# A Study on Link Functions for Modelling and Forecasting Old-Age Survival Probabilities of Australia and New Zealand

## Abstract

**:**

## 1. Introduction

## 2. Survival Rate Projection Models

## 3. Fitting Performances

- (1)
- For the first regression structure, the MAPE values are often smaller when the response is ${}_{n}p{}_{{x}_{0},t}$. By contrast, for the second regression structure, the MAPEs are much smaller when the response is ${({}_{n}p{}_{{x}_{0},t})}^{{\scriptscriptstyle \frac{1}{n}}}$.
- (2)
- When the response is ${}_{n}p{}_{{x}_{0},t}$, the MAPE values from the first regression structure are clearly smaller. However, when the response is ${({}_{n}p{}_{{x}_{0},t})}^{{\scriptscriptstyle \frac{1}{n}}}$, the situation is mostly reversed.
- (3)
- For the complementary log-log model structure with the first regression form, the MAPE remains the same regardless of the response (i.e., ${}_{n}p{}_{{x}_{0},t}$ or ${({}_{n}p{}_{{x}_{0},t})}^{{\scriptscriptstyle \frac{1}{n}}}$). The underlying reason is that $\mathrm{ln}(-\mathrm{ln}({({}_{n}p{}_{{x}_{0},t})}^{{\scriptscriptstyle \frac{1}{n}}}))={a}_{x}+{b}_{x}\text{}{k}_{t}$ is indeed equivalent to $\mathrm{ln}(-\mathrm{ln}({}_{n}p{}_{{x}_{0},t}))$$={a}_{x}+{b}_{x}\text{}{k}_{t}+\mathrm{ln}n$$={a}_{x}^{\ast}+{b}_{x}\text{}{k}_{t}$, where ${a}_{x}^{\ast}={a}_{x}+\mathrm{ln}n$. Hence, they produce the same fitted values of ${}_{n}p{}_{60,t}$ and so the same MAPEs.
- (4)
- Overall, the combination of the response ${({}_{n}p{}_{{x}_{0},t})}^{{\scriptscriptstyle \frac{1}{n}}}$ and the second regression structure (i.e., ${k}_{t,1}+{k}_{t,2}(x-\overline{x})$$+{k}_{t,3}({(x-\overline{x})}^{2}-{\sigma}^{2})$) gives the better MAPEs. In particular, the gevmin model structure, based on the newly proposed gevmin link, leads to the smallest MAPEs consistently for all the populations (0.73, 0.93, 1.20, 1.68) considered.
- (5)
- The best gevmin model structures noted above outperform the more traditional approaches of the Lee–Carter and CBD models in modelling the mortality rates (with MAPEs of 1.39 to 3.02).

## 4. Forecasting Performances

- (1)
- Out of the 16 cases (4 fitting periods × 2 countries × 2 sexes), the gevit and gevmin model structures produce the three lowest MAPEs in 12 cases. Their performances are the most consistent ones amongst all the candidates.
- (2)
- For the gevit and gevmin model structures, the average MAPE is 6.10. For the probit, complementary log-log, and logit model structures, the average MAPE is 6.29. For the LC and CBD models, the average MAPE is 6.53. For the naive random walk model, the average MAPE is 7.95.
- (3)
- The MAPE values tend to be lower for females (4.04 on average) than for males (8.98 on average).
- (4)
- The MAPE values tend to be lower for Australia (5.65 on average) than for New Zealand (7.36 on average).
- (5)
- The naive random walk model leads to the highest MAPE values in 10 cases.

## 5. The Effect of Economic Growth

## 6. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Estimated shape parameter values of gevit model structure fitted to Australian and New Zealand data (left value—first regression structure; right value—second regression structure).

Fitting Period | Australian Females | Australian Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

1970–2017 | −0.59/−0.32 | −0.85/−0.31 | −0.60/−0.39 | −0.93/−0.30 |

1970–1999 | −0.66/−0.35 | −0.99/−0.42 | −0.62/−0.45 | −1.08/−0.40 |

1970–1989 | −0.57/−0.34 | −0.89/−0.37 | −0.62/−0.43 | −1.06/−0.37 |

Fitting Period | New Zealand Females | New Zealand Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

1970–2013 | −0.58/−0.34 | −0.90/−0.31 | −0.53/−0.41 | −0.86/−0.35 |

1970–1999 | −0.54/−0.37 | −0.83/−0.37 | −0.48/−0.45 | −0.86/−0.45 |

1970–1989 | −0.55/−0.36 | −0.86/−0.36 | −0.53/−0.43 | −0.91/−0.38 |

**Table A2.**sMAPE values (%) on projected survival probabilities at age 60 using Australian and New Zealand data for different fitting periods.

Australian Females | Australian Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 5.79 | 1.87 | 2.89 | 2.90 | 17.67 | 11.25 | 2.76 | 1.67 |

log-log | 6.91 | 2.13 | 2.60 | 1.98 | 18.82 | 12.39 | 3.64 | 1.63 |

logit | 6.71 | 2.03 | 2.58 | 2.70 | 18.59 | 12.16 | 3.42 | 2.06 |

gevit | 4.75 | 2.52 | 3.91 | 3.68 | 16.27 | 10.01 | 2.41 | 2.04 |

gevmin | 4.75 | 2.35 | 3.67 | 3.49 | 16.73 | 10.40 | 2.41 | 1.74 |

LC | 3.92 | 3.08 | 3.00 | 2.29 | 11.72 | 8.07 | 4.13 | 3.33 |

CBD | 4.90 | 1.89 | 3.10 | 3.32 | 18.49 | 12.58 | 3.82 | 2.34 |

MRW | 9.88 | 4.72 | 2.52 | 2.03 | 23.17 | 17.20 | 8.55 | 5.87 |

New Zealand Females | New Zealand Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 5.06 | 5.21 | 3.41 | 2.90 | 15.20 | 9.25 | 10.79 | 9.36 |

log-log | 5.56 | 4.76 | 3.68 | 3.49 | 15.75 | 9.79 | 11.41 | 9.90 |

logit | 5.44 | 4.84 | 3.60 | 3.35 | 15.63 | 9.68 | 11.27 | 9.78 |

gevit | 4.97 | 5.65 | 3.55 | 2.95 | 14.36 | 8.57 | 10.16 | 8.97 |

gevmin | 4.98 | 5.67 | 3.55 | 2.93 | 14.64 | 8.83 | 10.35 | 9.04 |

LC | 6.37 | 6.21 | 4.92 | 3.35 | 18.57 | 10.26 | 9.43 | 6.82 |

CBD | 5.64 | 7.00 | 3.78 | 2.91 | 16.41 | 9.92 | 11.40 | 8.73 |

MRW | 7.19 | 3.15 | 3.80 | 3.57 | 18.79 | 12.51 | 13.26 | 11.39 |

**Table A3.**sMAPE values (%) on projected life expectancies at age 60 using Australian and New Zealand data for different fitting periods.

Australian Females | Australian Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 1.69 | 0.69 | 1.11 | 1.19 | 5.51 | 3.75 | 0.84 | 0.55 |

log-log | 2.02 | 0.71 | 0.87 | 1.02 | 6.10 | 4.33 | 1.35 | 0.39 |

logit | 1.99 | 0.71 | 0.89 | 1.04 | 6.03 | 4.27 | 1.29 | 0.40 |

gevit | 1.25 | 0.95 | 1.59 | 1.50 | 4.44 | 2.74 | 0.66 | 1.02 |

gevmin | 1.35 | 0.81 | 1.41 | 1.37 | 4.98 | 3.28 | 0.66 | 0.77 |

LC | 2.07 | 0.74 | 0.79 | 0.81 | 6.33 | 4.22 | 1.56 | 0.70 |

CBD | 2.25 | 0.79 | 0.78 | 0.99 | 6.63 | 4.63 | 1.70 | 0.52 |

MRW | 1.34 | 0.79 | 1.41 | 1.47 | 5.20 | 3.47 | 0.74 | 0.68 |

New Zealand Females | New Zealand Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 2.88 | 0.75 | 1.80 | 1.18 | 7.49 | 4.41 | 4.19 | 2.22 |

log-log | 2.98 | 0.81 | 1.85 | 1.15 | 7.79 | 4.71 | 4.42 | 2.36 |

logit | 2.97 | 0.80 | 1.85 | 1.15 | 7.75 | 4.67 | 4.40 | 2.35 |

gevit | 2.67 | 0.78 | 1.65 | 1.16 | 6.89 | 3.88 | 3.78 | 1.93 |

gevmin | 2.74 | 0.75 | 1.72 | 1.19 | 7.16 | 4.14 | 3.98 | 2.10 |

LC | 3.57 | 0.92 | 2.41 | 1.13 | 9.87 | 5.46 | 4.64 | 2.27 |

CBD | 3.18 | 0.80 | 1.80 | 1.17 | 7.70 | 4.93 | 4.47 | 2.57 |

MRW | 3.19 | 0.80 | 1.76 | 1.15 | 7.51 | 4.37 | 4.06 | 2.03 |

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1 | In future research, if mortality and social data are available at finer levels such as social deprivation groups and subpopulations, other indicators like income, education, and smoking status can be added to the analysis. |

**Figure 4.**Trends of ${k}_{t}$ and ${k}_{t,1}$ to ${k}_{t,3}$ based on five link functions for non-annualised survival probabilities of Australian females.

**Figure 5.**Trends of life expectancy at age 60, national GDP, and g

_{t}of Australia and New Zealand.

**Figure 6.**Trends of ${k}_{t}$ and age-sensitivity to GDP ${c}_{x}$ based on gevit and gevmin link functions for annualised and non-annualised survival probabilities of Australian females and males.

**Table 1.**MAPE values (%) on fitted survival probabilities at age 60 from fitting 22 models to Australian and New Zealand data (1970–2017 and 1970–2013 respectively).

Model | Australian Females | Australian Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

$\mathrm{probit}-{k}_{t}$ | 1.24 | 1.47 | 2.03 | 2.53 |

$\mathrm{log}\text{-}\mathrm{log}-{k}_{t}$ | 1.59 | 1.59 | 2.75 | 2.75 |

$\mathrm{logit}-{k}_{t}$ | 1.39 | 1.59 | 1.94 | 2.72 |

$\mathrm{gevit}-{k}_{t}$ | 1.11 | 1.09 | 1.91 | 2.00 |

$\mathrm{gevmin}-{k}_{t}$ | 1.12 | 1.14 | 1.82 | 1.87 |

$\mathrm{probit}-{k}_{t,1}$$-{k}_{t,3}$ | 4.38 | 1.14 | 5.76 | 1.12 |

$\mathrm{log}\text{-}\mathrm{log}-{k}_{t,1}$$-{k}_{t,3}$ | 7.20 | 2.17 | 11.74 | 1.80 |

$\mathrm{logit}-{k}_{t,1}$$-{k}_{t,3}$ | 8.49 | 1.96 | 11.76 | 1.60 |

$\mathrm{gevit}-{k}_{t,1}$$-{k}_{t,3}$ | 2.71 | 0.85 | 3.48 | 1.01 |

$\mathrm{gevmin}-{k}_{t,1}$$-{k}_{t,3}$ | 2.95 | 0.73 | 3.71 | 0.93 |

Lee-Carter | 1.39 | 2.42 | ||

CBD | 1.62 | 1.57 | ||

Model | New Zealand Females | New Zealand Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

$\mathrm{probit}-{k}_{t}$ | 2.23 | 2.61 | 2.76 | 3.11 |

$\mathrm{log}\text{-}\mathrm{log}-{k}_{t}$ | 2.82 | 2.82 | 3.27 | 3.27 |

$\mathrm{logit}-{k}_{t}$ | 2.36 | 2.81 | 2.77 | 3.24 |

$\mathrm{gevit}-{k}_{t}$ | 2.06 | 2.02 | 2.70 | 2.73 |

$\mathrm{gevmin}-{k}_{t}$ | 2.09 | 2.11 | 2.66 | 2.70 |

$\mathrm{probit}-{k}_{t,1}$$-{k}_{t,3}$ | 4.80 | 1.45 | 5.95 | 1.86 |

$\mathrm{log}\text{-}\mathrm{log}-{k}_{t,1}$$-{k}_{t,3}$ | 8.21 | 2.24 | 12.65 | 2.35 |

$\mathrm{logit}-{k}_{t,1}$$-{k}_{t,3}$ | 9.22 | 2.06 | 12.19 | 2.20 |

$\mathrm{gevit}-{k}_{t,1}$$-{k}_{t,3}$ | 3.09 | 1.26 | 3.53 | 1.71 |

$\mathrm{gevmin}-{k}_{t,1}$$-{k}_{t,3}$ | 3.38 | 1.20 | 3.71 | 1.68 |

Lee-Carter | 2.44 | 3.02 | ||

CBD | 2.00 | 2.15 |

**Table 2.**MAPE values (%) on projected survival probabilities at age 60 using Australian and New Zealand data for four different fitting periods.

Australian Females | Australian Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 5.25 | 1.87 | 3.03 | 3.03 | 14.65 | 9.92 | 2.70 | 1.65 |

log-log | 6.08 | 2.05 | 2.65 | 2.02 | 15.45 | 10.79 | 3.49 | 1.54 |

logit | 5.95 | 1.97 | 2.64 | 2.77 | 15.30 | 10.63 | 3.30 | 1.97 |

gevit | 4.43 | 2.61 | 4.21 | 3.94 | 13.59 | 8.91 | 2.42 | 2.08 |

gevmin | 4.42 | 2.44 | 3.96 | 3.72 | 13.98 | 9.26 | 2.40 | 1.77 |

LC | 3.96 | 3.31 | 3.24 | 2.41 | 10.65 | 7.51 | 3.97 | 3.20 |

CBD | 4.62 | 1.87 | 3.28 | 3.51 | 15.45 | 11.01 | 3.75 | 2.39 |

MRW | 8.61 | 4.44 | 2.51 | 2.05 | 18.05 | 14.03 | 7.70 | 5.50 |

New Zealand Females | New Zealand Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 4.78 | 5.70 | 3.34 | 2.77 | 13.23 | 8.48 | 9.53 | 8.15 |

log-log | 5.13 | 5.04 | 3.53 | 3.23 | 13.65 | 8.89 | 9.97 | 8.47 |

logit | 5.04 | 5.15 | 3.47 | 3.12 | 13.57 | 8.81 | 9.88 | 8.41 |

gevit | 4.79 | 6.49 | 3.54 | 2.88 | 12.57 | 7.95 | 9.03 | 7.94 |

gevmin | 4.80 | 6.47 | 3.53 | 2.85 | 12.81 | 8.17 | 9.20 | 7.97 |

LC | 6.60 | 7.28 | 5.09 | 3.56 | 15.95 | 9.77 | 8.70 | 6.26 |

CBD | 5.66 | 8.10 | 3.77 | 2.90 | 14.09 | 9.19 | 9.99 | 7.77 |

MRW | 6.61 | 3.28 | 3.60 | 3.37 | 15.60 | 10.81 | 11.25 | 9.74 |

**Table 3.**MAPE values (%) on projected life expectancies at age 60 using Australian and New Zealand data for four different fitting periods.

Australian Females | Australian Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 1.67 | 0.69 | 1.12 | 1.20 | 5.34 | 3.68 | 0.84 | 0.55 |

log-log | 2.00 | 0.71 | 0.88 | 1.03 | 5.90 | 4.23 | 1.34 | 0.39 |

logit | 1.97 | 0.71 | 0.90 | 1.05 | 5.83 | 4.17 | 1.28 | 0.40 |

gevit | 1.24 | 0.95 | 1.61 | 1.52 | 4.34 | 2.70 | 0.66 | 1.03 |

gevmin | 1.34 | 0.82 | 1.42 | 1.39 | 4.85 | 3.22 | 0.66 | 0.77 |

LC | 2.04 | 0.74 | 0.79 | 0.82 | 6.11 | 4.13 | 1.54 | 0.70 |

CBD | 2.23 | 0.78 | 0.78 | 1.00 | 6.39 | 4.52 | 1.69 | 0.51 |

MRW | 1.33 | 0.79 | 1.43 | 1.49 | 5.05 | 3.40 | 0.74 | 0.68 |

New Zealand Females | New Zealand Males | |||||||

Model | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 | 1970–1989 | 1970–1994 | 1970–1999 | 1970–2004 |

probit | 2.83 | 0.75 | 1.78 | 1.17 | 7.16 | 4.29 | 4.09 | 2.20 |

log-log | 2.93 | 0.81 | 1.83 | 1.14 | 7.43 | 4.57 | 4.31 | 2.33 |

logit | 2.92 | 0.80 | 1.83 | 1.15 | 7.39 | 4.53 | 4.29 | 2.32 |

gevit | 2.63 | 0.78 | 1.64 | 1.15 | 6.60 | 3.78 | 3.70 | 1.91 |

gevmin | 2.70 | 0.75 | 1.70 | 1.18 | 6.86 | 4.03 | 3.90 | 2.07 |

LC | 3.50 | 0.91 | 2.38 | 1.12 | 9.32 | 5.28 | 4.52 | 2.24 |

CBD | 3.12 | 0.80 | 1.78 | 1.16 | 7.34 | 4.78 | 4.36 | 2.54 |

MRW | 3.13 | 0.79 | 1.74 | 1.14 | 7.18 | 4.25 | 3.97 | 2.00 |

**Table 4.**MAPE values (%) on fitted survival probabilities at age 60 from fitting 2 models with real GDP per capita as explanatory variable to Australian and New Zealand data (1970–2017 and 1970–2013 respectively).

Model | Australian Females | Australian Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

with GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 0.86 | 0.79 | 1.69 | 1.61 |

$\mathrm{gevmin}-{k}_{t}$ | 0.87 | 0.73 | 1.66 | 1.41 |

without GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 1.11 | 1.09 | 1.91 | 2.00 |

$\mathrm{gevmin}-{k}_{t}$ | 1.12 | 1.14 | 1.82 | 1.87 |

Model | New Zealand Females | New Zealand Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

with GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 1.76 | 1.37 | 2.54 | 2.55 |

$\mathrm{gevmin}-{k}_{t}$ | 1.70 | 1.40 | 2.52 | 2.50 |

without GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 2.06 | 2.02 | 2.70 | 2.73 |

$\mathrm{gevmin}-{k}_{t}$ | 2.09 | 2.11 | 2.66 | 2.70 |

**Table 5.**MAPE values (%) on projected survival probabilities at age 60 from using 2 models with real GDP per capita as explanatory variable on Australian and New Zealand data (left value—fitting period 1970–1989; right value—fitting period 1970–1999).

Model | Australian Females | Australian Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

with GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 4.31/4.67 | 4.12/6.21 | 11.67/3.35 | 11.65/4.79 |

$\mathrm{gevmin}-{k}_{t}$ | 3.99/2.88 | 4.06/6.74 | 12.27/3.80 | 11.79/4.20 |

without GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 4.89/5.62 | 5.21/8.02 | 12.50/2.81 | 9.41/7.21 |

$\mathrm{gevmin}\text{}-\text{}{k}_{t}$ | 5.24/5.45 | 5.83/8.08 | 12.49/2.14 | 9.53/5.97 |

Model | New Zealand Females | New Zealand Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

with GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 6.79/4.18 | 5.94/4.34 | 12.93/8.67 | 12.47/7.00 |

$\mathrm{gevmin}-{k}_{t}$ | 6.53/3.71 | 6.37/4.94 | 12.60/8.67 | 12.64/6.95 |

without GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 7.38/5.32 | 7.51/5.36 | 13.19/8.60 | 12.34/7.52 |

$\mathrm{gevmin}-{k}_{t}$ | 7.25/5.23 | 6.68/5.73 | 13.00/8.51 | 12.49/7.53 |

**Table 6.**MAPE values (%) on projected life expectancies at age 60 from using 2 models with real GDP per capita as explanatory variable on Australian and New Zealand data (left value—fitting period 1970–1989; right value—fitting period 1970–1999).

Model | Australian Females | Australian Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

with GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 2.16/1.78 | 1.61/3.34 | 5.81/0.91 | 3.07/3.90 |

$\mathrm{gevmin}-{k}_{t}$ | 2.07/1.29 | 1.55/3.18 | 5.56/0.58 | 4.11/2.83 |

without GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 2.55/2.04 | 2.27/3.48 | 7.16/1.49 | 4.94/4.96 |

$\mathrm{gevmin}-{k}_{t}$ | 2.76/1.76 | 2.24/2.83 | 7.35/0.94 | 5.74/2.79 |

Model | New Zealand Females | New Zealand Males | ||

Non-Annualised | Annualised | Non-Annualised | Annualised | |

with GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 3.71/1.87 | 2.61/1.14 | 7.91/4.23 | 7.03/2.20 |

$\mathrm{gevmin}-{k}_{t}$ | 3.70/1.76 | 2.67/1.03 | 7.38/4.11 | 7.34/2.72 |

without GDP | ||||

$\mathrm{gevit}-{k}_{t}$ | 4.01/2.32 | 3.57/1.97 | 8.28/4.67 | 7.65/3.87 |

$\mathrm{gevmin}-{k}_{t}$ | 4.00/2.27 | 3.37/2.08 | 8.16/4.63 | 7.86/4.16 |

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**MDPI and ACS Style**

Liu, J.J.
A Study on Link Functions for Modelling and Forecasting Old-Age Survival Probabilities of Australia and New Zealand. *Risks* **2021**, *9*, 11.
https://doi.org/10.3390/risks9010011

**AMA Style**

Liu JJ.
A Study on Link Functions for Modelling and Forecasting Old-Age Survival Probabilities of Australia and New Zealand. *Risks*. 2021; 9(1):11.
https://doi.org/10.3390/risks9010011

**Chicago/Turabian Style**

Liu, Jacie Jia.
2021. "A Study on Link Functions for Modelling and Forecasting Old-Age Survival Probabilities of Australia and New Zealand" *Risks* 9, no. 1: 11.
https://doi.org/10.3390/risks9010011