I have a problem that I can't solve. I want to do a linear regression on $y = a + be^{ct}+\epsilon$ where $\epsilon$ is normally distributed with mean 0 and constant variance.
$t$ is time starting at 0.
Taking the log would lead to horrible things happening with the $a$.
My intentions are to use it to model a population where $a$ individuals are immortal (for argument's sake) and $b$ individuals die with some rate $c$.
How would I regress across some time data for $n$ populations to get ML estimates of $a, b$ and $ c$ in R ideally although a theoretical explanation would also work?
In extension, how would this be modeled for $y = ae^{dt} + be^{ct}+\epsilon$?
You may assume that $n>100$ and that measurements may be taken as often as is necessary.
