Keyfitz entropy: investigating some mathematical properties and its application for estimating survival function in life table

Abstract

Keyfitz entropy index is a new indicator that measures the sensitivity of life expectancy to a change in mortality rate. Understanding the characteristics of this indicator can significantly help life table studies in survival analysis. In this paper, we take a closer look at some mathematical properties of Keyfitz entropy index. First, using theoretical studies we show that in some cases this index belongs to the interval [0, 1] and in other cases, it is greater than 1. We also provide two inequalities for Keyfitz entropy using Shannon entropy and pth central moments of random variables. Then, we present an empirical value for it. This value can be useful and provides initial information about Keyfitz entropy value to the researcher, especially before estimating the population survival function with common parametric and nonparametric methods. Second, we propose a new nonparametric method for estimating the survival function in life table using information theory which applies existing information from the population, such as average and moments. The survival function estimated by this method provides the maximum value for Keyfitz entropy indicating the maximum sensitivity of life expectancy to changes in age-specific mortality rates. We also demonstrate that the survival function estimated by this method can be a powerful competitor to its counterparts which are estimated by common parametric and nonparametric methods.

Appendices

Appendix 1

In this section, we refer to some of the indicators presented about the life table uncertainty studies. (With the help of [30] literature.)

Among the indicators of uncertainty associated with survival studies are:

Wiener’s entropy index (1965) Wiener’s entropy index measures the amount of information revealed by the occurrence of a single event whose probability of occurrence is known [50]. When a person of age k dies, while the probability of death at age k (conditionally on reaching age k) is $d_{k}$, the amount of information learnt by the event “death at age k” is given by Wiener’s entropy $W(d_{k})=-{\mathrm{Log}}_{2}(d_{k})$.

Hill’s entropy index (1993) Hill’s entropy of the age at death is defined as ${\mathrm{HI}}_{k}=-\sum _{i=k}^{\omega }p_{i,k}{\mathrm{ln}}(p_{i,k})$ ([23]).

Shannon’s lifetime entropy index (Meyer’s entropy) (2019) Life tables allowed humans to shift from uncertainty to risk about the duration of life. Meyer and Ponthiere develop an indicator of risk about the duration of life whose metric has a concrete counterpart for the layman and makes that risk commensurable with the risk involved in more common situations (see [30].). They also propose to measure risk about the duration of life by means of Shannon’s lifetime entropy index defined to the base 2 as $H_{\mathrm{M}}=-\sum _{i=k}^{\omega }p_{i,k}\log _{2} p_{i,k},$ where $p_{i,k}$ is the probability of a life of (remaining) duration i for an individual of age k [34, 42]. $H_{\mathrm{M}}$ measures the mathematical expectation, along the life cycle, of the amount of information that is learnt from the event “death at a particular age $i \ge k$.” That index quantifies the risk relative to the duration of life (or, similarly, the risk about the age at death) in terms of bits. Therefore, $H_{\mathrm{M}}=HI_{k}\log _{2}e$.

Appendix 2

In the following, we indicated how to form a life table’s entropy. For this purpose, consider $\mu (x)=\frac{f(x)}{S(x)}$ be the force of mortality at age x that f(x) is a density function and S(x) is the survival function of x. According to this ratio and features of f(x) and S(x), the probability of surviving from birth to age x, also can be expressed as $S_{x}[\mu (s)]={\mathrm{exp}}(-\int _{0}^{x}\mu (s){\mathrm{d}}s)$. Now according to above S(x), $e_{x}$ is obtained using $e_{x}[\mu (s)]=\int _{x}^{\infty }{\mathrm{exp}}(-\int _{0}^{a}\mu (s){\mathrm{d}}s){\mathrm{d}}a$ and therefore $e_{0}[\mu (s)]$ is life expectancy at birth (see more details [18]). Now, similar to what Keyfitz proposed, consider a relative increase $\epsilon >0$ in $\mu$ at all ages [27]. Then, the new mortality function is $(1+\epsilon )\mu (s)$ (that $\frac{\delta \mu }{\mu }=\epsilon$), the new probability of surviving from birth to age x, the new life expectancy at age x and life expectancy at birth are, respectively, $S_{x}[(1+\epsilon )\mu (s)]=(S_{x}[\mu (s)])^{1+\epsilon }$, $e_{x}[(1+\epsilon )\mu (s)]=\int _{x}^{\infty }S(a)^{1+\epsilon }{\mathrm{d}}a$ and $e_{0}[(1+\epsilon )\mu (s)]=\int _{0}^{\infty }S(a)^{1+\epsilon }{\mathrm{d}}a$. Keyfitz calculate $\frac{\mathrm{d}e_{0}}{\mathrm{d}\epsilon }\rfloor _{\epsilon =0}$ [26] because they expected that a relative increase in mortality should result in a relative reduction in life expectancy, so they considered $\epsilon$ finite but small, to achieve the following approximation:

$$\begin{aligned} \frac{\bigtriangleup e_{0}}{e_{0}}\approx \left( \frac{\int _{0}^{\infty }S(x){\mathrm{ln}} S(x){\mathrm{d}}x}{\int _{0}^{\infty }S(x){\mathrm{d}}x}\right) \epsilon . \end{aligned}$$

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Because $0 \le S(x) \le 1$, the sign in parentheses is negative. Accordingly, the negative of the expression in above parentheses is known as the entropy of the life table and is customarily denoted by $H_{\mathrm{K}}$.

Appendix 3: discrete approximations

The following approximation formulas can be used to calculate $e_{0}$ and $e^{\dagger }$ [18]:

$$\begin{aligned} e(0,t)&= \int _{0}^{\infty }S(a,t){\mathrm{d}}a \cong \frac{1}{l(0,t)}\sum _{x=0}^{\omega }L(x,t),\nonumber \\ e^{\dagger }(0,t)&= \int _{0}^{\infty }S(a,t){\mathrm{ln}}S(a,t){\mathrm{d}}a\nonumber \\&\cong \frac{1}{l(0,t)}\sum _{y=0}^{\omega }d(y,t)\left[ \frac{e(y,t)+e(y+1,t)}{2}\right] , \end{aligned}$$

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where l(0, t), L(x, t), d(x, t), and e(x, t) correspond to the following life table values at age x, time t: radix at age 0, person-years lived, deaths, and life expectancy. Also, we have:

$$\begin{aligned} E(-X{\mathrm{ln}}S(X))&= e(0,t)+e^{\dagger }(0,t) \\&\cong \frac{1}{2l(0)}\sum _{y=0}^{\omega }\big ([e(y,t)+e(y+1,t)]d(y,t)+L(y,t)\big ). \end{aligned}$$

Also, Brocket has shown that $H_{\mathrm{Sh}}=E(-{\mathrm{ln}}f_{X}(X)) \cong -\sum _{x=0}^{\omega }q(x){\mathrm{ln}} q(x)$( [4]).

Appendix 4

The principle of maximum entropy (ME) Using Lagrange and as respects to $\int f(x){\mathrm{d}}x=1$ for probability distribution f that is satisfied in condition $\varLambda _{i}=\int _{s} f(x) x^{i}{\mathrm{d}}x, 1\le i<m$, f(x) will distribute so that have maximum entropy uniquely, in this way that $J(f)=-\int f(x){\mathrm{ln}}f(x)+\varLambda _{0}\int f(x)+\sum _{i=1}^{m}\varLambda _{i}\int f(x)x^{i}$, now by the differential of J(f) based on f(x) and equal zero we have $f(x)={\mathrm{exp}}(\varLambda _{0}-1+\sum _{i=1}^{m}\varLambda _{i}x^{i})$ [19].