I have a given PMF, $f_X(x)$, and am trying to create a fitted PMF, $g_X(x)$, that comes "as close as possible" to it, but am not sure what to use as a measure of fit. Simply minimizing standard error feels wrong, since it can produce fits that dramatically alter the relative probability of low values of $f_X$.
Currently I'm using $$\sum_{x\in X} \frac{\left[f_X(x)-g_X(x)\right]^2}{f_X(x)}$$ as my error measure, but that's just a guess that seems to give reasonable looking results.
Are there more suitable or standard error measures used for fitting PMFs?