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I am testing the significance of observed data sampled ($n\approx50$) using a multinomial distribution with known probabilities (with ~20 categories). Given the probability of observing the sample $P_{sam}$, I would like to compute the p-value, which is defined as the sum of all probabilities lower than this observed $P_{sam}$ i.e. $Sum\{p \mid p < P_{sam}, p \in Multinomial-distribution\}$.

One way to find this is to generate a large number of random samples from the given distribution and calculate the above sum using this set. But I am wondering if there exist any exact results and whether there are any good approximations which are less computationally intensive.

Thanks!

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    $\begingroup$ The $p$-value is always related to a hypothesis test. What is the null hypothesis ? $\endgroup$ Commented Oct 9, 2012 at 20:49
  • $\begingroup$ Sorry for the late reply, but the null hypothesis is that the observed samples are drawn from a multinomial distribution characterized by known frequencies of ~20 categories. $\endgroup$ Commented Mar 17, 2013 at 22:17

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You say you know the probability of each side of the die and there are about 20 sides. Let's denote the sides $i$ for $ i = 1, 2, ..., 20$. So your answer is exactly the definition-- $p$-value$(i_0) =$ $$ \sum_{ \{ i : p(i) \leq p(i_0)\} } p(i). $$

Note that I disagree slightly with your definition. I think the $<$ in your sum should be $\leq$.

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If you had a bigger set, you could perform Pearson's test https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test (go to "Calculating the test-statistic") Essentially, it approximates the multinomial distribution as multivariate Gaussian, from that, the p-value can be computed analytically.

With only 50 samples though I don't think there are better option than the one you mentionned (Monte-Carlo).

But if you found a better idea in ten years, I would be interested

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