# Model selection for nonlinear regression of a Gaussian CDF mixture distribution

I have a number of distributions which I want to fit to a CDF that is comprised of one or more Gaussian CDFs. I was able to use weighted least squares regression to find the best fit parameters for each regression depending on the number of modes I give the model (e.g. each Gaussian CDF has three parameters, mu, sigma, and a scaling factor, that are summed to create my total function). In addition, it is reasonable that the CDF may not actually approach 1. If the fit places the CDF between approximately 0.8 and 1.2, it would be physically justifiable. I've included example plots of my 1, 2, and 3 mode fits for a distribution (Ignore scale on left of plots).

I am trying to figure out if there is a good way to determine if one model is better than another without overfitting. My thought was to try using AIC, but I'm not sure the methods I'm using are the correct way of thinking about this problem or not.

Currently, I am minimizing the weighted least squares by minimizing: $$\sum_i [w_i * (f(x_i) - y_i)^2]$$ Where $w_i$ is the weight for each datapoint. I'm not getting confused though, as what I'm finding seems to indicate that log-likelihood would be something like: $$log L = \sum_i \frac{w_i * (f(x_i) - y_i)^2}{2\sigma^2}$$ Is this actually the log-likelihood? Why is it not being maximized? And additionally, I'm trying to understand how this fits in to AIC or AICc in order to penalize the additional parameters with the multiple mode fits. Does anyone have suggested resources for better understanding model selection for nonlinear regression? Most of the information I have found has been on Maximum Likelihood Estimation for probability distributions rather than this sort of sigmoid curve fitting.

• Note that if $\sigma^2$ doesn't vary across $i$, it won't affect the maximization, so the weighted least squares results would be the same as the log-likelihood results. – jbowman Mar 11 '14 at 20:45