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This isn't my original problem, but I'm posting a simplified version and then I'll try to see if I can apply it myself to my original problem. (yay for self-learning)

Suppose I have a sample and I want to test if the data has been drawn from a normal distribution, but $\mu$ and $\sigma$ are unknown. How am I supposed to compute the expected number of counts in each bin if I don't know these parameters? [and also, which bins should I use, since $\mu$ can stretch anywhere from $-\infty$ to $\infty$] I was thinking maybe we could use the MLE, but that seems bad because I think it will lead to an artificially low test statistic [because of something along the lines of "MLE is computed from your data so the fit would be artificially better"].

Sycorax
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renrenthehamster
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  • I answered this question (perhaps in more detail than you care for!) at http://stats.stackexchange.com/questions/16921/how-to-understand-degrees-of-freedom/17148#17148. To summarize, for the $\chi^2$ distribution to be valid you *must* use MLE and you must base the MLE on the counts rather than the raw data, as well as use sufficient amounts of data. The lowering of the test statistic is compensated for by reducing the degrees of freedom. – whuber Jul 02 '14 at 15:22
  • One can still use $\chi^2_{k-1}$ and $\chi^2_{k-p-1}$ as bounds on the distribution if using fully efficient estimation. Sometimes that's fairly satisfactory. – Glen_b Jul 03 '14 at 00:00

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