determining logistic curve parameters from 3 points of data

Question

I am trying to find the parameters, $L$, $k$, and $t_0$ that are present in a logistic function:

$y = \frac{L}{1+\exp\left(-k(t-t_0)\right)}$.

I have three points of data, namely: $(t,y) = (1,896.50),(2,1814.00),(3,1878.50)$.

Could I then solve the three resulting equations uniquely for the parameters $L,k,t_0$, or is this not possible, even numerically?

Thanks.

Answer 1

It should be solvable because you have three points and three parameters, and the points satisfy monotonicity. I get:

$$L = 1880.8216, k=3.3947179, t_0=1.0275293$$

I used Stata's nonlinear regression command for this, but you could do it by hand using Gauss-Newton or something similar. It should be relatively easy because the logistic function is differentiable everywhere.

There might be an analytical solution, but I don't know.