Measuring goodness-of-fit in a model that combines two distributions I have data with a double peak that I'm trying to model, and there's enough overlap between the peaks that I can't treat them independently. A histogram of the data might look something like this:

I've created two models for this: one uses two Poisson distributions, and the other uses two negative binomial distributions (to account for overdispersion). What's the appropriate way to tell which model fits the data more accurately?
My initial thought is that I could use a Kolmogorov-Smirnov test to compare each model to the data, then do a likelihood ratio test to see if one is a significantly better fit.  Does this make sense?  If so, I'm not exactly sure how to perform the likelihood  ratio test. Is chi-squared appropriate, and how many degrees of freedom do I have?
If it helps, some (very simplified) R code for the models might look something like this:
## inital data points
a <- read.table("data")

#create model data
model.pois = c(rpois(1000000,200),rpois(500000,250))
model.nb = c(rnbinom(1000000,200,0.5),rnbinom(500000,275,0.5)

#Kolmogorov-Smirnov test
#use ks.boot, since it's count data that may contain duplicate values
kpois = ks.boot(model.pois,a)
knb = ks.boot(model.nb,a)

#here's where I'd do some sort of likelihood ratio test
# . . .


Edit:  Here's an image that may explain the data and the distributions I'm fitting better.  It's totally clear from the visualization that the second model (using the negative binomial dist to account for overdispersion) is a better fit. I'd like to show this quantitatively, though.

(red - data, green - model)
 A: You can't compare them directly since the Negative Binomial has more parameters. Indeed the Poisson is "nested" within the Negative Binomial in the sense that it's a limiting case, so the NegBin will always fit better than the Poisson. However, that makes it possible to consider something like a likelihood ratio test but the fact that the Poisson is at the boundary of the parameter space for the negative binomial may affect the distribution of the test statistic.
In any case, even if the difference in number of parameters wasn't a problem, you can't do K-S tests directly because you have estimated parameters, and K-S is specifically for the case where all parameters are specified. Your idea of using the bootstrap deals with this issue, but not the first one (difference in number of parameters)
I'd also be considering smooth tests of goodness of fit (e.g. see Rayner and Best's book), which, for example, can lead to a partition the chi-square goodness of fit test into components of interest (measuring deviations from the Poisson model in this case) - taken out to say fourth order or sixth order, this should lead to a test with good power for the NegBin alternative.
(Edit: You could compare your poisson and negbin fits via a chi-squared test but it will have low power. Partitioning the chi-square and looking only at say the first 4-6 components, as is done with smooth tests might do better.)
A: You can use a metric such as Mean Squared Error between actual vs predicted values to compare the two models.
