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As the title:

I have a discrete distribution with a small number of categories. I want to test if it is compatible with a parameterized distribution. If I were to do a chi-squared or G-test, I should account for the fact that I'm estimating the parameter when computing the p-value from the test statistic.

I don't want to use these tests because I've been warned that they are not appropriate when categories have fewer than around 5 counts observed or expected, and my counts are nowhere near uniform, so categories are routinely small. But because I do have few categories, the multinomial test is completely feasible in terms of computation time. However, I'm comparing against a family of parameterized distributions, and all the presentations have been against a pre-specified distribution.

Is the multinomial test still appropriate here? Is correcting for estimating the parameter necessary or even possible?

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