Parameters of Gompertz law

Question

I need to find the best parameters $B,c$ so that Gompertz law could be good aproximation of life tables. By the Gompertz law force of mortality is given by $\mu(t)=Bc^t$. We know that $\mu(x)=\frac{l'_x}{l_x}=\frac{d}{dx}ln(l_x)$. So basically: $$l_x=\exp(\int u(x)dx)=\exp(\frac{Bc^t}{ln(c)}).$$ What should I do next? I could try just picking some values from life tables and then hope that $B$ and $c$ will eventually come from the equations but I feel it won't lead me to anything. I don't know any tools to do such things and I'm clueless. Could someone give me some hints?

Answer 1

Gompertz in its basic form is $y=k_\infty e^{-e^{-(x-\tau)/u}}$. Where $\tau$ is the location of the inflection point of the curve, $k_\infty$ is the value of y at infinite time. $u$ is the value of the function at the inflection point divided by the derivative of the function at the inflection point. It can be shown the value of the function at the inflection point is $k_\infty/e=k_\tau$. And u=$k_\tau/k_\tau'.$

This implies $\frac{dy}{dt}=\frac{y}{u}e^{-(t-\tau)/u}=-\frac{y}{u}\ln{\frac{y}{k_\infty}}$

So $\frac{d (\ln y)}{dt}=\frac{-1}{u}\ln{y} +\frac{1}{u} \ln{k_\infty}$

So if you know the function value and increments of y at each x value, essentially the first derivative, then you can find two of the parameters by linear regression. Once you know $u$ and $k_\infty$, then you can also use linear regression to find $\tau$, though that is often pretty clear from the data, just where the first derivative data flatlines.