I have a multi-parameter bivariate function, say $f(i,j)$ that I want to use to predict the entries of a matrix $M(i,j)$, the entries of which are binomial probabilities based on varying sample sizes, say $N(i,j)$. (I can easily limit the $N(i,j)$'s to be employed to large sizes, say greater than $1000$.)
What would be an appropriate objective function to employ in the fitting process, to use for the estimation of the parameters of $f(i,j)$?
I've considered using the sum of the absolute values of the deviations from the model weighed by the square root of the ratio of the sample size, N(i,j), divided by the product $p (1-p)$, where $p$ is the predicted value of the probability. Perhaps, some version of chi-squared would be appropriate.