# Statistic for goodness of fit to a multidimensional distribution with geometric tails?

Essentially a reference request, since someone must already have studied this...

I have a model which predicts some probabilities $p_{mn}$, which tends to decay geometrically for large $m$ and $n$. I also have some data $f_{mn}$ of counts when $(m,n)$ occurs. Is there a standard goodness of fit, preferably which reports a $p$-value, suitable for a biological audience?

Personally, I've been calculating the $G$-value: $$G = 2 \sum_{mn} f_{mn} \log\left(\frac{f_{mn}}{N p_{mn}}\right)$$ where $N$ is the sum of $f_{mn}$, and simulating the distribution by Monte-Carlo, and deriving a $p$-value that way. However, given the lack of real analytical understanding, even I don't completely trust it. Is there literature on the behaviour of this statistic? Notice that this converges to the standard $G$-test for a multinomial distribution (finite range for $m$ and $n$) and for large $f_{mn}$ it converges to $\chi^2$-test, so really it is the infinite range which is novel.

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