# Extensions on the Hatzopoulos–Sagianou Multiple-Components Stochastic Mortality Model

^{*}

## Abstract

**:**

## 1. Introduction

- We extend the stochastic mortality model HS formulated in terms of ${q}_{t,x}$, using generalised linear models and by adopting various link functions. We illustrate through experimental results that the HS model remains robust and consistent under all modelling variations.
- We introduce a new set of link functions, with a particular focus on heavy-tailed distributions, and we evaluate their applicability in the context of mortality through the HS extensions. This approach leads to the definition of a new estimation methodology for the HS model. To the best of our knowledge, it is the first time that those link functions are evaluated in the mortality modelling domain.
- We compare the efficiency of the new model extensions versus the established mortality models in fitting and forecasting modes. For the latter case, we use an out-of-sample approach to assess the prediction ability of each model by using the Random Walk with Drift (RWD) model and optimum Arima models based on the Bayesian Information Criterion (BIC) test.
- We highlight the lessons learnt to inform the community about the adoption of the various link functions in the models’ estimation methods having witnessed the beneficial impact of this approach to our model’s efficacy.

## 2. Preliminaries

#### 2.1. Data and Notation

#### 2.2. GLM Framework

**Y**, whose components are independently distributed with means $\mu $, is considered to originate from certain distributions belonging to the exponential family. This is a wide range of probability distributions, such as the Binomial, Poisson, Gamma and Normal distributions. The explanatory variables,

**X**, define the mean, $\mu $, of the distribution via:

**Y**, where

**Y**is a vector of response variables. $\mathbf{X}\mathit{\beta}$ is the linear predictor, a linear combination of parameters $\mathit{\beta}$ whose values are usually unknown and have to be estimated from the data. $\mathbf{X}$ is a matrix of explanatory variables, and $\mathit{\beta}$ is a vector of parameters. g is the link function.

**X**represents coefficients of the linear combination.

- The random component—refers to the probability distribution of the response variable (
**Y**). The components of**Y**have generated from an exponential family of probability distributions. - The systematic component—specifies the explanatory variables (
**X**) in the model producing the so-called linear predictor $\eta =\mathbf{X}\mathit{\beta}$. - The link function, g—specifies the connection between the random and systematic components. Specifically, it denotes how the expected value of the response relates to the linear predictor of explanatory variables, e.g., $\eta =g(\mu )$.

- The random component: the numbers of deaths ${D}_{t,x}$ follow the Poisson distribution with mean ${E}_{t,x}^{c}\times {m}_{t,x}$, or the Binomial distribution with mean ${E}_{t,x}^{0}\times {q}_{t,x}$, so that$${D}_{t,x}\sim Poisson({E}_{t,x}^{c}\times {m}_{t,x})$$$${D}_{t,x}\sim Binomial({E}_{t,x}^{0},{q}_{t,x})$$
- The systematic component: the effects of age x, calendar year t and cohort $c=t-x$ are captured through a predictor ${\eta}_{t,x}$ given by:$${\eta}_{t,x}={\alpha}_{x}+\sum _{i=1}^{p}{\beta}_{x}^{(i)}{\kappa}_{t}^{(i)}+\sum _{j=1}^{q}{\beta}_{x}^{\mathrm{c}(j)}{\gamma}_{c}^{(j)}$$
- The link function g associating the random component and the systematic component, so that$$g\left(\mathbb{E}\left(\frac{{D}_{t,x}}{{E}_{t,x}}\right)\right)={\eta}_{t,x}$$There are several link functions that can be used as suggested by (Currie 2016; Haberman and Renshaw 1996) for the context of mortality models, and (McCullagh and Nelder 1989) in the wider context of GLMs. As will be explained in detail in Section 3, in this paper, we extend the HS model and we formulate it in terms of ${q}_{t,x}$, using a wide set of canonical and non-canonical link functions.
- The set of parameter constraints: most of the stochastic mortality models have identifiability problems in the parameter estimation. In an effort to avoid this issue, a set of parameter constraints is required to ensure unique parameter estimates. Notably, in our case, the HS model does not need any parameter constraints as the model does not face identifiability problems during parameter estimation and always provides a unique solution due to its estimation methodology (Hatzopoulos and Sagianou 2020).

#### 2.3. Hatzopoulos–Sagianou (HS) Model

**Step #1**: GLM is applied, with the number of deaths being the response variable, ${D}_{t,x}$, for each calendar year t independently, by using the log link function, i.e., the canonical link function for the Poisson distribution, and we treat the logarithmic exposure variables (i.e., log(${E}_{t,x}^{c}$)) as an offset. The predictor structure is:

**Step #2:**This step aims to keep in the model only the factors explaining most of the data’s information in order to decrease the dimensionality of the problem. To do so, SPCA is applied to the covariance matrix of the GLM-estimated parameters B. Given the covariance matrix, $A=\mathrm{Cov}(B)$, as stated in (Luss and d’Aspremont 2006), the (dual) problem of defining a sparse factor, which will allow to capture the maximum amount of data’s variance, can be formulated as follows:

**Step #3:**(Hatzopoulos and Sagianou 2020) introduced a heuristic methodology based on UVR metric to define the optimal model structure, i.e., to incorporate the most important (i.e., informative) age–period and age–cohort components. It must be stated that the optimal model structure is achieved through the process of defining the optimal s value. In other words, the definition of the optimal model structure coincides with the definition of the optimal scalar s, which in turn controls the sparsity in SPCA. This process is driven by the UVR metric in order to converge to components that maximise the captured variance of the mortality data and to acquire distinct and significant stochastic components, which enable the attribution of an identified mortality trend to a unique age cluster. The aim is to decrease the number of age–period components from ${k}_{1}$ to p. The excluded components are then treated as residuals.

**Step #4:**The complete model structure is synthesised. Following the UVR-based method, after having estimated the significant (informative) age–period and age–cohort effects, we conclude to the final estimates of the log-graduated central mortality rates in age, period and cohort effects:

## 3. Methodology

#### 3.1. HS Model Using Off-the-Shelf Link Functions

- i
**logit:**$\eta =log\left(\frac{q}{1-q}\right)$- ii
**probit:**$\eta ={\mathsf{\Phi}}^{-1}(q)$ where $\mathsf{\Phi}(\xb7)$ is the Normal cumulative distribution function.- iii
**complementary log–log:**$\eta =log(-log(1-q)$)

**X**, is a matrix of orthonormal polynomials, so that:

#### 3.2. Introducing a New Form of Link Functions in the Mortality Modelling

- The cumulative distribution as link function, maps q, $0<q<1$, to $-\infty <{F}^{-1}(q;\xi ,\theta )<\infty $, so that:$$\phantom{\rule{-5.69046pt}{0ex}}{\eta}_{t}={F}^{-1}\left(\mathbb{E}({Q}_{t});\xi ,\theta \right)={F}^{-1}\left(\mathbb{E}\left(\frac{{D}_{t}}{{E}_{t}^{0}}\right);\xi ,\theta \right)={F}^{-1}\left({q}_{t};\xi ,\theta \right)={b}_{t}\xb7{L}^{T}$$$$\mathbb{E}\left(\frac{{D}_{t}}{{E}_{t}^{0}}\right)={q}_{t}=F({\eta}_{t};\xi ,\theta )\iff {q}_{t}=F({b}_{t}\xb7{L}^{T};\xi ,\theta )$$For instance, some distributions that can be used are: Normal, Logistic, Extreme value, Gumbel, etc.
- The cumulative distribution as link function, maps q, $0<q<1$, to $0<{F}^{-1}(q;\xi ,\theta )<\infty $, so the logarithmic form of the cumulative distribution is needed to map q, to $-\infty <log({F}^{-1}(q;\xi ,\theta ))<\infty $, the natural scale for regression, so that:$${\eta}_{t}=log({F}^{-1}(\mathbb{E}({Q}_{t});\xi ,\theta ))=log\left({F}^{-1}({q}_{t};\xi ,\theta )\right)={b}_{t}\xb7{L}^{T}$$$$\mathbb{E}\left(\frac{{D}_{t}}{{E}_{t}^{0}}\right)={q}_{t}=F(\mathrm{exp}({\eta}_{t});\xi ,\theta )\iff {q}_{t}=F(\mathrm{exp}({b}_{t}\xb7{L}^{T});\xi ,\theta )$$For instance, some distributions that can be used are: Generalised Pareto, Weibull, Fréchet, etc.
- The cumulative distribution as link function, maps q, $0<q<1$, to $0<{F}^{-1}(q;\xi ,\theta )<1$, so we need the logit of the cumulative distribution so that maps q, to $-\infty <\mathrm{logit}({F}^{-1}(q;\xi ,\theta ))<\infty $, the natural scale for regression, so that:$${\eta}_{t}=\mathrm{logit}({F}^{-1}(\mathbb{E}({Q}_{t});\xi ,\theta ))=\mathrm{logit}\left({F}^{-1}({q}_{t};\xi ,\theta )\right)=log\left(\frac{{F}^{-1}({q}_{t};\xi ,\theta )}{1-{F}^{-1}({q}_{t};\xi ,\theta )}\right)={b}_{t}\xb7{L}^{T}$$$$\phantom{\rule{-22.76228pt}{0ex}}\mathbb{E}\left(\frac{{D}_{t}}{{E}_{t}^{0}}\right)={q}_{t}=F\left(\frac{\mathrm{exp}({\eta}_{t})}{1+\mathrm{exp}({\eta}_{t})};\xi ,\theta \right)\iff {q}_{t}=F\left(\frac{\mathrm{exp}({b}_{t}\xb7{L}^{T})}{1+\mathrm{exp}({b}_{t}\xb7{L}^{T})};\xi ,\theta \right)$$For instance, Beta distribution can be used in this case.

#### 3.2.1. HS Revised Estimation Methodology

## 4. Application

#### 4.1. Evaluation Metrics

**Information Criteria:**A common practice in the mortality domain is the use of information criteria in order to evaluate the goodness-of-fit of mortality models that have a different number of parameters, while penalising models with more parameters. That is, we use AIC and BIC, which are defined as

**Percentage Error Tests:**We use percentage error tests to measure the difference between the estimator and what is estimated to measure the fitting quality of an estimator. Specifically, we apply the MSPE which measures the average of the squares of the errors. The MSPE is the second moment of the error and considers the variance of the estimator and its bias. An additional measure is the MAPE, which measures how close the estimations are to the actual values and quantifies the magnitude of the error and, thus, expresses the accuracy. For all ages from ${x}_{1}$ to ${x}_{a}$ and for all years ${t}_{1}$ to ${t}_{n}$, these measures are defined as follows:

**Unexplained Variance Ratio (UVR):**The UVR metric is used in order to capture and visualise the significant amount of variance captured by each of the model’s components and to illustrate the age clusters where this variance corresponds to. Therefore, we use this method with all the HS model extensions that take part in the evaluation to uncover the age clusters and their unexplained variance for each model’s component. In fact, this metric will help us understand whether the HS extensions are able to capture more information and reveal more fine-grained age–period and age–cohort components and ease the attribution of a mortality trend to a specific age cluster. In this respect, the UVR gives a qualitative overview of each model structure. For more details on the definition of the UVR metric, we prompt the interested reader to refer to (Hatzopoulos and Sagianou 2020).

#### 4.2. Evaluation Results

#### 4.2.1. Mortality Data and the Optimum Parameters

- Greece (GR), males, calendar years 1961–2013, individual ages 0–84.
- England and Wales (E&W), males, calendar years 1841–2016, individual ages 0–89.

#### 4.2.2. E&W Data Performance Analysis

#### E&W Quantitative Analysis

#### E&W Qualitative Analysis

#### 4.2.3. Greek Data Performance Analysis

#### GR Quantitative Analysis

#### GR Qualitative Analysis

#### 4.2.4. Out-of-Sample Results

## 5. Discussion and Conclusions

**${F}^{-1}(x;\xi ,\theta )$**for the mortality modelling domain, along with the necessary transformations to satisfy the condition that the CDF’s range should be mapped to the whole real line. We argue that the ability to use various link functions can lead to the better performance of the mortality models, as showcased through the use of the HS model in the context of this work.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIC | Akaike Information Criterion |

APC | Age–Period–Cohort |

BIC | Bayesian Information Criterion |

CBD | Cairns–Blake–Dowd |

CDF | Cumulative Distribution Function |

cll | Complementary Log–Log |

E&W | England and Wales |

GLM(s) | Generalised Linear Model(s) |

HS | Hatzopoulos–Sagianou |

LC | Lee–Carter |

lgt | logit |

MAPE | Mean Absolute Percentage Error |

MSPE | Mean Squared Percentage Error |

npar | Number of parameters |

PE | Percentage Error |

PL | Plat |

prbt | Probit |

prt | Generalised Pareto |

RH | Renshaw–Haberman |

RWD | Random Walk with Drift |

SPCA | Sparse Principal Component Analysis |

SPCs | Sparse Principal Components |

UVR | Unexplained Variance Ratio |

## Appendix A. Definition of User-Defined Link Functions in MATLAB

- The cumulative distribution as link function g, maps q, $0<q<1$, to $-\infty <{F}^{-1}(q;\xi ,\theta )<\infty $, so that:$$\eta ={F}^{-1}(q;\xi ,\theta )$$$$q=F\left(\eta ;\xi ,\theta \right)$$$$\frac{d\eta}{dq}={\left({F}^{-1}(q;\xi ,\theta )\right)}^{\prime}=\frac{1}{f({F}^{-1}(q;\xi ,\theta );\xi ,\theta )}$$
- The cumulative distribution as link function, g, maps q, $0<q<1$, to $0<{F}^{-1}(q;\xi ,\theta )<\infty $, so the logarithmic form of the cumulative distribution is needed to map q, to $-\infty <log({F}^{-1}(q;\xi ,\theta ))<\infty $, the natural scale for regression, so that:$$\eta =log\left({F}^{-1}(q;\xi ,\theta )\right)$$$$q=F\left(\mathrm{exp}(\eta );\xi ,\theta \right)$$$$\frac{d\eta}{dq}=\frac{1}{{F}^{-1}(q;\xi ,\theta )}\xb7{\left({F}^{-1}(q;\xi ,\theta )\right)}^{\prime}=\frac{1}{{F}^{-1}(q;\xi ,\theta )\xb7f({F}^{-1}(q;\xi ,\theta );\xi ,\theta )}$$
- The cumulative distribution as link function, g, maps q, $0<q<1$, to $0<{F}^{-1}(q;\xi ,\theta )<1$, so we need the logit of the cumulative distribution so that maps q, to $-\infty <\mathrm{logit}({F}^{-1}(q;\xi ,\theta ))<\infty $, the natural scale for regression, so that:$$\eta =\mathrm{logit}\left({F}^{-1}(q;\xi ,\theta )\right)=log\left(\frac{{F}^{-1}(q;\xi ,\theta )}{1-{F}^{-1}(q;\xi ,\theta )}\right)$$$$q=F\left(\frac{\mathrm{exp}(\eta )}{1+\mathrm{exp}(\eta )};\xi ,\theta \right)$$$$\frac{d\eta}{dq}=\frac{{\left({F}^{-1}(q;\xi ,\theta )\right)}^{\prime}}{{F}^{-1}(q;\xi ,\theta )\xb7(1-{F}^{-1}(q;\xi ,\theta ))}=\frac{1}{f\left({F}^{-1}(q;\xi ,\theta )\right)\xb7{F}^{-1}(q;\xi ,\theta )\xb7(1-{F}^{-1}(q;\xi ,\theta ))}$$

link = @(mu)where B contains the GLM-estimated parameters, etx0 contains the initial exposures, Lx is the matrix of the orthonormal polynomials and qtx is the probability of deaths.log(gpinv(mu, xi, theta)); derlink = @(mu) 1./(gpinv(mu,xi,theta).*gppdf(gpinv(mu,xi,theta),xi,theta)); invlink = @(eta) gpcdf(exp(eta),xi,theta); new_F = {link, derlink, invlink}; B = glmfit(Lx,qtx,’binomial’,’link’,new_F,’weights’,etx0,’constant’,’off’)

## Appendix B. Age–Period, Age–Cohort Components and Unexplained Variance Ratio Graphical Representations for E&W Dataset

## Appendix C. Age–Period, Age–Cohort Components and Unexplained Variance Ratio Graphical Representations for GR Dataset

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Dist | Link Name | Link Function $\mathit{\eta}=\mathit{g}(\mathit{\mu})=\mathit{X}\mathit{\beta}$ | Mean Function $\mathit{\mu}={\mathit{g}}^{-1}(\mathit{X}\mathit{\beta})$ |
---|---|---|---|

Binomial | Logit | $log\left(\frac{\mu}{1-\mu}\right)=\mathit{X}\mathit{\beta}$ | $\mu =\frac{\mathrm{exp}(\mathit{X}\mathit{\beta})}{1\text{}+\text{}\mathrm{exp}(\mathit{X}\mathit{\beta})}$ |

Cloglog | $log(-log(1-\mu )=\mathit{X}\mathit{\beta}$ | $\mu =1-\mathrm{exp}(-\mathrm{exp}(\mathit{X}\mathit{\beta}))$ | |

Probit | ${\mathsf{\Phi}}^{-1}(\mu )=\mathit{X}\mathit{\beta}$ | $\mu =\mathsf{\Phi}(\mathit{X}\mathit{\beta})$ | |

Inverse CDF | ${F}^{-1}(\mu ;\xi ,\theta )=\mathit{X}\mathit{\beta}$, if $-\infty <{F}^{-1}(x)<\infty $ | $\mu =F(\mathit{X}\mathit{\beta};\xi ,\theta )$ | |

$log({F}^{-1}(\mu ;\xi ,\theta ))=\mathit{X}\mathit{\beta}$, if $0<{F}^{-1}(x)<\infty $ | $\mu =F(\mathrm{exp}(\mathit{X}\mathit{\beta});\xi ,\theta )$ | ||

$\mathrm{logit}\left({F}^{-1}(\mu ;\xi ,\theta )\right)=\mathit{X}\mathit{\beta}$, if $0<{F}^{-1}\left(x\right)<1$ | $\mu =F\left(\frac{\mathrm{exp}\left(\mathit{X}\mathit{\beta}\right)}{1\text{}+\text{}\mathrm{exp}\left(\mathit{X}\mathit{\beta}\right)};\xi ,\theta \right)$ |

Country: E&W | Years: 1841–2016 | Ages: 0–89 (Years) | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | $\mathit{s}$ | ${\mathit{k}}_{\mathbf{1}}$ | $\mathit{p}$ | ${\mathit{k}}_{\mathbf{2}}$ | $\mathit{q}$ | ${\mathit{\xi}}^{\mathbf{t}}$ | ${\mathit{\theta}}^{\mathbf{t}}$ | ${\mathit{\xi}}^{\mathbf{c}}$ | ${\mathit{\theta}}^{\mathbf{c}}$ |

HS_log | 54.50 | 29 | 5 | 4 | 2 | - | - | - | - |

HS_lgt | 55.19 | 30 | 5 | 14 | 2 | - | - | - | - |

HS_cll | 6.40 | 30 | 5 | 14 | 2 | - | - | - | - |

HS_prbt | 9.26 | 30 | 5 | 10 | 2 | - | - | - | - |

HS_beta | 4.13 | 30 | 5 | 8 | 2 | 6.00 | 1.25 | 4.00 | 1.25 |

HS_prt | 71.45 | 23 | 6 | 4 | 1 | 16.50 | 1.00 | 11.50 | 1.00 |

Country: Greece | Years: 1961–2013 | Ages: 0–84 (Years) | |||||||
---|---|---|---|---|---|---|---|---|---|

Model | $\mathit{s}$ | ${\mathit{k}}_{\mathbf{1}}$ | $\mathit{p}$ | ${\mathit{k}}_{\mathbf{2}}$ | $\mathit{q}$ | ${\mathit{\xi}}^{\mathbf{t}}$ | ${\mathit{\theta}}^{\mathbf{t}}$ | ${\mathit{\xi}}^{\mathbf{c}}$ | ${\mathit{\theta}}^{\mathbf{c}}$ |

HS_log | 2.55 | 16 | 4 | 8 | 1 | - | - | - | - |

HS_lgt | 2.52 | 16 | 4 | 8 | 1 | - | - | - | - |

HS_cll | 2.58 | 16 | 4 | 8 | 1 | - | - | - | - |

HS_prbt | 0.21 | 16 | 5 | 8 | 1 | - | - | - | - |

HS_beta | 2.18 | 20 | 5 | 8 | 1 | 1.25 | 0.50 | 1.00 | 0.25 |

HS_prt | 3.37 | 20 | 5 | 8 | 1 | 4.00 | 1.00 | 7.00 | 1.00 |

Model | ${\mathit{k}}_{\mathbf{1}}$ | npar | Log-Likelihood | AIC | BIC | MSPE (%) | MAPE (%) |
---|---|---|---|---|---|---|---|

HS_log | 29 | 1598 | −135,970.84 | 275,137.68 | 287,394.81 | 0.329 | 3.696 |

HS_lgt | 30 | 1634 | −125,713.76 | 254,695.52 | 267,228.78 | 0.336 | 3.694 |

HS_cll | 30 | 1634 | −168,056.21 | 339,566.43 | 352,813.03 | 0.449 | 4.707 |

HS_prbt | 30 | 1622 | −115,111.76 | 233,467.52 | 245,908.73 | 0.346 | 3.693 |

HS_beta | 30 | 1343 | −129,916.62 | 262,519.24 | 272,820.45 | 0.381 | 3.945 |

HS_prt | 23 | 1492 | −113,143.64 | 229,271.28 | 240,715.35 | 0.346 | 3.596 |

**Table 5.**Results of the quantitative tests for well-known stochastic mortality models against HS_prt for E&W dataset.

Model | npar | Log-Likelihood | AIC | BIC | MSPE (%) | MAPE (%) |
---|---|---|---|---|---|---|

LC | 354 | −1,075,745.35 | 2,152,198.70 | 2,154,913.98 | 7.131 | 18.473 |

RH | 707 | −553,002.06 | 1,107,418.13 | 1,112,841.02 | 5.369 | 14.896 |

APC | 528 | −1,157,738.25 | 2,316,532.50 | 2,320,582.42 | 20.541 | 23.522 |

PL | 880 | −762,205.25 | 1,526,170.51 | 1,532,920.37 | 9.136 | 15.737 |

HS_log | 1598 | −135,970.84 | 275,137.68 | 287,394.81 | 0.329 | 3.696 |

HS_prt | 1492 | −113,143.64 | 229,271.28 | 240,715.35 | 0.346 | 3.596 |

Model | ${\mathit{k}}_{\mathbf{1}}$ | npar | Log-Likelihood | AIC | BIC | MSPE (%) | MAPE (%) |
---|---|---|---|---|---|---|---|

HS_log | 16 | 447 | −20,176.13 | 41,246.26 | 44,112.84 | 4.109 | 10.115 |

HS_lgt | 16 | 447 | −20,069.65 | 41,033.29 | 43,899.88 | 4.080 | 10.065 |

HS_cll | 16 | 447 | −20,133.42 | 41,160.85 | 44,027.43 | 4.104 | 10.079 |

HS_prbt | 16 | 516 | −19,569.21 | 40,170.43 | 43,479.50 | 4.078 | 10.074 |

HS_beta | 20 | 540 | −19,358.74 | 39,797.48 | 43,260.47 | 3.444 | 9.515 |

HS_prt | 20 | 540 | −19,353.29 | 39,786.59 | 43,249.58 | 3.449 | 9.498 |

**Table 7.**Results of the quantitative tests for well-known stochastic mortality models against HS_prt for GR dataset.

Model | npar | Log-Likelihood | AIC | BIC | MSPE (%) | MAPE (%) |
---|---|---|---|---|---|---|

LC | 221 | −21,213.44 | 42,868.89 | 44,286.15 | 4.287 | 11.379 |

RH | 441 | −18,633.10 | 38,148.19 | 40,976.30 | 3.790 | 9.460 |

APC | 272 | −21,317.67 | 43,179.33 | 44,923.65 | 5.405 | 12.715 |

PL | 378 | −19,113.68 | 38,983.36 | 41,407.45 | 5.395 | 10.371 |

HS_log | 447 | −20,176.13 | 41,246.26 | 44,112.84 | 4.109 | 10.115 |

HS_prt | 540 | −19,353.29 | 39,786.59 | 43,249.58 | 3.449 | 9.498 |

Model: HS_prt | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Country | Years | Ages | $\mathit{s}$ | ${\mathit{k}}_{\mathbf{1}}$ | $\mathit{p}$ | ${\mathit{k}}_{\mathbf{2}}$ | $\mathit{q}$ | ${\mathit{\xi}}^{\mathbf{t}}$ | ${\mathit{\theta}}^{\mathbf{t}}$ | ${\mathit{\xi}}^{\mathbf{c}}$ | ${\mathit{\theta}}^{\mathbf{c}}$ |

E&W | 1841–2006 | 0–89 | 56.58 | 29 | 6 | 4 | 1 | 15 | 1.00 | 11.50 | 1.00 |

GR | 1961–2003 | 0–84 | 1.55 | 20 | 3 | 8 | 1 | 5.50 | 1.00 | 11.50 | 1.00 |

**Table 9.**Results of the percentage error tests for predicted mortality rates of 10 years out-of-sample for well-known stochastic mortality models against HS_prt for E&W dataset.

MSPE (%) | MAPE (%) | |||
---|---|---|---|---|

RWD | ARIMA | RWD | ARIMA | |

HS_prt | 3.206 | 5.550 | 13.219 | 18.852 |

HS_log | 3.221 | 7.135 | 13.271 | 20.800 |

RH | 271.861 | 271.861 | 58.085 | 58.085 |

LC | 29.508 | 29.509 | 49.838 | 49.837 |

PL | 586.784 | 371.845 | 62.748 | 55.635 |

APC | 86.494 | 84.316 | 46.192 | 46.131 |

**Table 10.**Results of the percentage error tests for predicted mortality rates of 10 years out-of-sample for well-known stochastic mortality models against HS_prt for GR dataset.

MSPE (%) | MAPE (%) | |||
---|---|---|---|---|

RWD | ARIMA | RWD | ARIMA | |

HS_prt | 8.446 | 8.549 | 14.961 | 15.067 |

HS_log | 8.691 | 10.390 | 15.332 | 15.857 |

RH | 8.361 | 8.581 | 15.440 | 15.275 |

LC | 10.398 | 10.333 | 20.320 | 20.964 |

PL | 8.860 | 11.163 | 18.402 | 15.148 |

APC | 10.054 | 10.077 | 21.364 | 21.599 |

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**MDPI and ACS Style**

Sagianou, A.; Hatzopoulos, P.
Extensions on the Hatzopoulos–Sagianou Multiple-Components Stochastic Mortality Model. *Risks* **2022**, *10*, 131.
https://doi.org/10.3390/risks10070131

**AMA Style**

Sagianou A, Hatzopoulos P.
Extensions on the Hatzopoulos–Sagianou Multiple-Components Stochastic Mortality Model. *Risks*. 2022; 10(7):131.
https://doi.org/10.3390/risks10070131

**Chicago/Turabian Style**

Sagianou, Aliki, and Peter Hatzopoulos.
2022. "Extensions on the Hatzopoulos–Sagianou Multiple-Components Stochastic Mortality Model" *Risks* 10, no. 7: 131.
https://doi.org/10.3390/risks10070131