD | CBD-Poisson | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1981,2000]$ | Male | 0.4616 | 0.4848 | 0.3904 | 0.4467 | 0.4041 | 0.3644 | 0.3963 | 0.4065 | 0.3786 | 0.4061 |

Female | 0.2795 | 0.2842 | 0.2221 | 0.2512 | 0.2260 | 0.1996 | 0.2213 | 0.2304 | 0.2010 | 0.2299 | |

$[1986,2000]$ | Male | 0.4320 | 0.4872 | 0.3551 | 0.4073 | 0.3506 | 0.3522 | 0.3419 | 0.3631 | 0.3653 | 0.3618 |

Female | 0.2340 | 0.2699 | 0.2165 | 0.2244 | 0.1805 | 0.1940 | 0.1895 | 0.1803 | 0.1897 | 0.1800 | |

$[1991,2000]$ | Male | 0.4671 | 0.6129 | 0.3698 | 0.4625 | 0.3484 | 0.3423 | 0.3389 | 0.3660 | 0.3501 | 0.3616 |

Female | 0.2652 | 0.3721 | 0.2202 | 0.2510 | 0.1866 | 0.1888 | 0.1912 | 0.1961 | 0.1930 | 0.1954 | |

Average | 0.3566(9) | 0.4185(10) | 0.2957(7) | 0.3405(8) | 0.2827(4) | 0.2736(1) | 0.2799(3) | 0.2904(6) | 0.2796(2) | 0.2891(5) |

**Table 4.**MAFE, RMSFE and MAPFE values of forecast errors over the period $[2001,2010]$ for ages $[21,85]$.

(a) MAFE Values | |||||||||

$\mathit{MAFE}}_{[\mathbf{21},\mathbf{85}]$ | Bühlmann Methods | Regression Methods – RC | Regression Methods – FC | ||||||

Fitting Period | Gender | EW | MW | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1982,2000]$ | Male | 0.2348 | 0.2334 | 0.1404 | 0.1287 | 0.1379 | 0.1444 | 0.1361 | 0.1460 |

Female | 0.0930 | 0.0931 | 0.0816 | 0.0882 | 0.0843 | 0.0791 | 0.0780 | 0.0790 | |

$[1986,2000]$ | Male | 0.2170 | 0.2294 | 0.1329 | 0.1321 | 0.1306 | 0.1364 | 0.1358 | 0.1373 |

Female | 0.0918 | 0.0919 | 0.0782 | 0.0852 | 0.0909 | 0.0741 | 0.0805 | 0.0747 | |

$[1990,2000]$ | Male | 0.2355 | 0.2258 | 0.1399 | 0.1392 | 0.1369 | 0.1434 | 0.1422 | 0.1423 |

Female | 0.0954 | 0.0933 | 0.0836 | 0.0839 | 0.0879 | 0.0798 | 0.0818 | 0.0802 | |

Average | 0.1613(8) | 0.1612(7) | 0.1094(2) | 0.1096(4) | 0.1114(6) | 0.1095(3) | 0.1091(1) | 0.1099(5) | |

(b)RMSFEValues | |||||||||

$\mathit{RMSFE}}_{[\mathbf{21},\mathbf{85}]$ | Bühlmann Methods | Regression Methods – RC | Regression Methods – FC | ||||||

Fitting Period | Gender | EW | MW | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1982,2000]$ | Male | 0.4980 | 0.4948 | 0.2633 | 0.2342 | 0.2566 | 0.2756 | 0.2564 | 0.2799 |

Female | 0.1795 | 0.1795 | 0.1613 | 0.1884 | 0.1730 | 0.1540 | 0.1581 | 0.1541 | |

$[1986,2000]$ | Male | 0.4584 | 0.4861 | 0.2447 | 0.2387 | 0.2386 | 0.2564 | 0.2532 | 0.2591 |

Female | 0.1767 | 0.1772 | 0.1633 | 0.1781 | 0.1999 | 0.1484 | 0.1643 | 0.1502 | |

$[1990,2000]$ | Male | 0.4997 | 0.4765 | 0.2578 | 0.2574 | 0.2472 | 0.2704 | 0.2666 | 0.2668 |

Female | 0.1849 | 0.1802 | 0.1640 | 0.1761 | 0.1767 | 0.1567 | 0.1674 | 0.1570 | |

Average | 0.3329(8) | 0.3324(7) | 0.2091(1) | 0.2122(5) | 0.2153(6) | 0.2103(2) | 0.2110(3) | 0.2112(4) | |

(c)MAPFEValues | |||||||||

$\mathit{MAPFE}}_{[\mathbf{21},\mathbf{85}]$ | Bühlmann Methods | Regression Methods – RC | Regression Methods – FC | ||||||

Fitting Period | Gender | EW | MW | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1982,2000]$ | Male | 11.90 | 11.86 | 11.97 | 11.24 | 11.80 | 11.95 | 11.43 | 12.00 |

Female | 13.75 | 13.76 | 11.66 | 11.83 | 11.69 | 11.66 | 11.54 | 11.66 | |

$[1986,2000]$ | Male | 11.30 | 11.71 | 12.05 | 10.76 | 11.73 | 11.86 | 10.72 | 11.90 |

Female | 13.71 | 13.72 | 11.52 | 11.82 | 11.89 | 11.56 | 11.73 | 11.55 | |

$[1990,2000]$ | Male | 11.93 | 11.60 | 10.81 | 9.81 | 10.60 | 10.57 | 9.71 | 10.52 |

Female | 13.83 | 13.77 | 11.84 | 11.79 | 12.08 | 12.05 | 11.83 | 11.99 | |

Average | 12.73(7) | 12.74(8) | 11.64(6) | 11.21(2) | 11.63(5) | 11.61(4) | 11.16(1) | 11.60(3) |

**Table 5.**MAFE, RMSFE and MAPFE values of forecast errors over the period $[2001,2010]$ for ages $[56,85]$.

(a) MAFE Values | |||||||||

$\mathit{MAFE}}_{[\mathbf{56},\mathbf{85}]$ | Bühlmann Methods | Regression Methods – RC | Regression Methods – FC | ||||||

Fitting Period | Gender | EW | MW | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1982,2000]$ | Male | 0.3599 | 0.3503 | 0.3272 | 0.3012 | 0.3210 | 0.3262 | 0.3036 | 0.3255 |

Female | 0.1686 | 0.1623 | 0.1717 | 0.1633 | 0.1735 | 0.1711 | 0.1595 | 0.1709 | |

$[1986,2000]$ | Male | 0.3233 | 0.3430 | 0.2893 | 0.2958 | 0.2886 | 0.2946 | 0.2991 | 0.2937 |

Female | 0.1481 | 0.1539 | 0.1534 | 0.1601 | 0.1617 | 0.1495 | 0.1573 | 0.1511 | |

$[1990,2000]$ | Male | 0.3745 | 0.3641 | 0.2958 | 0.2934 | 0.2937 | 0.2999 | 0.2954 | 0.2973 |

Female | 0.1670 | 0.1646 | 0.1617 | 0.1616 | 0.1613 | 0.1601 | 0.1625 | 0.1615 | |

Average | 0.2569(8) | 0.2564(7) | 0.2332(3) | 0.2293(1) | 0.2333(4) | 0.2336(6) | 0.2296(2) | 0.2334(5) | |

(b) MAPFE Values | |||||||||

$\mathit{RMSFE}}_{[\mathbf{56},\mathbf{85}]$ | Bühlmann Methods | Regression Methods – RC | Regression Methods – FC | ||||||

Fitting Period | Gender | EW | MW | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1982,2000]$ | Male | 0.5411 | 0.5261 | 0.4670 | 0.4213 | 0.4524 | 0.4700 | 0.4305 | 0.4679 |

Female | 0.2358 | 0.2282 | 0.2368 | 0.2242 | 0.2366 | 0.2389 | 0.2202 | 0.2381 | |

$[1986,2000]$ | Male | 0.4852 | 0.5159 | 0.4065 | 0.4138 | 0.3987 | 0.4221 | 0.4224 | 0.4185 |

Female | 0.2120 | 0.2178 | 0.2151 | 0.2235 | 0.2271 | 0.2089 | 0.2192 | 0.2107 | |

$[1990,2000]$ | Male | 0.5636 | 0.5472 | 0.4139 | 0.4130 | 0.4072 | 0.4291 | 0.4184 | 0.4195 |

Female | 0.2338 | 0.2307 | 0.2243 | 0.2246 | 0.2236 | 0.2217 | 0.2257 | 0.2232 | |

Average | 0.3786(8) | 0.3777(7) | 0.3273(4) | 0.3201(1) | 0.3243(3) | 0.3318(6) | 0.3227(2) | 0.3297(5) | |

(c)MAPFEValues | |||||||||

$\mathit{MAPFE}}_{[\mathbf{56},\mathbf{85}]$ | Bühlmann Methods | Regression Methods – RC | Regression Methods – FC | ||||||

Fitting Period | Gender | EW | MW | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ | $\mathit{SEM}$ | $\mathit{MEM}$ | $\mathit{EEM}$ |

$[1982,2000]$ | Male | 9.53 | 9.34 | 9.48 | 9.17 | 9.54 | 9.29 | 8.97 | 9.31 |

Female | 9.93 | 9.72 | 9.98 | 9.81 | 10.36 | 9.65 | 9.43 | 9.69 | |

$[1986,2000]$ | Male | 8.82 | 9.20 | 8.78 | 8.97 | 8.99 | 8.61 | 8.85 | 8.66 |

Female | 9.23 | 9.45 | 9.14 | 9.42 | 9.62 | 8.84 | 9.26 | 8.98 | |

$[1990,2000]$ | Male | 9.82 | 9.61 | 8.85 | 8.74 | 9.00 | 8.62 | 8.74 | 8.78 |

Female | 9.88 | 9.81 | 9.49 | 9.33 | 9.46 | 9.32 | 9.37 | 9.48 | |

Average | 9.54(8) | 9.52(7) | 9.29(5) | 9.24(4) | 9.50(6) | 9.06(1) | 9.10(2) | 9.15(3) |

**Table 6.**MAFE and RMSFE values (ranking order in brackets) for a 10-year forecasted life insurance, a pure endowment and a life annuity for males and females of ages 55–74 during 2001–2010.

(a) Life Insurance | ||||||||||

$MAF{E}_{avg}$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | |||||||

Gender | LC | LC-Poisson | CBD | CBD-Poisson | $SEM$ | $MEM$ | $EEM$ | $SEM$ | $MEM$ | $EEM$ |

Male | 1.6019(8) | 1.5640(7) | 1.7151(10) | 1.6794(9) | 1.5000(6) | 1.4169(2) | 1.4924(5) | 1.4735(3) | 1.3932(1) | 1.4741(4) |

Female | 1.0264(6) | 1.0269(7) | 1.2141(10) | 1.1079(9) | 1.0262(5) | 0.9317(2) | 1.0346(8) | 0.9898(3) | 0.8840(1) | 0.9910(4) |

$RMSF{E}_{avg}$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | |||||||

Gender | LC | LC-Poisson | CBD | CBD-Poisson | $SEM$ | $MEM$ | $EEM$ | $SEM$ | $MEM$ | $EEM$ |

Male | 1.8423(8) | 1.8043(7) | 1.9401(10) | 1.9089(9) | 1.7125(6) | 1.6143(2) | 1.7043(5) | 1.6871(3) | 1.5989(1) | 1.6875(4) |

Female | 1.2320(8) | 1.2294(7) | 1.4023(10) | 1.2918(9) | 1.2133(5) | 1.0965(2) | 1.2215(6) | 1.1744(3) | 1.0494(1) | 1.1756(4) |

(b) Pure Endowment | ||||||||||

$MAF{E}_{avg}$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | |||||||

Gender | LC | LC-Poisson | CBD | CBD-Poisson | $SEM$ | $MEM$ | $EEM$ | $SEM$ | $MEM$ | $EEM$ |

Male | 1.1439(8) | 1.1139(7) | 1.2417(10) | 1.2044(9) | 1.0722(6) | 1.0153(2) | 1.0681(5) | 1.0512(3) | 0.9942(1) | 1.0518(4) |

Female | 0.7181(7) | 0.7192(8) | 0.8894(10) | 0.7923(9) | 0.7340(5) | 0.6717(2) | 0.7463(6) | 0.7026(3) | 0.6297(1) | 0.7038(4) |

$RMSF{E}_{avg}$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | |||||||

Gender | LC | LC-Poisson | CBD | CBD-Poisson | $SEM$ | $MEM$ | $EEM$ | $SEM$ | $MEM$ | $EEM$ |

Male | 1.3274(8) | 1.2975(7) | 1.4104(10) | 1.3786(9) | 1.2347(6) | 1.1659(2) | 1.2303(5) | 1.2150(3) | 1.1535(1) | 1.2154(4) |

Female | 0.8745(8) | 0.8717(7) | 1.0310(10) | 0.9319(9) | 0.8774(5) | 0.7968(2) | 0.8889(6) | 0.8440(3) | 0.7552(1) | 0.8451(4) |

(c) Life Annuity | ||||||||||

$MAF{E}_{avg}$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | |||||||

Gender | LC | LC-Poisson | CBD | CBD-Poisson | $SEM$ | $MEM$ | $EEM$ | $SEM$ | $MEM$ | $EEM$ |

Male | 5.4466(8) | 5.2602(7) | 6.3032(10) | 5.7857(9) | 5.1642(6) | 4.9260(2) | 5.1561(5) | 5.0331(3) | 4.7893(1) | 5.0369(4) |

Female | 2.9140(7) | 2.9361(8) | 4.4471(10) | 3.5932(9) | 3.1527(5) | 2.9479(2) | 3.2151(6) | 2.9656(3) | 2.7024(1) | 2.9721(4) |

$RMSF{E}_{avg}$ | Mortality Models | Random Coefficients (RC) | Fixed Coefficients (FC) | |||||||

Gender | LC | LC-Poisson | CBD | CBD-Poisson | $SEM$ | $MEM$ | $EEM$ | $SEM$ | $MEM$ | $EEM$ |

Male | 6.6138(7) | 6.4342(8) | 7.2583(10) | 6.8729(9) | 6.1786(6) | 5.8730(2) | 6.1608(5) | 6.0681(3) | 5.7919(1) | 6.0704(4) |

Female | 3.7013(7) | 3.7218(8) | 5.1510(10) | 4.3300(9) | 3.9187(5) | 3.6344(2) | 3.9846(6) | 3.7084(3) | 3.3878(1) | 3.7155(4) |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bozikas, A.; Pitselis, G.
Credible Regression Approaches to Forecast Mortality for Populations with Limited Data. *Risks* **2019**, *7*, 27.
https://doi.org/10.3390/risks7010027

**AMA Style**

Bozikas A, Pitselis G.
Credible Regression Approaches to Forecast Mortality for Populations with Limited Data. *Risks*. 2019; 7(1):27.
https://doi.org/10.3390/risks7010027

**Chicago/Turabian Style**

Bozikas, Apostolos, and Georgios Pitselis.
2019. "Credible Regression Approaches to Forecast Mortality for Populations with Limited Data" *Risks* 7, no. 1: 27.
https://doi.org/10.3390/risks7010027