Linear and Trigonometric Force of Mortality

Studying for Exam LTAM, Part 1.11

Survival function for a continuous lifetime random random variable with a trigonometric (sinusoidal) force of mortality.

In the last post, “Transformed Power Function Survival Models”, we explored the consequences of assuming certain forms of the survival function for a continuous survival random variable $T_{x}$ .

Here we change our perspective. Instead of starting with a reasonable survival function, we start with a reasonable force of mortality.

What should a reasonable force of mortality look like? Let’s start by looking at the examples we have considered so far.

(Uniform Distribution — De Moivre’s Law) The force of mortality (FM) is $\mu_{t}=\frac{1}{\omega-t}$ for $0\leq t<\omega$ . This function is positive and increasing for all $0\leq t<\omega$ . It also has a vertical asymptote at $t=\omega$ .
(Exponential Distribution — Constant Force) The FM is $\mu_{t}=\lambda>0$ for all $t\geq 0$ . This function is a positive constant function. Its graph is a horizontal line above the horizontal axis.
(Triangular Survival Models) The FM is a complicated piecewise function. However, its graph is similar to the uniform distribution graph above. It is positive, increasing, and has a vertical asymptote at $t=\omega$ .
(Gompertz-Makeham Survival Model) The FM is $\mu_{t}=A+Bc^{t}$ for $t\geq 0$ . The constants satisfy $A>0$ , $B>0$ , and $c>1$ . This is a positive increasing function. It grows “very fast” (exponentially) for “large” $t$ , but not so fast as to have a vertical asymptote. This is an example of a model where we start with the FM and derive the other functions from it.
(Transformed Power Function Survival Models) There were two models we considered here. Model $H$ has FM $\mu_{t}=\frac{p}{\omega-t}$ for $0\leq t<\omega$ and $p>0$ . This reduces to a uniform distribution when $p=1$ . Model $V$ has FM $\mu_{t}=\frac{pt^{p-1}}{\omega^{p}-t^{p}}$ for $0\leq t<\omega$ and $p>0$ . These functions are both positive and increasing with a vertical asymptote at $t=\omega$ .

New Models: Linear and Trigonometric Force of Mortality

All the models above have a positive FM, which makes good sense. All of them except for #2 have an FM that strictly increases. And all of them except for #2 and #4 have a vertical asymptote at some value $t=\omega$ .

So, certainly a linear function of the form $\mu_{t}=A+Bt$ would be a reasonable FM for some constants $A>0$ and $B>0$ .

But how about a trigonometric (sinusoidal) function? How about $\mu_{t}=A\cos(Bt+\phi)+C$ for some constant $\phi$ and some positive constants $A, B$ , and $C$ , with $A<C$ ? Could this be a reasonable FM? It is certainly positive, but it is not increasing and it does not have a vertical asymptote.

Since an FM is essentially an “instantaneous risk of death”, I think it’s not too unreasonable, at least in the short-term. Risk of death could vary throughout a given year, for example. Maybe the effect is not strong for human lifetimes, but it could be strong for appliances like furnaces or air conditioners.

Let’s do our usual exploration of the various model functions in these situations now. If nothing else, I think the trigonometric FM could yield interesting formulas and graphs.

Survival Functions

In general, a survival function (SF) can be obtained from an FM by the formula $\,_{t}p_{x}=\exp\left(-\displaystyle\int_{0}^{t}\mu_{x+s}\, ds\right)$ .

SF for Linear FM

For the linear FM model, this last formula produces $\,_{t}p_{x}=\exp\left(-\displaystyle\int_{0}^{t}(A+Bx+Bs)\, ds\right)=\exp\left(-(A+Bx)t-\frac{1}{2}Bt^{2}\right)$ .

The graph of this as a function of $t$ is shown below. You can see the ranges of values for $A$ and $B$ , as well as the attained age $x$ .

Survival function (SF) $\,_{t}p_{x}$ when the FM is linear $\mu_{t}=A+Bt$ .

Survival function (SF) $\,_{t}p_{x}$ when the FM is linear $\mu_{t}=A+Bt$ .

We see that increasing $A$ makes the graph become always concave up (like an exponential model). Increasing $B$ does not affect the concavity near $t=0$ (at least when $A$ and $x$ are small). The effect of increasing $x$ is to translate the original graph to the left while also vertically stretching it. This is because $\,_{t}p_{x}=\frac{S_{0}(x+t)}{S_{0}(x)},$ where $S_{0}(t)=_{t}p_{0}$ .

The first two properties can be confirmed by differentiation. The first derivative is $\frac{d}{dt}\left(e^{-(A+Bx)t-\frac{1}{2}Bt^{2}}\right)=e^{-(A+Bx)t-\frac{1}{2}Bt^{2}}\cdot (-(A+Bx)-Bt)<0$ for all $t\geq 0$ . And the second derivative $\frac{d^{2}}{dt^{2}}\left(\,_{t}p_{x}\right)$ can be shown (exercise) to be:

$e^{-(A+Bx)-Bt}\cdot \left(B^{2}t^{2}+(2B^{2}x+2AB)t+(A+Bx)^{2}-B\right)$ .

If $x=0$ and $A^{2}<B$ , then this last expression will be negative when $t$ is small (take the time to think about this). Hence, the graph above will be concave down for such $t$ . On the other hand, this last expression will be clearly positive when $t$ is sufficiently large. That is, the graph will eventually become concave up, no matter what $B$ is.

Though $\,_{t}p_{x}>0$ for all $t>0$ (there is no “limiting age” $\omega$ ), we could ask what values of $t$ will lead to $\,_{t}p_{x}$ being very small. For example, for what values of $t>0$ will we have $\,_{t}p_{0}<0.01$ ?

This can be solved with logarithms. We want $e^{-At-\frac{1}{2}Bt^{2}}<0.01$ . This is equivalent to $-At-\frac{1}{2}Bt^{2}<\ln(0.01)=-\ln(100),$ or $\frac{1}{2}Bt^{2}+At-\ln(100)>0$ . With the quadratic formula, this can be seen to have the solution $t>\frac{-A+\sqrt{A^{2}+2B\ln(100)}}{B}$ .

SF for Trigonometric FM

Let us simplify things a bit here and take $\phi=0$ . Then we are considering $\mu_{t}=A\cos(Bt)+C$ for $B>0$ and $0<A<C$ . The SF is then $_{t}p_{x}=\exp\left(-\displaystyle\int_{0}^{t}(C+A\cos(Bx+Bs))ds\right)$ . This simplifies to $_{t}p_{x}=\exp\left(-Ct-\frac{A}{B}(\sin(B(x+t))-\sin(Bx))\right)$ .

The animated graph for this shown below is one of my favorites. I especially like seeing what happens as $x$ increases — it’s mesmerizing! Note that I have fixed $A$ at $A=0.014$ .

*SF $\,_{t}p_{x}$ when the FM is trigonometric $\mu_{t}=A\cos(Bt)+C$ , with $A$ fixed at $A=0.014$ .*

*SF $\,_{t}p_{x}$ when the FM is trigonometric $\mu_{t}=A\cos(Bt)+C$ , with $A$ fixed at $A=0.014$ .*

While $_{t}p_{x}$ is not strictly periodic, we might say that it is quasi-periodically decreasing (I guess I just made up a new term). In fact, it seems that as $x$ increases, we eventually get the same graph over and over again. In other words, while not being a periodic function of the time variable $t$ , it is a periodic function of the attained age parameter $x$ !

This makes sense if you look at the formula again: $_{t}p_{x}=\exp\left(-Ct-\frac{A}{B}(\sin(B(x+t))-\sin(Bx))\right)$ . Since the sine function is periodic with period $2\pi$ , this function will be periodic in the attained age $x$ with period $\frac{2\pi}{B}$ . We can write that $_{t}p_{x+\frac{2\pi}{B}}=\,_{t}p_{x}$ for all values of $t$ and $x$ .

But it is not periodic in time $t$ because of the $-Ct$ term. Of course, survival functions can never be periodic functions of $t$ . In fact, they must always decrease (perhaps not always strictly) as $t$ increases.

Probability Density Functions

Recall that the probability density function (PDF) for the remaining lifetime $T_{x}$ of a person of attained age $x$ will be $f_{x}(t)=\,_{t}p_{x}\cdot \mu_{x+t}$ . Alternatively, we could use the formula $f_{x}(t)=-\frac{d}{dt}\left(_{t}p_{x}\right)$ .

PDF for Linear FM

For $\mu_{t}=A+Bt$ , we saw above that $_{t}p_{x}=\exp\left(-(A+Bx)t-\frac{1}{2}Bt^{2}\right)$ . Therefore, the PDF is $f_{x}(t)=(A+B(x+t))\cdot \exp\left(-(A+Bx)t-\frac{1}{2}Bt^{2}\right)$ .

Here’s an animation of the graph of this PDF.

*Probability density function (PDF) $f_{x}(t)$ when the FM is linear $\mu_{t}=A+Bt$ .*

*Probability density function (PDF) $f_{x}(t)$ when the FM is linear $\mu_{t}=A+Bt$ .*

Consistent with the animation of the SF for the linear FM above, this PDF looks more and more exponential as $A$ increases. On the other hand, it does not look exponential as $B$ increases (at least when $A$ is small and $x=0$ ). These observations can be confirmed with the same derivative calculations as above, since $f_{x}(t)=-\frac{d}{dt}\left(_{t}p_{x}\right)$ .

PDF for Trigonometric FM

For $mu_{t}=A\cos(Bt)+C$ , we saw above that $\,_{t}p_{x}=\exp\left(-Ct-\frac{A}{B}(\sin(B(x+t))-\sin(Bx))\right)$ . Therefore, the PDF is $f_{x}(t)=(A\cos(Bt)+C)\cdot \exp\left(-Ct-\frac{A}{B}(\sin(B(x+t))-\sin(Bx))\right)$ .

This animation is also one of my favorites. Once again, $A$ is fixed at $A=0.014$ . Enjoy — and think about it!

*PDF $f_{x}(t)$ when the FM is trigonometric $\mu_{t}=A\cos(Bt)+C$ , where $A$ is fixed at $A=0.014$ .*

*PDF $f_{x}(t)$ when the FM is trigonometric $\mu_{t}=A\cos(Bt)+C$ , where $A$ is fixed at $A=0.014$ .*

Once again this function is periodic in the attained age $x$ with period $\frac{2\pi}{B}$ .

Expected Remaining Lifetime (Complete and Curtate)

Recall that the complete expectation of life is $\stackrel{\circ}e_{x}=\displaystyle\int_{0}^{\infty}\,_{t}p_{x}dt$ and the curtate expectation of life is $e_{x}=\displaystyle\sum_{k=1}^{\infty}\,_{k}p_{x}$ . The first of these is the mean (expected value) of the remaining lifetime random variable $T_{x}$ while the second is the mean of $K_{x}=\lfloor T_{x}\rfloor$ .

For both models, both the integral and the infinite series cannot be computed in closed form. However, they can be approximated with software and graphed.

Linear Model FM

The graph below shows $\stackrel{\circ}e_{x}$ for the linear model when $A=0.0001$ and $B=0.0005$ . We would find the graph of $e_{x}$ is similar, but a bit lower.

The graph of the complete expectation of life $\stackrel{\circ}e_{x}$ when $\mu_{t}=A+Bt$ , $A=0.0001$ , and $B=0.0005$ .

To me, the most interesting thing about this graph is that it doesn’t decrease as fast as you might think. It even seems that it might be “leveling off” toward a horizontal asymptote at some positive (nonzero) value. That would be unexpected if it is true, especially since the graph of $\,_{t}p_{x}$ goes to zero in a seemingly rapid way further above. I’m pretty sure it is not true, but you might want to see if you can prove it is not true as a challenge problem.

Trigonometric Model FM

The graph below shows $\stackrel{\circ}e_{x}$ for the trigonometric model when $A=0.014$ , $B=1$ , and $c=0.025$ . We would find the graph of $e_{x}$ is similar, but a bit lower.

*The graph of the complete expectation of life $\stackrel{\circ}e_{x}$ when $\mu_{t}=A\cos(Bt)+C$ , $A=0.014$ , $B=1$ , and $C=0.025$ .*

Wow! It looks like this one is periodic! Is it a cosine wave? No, it starts near its average value, not its peak. Is it a sine wave? That seems difficult to say just looking at the picture.

On the other hand, it seems impossible that it would be a pure sine wave. Its formula is just too complicated: $\stackrel{\circ}e_{x}=\displaystyle\int_{0}^{\infty}\exp\left(-Ct-\frac{A}{B}(\sin(B(x+t))-\sin(Bx))\right)\, dt$ .

In spite of the fact that this formula is complicated, it is clearly periodic with period $\frac{2\pi}{B}$ in the attained age parameter $x$ . This is pretty unrealistic for human lifetimes, but it sure is interesting. On the other hand, maybe it is a decent model, at least in the short-term, for lifetimes of appliances such as furnaces or air conditioners. That’s because these appliances are subject to seasonal stressors which could make $\mu_{t}$ “almost” periodic.

Past Research Project

In fact, I was so interested in this problem that I did a research project on it with a former student. We found a formula for $\stackrel{\circ}e_{x}$ without an integral sign, but it was very complicated. It ended up involving a “nonelementary” function, called a modified Bessel function of the first kind. We also were able to represent this function as a Fourier series. This provided a mathematically useful framework for estimating the (rough) amplitude of $\stackrel{\circ}e_{x}$ in terms of the parameters $A$ , $B$ , and $c$ .

The computations were quite tricky, but is was amazing that it all worked out in the end. I may do a series of blog posts about it sometime.