One-way table largest proportion test Suppose I've a one-way table with three categories (A, B and C), and let $p_a$, $p_b$ and $p_c$ be the true proportion of observations in each category, i.e. $p_a+p_b+p_c=1$. How can I conduct a statistical test on the hypothesis "$p_a$ is the largest among the three", i.e. $p_a > p_b$ and $p_a > p_c$?
Thanks!
 A: You are looking for something called a "simple tree ordering test". There are quite a few methods for this in the order restricted inference literature, though I am not sure how much of it is available in software. A quick Google search lead me to a paper that seems applicable.
Peddada, S. D., Prescott, K. E. and Conaway, M. (2001), Tests for Order Restrictions in Binary Data. Biometrics, 57: 1219–1227.
Edit
Based on the comments, it appears I misunderstood the original question. It is still a simple tree ordering, which is a topic of ordered restricted statistical inference, but the reference addresses the comparison of independent binomial probabilities. I have found another paper,
Nettleton, D (2009), 'Testing for the supremacy of a multinomial cell probability', Journal of the American Statistical Association. Vol. 104, Pages 1052-1059, that describes exactly your situation. I think the author has a preprint posted on his webpage, and you might be able to get most of the info from there.
A: I don't see why this is so complicated.  Why can't you simply do the two hypothesis tests and apply a mutliplicity adjustment and conclude that $p_a$ is the largest if the adjusted $p$-value is less than say $0.05$?  I don't think the fact that the tests are dependent matters.
