Statistic for goodness of fit to a multidimensional distribution with geometric tails? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:22:01Z http://mathoverflow.net/feeds/question/98655 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98655/statistic-for-goodness-of-fit-to-a-multidimensional-distribution-with-geometric-t Statistic for goodness of fit to a multidimensional distribution with geometric tails? genneth 2012-06-02T13:14:09Z 2012-06-02T13:14:09Z <p>Cross post from Stats.SE: <a href="http://stats.stackexchange.com/questions/29601/statistic-for-goodness-of-fit-to-a-multidimensional-distribution-with-geometric" rel="nofollow">http://stats.stackexchange.com/questions/29601/statistic-for-goodness-of-fit-to-a-multidimensional-distribution-with-geometric</a></p> <p>Essentially a reference request, since <em>someone</em> must already have studied this...</p> <blockquote> <p>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?</p> <p>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.</p> </blockquote>