This is a project of making lists of cohort- and country-wise distribution functions of human lifetime.
What is "SEM"?
We assume that human being comes into the world with "Survival Energy," which is unobservable, but you may also believe the existence 🙂 We further believe that a stochastic process can approximate the energy process, calling it "SEM" (survival energy model).
The SEM is a new "structural approach" to describe
"Why does a human die?",
which is proposed by Shimizu et al. (2020), ASTIN Bulletin. 51 (1), 191-219. ***Erratum***(on Theorem 3.2) and Shimizu et al. (2023), ASTIN Bulletin, 53, (2), 377-391.
(See also Shimizu et al. (2022): SSRN:4127900)
Based on the SEM, we can predict the long future mortality so well in the form of the distribution function of a human lifetime (which we call "mortality function"), and the form is very convenient for many (statistical) computations in actuarial management in practice.
This SEM project introduces a new methodology to predict the cohort-wise mortality functions. This page gives predicted mortality functions for selected cohorts and countries(now, Japan only). We hope you can enjoy their advantages in, e.g., actuarial practice and demographic researchers, among others.
We are currently proposing two types of SEM: ID-SEM and IG-SEM
ID-SEM: Survival Energy process is modeled by (time-) Inhomogeneous-Diffuison (ID) processes.
IG-SEM: Survival Energy process is modeled by Inverse-Gaussian (IG) processes.
- See "What is SEM?" for a brief introduction to SEM.
- 日本語の解説はここ: What is SEM?
Let $X^c=(X^c_t)_{t\ge 0}$ be a SEM, the lifetime distribution is given by the first passage time of $X^c$ to zero:
$$\tau^c = \inf \{ t>0\,:\, X^c_t < 0\}$$
Then the "mortality function" is given as follows:
$$ q_c^\star(t) = \mathbb{P}(\tau^c \le t),\qquad t\ge 0,$$
where *=ID as ID-SEM, and *=IG as IG-SEM.
Data Processing from "HMD-format" to "SEM-format"
R-function for data processing is here (semData.r) , which outputs cohort-wise mortality data in csv-format. Please read the instruction (usage) after the function code of "semData" in that r.file:
- Download data: "Life tables, Females/Males, 1x1" (including the header) from Human Mortality Database (HMD) and save them to a text file.
- Specify the path to the above text file as "hmd_file", and the path to a directory as "sem_folder" as in the instruction.
- Excute the semData.r in R, and follow its instruction.
- Then, csv files of cohort-wise mortality data are exported in the directory specified at "sem_form".
ID-SEM
$$q_c^{ID}(t,\theta_c)=1-\int_0^{\infty}\left(\frac{1}{2\sqrt{\pi S(t,\theta_c)}}\exp{\left[-\frac{(z-x_c-M(t,\theta_c))^2}{4S(t,\theta_c)}\right]}-{\rm{e}}^{-\kappa_cx_c}\frac{1}{2\sqrt{\pi S(t,\theta_c)}}\exp{\left[-\frac{(z+x_c-M(t,\theta_c))^2}{4S(t,\theta_c)}\right]}\right)dz,$$
\[x_c=1000,~~T=70,~~M(t,\theta_c)=\int_0^t(\alpha_c+\beta_c\exp{[\gamma_c(s-T)]}\mathbb{1}_{{s>T}})ds=\kappa_c S\]
IG-SEM
$$q_c^{IG}(t,\theta_c)=\Phi[\sqrt{\frac{\sigma_c}{x_c}}(\Lambda_{\theta_c}(t)-x_c)]-\mathrm{e}^{2\sigma_c\Lambda_{\theta_c}(t)}\Phi[-\sqrt{\frac{\sigma_c}{x_c}}(\Lambda_{\theta_c}(t)+x_c)],$$
\[\Phi(x) = \int_{-\infty}^x e^{-\frac{z^2}{2}}\, dz,~\sigma_c>0,~~ x_c=1000,~~\Lambda_{\theta_c}(t)=\mathrm{e}^{a_c t}+b_c t-1\]
Cohort-wise mortality functions (distribution functions of lifetime)
※If you have a request to update countries or cohorts, please contact us: shimizu(at)waseda.jp