Estimating two constants from fitted equation

Question

I have an equation $A=\alpha[1-\exp(-\beta\cdot B)]$ where $A$ and $B$ are known 20x1 column vectors and $\alpha$ and $\beta$ are unknown constants. I'm probably missing something really simple but I'm struggling to find a way to estimate the values of $\alpha$ and $\beta$. The problem I have is that $A$ is not exact so I can't simply use simultaneous equations and instead need to find the best estimate of $\alpha$ and $\beta$ (I have a rough idea what these values should be).

My first thought was to use a likelihood estimator but I don't know the distribution of $A$, although $A_i=\frac{1}{n}\Sigma Y_i$, and I believe $Y_i\sim \text{Pois}(A_i)$

Can someone point me in the right direction?

Thanks, James

Answer 1

This problem needs to be solved numerically. You can use a non-linear least-squares solver, because what you have is an overdetermined system of non-linear equations. There are 20 equations for 2 unknowns. The least-squares solver will minimize the quadratic error measure

$$e=\sum_{i=0}^{19}[A_i-\alpha(1-e^{-\beta B_i})]^2$$

where $A_i$ and $B_i$ are the components of the two vectors $A$ and $B$.

Answer 2

As answered by Matt L., you face a nonlinear least square problem. This is not difficult but requires some reasonable estimates (you tell that you already have a rough idea of them).

If you have a scatter plot of your data, you probably know already that $\beta$ is positive. Then an estimate of of $\alpha$ is the asymptotic value of the dependent variable. Concerning $\beta$, you can estimate it from the $B_{*}$ corresponding more or less to the value of $A$ closest to $\alpha /2$; this gives $\beta=log(2)/B_{*}$. From here, you can start.