Extensions on the Hatzopoulos&ndash;Sagianou Multiple-Components Stochastic Mortality Model

Aliki Sagianou; Peter Hatzopoulos

doi:10.3390/risks10070131

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

Sagianou, Aliki;Hatzopoulos, Peter

2022-06-21 00:00:00

risks Article Extensions on the Hatzopoulos–Sagianou Multiple-Components Stochastic Mortality Model Aliki Sagianou * and Peter Hatzopoulos Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, 83200 Samos, Greece; xatzopoulos@aegean.gr * Correspondence: asagianou@aegean.gr Abstract: In this paper, we present extensions of the Hatzopoulos–Sagianou (2020) (HS) multiple- component stochastic mortality model. Our aim is to thoroughly evaluate and stress test the HS model by deploying various link functions, using generalised linear models, and diverse distributions in the model’s estimation method. In this work, we differentiate the HS approach by modelling the number of deaths using the Binomial model commonly employed in the literature of mortality modelling. Given this, new HS extensions are derived using the off-the-shelf link functions, namely the complementary log–log, logit and probit, while we also reform the model by introducing a new form of link functions with a particular focus on the use of heavy-tailed distributions. The above-mentioned enhancements conclude to a new methodology for the HS model, and we prove that it is more suitable than those used in the literature to model the mortality dynamics. In this regard, our work offers an extensive experimental testbed to scrutinise the efﬁciency, explainability and capacity of the HS model extensions using both the off-the-shelf and the newly introduced form of link functions over datasets with different characteristics. The introduced HS extensions bring an improvement by approximately 16% to the model’s goodness-of-ﬁt, while they uncover more ﬁne-grained age clusters. In addition, we compare the performance of the HS extensions Citation: Sagianou, Aliki, and Peter against other well-known mortality models, both under ﬁtting and forecast modes. The results Hatzopoulos. 2022. Extensions on the reﬂect the advantageous features of the HS extensions to deliver a highly informative structure and Hatzopoulos–Sagianou Multiple- enable the attribution of an identiﬁed mortality trend to a unique age cluster. The above-mentioned Components Stochastic Mortality improvements enable mortality analysts to perform an in-depth and more detailed investigation of Model. Risks 10: 131. https:// mortality trends for speciﬁc age clusters and can contribute to the attempts of academia and industry doi.org/10.3390/risks10070131 to tackle the uncertainties and risks introduced by the increasing life expectancy. Academic Editors: Georgios Pitselis, Apostolos Bozikas, Ioannis Badounas Keywords: mortality modelling; generalised linear models; link functions; heavy-tailed distributions; and Séverine Arnold longevity risk; solvency II Received: 28 February 2022 Accepted: 17 June 2022 Published: 21 June 2022 1. Introduction Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in The continuous increase in life expectancy observed in recent years, especially in published maps and institutional afﬁl- developed countries, as well as the actuarial risks raised due to the aging population iations. have triggered scientists and actuaries to develop appropriate models in order to model and project mortality rates. Although the increase in life expectancy is a bright sign of social, medical and scientiﬁc progress, it poses challenges to governments, private pension plans and life insurers, as it signiﬁcantly affects the cost of pensions. To this end, the Copyright: © 2022 by the authors. problem of predicting the future mortality rates has been a subject of great interest for both Licensee MDPI, Basel, Switzerland. governments and the scientiﬁc community. This article is an open access article The analysis of the historical evolution of mortality rates is performed using stochastic distributed under the terms and mortality models. In this context, several research endeavours were committed to deliver a conditions of the Creative Commons number of age–period–cohort models (Cairns et al. 2009; Currie 2006; Hatzopoulos and Attribution (CC BY) license (https:// Sagianou 2020; Hatzopoulos and Haberman 2011; Lee and Carter 1992; Plat 2009; Renshaw creativecommons.org/licenses/by/ and Haberman 2006). Consequently, several attempts have been made to document the 4.0/). Risks 2022, 10, 131. https://doi.org/10.3390/risks10070131 https://www.mdpi.com/journal/risks Risks 2022, 10, 131 2 of 30 similarities among these models (Hunt and Blake 2021), as well as to compare and present their disadvantages and advantages (Booth and Tickle 2008; Cairns et al. 2009; Renshaw and Haberman 2006). Hunt and Blake (2021) focused on the structure of mortality models and provided a generalised age–period–cohort model structure which reﬂects most of the mortality models. Moreover, for the ease of use of these models, several software implementations and packages have been developed and made available to model the mortality in the R environment. For instance, and as reported in (Millossovich et al. 2018), the demography package (Hyndman et al. 2015) capitalises on the (Lee and Carter 1992) model and its variants presented in (Booth et al. 2002; Hyndman and Ullah 2007; Lee and Miller 2001). The ilc package (Butt et al. 2014) offers the Renshaw and Haberman model and the Lee–Carter model under a Poisson regression framework. The LifeMetrics R functions (Coughlan et al. 2007) consider the Cairns–Blake–Dowd (CBD) models and their extensions introduced in (Cairns et al. 2009), the Lee–Carter model (using Poisson maximum likelihood), the age–period–cohort model (Currie 2006; Osmond 1985) and the Renshaw and Haberman model. Regardless of the peculiarities of each implementation, mortality models aim to analyse mortality by decomposing the mortality rates in the dimensions of age, period and cohort (or year of birth). The aforementioned dimensions enable the analysis of mortality rates of individuals as they age, aid to the understanding of medical and social progress and act as enablers to shed light on the lifelong mortality effects. In fact, according to a recent study by Buxbaum et al. (2020), 44% of the improvement in life expectancy was attributable to public health, while pharmaceuticals were the second-leading cause of gain in life years at 35%. Mortality improvements pose a threat to the solvency of ﬁnancial institutions. In this direction, mortality models can be used to support the analysis, modelling and management of the inherent market risks for the sake of safeguarding the solvency of organisations. According to the literature, a generic formula (see Equation (1)) is deﬁned and can rep- resent the majority of stochastic mortality models in line with the principles of generalised linear models (GLMs) (Currie 2016; Hunt and Blake 2021; Millossovich et al. 2018), as it will be explained in detail in Sections 2.1 and 2.2. The complete documentation of the principles, the theory and application of GLMs can be found in (McCullagh and Nelder 1989). In short, in the context of GLMs, the results of the dependent variables are 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 GLMs are a generalisation of the classical linear models. More speciﬁcally, the GLMs generalise linear regression by allowing the linear model to connect to the response variable using a link function. Thus, the latter enables the connection of the linear predictor and the mean of the distribution function. In general, various link functions can be used, but as is the case in the GLM literature, the so-called canonical link functions are the prominent ones used in the mortality literature by stochastic mortality models. The canonical link functions are the log and the logit for the Poisson and Binomial distributions, respectively. Despite the canonical ones, the complementary log–log and probit link functions have also been used for the Binomial distribution. In the context of GLMs, some requirements must be taken into account in order to select the proper link function. The same applies for mortality models. Early attempts in the mortality modelling ﬁeld consider as a requirement that the data should be transformed in order to obtain an approximately linear predictor structure (Hunt and Blake 2021). To achieve this, one has a wide range of options of link functions to utilise. Notably, there are several studies in the wild that use and experiment on various link functions in order to more accurately model either mortality rates or death probabilities and even to extend already known or introduce new ﬂavours of stochastic mortality models (Currie 2016; Haberman and Renshaw 1996). In this line of thought, StMoMo, a well-established mortality modelling R package, gives the ability to parameterise the models’ estimation process. However, it limits the choices only to the use of the canonical link functions, while Risks 2022, 10, 131 3 of 30 GLM-based estimation could be performed under various link function options and there is no a priori reason why other links should not work well. Motivated by this fact, our work aims to explore and expand the range of choices of link functions used in the context of mortality modelling. To do so, we conduct our investigation using the (Hatzopoulos and Sagianou 2020) (hereafter called HS) multiple-component mortality model, and we introduce a new form of link functions that extend the model and its capacity by offering new HS ﬂavours. In view of the above, this work extends the stochastic mortality model of the HS, which was ﬁrstly introduced using central mortality rates, m , under the Poisson distribution t,x and log link function, by formulating it in terms of conditional probabilities of death, q , t,x and the Binomial distribution using a wide set of link functions. In fact, the extension of the HS model is twofold. On the one hand, we adapt the HS to be modelled using off-the-shelf, i.e., well-known, link functions in the literature, namely the logit, complementary log–log and probit. This approach is in line with the rationale used in tools such as StMoMo for ratifying models both in the Poisson and the Binomial cases, while it calls forth the HS model to prove its capacity on a new experimental testbed. On the other hand, we introduce a new form of link functions in an effort to further improve the goodness-of-ﬁt and the explainability of the model, and we evaluate their applicability in the context of mortality through the HS extensions. A particular focus is given to the use of heavy-tailed distributions. From a modelling perspective, heavy-tailed distributions are important when extreme events must be part of the model. For example, although the probability of shock cases, such as mortality during World War II or the COVID-19 pandemic, is small, the magnitude of the impact is so large that these events are vital in capturing the mortality dynamics in an accurate manner, i.e., to ensure that the high variability due to the extreme events (e.g., k in Figure A1) can be explained by the model. To the best of our knowledge, it is the ﬁrst time that those link functions are tested in the mortality domain. Therefore, we build a multidimensional testbed, and we showcase that the HS retains its robust characteristics, i.e., (i) the identiﬁcation of the signiﬁcant incor- porated period and cohort effects, (ii) the identiﬁcation of distinct stochastic components and (iii) the attribution of an identiﬁed mortality trend to a unique age cluster. We assess the capabilities of the HS extensions using the data of males from England and Wales and also Greece provided by the (Human-Mortality-Database n.d.). In short, the contributions of this work are summarised in the following points: • We extend the stochastic mortality model HS formulated in terms of q , using gener- t,x alised linear models and by adopting various link functions. We illustrate through experimental results that the HS model remains robust and consistent under all mod- elling variations. • We introduce a new set of link functions, with a particular focus on heavy-tailed dis- tributions, 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 mortal- ity 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 var- ious link functions in the models’ estimation methods having witnessed the beneﬁcial impact of this approach to our model’s efﬁcacy. The rest of this paper is organised as follows: In Section 2, we introduce the notation and we provide background information for the GLM framework and the HS mortality model, with a particular focus on the original estimation methodology used. In Section 3, we present the newly introduced methodology and the updated estimation method for the Risks 2022, 10, 131 4 of 30 HS extensions using the off-the-shelf and the new form of link functions. Section 4 outlines the evaluation approach and the performance results of the HS extensions. Section 5 provides an overall discussion on the ﬁndings and concludes. 2. Preliminaries In this section, we elaborate on the notation, we provide an overview of the GLM framework and we brieﬂy present the Hatzopoulos–Sagianou (HS) multiple-component stochastic mortality model which will be used throughout this work. More speciﬁcally, we outline some background information about the HS model and its estimation method to ease the reader in following the extensions described later in Section 3. The notation and data used in the HS model are aligned with the common practices in the mortality domain. What differentiates its structure is the consideration of multiple age–period and age–cohort components which enables the model to capture the mortality trend of multiple and distinct age clusters. The HS model uses a semiparametric estimation method based on two methodologies, namely generalised linear models (GLMs) and a sparse principal com- ponent analysis (SPCA). An additional methodology, driven by the unexplained variance metric, contributes to the deﬁnition of the optimal model structure for including the most important age–period and age–cohort components. The aforementioned methodologies are summarised in the following sections before we proceed to the documentation of the necessary adaptations of the model in order to introduce a new form of link functions and test their applicability to the HS in Section 3. Note that the following sections of- fer a synopsis of the estimation method of the model. We prompt the reader to refer to (Hatzopoulos and Sagianou 2020) to discover the mathematical details behind the formula- tion of the model. 2.1. Data and Notation Throughout this work, the data include the number of deaths, D , the central ex- t,x c 0 posures to risk, E , and the corresponding initial exposures to risk, E , at age x last t,x t,x birthday during calendar year t. The data take the form of a rectangular structure (t, x) over a unit range of individual calendar years t (t , . . . , t ) and individual ages, x, last 1 n birthday (x , . . . , x ). That is, the probability that a person at age x and calendar year t 1 a will die within one year is deﬁned as q = D /E . The crude central mortality rate for t,x t,x t,x c c any age x and calendar year t is deﬁned as m = D /E . E is often derived based t,x t,x t,x t,x on an approximation of the number of those aged x years old during their last birthday half way through year t, or based on an estimation of the average of said population at the beginning and end of that year. The E is deﬁned as the population size on 1 January, at t,x age x in calendar year t. We treat D as independent Poisson or Binomial realisations, i.e., t,x D follows the Poisson distribution with mean E m , or the Binomial distribution t,x t,x t,x with mean E q . t,x t,x D Poisson(E m ) t,x t,x t,x or D Binomial(E , q ) t,x t,x t,x An age–period–cohort stochastic mortality model can take the form suggested by the following generic formula (Hunt and Blake 2021), reﬂecting the methodology of generalised linear models (McCullagh and Nelder 1989): p q (i) (i) c(j) (j) h = a + b k + b g (1) t,x x å x å x c i=1 j=1 h = g(E(D /E )) is the link function used to transform the response variable t,x t,x t,x (which is a measure of mortality rates) at age x and for year t into a form suitable for modelling and link it to the proposed predictor structure (Millossovich et al. 2018). The Risks 2022, 10, 131 5 of 30 term a represents the main age proﬁle of mortality by age. b reﬂects the age effect for x x each period component, while b is the age effect for each cohort component. The term k refers to the period-related effects and corresponds to the mortality trend. The term g t c reﬂects the cohort-related effects, where c = t x. p(> 1) and q(> 0) are indexes referring to the period and cohort components included in the model structure. 2.2. GLM Framework In a generalised linear model (see, e.g., (McCullagh and Nelder 1989)), each outcome of a random variable Y, whose components are independently distributed with means m, 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, m, of the distribu- tion via: E(Y) = m = g (Xb) where E(Y) is the expected value of Y, where Y is a vector of response variables. Xb is the linear predictor, a linear combination of parameters b whose values are usually unknown and have to be estimated from the data. X is a matrix of explanatory variables, and b is a vector of parameters. g is the link function. The model’s independent variables are incorporated through the linear predictor (h). The link function relates it to the expected value of the data and it is expressed as a linear combination of the unknown parameters b. The matrix of independent variables X represents coefﬁcients of the linear combination. The linear predictor and the mean of the distribution function are related via the link function. A number of considerations need to be taken into account when choosing the most appropriate link function. The exponential of the response’s density function determines a well-deﬁned canonical link function. Although, for algorithmic purposes, in some cases, it makes sense to use a non-canonical link function. Therefore, there are three components to any GLM (McCullagh and Nelder 1989): • 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 produc- ing the so-called linear predictor h = Xb. • 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., h = g(m). Hence, as is the case for the generalised linear models, the same applies to the stochas- tic mortality models. The corresponding four components in a stochastic mortality model are (Millossovich et al. 2018): 1. The random component: the numbers of deaths D follow the Poisson distribution t,x c 0 with mean E m , or the Binomial distribution with mean E q , so that t,x t,x t,x t,x D Poisson(E m ) t,x t,x t,x or D Binomial(E , q ) t,x t,x t,x 2. The systematic component: the effects of age x, calendar year t and cohort c = t x are captured through a predictor h given by: t,x p q (i) (i) c(j) (j) h = a + b k + b g t,x x x x c å å i=1 j=1 Risks 2022, 10, 131 6 of 30 3. The link function g associating the random component and the systematic component, so that t,x g E = h t,x 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 , using a wide set t,x of canonical and non-canonical link functions. 4. The set of parameter constraints: most of the stochastic mortality models have identiﬁa- bility 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 In our previous publication, (Hatzopoulos and Sagianou 2020), we proposed a dynamic multiple-component model that includes p age–period and q age–cohort non-predefined effects. The aim of this section is to overview that work, as a more detailed analysis is out of the scope of this paper. We prompt the interested reader to refer to (Hatzopoulos and Sagianou 2020) for more details on the estimation process and model per se. The HS model has the following form: p q (i) (i) c(j) (j) log(m ) = a + b k + b g t,x x å x å x c i=1 j=1 The number of age–period and age–cohort effects vary depending on the experimental dataset, i.e., the intrinsic mortality peculiarities of the examined population for a given time frame. The estimation methodology for deﬁning the most important p age–period and q age-–cohort components is brieﬂy described below. The data incorporate the number of deaths (D ), with the respective central exposure t,x to risk (E ), for each calendar year t = t , ..., t and for each age x = x , ..., x last birthday. 1 n 1 a t,x The deaths, D , follow a Poisson distribution with mean E m . These data are used t,x t,x t,x for each calendar year t in order to model the mortality dynamics and consequently to estimate the age–period and age–cohort effects. As described in (Hatzopoulos and Sagianou 2020), the model structuring method is unfolded in four steps, each of which utilises a method which brings to the model its competitive advantages as will be described later. 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 )) as an offset. The predictor structure is: t,x c c h = log(E(D )) = log(E m ) = log(E ) + log(m ) t,x t,x t,x t,x t,x t,x Therefore, treating age as an explanatory variable, the linear predictor is produced using an orthonormal polynomial structure in age effects for each calendar year indepen- dently. That is, the following predictor structure is deﬁned as: log(m ˆ ) = b L (2) t t where, b corresponds to the vector of the GLM-estimated parameters for calendar year t. Given this, a random (asymptotically normal) matrix B = fb g, k = 1, . . . , k , of t,k1 1 order (n k ), includes the GLM-estimated parameters for all calendar years t = t , . . . , t . 1 1 n Risks 2022, 10, 131 7 of 30 L = fl g, k = 1, . . . , k , is a (a k )-dimensional matrix of orthonormal polynomials 1 1 x,k1 with degree of k 1 for age x. 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 = 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: max min l ( A + U), s.t. jU j s (3) i j where s is a scalar that deﬁnes the sparsity. The deﬁnition of the “optimum” s value and, consequently, the most important p age–period (and corresponding q age–cohort components) is achieved using the unexplained variance ratio (UVR) approach. 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 to p. The excluded components are then treated as residuals. Overall, the HS estimation methodology takes advantage of the SPCA and the UVR criterion, not only for pinpointing the informative p components but also for uncovering the age clusters that signiﬁcantly contribute to the mortality trend of each component. This approach results in the following model: (i) (i) ˆ ˜ log(m ) = a + b k + # (4) t,x x x t,x i=1 where the residuals # at age x in year t denote the deviation of the model as a result of t,x the excluded sparse principal components (SPCs). The estimation process for the age–cohort components follows a similar approach to that for the age–period ones. However, there is no need to deﬁne a sparsity factor, as the cohort structure is deﬁned using principal component analysis. Thus, the UVR method determines the optimal number q of age–cohort components by pinpointing those that, again, reveal unique age clusters. 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: p q (i) (i) c(j) (j) log(m ) = a + b k + b g t,x,c x x x c å å i=1 j=1 for the calendar years t = t , . . . , t , ages x = x , . . . , x and cohorts c = t x = c , . . . , c . 1 n 1 a 1 n 3. Methodology In Section 2, we outlined the methodology followed by the HS model as it will serve as the basis for elaborating on the extensions introduced in the context of this paper. In addition, we provided an overview of the fundamental principles of the GLM methodology, Risks 2022, 10, 131 8 of 30 which contribute to the formation of the presented methodology and the extensions of the HS model. Based on this, we proceed to the documentation of the extended form of the HS model, formulated in terms of q (while the former was in terms of m ), and we t,x t,x adapt the HS to be modelled using: (i) off-the-shelf link functions in the literature, namely the logit, complementary log–log and probit, and (ii) by introducing a new form of link functions in an effort to further improve the goodness-of-ﬁt of the model and to evaluate their applicability in the context of mortality through the HS extensions. The aforementioned cases are described in Sections 3.1 and 3.2, respectively. 3.1. HS Model Using Off-the-Shelf Link Functions We extend the HS model through the formulation in terms of q and by adopting the t,x estimation methodology described in Section 2.3 making the appropriate conﬁgurations. Under this prism, the data include the number of deaths (D ) and the initial exposure to t,x risk (E ), for t = t , ..., t and for x = x , ..., x . n a 1 1 t,x We treat d as a random variable, D and the initial exposures E as ﬁxed, then D t,x t,x t,x t,x 0 t,x follows a Binomial distribution with mean E q and let Q = be the estimator of t,x t,x t,x t,x t,x q = . In the case of the Binomial distribution, the following three link functions can t,x t,x be used: i logit: h = log 1q ii probit: h = F (q) where F() is the Normal cumulative distribution function. iii complementary log–log: h = log( log(1 q)) Thus, and according to (McCullagh and Nelder 1989) (Section 2.2.3) and (Currie 2016), we have 0 < q < 1 and a link should satisfy the condition that it maps the interval (0, 1) onto the whole real line (e.g., ¥ < logit(q) < ¥). At this point, we follow the four steps mentioned in Section 2.3 to model the age– period and age–cohort components. Therefore, a GLM approach is applied for each calendar year t independently, with dependent variable D , binomial error, E weights, t,x t,x the corresponding link function and linear predictor h = b X. We note that the exposures are introduced into the Poisson model as an offset, but into the Binomial model as a weight, and the model matrix, X, is a matrix of orthonormal polynomials, so that: !! D q t,x t h = logit(E(Q )) = logit E = log = b L t t t 1 q t,x or 1 T h = probit(E(Q )) = F (q ) = b L t t t t or eta = cloglog(E(Q )) = log( log(1 q )) = b L t t t t where b corresponds to the vector of the GLM-estimated parameters for calendar year t. Given this, a random (asymptotically normal) matrix B = fb g, k = 1, . . . , k , of t,k1 1 order (n k ), includes the GLM-estimated parameters for all calendar years t = t , . . . , t . 1 1 L = fl g, k = 1, . . . , k , is a (a k )-dimensional matrix of orthonormal polynomials x,k1 1 1 with degree of k 1 for age x. The optimal degree k 1 of the orthonormal polynomials 1 1 is deﬁned by using a variety of statistical tests. Then, we continue with the next two steps by applying the SPCA to the associated covariance matrix of the GLM-estimated parameters B and, in tandem with the UVR approach, we come to the following age–period stochastic mortality model: (i) (i) h = logit(q ) = a ˜ + b k + # t,x t,x x å x t,x i=1 Risks 2022, 10, 131 9 of 30 In a similar way, we deploy the rest of the link functions, i.e., the probit and cloglog, in order to deﬁne the corresponding HS variations for the age–period part of the model. Finally, the age–cohort components are estimated in a way similar to that of the age– period components, and then the ﬁnal form of the estimates of the age–period–cohort stochastic mortality model is deﬁned as: p q (i) (i) c(j) (j) h = logit(q ˆ ) = a + b k + b g t,c,x t,c,x x å x å x c i=1 j=1 for the calendar years t = t , . . . , t , ages x = x , . . . , x and cohorts c = t x = c , . . . , c . 1 n 1 a 1 n In a similar way, one can deploy the rest of the off-the-shelf link functions. 3.2. Introducing a New Form of Link Functions in the Mortality Modelling Triggered by the probit case, where the cumulative distribution function of the stan- dard normal distribution is used, we generalise this rationale as the inverse of any con- tinuous cumulative distribution function can be used for the link, as long as a proper transformation is given to satisfy the condition that the CDF’s range should be mapped to the whole real line. As is the case for the probit, the same applies to the newly proposed form of link functions. We treat d as a random variable, D and the initial exposures E as ﬁxed, t,x t,x t,x 0 t,x then D follows a Binomial distribution with mean E q and let Q = be t,x t,x t,x t,x t,x t,x the estimator of q = . As noted in Section 3.1, we have 0 < q < 1. Thus, the t,x t,x t,x link function that we choose should satisfy the condition that it maps the interval (0, 1) to the whole real line. Following this rationale, there are the following three categories to be considered and provide a transformation method, depending on the cumulative distribution chosen: 1. The cumulative distribution as link function, maps q, 0 < q < 1, to¥ < F (q; x , q) < ¥, so that: 1 1 1 T h = F (E(Q ); x, q) = F E ; x, q = F (q ; x, q) = b L t t t t or E = q = F(h ; x, q) , q = F(b L ; x, q) t t t t For instance, some distributions that can be used are: Normal, Logistic, Extreme value, Gumbel, etc. 2. The cumulative distribution as link function, maps q, 0 < q < 1, to 0 < F (q; x , q) < ¥, so the logarithmic form of the cumulative distribution is needed to map q, to ¥ < log(F (q; x, q)) < ¥, the natural scale for regression, so that: 1 1 T h = log(F (E(Q ); x, q)) = log F (q ; x, q) = b L t t t t or E = q = F(exp(h ); x, q) , q = F(exp(b L ); x, q) t t t t For instance, some distributions that can be used are: Generalised Pareto, Weibull, Fréchet, etc. 3. The cumulative distribution as link function, maps q, 0 < q < 1, to 0 < F (q; x , q) < 1, so we need the logit of the cumulative distribution so that maps q, to ¥ < logit(F (q; x, q)) < ¥, the natural scale for regression, so that: Risks 2022, 10, 131 10 of 30 F (q ; x , q) 1 1 T h = logit(F (E(Q ); x, q)) = logit F (q ; x , q) = log = b L t t t t 1 F (q ; x, q) or D exp(h ) exp(b L ) t t t E = q = F ; x, q , q = F ; x , q t t 0 T 1 + exp(h ) 1 + exp(b L ) E t For instance, Beta distribution can be used in this case. In this work, a particular focus is given to the use of heavy-tailed distributions. From a modelling perspective, heavy-tailed distributions are important when extreme events must be part of the model. For example, although the probability of shock cases, such as mortality during World War II or the COVID-19 pandemic, is small, the magnitude of the impact is so large that these events are vital in capturing the mortality dynamics in an accurate manner. In this regard, as will be showcased in Section 4, we document the use of the Generalised Pareto and Beta distributions through the corresponding extensions of the HS model. Table 1 summarises the transformations of the HS model along with key characteristics. Table 1. Link functions transformations of the HS model. Dist Link Name Link Function h = g(m) = Xb Mean Function m = g (Xb) m exp(Xb) Logit log = Xb m = 1m 1+exp(Xb) Cloglog log( log(1 m) = Xb m = 1 exp( exp(Xb)) Probit F (m) = Xb m = F(Xb) F (m; x, q) = Xb, if ¥ < m = F(Xb; x , q) F (x) < ¥ log(F (m; x, q)) = Xb, if 0 < m = F(exp(Xb); x, q) Inverse CDF F (x) < ¥ exp(Xb) logit(F (m; x, q)) = Xb, if 0 < m = F ; x , q 1+exp(Xb) F (x) < 1 Based on the analysis given above, one needs to estimate the parameters of each possible distribution. That is, in the context of the HS model, the estimation methodology deﬁned in Section 2.3 needs to be revised in order to incorporate the estimation process of the parameters of each possible distribution. This revised method is described in the following subsection. 3.2.1. HS Revised Estimation Methodology The revised estimation process follows the rationale and the methodology steps analysed in (Hatzopoulos and Sagianou 2020) by making appropriate conﬁgurations. The ultimate goal is the deﬁnition of the age–period and age–cohort effects, but this time, by including the task of estimating the parameters of the adopted distribution. In this section, we opt to showcase how the estimation process is updated using the 2nd case of the new form of the link function, namely the logarithm of the inverse cumulative distribution, as documented in Section 3.2, i.e., ¥ < log(F (q; x, q)) < ¥, where F is a cumulative distribution (e.g., Generalised Pareto). The same process can also be followed for the rest of the identiﬁed cases in Section 3.2, but we opt to document one of the cases for simplicity reasons. Binomial Risks 2022, 10, 131 11 of 30 In this regard, the estimation of the parameters of the adopted distribution requires the estimation of the (x, q) pairs for both the age–period and the age–cohort part of the t t c c model, referring to them as (x , q ) and (x , q ), respectively. Hence, regarding the age–period effects, before we follow the four steps of the estima- tion methodology described in Section 2.3, we ﬁrst need to determine the parameters of the selected distribution and the respective link function. For ﬁtting the probabilities of deaths, q , to the chosen cumulative distribution, we t,x t t need to pinpoint a particular pair (x , q ) of parameter values from the set of all possible pairs, so that the q ˆ will be estimated as close as possible to the observed data. To build an t,x unambiguous approach, this parameter estimation process must be driven by a measure of distance or discrepancy between the observed q and the ﬁtted q ˆ (e.g., classical t,x t,x least square) or by maximising the model’s log-likelihood. In our case, we opt for the log-likelihood maximisation approach given by: ( ! ! ) ˆ 0 E d d t,x t,x t,x 0 t,x L(d , d ) = d log + (E d ) log + log (5) t,x t,x åå t,x t,x t,x 0 0 E E d t,x t t,x t,x where (i) (i) 0 t t d = E F(exp(a ˜ + b k ); x , q ) (6) t,x t,x x å x i=1 t t and the pair of parameters (x , q ) is the one that maximises the aforementioned measure. The ﬁtted q ˆ are estimated by using the GLM approach. Thus, a GLM approach t,x is applied for each calendar year t independently, with dependent variable D , and the t,x initial exposures E are handled as a weight. By using the appropriate link function, the t,x following predictor structure is deﬁned: ! !! t,x 1 t t 1 t t h = log(F (E(Q ); x , q )) = log F E ; x , q t t t,x (7) 1 t t T = log F (q ; x , q ) = b L t t where b corresponds to the vector of the GLM-estimated parameters for calendar year t, t t having chosen the optimum parameter values (x , q ). Given this, a random (asymptotically normal) matrix B = fb g, k = 1, . . . , k , of order (n k ), includes the GLM-estimated 1 1 t,k1 parameters for all calendar years t = t , . . . , t . L = fl g, k = 1, . . . , k , is a (a k )- 1 n x,k1 1 1 dimensional matrix of orthonormal polynomials of k 1 degree for age x. The optimal degree k 1 of the orthonormal polynomials is deﬁned by using a variety of statistical tests. According to the steps described in Section 2.3, SPCA is applied to the covariance matrix of B to decrease the dimensionality of the problem. Note that we apply the UVR approach, not only to deﬁne the “optimum” s value for the sparsity but also to deﬁne the most signiﬁcant p age–period components. In such a manner, the number of age–period components is decreased from k to p. Thus, the age–period form (logarithm of the inverse cumulative distribution in this case) of the model is as follows: (i) (i) 1 t t log(F (q ; x , q )) = a ˜ + b k + # (8) t x t,x å x i=1 where the disturbance term # N(0, v ) (or residuals) is the error component at age x in t,x x year t and reﬂects the deviation of the model due to the excluded components. Regarding the age–cohort components, the estimation process is, in a way, similar to that for the age–period components. Therefore, our data, D , E and q , are processed t,x t,x t,x as cohorts. Risks 2022, 10, 131 12 of 30 Thus, the GLM approach is applied for each cohort c = t c = c , ..., c , with 1 nc dependent variable D . Note that the exposures are introduced as a weight, and the p c,x age–period effects (Equation (8)) are handled as an offset, so that: 1 c c c T h = log(F (q ; x , q )) = b (L ) + offset (9) c c c (i) (i) 1 t t offset = log(F (q ˆ ; x , q )) = a ˜ + b k t x x å t i=1 where b corresponds to the vector of the GLM-estimated parameters for each cohort c. Given this, a random (asymptotically normal) matrix B = fb g, k = 1, . . . , k , of order c,k1 (n k ), includes the GLM-estimated parameters for all cohorts c = c , . . . , c . L = c n 2 1 fl g, for k = 1, . . . , k , is an (a k )-dimensional matrix of orthonormal polynomials 2 2 x,k1 c c with k 1 degree for age x. Note that the pair of optimum parameter values (x , q ) for the age–cohort components are deﬁned by following the same approach as described in the age–period case, i.e., by maximising the log-likelihood of the model. To do so, this process is driven again by Equation (5), but d and E are processed as cohorts and d is t,x c,x t,x deﬁned as: ! ! c(j) (j) 0 c c c d = E F exp a ˜ + b g + offset ; x , q c,x (10) c,x x å x c j=1 The optimum degree k 1 of the orthogonal polynomials L is achieved using the same statistical tests as was the case for the age–period effect estimation methodology. Finally, we apply eigenvalue decomposition to the matrix of the GLM-estimated c c parameters B and, in tandem with the UV R approach described in (Hatzopoulos and Sagianou 2020), which is used to determine the optimum number of age–cohort compo- nents, we come to the following age–cohort stochastic mortality model: c(j) (j) 1 c c c h = log(F (q ; x , q )) = a ˜ + b g + # (11) c,x c,x å x c c,x j=1 Hence, by combining Equations (8) and (11), the ﬁnal form of the age–period–cohort stochastic mortality model is deﬁned as: p q (i) (i) c(j) (j) 1 t t 1 c c h = log(F (q ˆ ; x , q ) + log(F (q ˆ ; x , q ) = a + b k + b g (12) t,c,x t,x c,x x å x å x c i=1 j=1 for the calendar years t = t , . . . , t , ages x = x , . . . , x and cohorts c = t x = c , . . . , c . n a n 1 1 1 c According to the aforementioned method, after estimating the signiﬁcant age–period and age–cohort effects utilising the SPCA, UVR and GLM approaches, we conclude to the ﬁnal estimates of the probabilities of death in age, period and cohort effects. It becomes clear that the UVR plays an important role in the formation of the model. That is, the UVR takes the form given in Equation (13), specifically for the case log(F (q; x, q)), where F is the chosen cumulative distribution. We prompt the reader to refer to (Hatzopou- los and Sagianou 2020) for more information on the use of the UVR metric. 1 t t In short, the aim is to measure the variance of the time series of the log(F (q ; x , q )) t,x and the variance of the error between the estimated probabilities of death and the actual ones (for each age x and each component i). The ratio of these two variances for successive components deﬁnes the UV R (i) value. Based on the updated estimation methodology described in this section, the UVR metric is given in Equation (13). Risks 2022, 10, 131 13 of 30 (1) (1) 1 t t Var log(F (q ;x ,q ))a b k > t,x x x > , i = 1 1 t t > Var log(F (q ;x ,q ))a t,x x UV R (i) = (13) > (1) (1) (i) (i) 1 t t Var log(F (q ;x ,q ))a b k b k t,x x > x x t t , i = 2, . . . , p (1) (1) 1 t t Var log(F (q ;x ,q ))a b k t,x x x In a similar way, one can adapt the HS model to the rest of the link functions docu- mented in Table 1. Implementation-wise, our model and the proposed extensions have been developed in MATLAB, which enables a user to deﬁne their own link functions. Taking advantage of this functionality, we ﬁt the Binomial model with new link for the probabilities of death q t,x using the following command: B = glmﬁt(L , q , ”binomial”, ”link”, new_F, ”weights”, E , ”constant”, ”off”) x t,x t,x Details of the user-deﬁned link new_F are given in Appendix A. 4. Application In this section, we elaborate on the experimental approach followed to evaluate the performance of the HS mortality model using the newly proposed methodology presented in Section 3. In order to offer a fair and straightforward comparison to the performance of the HS model, we opt for the same evaluation metrics and datasets as in (Hatzopoulos and Sagianou 2020). In addition, we perform a comparison with other well-known mortality models of the literature, namely the (Lee and Carter 1992) (LC), (Renshaw and Haberman 2006) (RH), (Currie 2006) (APC) and (Plat 2009) (PL) models. This section also describes the datasets and the experimental parameters of the HS model extensions and analyses the results of the evaluation through a comparative analysis. Moreover, we evaluate the models in terms of forecast. That is, we provide a back- testing framework to forecast 10 years ahead in order to compare out-of-sample estimations against the actual values. Based on this, we measure the efﬁciency of the models using the MSPE and the MAPE metrics. For providing an objective and unbiased evaluation, we forecast the k and the g indexes using two different approaches, namely the Random t c Walk with Drift (RWD) (Arima(0,1,0) with drift), and the appropriate Arima model based on Bayesian Information Criterion (ARIMA). It has to be noted that our evaluation considers two experimental setups to evaluate the capacity of the HS extensions on datasets having both long and short periods. More speciﬁcally, apart from the E&W dataset that offers an extremely long ﬁtting period, we apply the models to the Greek dataset, which begins in 1961 and contrasts to the rather long periods of the E&W mortality data that start in 1841. The motivation behind this approach lies in the fact that the utilisation of a different link function and distribution could make the difference to the ﬁtting process and deliver a more accurate model ﬁt, as the link function and the distribution could aid in capturing the intrinsic characteristics of the data in a more precise way. 4.1. Evaluation Metrics We have chosen a set of quantitative criteria in an effort to approach the problem from different angles. Both Information Criteria and Percentage error tests are used to evaluate the ﬁtting performance of the models. Speciﬁcally, we use the Bayesian Information Criterion (BIC), the Akaike Information Criterion (AIC), the Mean Squared Percentage Error (MSPE) and the Mean Absolute Percentage Error (MAPE). Additionally, when it comes to the qualitative analysis of the results, we adopt the unexplained variance ratio (UVR) which is used to reveal the magnitude of the captured information by each age– period and age–cohort component. Risks 2022, 10, 131 14 of 30 Information Criteria: A common practice in the mortality domain is the use of in- formation 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 AIC = 2l + 2 npar and BIC = 2l + npar log N where N is the number of observations, l is the log-likelihood and npar is the number of parameters being estimated, and is calculated based on the following formula: n par = 2 + (k + k ) + p (n + k ) + q (n + k ) 2 c 2 1 1 where n is the index of last calendar year (i.e., t = t , . . . , t ) and n is the index of the last n c cohort (i.e., c = c , . . . , c ) 1 n Percentage Error Tests: We use percentage error tests to measure the difference be- tween 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 to x and for all years t to t , these measures are deﬁned as 1 a 1 n follows: 1 q ˆ q 1 jq ˆ q j t,x t,x t,x t,x MSPE = and MAPE = åå åå t x q t x q t,x t,x t x t x where q ˆ is the estimated central mortality rate and q is the observed crude central mortal- t,x t,x ity rate. 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 In this section, we present the data and the optimum parameters of the HS model and its extensions used to evaluate the proposed methodology described in Section 3. Sections 4.2.2 and 4.2.3 document the results and proceed with the comparative analysis. 4.2.1. Mortality Data and the Optimum Parameters The historic mortality data used are the same presented in (Hatzopoulos and Sagianou 2020) for the purpose of making our results directly comparable. We have selected datasets of both long and short periods to evaluate the behaviour of all extensions of the HS model in an effort to validate its modelling capacity both in the case of highly informative and non-informative data. The latter are provided by the (Human-Mortality-Database n.d.) and (Eurostat n.d.), having the following characteristics: • 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. Tables 2 and 3 present the parameters used for ﬁtting each model structure for each of the aforementioned datasets. The tables contain the number of optimum parameters of Risks 2022, 10, 131 15 of 30 the GLM polynomials, k (Equation (7)) and k (Equation (9)), for the age–period and age– 1 2 cohort effects, respectively. In addition, the tables document the optimum s values, while p and q denote the optimum number of the period (Equation (8)) and cohort (Equation (11)) t t c c components, according to the approach described in Section 3. x , q and x , q are the parameters (see Section 3.2.1) of the adopted distribution for the age–period and the age– cohort part of the model, respectively. Table 2. Optimum parameters for each HS model extension structure for the E&W dataset. Country: E&W Years: 1841–2016 Ages: 0–89 (Years) t t c c Model s k p k q x q x q 1 2 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 Table 3. Optimum parameters for each HS model extension structure for the GR dataset. Country: Greece Years: 1961–2013 Ages: 0–84 (Years) t t c c Model s k p k q x q x q 1 2 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 In our application, we make use of the Generalised Pareto and Beta distributions to offer the HS_prt and the HS_beta model extensions, respectively. The former utilises the probability density function for the Generalised Pareto distribution with shape parameter x 6= 0, scale parameter q, as given in Equation (14), based on the MATLAB implementation given in (MATLAB n.d.b). 1 x (14) f (xjx, q) = 1 + x q q and the respective cumulative distribution function is deﬁned as: (15) F(xjx, q) = 1 1 + x In our implementation, we set the scale parameter q = 1. Equation (14) can take the form of the Lomax distribution as a special case of the Generalised Pareto distribution, with 1 l x = and q = . In that case, the probability density function is a a 1a a x (16) f (xja, l) = 1 + l l 1 q where a = and l = . x x For the Beta extension, we make use of the probability density function of Equa- tion (17), with ﬁrst shape parameter x and second shape parameter q, following the MAT- LAB implementation given in (MATLAB n.d.a). Risks 2022, 10, 131 16 of 30 q1 x1 f (xjx, q) = x (1 x) (17) B(x, q) and the respective cumulative distribution function is deﬁned as: B(xjx, q) F(xjx, q) = (18) B(x, q) where B(x, q) is the beta function and B(xjx, q) is the incomplete gamma function. 4.2.2. E&W Data Performance Analysis E&W Quantitative Analysis Table 4 summarises the quantitative results that reﬂect the efﬁcacy of the HS model and its extensions for the ﬁtting process in terms of the log-likelihood-based tests (AIC, BIC) and PE tests (MSPE, MAPE). The best scores are highlighted in bold, with the HS_prt extension being dominant among them. Table 4. Results of the quantitative tests for E&W mortality dataset. Model k npar Log- AIC BIC MSPE MAPE Likelihood (%) (%) 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 A key observation that clearly supports the motivation of this work is that the HS model under the binomial form and q modelling reached a higher level of performance. t,x More speciﬁcally, four out of the ﬁve extensions performed better than the original HS_log model, showing better goodness-of-ﬁt through the log-likelihood, AIC and BIC metrics and a slight improvement for the MAPE. Notably, even though the BIC is destined to penalise complex structures, the HS_lgt and HS_prbt extensions score a better BIC even when having more parameters than the HS_log, meaning that the additional ones contribute substantially to the improvement of ﬁt. In fact, the better performance of the HS_lgt and HS_prbt will be noted also on the qualitative results later in this section. Among the HS extensions based on the off-the-shelf link functions, the HS_prbt is the one achieving the best performance under all metrics, having also less parameters than the HS_lgt and HS_cll. On the downside, the HS_cll extension had the lowest performance among the collected results. When it comes to the structure of the models of the HS_lgt, HS_cll and HS_prbt extensions, according to Table 2, the number of the generated p and q components remains the same as those of the HS_log. However, a slight deviation is noted for the k and k 1 2 factors, which also justiﬁes the increased number of parameters noted in Table 2. Overall, as will be shown in the qualitative analysis, the HS extensions remain consistent when it comes to the identiﬁed p and q components and the corresponding age ranges which are revealed. When it comes to the quantitative results of the HS extensions based on the new form of link functions, i.e., the HS_beta and HS_prt cases, the latter is the one dominating among all, while the former is ranked around the performance achieved for the HS_lgt, HS_cll and HS_prbt cases. According to Table 4, the HS_beta incorporates the lowest number of parameters, resulting in a more simple model structure, but this is reﬂected as an inefﬁciency in the rest of the qualitative metrics, implying that the HS_beta cannot capture the E&W data dynamics in an efﬁcient manner. When it comes to the incorporated p and q Risks 2022, 10, 131 17 of 30 components, the HS_beta still remains consistent, as will be discussed in the qualitative analysis, but fails to make a difference in terms of the quantitative metrics. Notably, the case of the HS_prt extension is of special interest as it outperforms the rest of the extensions and the original HS_log for all metrics, apart from a negligible deﬁciency on the MSPE. More speciﬁcally, contrary to the HS_log, the HS_prt has an improvement of 16.24%, 16.67% and 16.79% for the BIC, AIC and log-likelihood metrics, respectively, while the MAPE is improved by 2.7%. According to Table 4, the HS_prt comes with a decreased number of parameters and a smaller number of the optimum parameters of the GLM orthonormal polynomials (k ). Hence, the HS_prt extension concludes to a simpler model which also produces lower error in the ﬁtting process. In addition, according to the p and q parameters in Table 2, the HS_prt has identiﬁed one age–period component more (i.e., 6) and one age–cohort component less (i.e., 1). Such a difference may sound minor, but as will be documented in the qualitative analysis, it is quite substantial for the explainability of mortality. Moreover, the results of the quantitative evaluation of the well-known stochastic mortality models are given in Table 5. According to the table, and having selected the best model from the extensions in Table 4, we notice that the HS_prt not only performs better compared to the HS_log but also achieves the best scores among the rest of the well-known stochastic mortality models. 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 E&W Qualitative Analysis The results of the qualitative analysis aim to complement the numeric ones presented in Table 4. To do so, Appendix B offers the graphical representation of the age–period and age–cohort components for both the off-the-shelf HS extensions and those following the new form of link function. The graphical representation reveals the identiﬁed mortality trends and eases the attribution of a trend to a speciﬁc age cluster. Figures A1 and A2 provide a graphical representation of the age–period components and the corresponding UVR values for all the HS extensions. One can observe that the HS extensions remain consistent with the HS_log and reveal the same age clusters. However, in some cases, according to the UVR values, the HS extensions under the Binomial form can capture more information from the mortality data. More speciﬁcally, according to UV R (1), the HS_prbt, HS_beta and HS_prt model structures are able to grasp more information regarding the 60+ ages. In fact, this reﬂects a slightly different mortality trend in the k factor for the HS_prt. The k for the HS_prbt and HS_beta is factorised in a different way than the other cases, that is why the corresponding graphs seem to be dislocated. Despite this fact, the HS_prbt and HS_beta generate k graphs which are almost identical to the HS_prt case. UV R (3) and UV R (4) refer to the 20-25 and 30-40 age clusters, respectively. The x x lower UVR values achieved by the HS_prbt structure, which are illustrated in Figures A1 and A2, conﬁrm the good performance advocated by the quantitative per- formance reported in Table 4. When it comes to Figure A2, the HS extensions have revealed a more clear age cluster compared to the HS_log. More speciﬁcally, according to UV R (5), the HS_log had initially x Risks 2022, 10, 131 18 of 30 identiﬁed a mixed age cluster for the 10–20 and 20–25 age ranges. In this work, all the HS extensions clearly attribute the k mortality trend to the 10–20 age range, while the 20–25 age cluster is clearly addressed by the 3rd age–period component. Overall, the HS_prt structure is of special interest. Again, one can notice that the HS model for the Generalised Pareto case remains consistent as, in general terms, the identiﬁed age–period and age–cohort components reveal ﬁne-grained age clusters and ease the attribution of a mortality trend to distinct age ranges. In fact, the most important outcome of the application for the HS_prt is the identiﬁcation of one additional and distinct age–period component. UV R (6) is a new informative component that reveals the mortality trend for the 25–30 age range. As aforementioned, the focal point of UV R (3) is the 20–25 age range, while a signiﬁcant and clear trend for the 10–20 age cluster is revealed in k as advocated by UV R (5). In fact, in the case of the HS_log, the 10–20 cluster was included in the captured variance of the 5th age–period component along with the variance of the 20–30 age range. Considering these facts, the HS_prt model generated more ﬁne-grained age clusters, and it eases the attribution of a mortality trend to unique age clusters, contributing signiﬁcantly to the explainability of the model. This behaviour justiﬁes the improvement of around 16% in the quantitative metrics in contrast to the HS_log. Moreover, in Figure A3, one can notice that for the HS_prt, only one age–cohort component is identiﬁed. This is a normal outcome, as the age–period modelling process has captured most of the data variance and resulted in one additional age–period component (six in total). As a result, the residuals bear the information for the identiﬁcation of one age–cohort component. Generally, this modelling behaviour results in a less complex model structure which incorporates less parameters as shown in Table 2. Overall, when it comes to the off-the-shelf HS extensions, the HS_prbt case seems to perform better both in the qualitative and quantitative evaluation results. When using the new form of link function, the HS_prt case is the one that stands out among all cases. 4.2.3. Greek Data Performance Analysis Based on Table 3, the Greek dataset covers a shorter period (from 1961) in contrast to the rather long period for the E&W data. One could say that it is reasonable for multiple component models, such as the HS model and the introduced extensions, to achieve good results. That is, in order to verify that the better performance of the HS extensions (as happened for the case of the E&W data) is not solely attributed to the extremely long ﬁtting periods, we also conduct the evaluation using the shorter GR dataset, and we assess the consistency and performance of the Binomial HS extensions with respect to the original Poisson model (HS_log) presented in (Hatzopoulos and Sagianou 2020). In fact, according to the (Human-Mortality-Database n.d.) and as reported in the (HMD-Greek-data n.d.), the GR dataset contains inconsistencies and is of poor quality. Hence, due to the limited time period of the GR data and the low quality, we form a more challenging basis for a model to thrive. In addition, we aim to prove through experimental results that the model extensions inherit the beneﬁcial characteristics of the HS model and can perform even better through the use of the suggested transformations of our methodology, being at the same time consistent with the identiﬁcation of mortality trends for unique age clusters. GR Quantitative Analysis Following the same approach as in Section 4.2.2, Table 6 summarises the quantitative results that reﬂect the efﬁcacy of the HS model and its extensions for the ﬁtting processes. The best scores are highlighted in bold, with the HS_prt being the dominant and the HS_beta being very close to the HS_prt. Risks 2022, 10, 131 19 of 30 Table 6. Results of the quantitative tests for the Greek mortality dataset. Model k npar Log- AIC BIC MSPE MAPE Likelihood (%) (%) 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 As advocated by the quantitative results, also for the case of the short GR dataset, the HS model under the binomial form and q modelling achieved better performance t,x scores. More speciﬁcally, all four extensions performed better than the original HS_log model, showing better goodness-of-ﬁt through the log-likelihood, AIC and BIC metrics and a slight improvement for the MAPE. However, it must be mentioned that the improvement is not as signiﬁcant as the one achieved for the E&W case. Taking a closer look at the performance of the HS extensions for the off-the-self link functions, the HS_prbt is the one that stands out in contrast to the HS_lgt and HS_cll, which achieve quite similar performances with the initial HS_log. In fact, the HS_prbt has a negligible improvement for the MSPE and the MAPE, but the improvement in the log-likehood, AIC and BIC needs to be highlighted. As can be seen, the HS_prbt is based on 516 parameters, leading one to expect higher BIC and AIC scores due to the more complex structure. Though, despite the higher number of parameters, the HS_prbt structure achieves a better BIC, meaning that the additional parameters contribute substantially to the improvement of ﬁt. In fact, the better performance of the HS_prbt is more evident in the qualitative results which are analysed later in this section. According to Table 3, the number of the generated p and q components of the HS_lgt and HS_cll extensions remain the same as those of the HS_log (i.e., 4) and so does the k factor. However, for the HS_prbt, a new additional age–period component is identiﬁed; thus, p = 5, and as will be noted in the qualitative analysis, this component contributes to the explainability of the model. A case of special interest is the application of the HS_prt, for which the new form of link functions led to a notable improvement of the model. More speciﬁcally, Table 6 reports that the HS_prt achieves a signiﬁcant improvement for all the evaluation metrics, apart from the MSPE, for which the HS_beta is slightly better. More speciﬁcally, contrary to the HS_log, the HS_prt has an improvement of 1.96%, 3.54% and 4.25% for the BIC, AIC and log-likelihood metrics, respectively, while the MSPE is improved by 16.06% and the MAPE by 6.10%. Even though the magnitude of improvement is less than the improvement for the case of the E&W data, one needs to consider the challenging nature of the Greek mortality data, where the room for improvement is rather limited. In addition, the HS_beta achieves almost identical performance to the HS_prt and, without a doubt, it is also a competitive extension for the GR case. The HS_log model was slightly lacking in performance when compared with the Renshaw and Haberman (2006) and Plat (2009) models, speciﬁcally in the case of the Greek data, as reported by the comparative study given in (Hatzopoulos and Sagianou 2020) and as can be seen in Table 7. By utilising the proposed methodology, the HS_prt and HS_beta extensions have signiﬁcantly improved the goodness-of-ﬁt. In addition, according to the p and q parameters in Table 3, the HS_prt and HS_beta have identiﬁed one age– period component more (i.e., 5), as was the case with the HS_prbt, and this contributes substantially to the explainability of mortality. The results of the quantitative evaluation against the well-known stochastic mortality models LC, RH, APC and PL are given in Table 7. According to the table, and having selected the best model from the extensions in Table 6, we note that the HS_prt is the model variation that achieves a similar performance to the RH model in terms of the quantitative Risks 2022, 10, 131 20 of 30 tests. However, the RH can be questioned for its robustness, as it faces difﬁculties to converge and presents unstable behaviour (Hunt and Villegas 2015). 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 It must be noted that the HS_prt achieved a signiﬁcant improvement in the case of the E&W data, contrary to the GR. As discussed before, the Generalised Pareto belongs to the heavy-tailed distributions family. That is, it is expected to perform better when extreme values are part of the data. As can be seen from the k ﬁgures in Appendices B and C, the E&W data include several high peaks in contrast to the GR. That is why the HS_prt seems to have a greater performance improvement in the E&W case. GR Qualitative Analysis The results of the qualitative analysis complement the numeric ones presented in Table 6 by explaining the UVR for the identiﬁed age clusters. The graphical representations of the identiﬁed age–period and age–cohort components of the Greek dataset are given in Appendix C. Figures A4 and A5 provide a graphical representation of the age–period components and the corresponding UVR values for the HS extensions. As a general remark, one might note that all the extensions remain consistent, but the HS_prbt, HS_beta and HS_prt identiﬁed an extra (5th) age–period component. Overall, slight differences in the UVR graphs and the mortality trends can be noticed among the other extensions. The HS_prbt is of special interest as its structure leads to the deﬁnition of more ﬁne- grained age clusters. More speciﬁcally, focusing on Figure A4, UV R (1) reveals that the HS_prbt is able to grasp more information regarding the 60+ ages. UV R (4) is a component that reﬂects the difﬁculty to identify clear trends on the Greek mortality data. As can be seen, the interpretation of the cluster is quite difﬁcult, as for all model extensions, there is no strong indication of the age cluster which is being highlighted. On the bright side, the HS_prbt is able to identify a quite clear cluster and the corresponding mortality trend for the 30–40 age cluster. In addition, UV R (2) shows a clearer concentration of the explained variance to the ages around 80, in contrast to the rest of the extension which makes an attribution to the 70–80 cluster. In fact, the HS_prbt has identiﬁed a clear cluster for the ages of 70 in UV R (5) of Figure A5. Hence, it is evident that the HS_prbt structure achieved a clean cut in the 70–80 cluster, and it eases the attribution of a trend to a speciﬁc age cluster. In addition, despite the different structure achieved for the age–period components (i.e., 5 instead of 4), UV R (1) in Figure A6 remains consistent and still captures considerable variance of the data. When it comes to the HS_prt, as advocated by the quantitative results, it achieved a noticeable improvement from the HS_log case, and this is actually reﬂected in the ﬁgures of Appendix C. UV R (2) for the HS_prt can signiﬁcantly explain the variance of the 80s age range, while the focal point of UV R (3) is on the 20s. The middle ages of 40–50 are addressed by UV R (4), and ﬁnally, a newly identiﬁed cluster for the 70s is highlighted in UV R (5). Almost the same behaviour can be seen for the HS_beta extension. The most important outcome of the application for the HS_prt and HS_beta, as was the case for the HS_prbt, is the identiﬁcation of the 5th age–period component. This was not initially Risks 2022, 10, 131 21 of 30 uncovered by the HS_log model. Thus, the addition of the new component justiﬁes the improvements noted for the quantitative performance metrics. The HS_prt model remains consistent as the identiﬁed age–period and age–cohort components reveal ﬁne-grained age clusters and ease the attribution of a mortality trend to distinct age ranges, even when operating over the non-informative Greek dataset. When it comes to the age–cohort components, Figure A6 shows that for the HS extensions, UV R (1) remains consistent and still captures a considerable variance of the data for all cases. However, due to the existence of the 5th age–period component for the HS_prbt, HS_beta and HS_prt, the age–cohort differs slightly from that of the HS_log, HS_cll and HS_lgt. Nonetheless, the age–cohort components are still informative, they denote a clear age cluster and contribute to the explainability of the model. 4.2.4. Out-of-Sample Results The results of the quantitative analysis for the forecasting process complement the comparative evaluation. In order to evaluate the HS_prt extension under the out-of-sample mode, the model needs to be ﬁt to a shorter dataset, excluding the years to be forecasted. This implies that the ﬁtting and estimation processes will be executed again in order to deﬁne the necessary model parameters. The parameters of the HS_prt for the E&W and GR datasets for the shorter periods are given in Table 8. Table 8. Optimum parameters for HS_prt structure for the E&W and GR shorter datasets. Model: HS_prt t t c c Country Years Ages s k p k q x q x q 1 2 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 According to the results of Tables 9 and 10, the HS_prt model under the RWD and ARIMA achieves lower error scores and demonstrates a consistent behaviour also under the forecasting mode. More speciﬁcally, for the E&W case, the difference in the MSPE and MAPE is considerable, comparing to the other well-known models. This is not only attributed to the high goodness-of-ﬁt performance of the model, which provides a solid base to be extrapolated, but also due to the fact that the HS model deﬁnes multiple components which reveal the clear mortality trends of unique age clusters. On the contrary, the unreasonably high error values of the RH and PL models come, to a certain extent, as a result of their unstable behaviour during the ﬁtting process and the limitations of their structure and estimation process. The results for the GR dataset in Table 10 show overall a more consistent behaviour for all models. The HS_log was slightly lacking in performance in contrast to the PL and RH models. However, the newly derived HS_prt extension was able to outperform the rest of the models both under the MSPE and the MAPE metrics in the RWD and ARIMA approaches. Overall, it seems that the HS model, due to its structure, i.e., it is a multiple-component model which reveals unique age clusters, and the notable goodness-of-ﬁt performance achieved by utilising the newly introduced methodology, delivers an informative—and close to the actual data—basis that can lead the out-of-sample process to a more precise prediction. Risks 2022, 10, 131 22 of 30 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 5. Discussion and Conclusions This section offers a discussion over our experimental results and highlights the lessons learnt by the adoption of the various link functions in a model’s estimation methods, having witnessed the beneﬁcial impact of the proposed methodology to our model’s efﬁcacy. Our aim was to extend the HS model and investigate its behaviour when formulated in terms of q , using generalised linear models and by adopting various link functions. Our t,x motivation originates from similar endeavours in the literature (Currie 2016; Haberman and Renshaw 1996) and is aligned with the capabilities offered by well-known tools, such as StMoMo (Millossovich et al. 2018)). To the best of our knowledge, it is the ﬁrst time that a systematic approach is documented for the adoption of this new form of link functions, F (x; x , q) 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. An additional point to be highlighted is the integration of heavy-tailed distributions (Generalised Pareto in our case). The motivation is two-fold. On the one side, the proposed methodology of Section 3 enables the mortality analyst to apply such kinds of distributions in a mortality model in case the intrinsic characteristics of the mortality data ﬁt better to a heavy-tailed distribution proﬁle, i.e., when extreme events must be part of the modelling. In this context, the proposed methodology eases an analyst to adapt a model depending on the data, the time period and the case to be analysed, without being limited only to the off-the-shelf link functions and distributions. Based on our results for the k graphs given in the appendices, one can observe that the long period of the E&W data includes several mortality trend peaks (extreme values) as a result of the mortality during critical Risks 2022, 10, 131 23 of 30 events of the 20th century. However, the Greek mortality data and the respective k graphs do not imply the presence of extreme events and outliers. Thus, despite the fact that the improvement on the use of the new form of link functions and different distributions is evident based on the quantitative and qualitative results, the improvement is greater for the E&W for which we ﬁnally applied the Generalised Pareto heavy-tailed distribution to better ﬁt to the nature of the data. Even though the rationale behind the use of the heavy-tailed distribution in the mortality realm is valid, we need to note that this outcome needs to be further evaluated. In fact, we aim to continue in this line of research and to explore in our future endeavours the behaviour of such distributions in mortality modelling. In addition, it has to be noted that the adaptation of a mortality model to the new form of link functions implies an additional step in the model’s estimation process, as explained in detail in Section 3.2, in order to estimate the x and q parameters. In terms of the codebase, this requires the addition of an extra, but simple routine, which can tackle this problem. Overall, the complexity of the process is slightly increased but we argue that the potential beneﬁt outweighs the added complexity. The above-mentioned highlights are advocated by our experimental results, while the efﬁcacy of the HS model extensions support the motivation of this work. In fact, the transition to the use of the Binomial distribution has improved the performance of the HS model under the vast majority of the adopted link functions. Notably, the performance improvement for the reference model of the HS_log was even greater in the case of the HS_prt extension for the E&W data, where we noted an improvement of 16.24%, 16.67% and 16.79% for the BIC, AIC and log-likelihood metrics, respectively, while the MAPE improved by 2.7%. In the case of the short GR dataset, the HS_prt extension was the one that outperformed the rest, achieving a similar performance with the HS_beta, proving again that the selection of the correct distribution and link function can further boost the performance of a mortality model even in cases where we need to operate over a poor and non-informative dataset. During our experiments, we applied several distributions, including the Generalised Extreme value, Gumbel, Fréchet, Weibull and Burr, and we experimented with other various datasets (e.g., French data). In order to keep the length of the paper within reasonable limits, we chose to present the best results. However, our methodology enables an analyst to apply virtually any distribution in the modelling process. In addition, we offered a comparative analysis among the HS extensions and other well-established mortality models of the literature, both under the ﬁtting and forecasting modes. The results showed that the HS_prt achieved the best goodness-of-ﬁt performance, while under the forecasting mode, the new HS_prt extension improved the position of the HS and outperformed the rest of the models in terms of prediction accuracy. In addition, we need to note that the HS improvement through the introduced exten- sions is not solely reﬂected in the quantitative results, but most importantly, it is reﬂected in the qualitative ones. The HS model was able to improve its performance in the identiﬁca- tion of more ﬁne-grained age–period and age–cohort components, and thus prove that the proposed methodology retains the model’s efﬁcacy and consistency, and it can boost even more its explainability through the attribution of a mortality trend to unique age clusters. This feature is one of the unique characteristics of the HS model in contrast to other models of the literature. Overall, our application advocates that different mortality data imply the need for a different model transformation in order to increase the goodness-of-ﬁt, capture the data dynamics and uncover interesting characteristics in the mortality trend of individual age clusters. Our results show that even a top-notch mortality model, such as the HS, has room for improvement, as the adaptation with an appropriate link function can lead to even better qualitative and quantitative performance. This work complements the series of work for positioning the HS model and its extensions among the other well-established models of the literature. As future work, we aim to investigate machine learning-based approaches in mortality modelling (Deprez et al. Risks 2022, 10, 131 24 of 30 2017; Levantesi and Pizzorusso 2019) that could beneﬁt our approach and to explicitly evaluate their explainability. Author Contributions: Conceptualisation, A.S. and P.H.; methodology, A.S.; software, A.S.; val- idation, P.H.; investigation, A.S.; data curation, A.S.; writing—original draft preparation, A.S.; writing—review and editing, P.H.; visualisation, A.S.; supervision, P.H.; project administration, A.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the “Ypatia” scholarship of the University of the Aegean. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. Abbreviations The following abbreviations are used in this manuscript: 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 We consider the class of models for the probability of deaths with Binomial errors and a user-deﬁned link function. We suppose D B(E, q) have the Binomial distribution and let Q = D/E be the random variable corresponding to q. Here, q = E(Q) is the mean of Q, E is the initial exposure and h is a linear function of the explanatory variables. A user-deﬁned link in MATLAB requires three functions: the link function itself, i.e., h as a function of q, the inverse link function, i.e., q as a function of h, and the derivative of h with respect to q. As deﬁned in Section 3.2, we reported three cases depending on the cumulative distribution chosen. 1. The cumulative distribution as link function g, maps q, 0 < q < 1, to ¥ < F (q; x, q) < ¥, so that: h = F (q; x, q) Risks 2022, 10, 131 25 of 30 q = F(h; x, q) dh 1 = F (q; x, q) = dq f (F (q; x, q); x, q) 2. The cumulative distribution as link function, g, maps q, 0 < q < 1, to 0 < F (q; x , q) < ¥, so the logarithmic form of the cumulative distribution is needed to map q, to ¥ < log(F (q; x, q)) < ¥, the natural scale for regression, so that: h = log F (q; x , q) q = F(exp(h); x , q) dh 1 1 = F (q; x, q) = 1 1 1 dq F (q; x, q) F (q; x, q) f (F (q; x, q); x, q) 3. The cumulative distribution as link function, g, maps q, 0 < q < 1, to 0 < F (q; x , q) < 1, so we need the logit of the cumulative distribution so that maps q, to ¥ < logit(F (q; x, q)) < ¥, the natural scale for regression, so that: F (q; x, q) h = logit F (q; x, q) = log 1 F (q; x, q) exp(h) q = F ; x, q 1 + exp(h) F (q; x , q) dh 1 = = 1 1 1 1 1 dq F (q; x , q) (1 F (q; x, q)) f F (q; x, q) F (q; x, q) (1 F (q; x , q)) Note that there is no need to deﬁne the derivative of any cumulative distribution, as this can be achieved by using the probability distribution function as described below. By using the chain rule: f ( f (x)) = x and therefore d d f ( f (x)) = x = 1 (A1) dx dx and also by chain rule 1 0 1 1 0 f ( f (x)) = f ( f (x)) ( f ) (x) (A2) dx Thus, by (A1) and (A2), we have 0 1 1 0 f ( f (x)) ( f ) (x) = 1 1 0 ( f ) (x) = 0 1 f ( f (x)) Thus, in our case, by replacing f with F, we have 1 1 1 0 (F ) (x) = = 0 1 1 F (F )(x) f (F (x)) In addition, the MATLAB code for the user-deﬁned link function is: l i n k = @(mu) l o g ( g p i n v (mu, x i , t h e t a ) ) ; d e r l i n k = @(mu) 1 . / ( g p i n v (mu, x i , t h e t a ) . gppdf ( g p i n v (mu, x i , t h e t a ) , x i , t h e t a ) ) ; * Risks 2022, 10, 131 26 of 30 i n v l i n k = @( e t a ) g p c d f ( exp ( e t a ) , x i , t h e t a ) ; new_F = { l i n k , d e r l i n k , i n v l i n k } ; B = g l m f i t ( Lx , q t x , ’ b i n o m i a l ’ , ’ l i n k ’ , new_F , ’ w e i g h t s ’ , e t x 0 , ’ c o n s t a n t ’ , ’ o f f ’ ) 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. Appendix B. Age–Period, Age–Cohort Components and Unexplained Variance Ratio Graphical Representations for E&W Dataset 1 1 HS.1 HS.2 HS.3 UVR (1) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1850 1900 1950 2000 0 20 40 60 80 2 2 HS.4 HS.5 HS.6 UVR (2) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1850 1900 1950 2000 0 20 40 60 80 HS.7 3 HS.8 3 HS.9 UVR (3) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1850 1900 1950 2000 0 20 40 60 80 HS.10 4 HS.11 4 HS.12 UVR (4) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1850 1900 1950 2000 0 20 40 60 80 Figure A1. First to fourth age-period component for E&W dataset. EW EW EW EW -0.2 -0.1 0.0 0.1 0.2 -0.3 -0.1 0.1 -0.15 -0.05 0.05 0.15 0.00 0.10 0.20 -2 -1 0 1 2 -1.0 0.0 1.0 2.0 -4 -2 0 2 -15 -5 0 5 10 0.2 0.6 1.0 1.4 0.0 0.5 1.0 1.5 0.0 0.4 0.8 1.2 0.0 0.5 1.0 1.5 2.0 Risks 2022, 10, 131 27 of 30 HS.13 5 HS.14 HS.15 UVR (5) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1850 1900 1950 2000 0 20 40 60 80 HS.16 6 HS.17 6 HS.18 UVR (6) b k x x t pareto 0 20 40 60 80 1850 1900 1950 2000 0 20 40 60 80 Figure A2. Fifth and sixth age-period component for E&W dataset. HS.19 c(1) HS.20 (1) HS.21 c(1) UVR b g x x c log logit cloglog probit beta pareto 0 20 40 60 80 1880 1900 1920 1940 0 20 40 60 80 c(2) (2) HS.22 HS.23 HS.24 c(2) UVR b g x x c log logit cloglog probit beta 0 20 40 60 80 1880 1900 1920 1940 0 20 40 60 80 Figure A3. First and second age-cohort component for E&W dataset. EW EW EW EW -0.2 0.0 0.2 0.4 -0.10 0.00 0.10 0.20 -0.3 -0.1 0.1 0.2 -0.2 0.0 0.2 -0.2 0.0 0.2 -0.6 -0.2 0.2 0.6 -2 0 2 4 6 -0.5 0.5 1.5 2.5 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.2 0.6 1.0 1.4 0.5 1.0 1.5 2.0 Risks 2022, 10, 131 28 of 30 Appendix C. Age–Period, Age–Cohort Components and Unexplained Variance Ratio Graphical Representations for GR Dataset HS.25 1 HS.26 1 HS.27 UVR (1) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1960 1980 2000 0 20 40 60 80 2 2 HS.28 HS.29 HS.30 UVR (2) b k x log logit cloglog probit beta pareto 0 20 40 60 80 1960 1980 2000 0 20 40 60 80 3 3 HS.31 HS.32 HS.33 UVR (3) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1960 1980 2000 0 20 40 60 80 HS.34 4 HS.35 4 HS.36 UVR (4) b k x x t log logit cloglog probit beta pareto 0 20 40 60 80 1960 1980 2000 0 20 40 60 80 Figure A4. First to fourth age-period component for GR dataset. HS.37 5 HS.38 5 HS.39 UVR (5) b k x x t probit beta pareto 0 20 40 60 80 1960 1980 2000 0 20 40 60 80 Figure A5. Fifth age-period component for GR dataset. GR GR GR GR GR -0.2 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2 -0.2 0.0 0.2 -0.1 0.1 0.2 0.3 0.00 0.10 0.20 0.30 -0.6 -0.2 0.2 0.6 -0.5 0.0 0.5 -1.5 -1.0 -0.5 0.0 0.5 -1.5 -1.0 -0.5 0.0 0.5 -4 -2 0 2 4 0.5 1.0 1.5 0.6 0.8 1.0 1.2 0.4 0.8 1.2 0.2 0.6 1.0 0.0 0.4 0.8 1.2 Risks 2022, 10, 131 29 of 30 HS.40 c(1) HS.41 (1) HS.42 c(1) UVR b g x x c log logit cloglog probit beta pareto 0 20 40 60 80 1880 1900 1920 1940 0 20 40 60 80 Figure A6. Age-cohort component for GR dataset. References Booth, Heather, and Leonie Tickle. 2008. Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science 3: 3–43. [CrossRef] Booth, Heather, John Maindonald, and Len Smith. 2002. Applying lee-carter under conditions of variable mortality decline. Population Studies 56: 325–36. 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ISSN: 2227-9091
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Risks – Multidisciplinary Digital Publishing Institute

Published: Jun 21, 2022

Keywords: mortality modelling; generalised linear models; link functions; heavy-tailed distributions; longevity risk; solvency II

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

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Extensions on the Hatzopoulos&ndash;Sagianou Multiple-Components Stochastic Mortality Model

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

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