# A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach

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## Abstract

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## 1. Introduction

## 2. A New Class of Counting Distributions on the Set of Nonnegative Integers ${\mathbb{N}}_{\mathbf{0}}$

#### 2.1. NEFs—Some Preliminaries

#### 2.2. The Mean Value Parametrization of NEFs

**Remark**

**1.**

#### 2.3. Polynomial VFs of Counting NEFs Supported on the Set of Nonnegative Integers ${\mathbb{N}}_{0}$

#### 2.4. A New Class of Polynomial VFs—The ABM NEFs

- It is a class of counting distributions supported on the non-negative integers;
- It is overdispersed as $V\left(m\right)/m>1$;
- It allows a mean value parameterization in a closed form;
- It is infinitely divisible, which allows the construction of an exponential dispersion model (EDM) with dispersion parameter space equal to ${\mathbb{R}}^{+}$. EDMs are used to describe the error distribution in generalized linear models (see Jorgensen 1987, 1997);
- ${p}_{1}$ is an unknown parameter to be estimated (see next section). ${p}_{2}\in {\mathbb{N}}_{0}$ is a parameter governing the particular model within the ABM class and is considered to be a decision variable (note that different values of ${p}_{2}$ determine different ABM NEFs). Accordingly, for given national datasets (i.e., those of US, Ireland and Ukraine), the goal will be to locate that value of ${p}_{2}$, which minimizes a respective RMSE (see in the sequel). However, due to the rather cumbersome and intractable structure of the ABM probabilities (or likelihood) in (6) and the fact that the larger the ${p}_{2}$, the larger the number of elements in the summands appearing in (6), no analytic solution for an optimal ${p}_{2}$ is feasible at all for achieving such a goal. Consequently, only numerical search algorithms are plausible. The search starts with ${p}_{2}=0$ (the Poisson NEF), ${p}_{2}=1$ (the negative binomial NEF), ${p}_{2}=2$ (the Abel NEF) and so on;
- As already noted, the ABM class ${\left\{ABM({p}_{1},{p}_{2})\right\}}_{{}^{{p}_{2}}\in {\mathbb{N}}_{0}}$ is composed of infinitely countable set of families of counting NEFs supported on the non-negative integers and thus can also be used to model real datasets by employing the classical frequency approach (and not only Bayesian). Indeed, the ABM class has been compared in Bar-Lev and Ridder (2021a, 2021b) with other common counting probability models (such as Poisson-inverse Gaussian distribution, new logarithmic distribution, an exponentiated discrete Lindley distribution) for various real count datasets stemming from automobile insurance claims, marketing, biometry, health, and social sciences (none of which is related to mortality projections). Members of the ABM counting class have shown superiority with respect to various metrics for goodness-of-fit tests (chi-squared test, Akaike information criterion (AIC), root-mean-square error (RMSE) and Kullback–Leibler divergence (KL)), and provided a much better fit for each of the datasets considered (more details can be found in Bar-Lev and Ridder 2021b).

## 3. ABM Based LC Model and its Bayesian Framework

**The prior distribution for ${\mathit{k}}_{\mathit{t}}$ and $\mathit{\theta}$**

**The prior distribution for ${\mathit{\beta}}_{\mathit{x}}$**

**The prior distribution for ${\mathit{\alpha}}_{\mathit{x}}$**

**The prior distribution for ${\mathit{p}}_{\mathbf{1}}$**

#### MH (Metropolis–Hastings) Algorithm for Estimating the Parameters $\alpha ,\beta ,k$ and ${p}_{1}$

**Estimation of ${\mathit{k}}_{\mathit{t}}$ using the MH algorithm**

- Draw ${k}_{t}^{*}$ from the proposal density function $N({k}_{t}^{\left(i\right)},{\sigma}_{t}^{2})$, such that ${\sigma}_{t}^{2}$ is assumed known;
- Calculate the following probability:$$\Psi \left({k}_{t}^{\left(i\right)},{k}_{t}^{*}\right)=min\left(1,\frac{f({k}_{t}^{*}\mid D,\alpha ,\beta ,{k}_{-t}^{\left(i\right)},\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})}{f({k}_{t}^{\left(i\right)}\mid D,\alpha ,\beta ,{k}_{-t}^{\left(i\right)},\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})}\right),$$
- Draw a value u from uniform probability function in range $U(0,1)$ and decide in accordance with the following formula:$$\left\{\begin{array}{c}if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}\le \phantom{\rule{4.pt}{0ex}}\Psi \left({k}_{t}^{\left(i\right)},{k}_{t}^{*}\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{then}\phantom{\rule{4.pt}{0ex}}{k}_{t}^{(i+1)}={k}_{t}^{*}\hfill \\ if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}>\phantom{\rule{4.pt}{0ex}}\Psi \left({k}_{t}^{\left(i\right)},{k}_{t}^{*}\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{then}\phantom{\rule{4.pt}{0ex}}{k}_{t}^{(i+1)}={k}_{t}^{\left(i\right)};\hfill \end{array}\right\}.$$
- Going over all values of t, we have:$${k}^{(i+1)}={\left({k}_{{t}_{min}}^{(i+1)},\cdots ,{k}_{t}^{(i+1)},{k}_{t+1}^{\left(i\right)},\cdots ,{k}_{{t}_{max}}^{\left(i\right)}\right)}^{\prime};$$
- Transforming ${k}^{(i+1)}$ and ${\alpha}^{\left(i\right)}$ to assure identifiably:$${k}^{(i+1)}-\overline{k}\to {k}^{(i+1)},{\alpha}^{\left(i\right)}+{\beta}^{\left(i\right)}\overline{k}\to {\alpha}^{\left(i\right)},$$$$\overline{k}=\frac{1}{T}\left(\sum _{j\le t}{k}_{j}^{(i+1)}+\sum _{j>t}{k}_{j}^{\left(i\right)}\right);$$
- Repeat steps 1 to 5.

**Estimation of ${\mathit{\beta}}_{\mathit{x}}$ using MH algorithm**

- Draw ${\beta}_{x}^{*}$ from the proposal density function $N({\beta}_{x}^{\left(i\right)},{\sigma}_{\beta}^{2})$, such that ${\sigma}_{\beta}^{2}$ is assumed to be known;
- Calculate the following probability:$$\Psi \left({\beta}_{x}^{\left(i\right)},{\beta}_{x}^{*}\right)=min\left(1,\frac{f({\beta}_{x}^{*}\mid D,\alpha ,{\beta}_{-x}^{\left(i\right)},k,\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})}{f({\beta}_{x}^{\left(i\right)}\mid D,\alpha ,{\beta}_{-x}^{\left(i\right)},k,\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})}\right),$$$${\beta}_{-x}={({\beta}_{xmin},\cdots ,{\beta}_{x-1},{\beta}_{x+1},\cdots .{\beta}_{xmax})}^{\prime}.$$
- Draw a value u from uniform probability function in range $U(0,1)$ and decide in accordance with the following formula:$$\left\{\begin{array}{c}if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}\le \phantom{\rule{4.pt}{0ex}}\Psi \left({\beta}_{x}^{\left(i\right)},{\beta}_{x}^{*}\right)\phantom{\rule{4.pt}{0ex}}then\phantom{\rule{4.pt}{0ex}}{\beta}_{x}^{(i+1)}={\beta}_{x}^{*}\hfill \\ if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}>\phantom{\rule{4.pt}{0ex}}\Psi \left({\beta}_{x}^{\left(i\right)},{\beta}_{x}^{*}\right)\phantom{\rule{4.pt}{0ex}}then\phantom{\rule{4.pt}{0ex}}{\beta}_{x}^{(i+1)}={\beta}_{x}^{\left(i\right)};\hfill \end{array}\right\}.$$
- Going over all values of x, we have:$${\beta}^{(i+1)}=\left({\beta}_{{x}_{min}}^{(i+1)},\cdots ,{\beta}_{x}^{(i+1)},{\beta}_{x+1}^{\left(i\right)},\cdots ,{\beta}_{{x}_{max}}^{\left(i\right)}\right){;}^{\prime}.$$
- Transforming ${k}^{(i+1)}$ and ${\beta}^{(i+1)}$ to assure identifiably:$$\frac{{\beta}^{(i+1)}}{{\beta}_{sum}}\to {\beta}^{(i+1)},{k}^{(i+1)}\times {\beta}_{sum}\to {k}^{(i+1)},$$$${\beta}_{sum}=\left(\sum _{j\le x}{\beta}_{j}^{(i+1)}+\sum _{j>x}{\beta}_{j}^{\left(i\right)}\right);$$
- Repeat steps 1 to 5.

**Estimation of ${\mathit{\alpha}}_{\mathit{x}}$ using MH algorithm**

- Draw ${\alpha}_{x}^{*}$ from the proposal density function $N({\alpha}_{x}^{\left(i\right)},{\sigma}_{\alpha}^{2})$, such that ${\sigma}_{\alpha}^{2}$ is assumed known;
- Calculate the following probability:$$\Psi \left({\alpha}_{x}^{\left(i\right)},{\alpha}_{x}^{*}\right)=min\left(1,\frac{f({\alpha}_{x}^{*}\mid D,{\alpha}_{-x}^{\left(i\right)},\beta ,k,\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})}{f({\alpha}_{x}^{\left(i\right)}\mid D,{\alpha}_{-x}^{\left(i\right)},\beta ,k,\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})}\right),$$$${\alpha}_{-t}={({\alpha}_{xmin},\dots ,{\alpha}_{x-1},{\alpha}_{x+1},\dots .{\alpha}_{xmax})}^{\prime};$$
- Draw a value u from uniform probability function in range $U(0,1)$ and decide in accordance with the following formula:$$\left\{\begin{array}{c}if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}\le \phantom{\rule{4.pt}{0ex}}\Psi \left({\alpha}_{x}^{\left(i\right)},{\alpha}_{x}^{*}\right)\phantom{\rule{4.pt}{0ex}}then\phantom{\rule{4.pt}{0ex}}{\alpha}_{x}^{(i+1)}={\alpha}_{x}^{*}\hfill \\ if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}>\phantom{\rule{4.pt}{0ex}}\Psi \left({\alpha}_{x}^{\left(i\right)},{\alpha}_{x}^{*}\right)\phantom{\rule{4.pt}{0ex}}then\phantom{\rule{4.pt}{0ex}}{\alpha}_{x}^{(i+1)}={\alpha}_{x}^{\left(i\right)};\hfill \end{array}\right\}.$$
- Receiving ${\alpha}^{(i+1)}$ in $(i+1)$th iteration as follows:$${\alpha}_{x}^{(i+1)}=\left({\alpha}_{{x}_{min}}^{(i+1)},\dots ,{\alpha}_{x}^{(i+1)},{\alpha}_{x+1}^{\left(i\right)},\dots ,{\alpha}_{{x}_{max}}^{\left(i\right)}\right);$$
- Repeat steps 1 to 4.

**Estimation of ${\mathit{p}}_{\mathbf{1}}$ using MH algorithm**

- Draw ${p}_{1}^{*}$ from the probability function $gamma({\alpha}_{{p}_{1}},{b}_{{p}_{1}})$, such that ${\alpha}_{{p}_{1}}$ and ${b}_{{p}_{1}}$ are hyperparameters and are assumed known;
- Calculate the following probability:$$\Psi \left({p}_{1}^{\left(i\right)},{p}_{1}^{*}\right)=min\left(1,\frac{f({p}_{1}^{*}\mid D,\alpha ,\beta ,k,\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2})}{f({p}_{1}^{\left(i\right)}\mid D,\alpha ,\beta ,k,\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2})}\right);$$
- Draw a value u from uniform probability function in range $U(0,1)$ and decide in accordance with the following formula:$$\left\{\begin{array}{c}if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}\le \phantom{\rule{4.pt}{0ex}}\Psi \left({p}_{1}^{\left(i\right)},{p}_{1}^{*}\right)\phantom{\rule{4.pt}{0ex}}then\phantom{\rule{4.pt}{0ex}}{p}_{1}^{(i+1)}={p}_{1}^{*}\hfill \\ if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}>\phantom{\rule{4.pt}{0ex}}\Psi \left({p}_{1}^{\left(i\right)},{p}_{1}^{*}\right)\phantom{\rule{4.pt}{0ex}}then\phantom{\rule{4.pt}{0ex}}{p}_{1}^{(i+1)}={p}_{1}^{\left(i\right)};\hfill \end{array}\right\}.$$
- Then receiving ${p}^{(i+1)}$ in $(i+1)$th iteration;
- Repeat steps 1 to 4.

**Estimation of $\mathit{\theta}\mathbf{,}{\mathit{\sigma}}_{\mathit{\alpha}}^{\mathbf{2}}\mathbf{,}{\mathit{\sigma}}_{\mathit{\beta}}^{\mathbf{2}}$ and ${\mathit{\sigma}}_{\mathit{w}}^{\mathbf{2}}$ using the Gibbs sampler**

## 4. Numerical Experiment

#### 4.1. Methods

- Predicting mortality rates $\left(\mu \right)$ by age. In other words, after model parameters were estimated, mortality rates were predicted for a given age across years. For instance, predicting mortality rates for those age 70 was carried out over the years beyond 2000;
- Predicting mortality rates $\left(\mu \right)$ by cohort. In other words, after model parameters were estimated, mortality rates were predicted for a cohort that was at a particular age at the beginning of the test period. For example, predicted mortality rates in 2001–2007 for a cohort aged 70 in 2001.

#### 4.2. Results

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1

- For $t={t}_{min}$, the marginal posterior probability function of the time index ${k}_{t}$ is:$$f({k}_{t}\mid D,\alpha ,\beta ,{k}_{-t},\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})\propto f({D}_{t}\mid \alpha ,\beta ,{k}_{t},{p}_{1})\times f({k}_{t}\mid \theta ,{\sigma}_{w}^{2})\times f({k}_{t+1}\mid {k}_{t},\theta ,{\sigma}_{w}^{2}),$$$$f({k}_{t}\mid \theta ,{\sigma}_{w}^{2})=exp\left(-\frac{1}{2{\sigma}_{w}^{2}}{\left({k}_{t}-\theta \right)}^{2}\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}$$$$f({k}_{t+1}\mid {k}_{t},\theta ,{\sigma}_{w}^{2})=exp\left(-\frac{1}{2{\sigma}_{w}^{2}}{\left({k}_{t}-{k}_{t-1}-\theta \right)}^{2}\right).$$
- For ${t}_{min}<t<{t}_{max}$, the marginal posterior probability function of the time index ${k}_{t}$ is:$$\begin{array}{c}f({k}_{t}\mid D,\alpha ,\beta ,{k}_{-t},\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})\propto f({D}_{t}\mid \alpha ,\beta ,{k}_{t},{p}_{1})\hfill \\ \hfill \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\times f({k}_{t}\mid {k}_{t-1},\theta ,{\sigma}_{w}^{2})\times f({k}_{t+1}\mid {k}_{t},\theta ,{\sigma}_{w}^{2}),\end{array}$$$$f({k}_{t}\mid {k}_{t-1},\theta ,{\sigma}_{w}^{2})=exp\left(-\frac{1}{2{\sigma}_{w}^{2}}{\left({k}_{t}-{k}_{t-1}-\theta \right)}^{2}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}$$$$f({k}_{t+1}\mid {k}_{t},\theta ,{\sigma}_{w}^{2})=exp\left(-\frac{1}{2{\sigma}_{w}^{2}}{\left({k}_{t+1}-{k}_{t}-\theta \right)}^{2}\right).$$
- For $t={t}_{max}$, the marginal posterior probability function of the time index ${k}_{t}$ is:$$f({k}_{t}\mid D,\alpha ,\beta ,{k}_{-t},\theta ,{\sigma}_{\alpha}^{2},{\sigma}_{\beta}^{2},{\sigma}_{w}^{2},{p}_{1})\propto f({D}_{t}\mid \alpha ,\beta ,{k}_{t},{p}_{1})\times f({k}_{t}\mid {k}_{t-1},\theta ,{\sigma}_{w}^{2}),$$$$f({k}_{t}\mid {k}_{t-1},\theta ,{\sigma}_{w}^{2})=exp\left(-\frac{1}{2{\sigma}_{w}^{2}}{\left({k}_{t}-{k}_{t-1}-\theta \right)}^{2}\right).$$

#### Appendix A.2

#### Appendix A.3

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**MDPI and ACS Style**

Awad, Y.; Bar-Lev, S.K.; Makov, U.
A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach. *Risks* **2022**, *10*, 111.
https://doi.org/10.3390/risks10060111

**AMA Style**

Awad Y, Bar-Lev SK, Makov U.
A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach. *Risks*. 2022; 10(6):111.
https://doi.org/10.3390/risks10060111

**Chicago/Turabian Style**

Awad, Yaser, Shaul K. Bar-Lev, and Udi Makov.
2022. "A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach" *Risks* 10, no. 6: 111.
https://doi.org/10.3390/risks10060111