# Determining 3 parameters from exponential regression

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.

• Is your question how to run a nonlinear regression i.e. are you stumped by something conceptual or the implementation?
– mkt
Nov 27, 2017 at 12:44
• It's the implementation. Sorry, should have said. How this could be done in R would be most useful but a theoretical explanation would work also. Nov 27, 2017 at 12:51