# Mortality Projections for Small Populations: An Application to the Maltese Elderly

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- “by mixing appropriately the mortality data obtained from other populations” (e.g., Ahcan et al. (2014));
- by creating a two-population mortality model, which considers jointly the mortality of a small population related to a larger one.

- Li and Lee (2005), who applied the Lee–Carter model, with the introduction of common factors, for a group of given population, in order to predict single mortality evolution;
- Cairns et al. (2011a), who introduced a Bayesian framework to jointly model two populations, referring to one of them as sub-population of the other one;
- Dowd et al. (2011), who proposed the gravity model for two populations in order to obtain coherent mortality forecasts;
- Jarner and Kryger (2011), who proposed a model for the Danish mortality (the Spread Adjusted InterNational Trend (SAINT) model) combining the mortality deterministic evolution of a basket of population with the stochastic evolution of the spread;
- D’Amato et al. (2014), who extended the Lee–Carter model in order to take into account the existence of dependence in mortality data across multiple populations;
- Villegas and Haberman (2014), who applied a relative modeling approach where the death rates of a subpopulation are modeled in relation to the death rates of a reference population; they considered different multiple population extensions of the Lee–Carter model and applied their approach in order to study and forecast socioeconomic mortality differentials across deprivation subgroups in England;
- Wan and Bertschi (2015), who proposed a two part model to fit Swiss historical data and make coherent forecasts, taking information from a larger population;
- Antonio et al. (2017), who developed a Li and Lee multi-population model to project Dutch and Belgian mortality evolution and measure the actuarial implication of their model;
- Chen et al. (2017), who proposed “the use of parametric bootstrap methods to investigate the finite sample distribution of the maximum likelihood estimator for the parameter vector of a stochastic mortality model”;
- Hunt and Blake (2017), who modeled the mortality rates of a pension scheme through an Age Period Cohort (APC) model that has the same form of the reference population model but is characterized by scaling factors that multiply period and cohort parameters and reduce or increase the dependence between the two models;
- Villegas et al. (2017), who developed a comprehensive comparative study of mortality models for two populations proposed in the literature and applied them to the case of a population of a pension scheme in order to measure the basis risk involved in longevity hedges;
- Wang et al. (2018), who proposed an approach based on a combination of data aggregation and mortality graduation applied to the empirical data from Taiwan and Taipei City.

## 2. The Model Structure and the Choice of the Reference Population

#### 2.1. The Model Structure

- being greater than the one for which we want to predict mortality;
- having a background trend similar to that of the smaller population’s mortality.

#### 2.2. Mortality Data

- firstly, the mortality rates derived from the death probabilities2;
- then, from these values and those of mid-year population, number of deaths are easily calculated.

#### 2.3. The Choice of the Reference Population

#### 2.4. An Alternative Approach for the Choice of the Reference Population

## 3. The Mortality Models Choice

#### 3.1. The Reference Population Mortality Model

#### 3.2. The Model Choice for the Spread

## 4. Mortality Projections

#### 4.1. Time Series Dynamics

#### 4.2. Mortality Projections

#### 4.3. Parameters Uncertainty

## 5. Conclusions

- to consider other methodologies for bootstrap as, for instance, the sieve bootstrap proposed by D’Amato et al. (2012) to capture the dependency risk produced by the presence of spatial dependence across age and time;
- to extend our analysis by considering other models for the reference and/or the small population (e.g., a two-part model obtained mixing models of the Cairns–Blake–Dowd family);
- to use the simulated death rates obtained with the proposed models in order to measure the longevity risk in the Maltese pension system;
- to test the effectiveness of the proposed models on other small populations, e.g., other small countries of the European continent.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Mortality rate projections at age 65, 75, 85 with Plat–LC model (best ARIMA processes) for (

**a**) female Maltese population, (

**b**) male Maltese population).

**Figure A2.**Mortality rate projections at age 65, 75, 85 with parameters uncertainly with Plat–LC model (best ARIMA processes) for (

**a**) female Maltese population, (

**b**) male Maltese population.

**Table A1.**Relative amplitude of prediction intervals for Maltese death rates in 2046 with Plat–LC model with different ARIMA processes.

Age | Females | Males | ||
---|---|---|---|---|

Plat–LC (MRWD) | Plat–LC (best ARIMA) | Plat–LC (MRWD) | Plat–LC (best ARIMA) | |

65 | 51.87% | 63.92% | 63.34% | 66.88% |

75 | 82.10% | 77.97% | 68.43% | 66.32% |

85 | 90.55% | 67.76% | 65.93% | 53.60% |

## References

- Ahcan, Ales, Darko Medved, Annamaria Olivieri, and Ermanno Pitacco. 2014. Forecasting mortality for small populations by mixing mortality data. Insurance: Mathematics and Economics 54: 12–27. [Google Scholar] [CrossRef]
- Antonio, Katrien, Sander Devriendt, Wouter de Boer, Robert de Vries, Anja De Waegenaere, Hok-Kwan Kan, Egbert Kromme, Wilbert Ouburg, Tim Schulteis, Erica Slagter, and et al. 2017. Producing the Dutch and Belgian mortality projections: A stochastic multi-population standard. European Actuarial Journal 7: 297–336. [Google Scholar] [CrossRef]
- Benjamin, Bernard, John H. Pollard, and Herbert W. Haycocks. 1993. The Analysis of Mortality and Other Actuarial Statistics. London: The Institute of Actuaries and The Faculty of Actuaries. [Google Scholar]
- Bongaarts, John. 2005. Long-range trends in adult mortality: Models and projections methods. Demography 42: 23–49. [Google Scholar] [CrossRef] [PubMed]
- Booth, Heather, Rob J. Hyndman, Leonie Tickle, and Piet de Jong. 2006. Lee–Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research 15: 289–310. [Google Scholar] [CrossRef]
- Bravo, Jorge, and Joana Malta. 2010. Estimating Life Expectancy in Small Population Areas. Geneva: United Nations Economic Commission for Europe. [Google Scholar]
- Brouhns, Natacha, Michel Denuit, and Ingrid Van Keilegom. 2005. Bootstrapping the Poisson Log-bilinear Model for Mortality Projection. Scandinavian Actuarial Journal 3: 212–24. [Google Scholar] [CrossRef]
- Cairns, Andrew, David Blake, and Kevin Dowd. 2008. Modelling and management of mortality risk: A review. Scandinavian Actuarial Journal 2: 79–113. [Google Scholar] [CrossRef]
- Cairns, Andrew, 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]
- Cairns, Andrew, David Blake, Kevin Dowd, Guy D. Coughlan, and Marwa Khalaf-Allah. 2011a. Bayesian Stochastic Mortality Modelling for Two Populations. Astin Bulletin 41: 29–59. [Google Scholar]
- Cairns, Andrew, David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, and Marwa Khalaf-Allah. 2011b. Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics 48: 355–67. [Google Scholar] [CrossRef]
- Chen, Liang, Andrew Cairns, and Torsten Kleinow. 2017. Small population bias and sampling effects in stochastic mortality modeling. European Actuarial Journal 7: 193–230. [Google Scholar] [CrossRef] [PubMed]
- Currie, Iain. 2006. Smoothing and forecasting mortality rates with P-splines. Talk given at the Institute of Actuaries, June 2006. Available online: http://www.macs.hw.ac.uk/~iain/research/talks/Mortality.pdf (accessed on 18 June 2018).
- D’Amato, Valeria, Steven Haberman, Gabriella Piscopo, and Maria Russolillo. 2012. Modelling dependent data for longevity projections. Insurance: Mathematics and Economics 51: 694–701. [Google Scholar] [CrossRef]
- D’Amato, Valeria, HSteven Haberman, Gabriella Piscopo, Maria Russolillo, and Lorenzo Trapani. 2014. Detecting common longevity trends by a multiple population approach. North American Actuarial Journal 18: 139–49. [Google Scholar] [CrossRef]
- Dowd, Kevin, Andrew Cairns, David Blake, Guy D. Coughlan, and Marwa Khalaf-Allah. 2011. A gravity model of mortality rates for two related populations. North American Actuarial Journal 15: 334–56. [Google Scholar] [CrossRef]
- Haberman, Steven, and Arthur Renshaw. 2011. A Comparative Study of Parametric Mortality Projection Models. Insurance: Mathematics and Economics 48: 35–55. [Google Scholar] [CrossRef]
- Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available online: www.mortality.orgorwww.humanmortality.de (accessed on 5 October 2018).
- Hunt, Andrew, and David Blake. 2017. Modelling Mortality for Pension Schemes. Astin Bulletin 47: 601–29. [Google Scholar] [CrossRef]
- Hunt, Andrew, and Andrés M. Villegas. 2015. Robustness and Convergence in the Lee-carter model with cohort effects. Insurance: Mathematics and Economics 64: 186–202. [Google Scholar] [CrossRef]
- ISTAT. 2001. Tavole di Mortalitá Della Popolazione Italiana per Provincia e Regione di Residenza. Available online: https://www.istat.it/it/files//2018/08/volume-tavole-mortalita-1998.pdf (accessed on 1 July 2018).
- ISTAT. 2015. Tavole di Mortalitá: Singole etá. Anno 2015. Available online: https://www.istat.it (accessed on 1 July 2018).
- ISTAT. 2016. Tavole di Mortalitá: Singole etá. Anno 2016. Available online: https://www.istat.it (accessed on 1 July 2018).
- Jarner, Søren Fiig, and Esben Masotti Kryger. 2011. Modelling Adult Mortality in Small Populations: The SAINT Model. Astin Bulletin 41: 377–418. [Google Scholar]
- Keyfitz, Nathan, and Hal Caswell. 2005. Applied Mathematical Demography. New York: Springer. [Google Scholar]
- Kleinow, Torsten. 2015. A common age effect model for the mortality of multiple populations. Insurance: Mathematics and Economics 63: 147–52. [Google Scholar] [CrossRef]
- 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]
- Li, Johnny Siu-Hang, Rui Zhou, and Mary Hardy. 2015. A step-by-step guide to building two population stochastic mortality models. Insurance: Mathematics and Economics 63: 121–34. [Google Scholar] [CrossRef]
- Li, Nan, and Ronald Lee. 2005. Coherent Mortality Forecasts for a Group of Populations: An Extension of the Lee–Carter Method. Demography 42: 575–94. [Google Scholar] [CrossRef]
- Plat, Richard. 2009. On stochastic mortality modeling. Insurance: Mathematics and Economics 45: 393–404. [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]
- Villegas, Andrés M., and Steven Haberman. 2014. On the Modeling an Forecasting of Socioeconomic Mortality Differentials: An Application to Deprivation and Mortality in England. North American Actuarial Journal 18: 168–93. [Google Scholar] [CrossRef]
- Villegas, Andrés M., Steven Haberman, Vladimir K. Kaishev, and Pietro Millossovich. 2017. A Comparative Study of Two-Population Models for the Assessment of Basis Risk in Longevity Hedges. Astin Bulletin 47: 631–79. [Google Scholar] [CrossRef]
- Villegas, Andrés M., Steven Haberman, Vladimir K. Kaishev, and Pietro Millossovich. 2018. StMoMo: An R Package for Stochastic Mortality Modeling. Journal of Statistical Software 84: 1–38. [Google Scholar] [CrossRef]
- Wan, Cheng, and Ljudmila Bertschi. 2015. Swiss coherent mortality model as a basis for developing longevity de-risking solutions for Swiss pension funds: A practical approach. Insurance: Mathematics and Economics 63: 66–75. [Google Scholar] [CrossRef]
- Wang, Hsin-Chung, Ching-Syang Jack Yue, and Chen-Tai Chong. 2018. Mortality Models and Longevity Risk for Small Populations. Insurance: Mathematics and Economics 78: 351–59. [Google Scholar] [CrossRef]
- Willets, Richard. 2004. The Cohort Effect: Insights and Explanations. British Actuarial Journal 10: 833–77. [Google Scholar] [CrossRef]

1 | Although the data on the Eurostat site are available for France, United Kingdom, Italy and Spain for the whole period 1999–2016, we considered it more convenient to use the data from HMD, immediately available in R thanks to the functions of the “StMoMo” package. |

2 | |

3 | Both AIC and BIC provide a trade-off between the quality of the fit and the parsimony of the model. The AIC formula is: $-2logL+2{N}_{p}$, where L is the maximized value of the likelihood function for the estimated model and ${N}_{p}$ is the number of free parameters to be estimated. The BIC is calculated as: $-2logL+{N}_{p}log\left({N}_{obs}\right)$, where ${N}_{obs}$ is the number of observations. |

4 | We have chosen these two models because they are the ones that turned out to have the lowest BIC value among the four applied directly to the Maltese population for both women and men. |

**Figure 1.**Age(-range)-specific female death rates: (

**a**) at age 60–64, (

**b**) at age 65–69, (

**c**) at age 70–74, (

**d**) at age 75–79, (

**e**) at age 80–84, (

**f**) at age 85–89.

**Figure 2.**Age(-range)-specific male death rates: (

**a**) at age 60–64, (

**b**) at age 65–69, (

**c**) at age 70–74, (

**d**) at age 75–79, (

**e**) at age 80–84, (

**f**) at age 85–89.

**Figure 7.**Estimated Spread parameters of the RH–LC model for (

**a**) female Maltese population and (

**b**) male Maltese population.

**Figure 8.**Estimated spread parameters of the Plat–LC model for (

**a**) female Maltese population and (

**b**) male Maltese population.

**Figure 13.**Mortality rate projections at age 65, 75, 85 with RH–LC model: (

**a**) female Maltese population, (

**b**) male Maltese population.

**Figure 14.**Mortality rate projections at age 65, 75, 85 with Plat–LC model: (

**a**) female Maltese population, (

**b**) male Maltese population.

**Figure 15.**Mortality rate projections at age 65, 75, 85 with parameters uncertainly with RH–LC model for (

**a**) female Maltese population, (

**b**) male Maltese population.

**Figure 16.**Mortality rate projections at age 65, 75, 85 with parameters uncertainly with Plat–LC model for (

**a**) female Maltese population, (

**b**) male Maltese population.

**Figure 17.**Mortality rate projections at age 65, 75, 85 with parameters uncertainly with Lee–Carter model for (

**a**) female Maltese population, (

**b**) male Maltese population.

**Figure 18.**Mortality rate projections at age 65, 75, 85 with parameters uncertainly with the APC model for (

**a**) female Maltese population, (

**b**) male Maltese population.

Model | Formula |
---|---|

LC | $log\left({m}_{x,t,r}\right)={\alpha}_{x}^{\left(r\right)}+{\beta}_{x}^{(1,r)}{\kappa}_{t}^{(1,r)}$ |

APC | $log\left({m}_{x,t,r}\right)={\alpha}_{x}^{\left(r\right)}+{\kappa}_{t}^{(1,r)}+{\gamma}_{t-x}^{\left(r\right)}$ |

RH | $log\left({m}_{x,t,r}\right)={\alpha}_{x}^{\left(r\right)}+{\beta}_{x}^{(1,r)}{\kappa}_{t}^{(1,r)}+{\beta}_{x}^{(0,r)}{\gamma}_{t-x}^{\left(r\right)}$ |

Plat | $log\left({m}_{x,t,r}\right)={\alpha}_{x}^{\left(r\right)}+{\kappa}_{t}^{(1,r)}+(\overline{x}-x){\kappa}_{t}^{(2,r)}+{\gamma}_{t-x}^{\left(r\right)}$ |

Model | Constraints |
---|---|

LC | ${\sum}_{x}{\beta}_{x}^{(1,r)}=1$, ${\sum}_{t}{\kappa}_{t}^{(1,r)}=0$ |

APC | ${\sum}_{t}{\kappa}_{t}^{(1,r)}=0$, ${\sum}_{t-x}{\gamma}_{t-x}^{\left(r\right)}=0$, ${\sum}_{t-x}(t-x){\gamma}_{t-x}^{\left(r\right)}=0$ |

RH | ${\sum}_{x}{\beta}_{x}^{(1,r)}=1$, ${\sum}_{t}{\kappa}_{t}^{(1,r)}=0$, ${\sum}_{t-x}{\gamma}_{t-x}^{\left(r\right)}=0$, ${\sum}_{t-x}(t-x-(\overline{t}-\overline{x})){\gamma}_{t-x}^{\left(r\right)}=0$ |

Plat | ${\sum}_{t}{\kappa}_{t}^{(1,r)}=0$, ${\sum}_{t}{\kappa}_{t}^{(2,r)}=0$, ${\sum}_{t-x}{\gamma}_{t-x}^{\left(r\right)}=0$, ${\sum}_{t-x}(t-x){\gamma}_{t-x}^{\left(r\right)}=0$, ${\sum}_{t-x}{(t-x)}^{2}{\gamma}_{t-x}^{\left(r\right)}=0$ |

LogLik | AIC | BIC | |
---|---|---|---|

LC | −3526.20(4) | 7204.40(4) | 7527.69(4) |

APC | −3062.91(3) | 6293.81(3) | 6651.13(3) |

RH | −2947.52(1) | 6121.03(2) | 6601.71(2) |

Plat | −2955.81(2) | 6111.62(1) | 6537.00(1) |

LogLik | AIC | BIC | |
---|---|---|---|

LC | −3505.45(4) | 7162.90(4) | 7486.20(4) |

APC | −3132.11(3) | 6432.22(3) | 6789.54(3) |

RH | −3017.30 (1) | 6260.61(2) | 6741.29(2) |

Plat | −3029.96 (2) | 6259.93(1) | 6685.31(1) |

Model | Formula |
---|---|

LC | $log\left({m}_{x,t,s}\right)-log\left({\widehat{m}}_{x,t,r}\right)={\alpha}_{x}^{\left(s\right)}+{\beta}_{x,i}^{(1,s)}{\kappa}_{t}^{(1,s)}$ |

APC | $log\left({m}_{x,t,s}\right)-log\left({\widehat{m}}_{x,t,r}\right)={\alpha}_{x}^{\left(s\right)}+{\kappa}_{t}^{(1,s)}+{\gamma}_{t-x}^{\left(s\right)}$ |

RH | $log\left({m}_{x,t,s}\right)-log\left({\widehat{m}}_{x,t,r}\right)={\alpha}_{x}^{\left(s\right)}+{\beta}_{x}^{(1,s)}{\kappa}_{t}^{(1,s)}+{\gamma}_{t-x}^{\left(s\right)}$ |

Plat | $log\left({m}_{x,t,s}\right)-log\left({\widehat{m}}_{x,t,r}\right)={\alpha}_{x}^{\left(s\right)}+{\kappa}_{t}^{(1,s)}+(\overline{x}-x){\kappa}_{t}^{(2,s)}+{\gamma}_{t-x}^{\left(s\right)}$ |

LogLik | ||||

Spread model | ||||

Trend model | LC | APC | RH | Plat |

LC | −1591.82 | −1592.24 | −1576.74 | −1586.54 |

APC | −1592.67 | −1592.42 | −1577.22 | −1585.77 |

RH | −1592.87 | −1593.39 | −1577.69 | −1586.09 |

Plat | −1592.17 | −1592.76 | −1577.18 | −1585.77 |

AIC | ||||

Spread model | ||||

Trend model | LC | APC | RH | Plat |

LC | 3335.64 | 3352.48 | 3379.47 | 3373.07 |

APC | 3337.34 | 3352.85 | 3380.44 | 3371.54 |

RH | 3337.74 | 3354.78 | 3381.38 | 3372.17 |

Plat | 3336.34 | 3353.51 | 3380.36 | 3371.54 |

BIC | ||||

Spread model | ||||

Trend model | LC | APC | RH | Plat |

LC | 3658.93 | 3709.80 | 3860.16 | 3798.45 |

APC | 3660.63 | 3710.17 | 3861.12 | 3796.92 |

RH | 3661.03 | 3712.10 | 3862.06 | 3797.56 |

Plat | 3659.64 | 3710.83 | 3861.05 | 3796.92 |

LogLik | ||||

Spread model | ||||

Trend model | LC | APC | RH | Plat |

LC | −1661.38 | −1653.98 | −1634.86 | −1639.59 |

APC | −1664.95 | −1652.46 | −1634.75 | −1638.62 |

RH | −1663.67 | −1652.29 | −1633.15 | −1638.54 |

Plat | −1663.33 | −1651.21 | −1632.79 | −1638.62 |

AIC | ||||

Spread model | ||||

Trend model | LC | APC | RH | Plat |

LC | 3474.77 | 3475.97 | 3495.72 | 3479.17 |

APC | 3481.89 | 3472.92 | 3495.50 | 3477.24 |

RH | 3479.34 | 3472.58 | 3492.31 | 3477.09 |

Plat | 3478.67 | 3470.43 | 3491.59 | 3477.24 |

BIC | ||||

Spread model | ||||

Trend model | LC | APC | RH | Plat |

LC | 3798.06 | 3833.29 | 3976.40 | 3904.56 |

APC | 3805.18 | 3830.24 | 3976.19 | 3902.62 |

RH | 3802.63 | 3829.91 | 3972.99 | 3902.47 |

Plat | 3801.96 | 3827.75 | 3972.27 | 3902.62 |

Females | Males | |
---|---|---|

RH-LC | 13.55% | 12.82% |

Plat–LC | 13.50% | 12.81% |

**Table 9.**BIC for different ARIMA processes for period effect in the RH model for the reference population.

Females | Males | ||||
---|---|---|---|---|---|

Process | d Order | Best ARIMA | BIC | Best ARIMA | BIC |

${\kappa}_{t}^{(1,r)}$ | d = 2 | (0,2,1) | 44.79 | (0,2,1) | 35.75 |

d = 1 | (0,1,0) | 44.74 | (0,1,0) | 38.39 | |

d = 0 | (3,0,0) | 60.45 | (3,0,0) | 51.96 |

**Table 10.**BIC for different ARIMA processes for period effect in the Plat model for the reference population.

Females | Males | ||||
---|---|---|---|---|---|

Process | d Order | Best ARIMA | BIC | Best ARIMA | BIC |

${\kappa}_{t}^{(1,r)}$ | d = 2 | (0,2,1) | −67.97 | (0,2,1) | −75.56 |

d = 1 | (0,1,0) | −76.22 | (0,1,0) | −82.26 | |

d = 0 | (3,0,0) | −65.16 | (3,0,0) | −72.50 | |

${\kappa}_{t}^{(2,r)}$ | d = 2 | (1,2,1) | −160.98 | (0,2,2) | −172.71 |

d = 1 | (1,1,0) | −177.08 | (0,1,0) | −189.37 | |

d = 0 | (1,0,0) | −186.76 | (1,0,0) | −197.21 |

**Table 11.**Relative amplitude of prediction intervals for female Maltese death rates in 2046 with different mortality models.

Age | LC | APC | RH–LC | Plat–LC |
---|---|---|---|---|

65 | 225.77% | 166.87% | 74.66% | 51.87% |

75 | 232.54% | 167.18% | 90.75% | 82.10% |

85 | 169.40% | 163.49% | 100.23% | 90.55% |

**Table 12.**Relative amplitude of prediction intervals for male Maltese death rates in 2046 with different mortality models.

Age | LC | APC | RH–LC | Plat–LC |
---|---|---|---|---|

65 | 291.50% | 209.40% | 76.47% | 63.34% |

75 | 319.79% | 205.04% | 80.76% | 68.43% |

85 | 192.27% | 206.97% | 71.84% | 65.93% |

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**MDPI and ACS Style**

Menzietti, M.; Morabito, M.F.; Stranges, M.
Mortality Projections for Small Populations: An Application to the Maltese Elderly. *Risks* **2019**, *7*, 35.
https://doi.org/10.3390/risks7020035

**AMA Style**

Menzietti M, Morabito MF, Stranges M.
Mortality Projections for Small Populations: An Application to the Maltese Elderly. *Risks*. 2019; 7(2):35.
https://doi.org/10.3390/risks7020035

**Chicago/Turabian Style**

Menzietti, Massimiliano, Maria Francesca Morabito, and Manuela Stranges.
2019. "Mortality Projections for Small Populations: An Application to the Maltese Elderly" *Risks* 7, no. 2: 35.
https://doi.org/10.3390/risks7020035