# Distribution of $i$th largest entry in multinomial sample

I have a $k$-class multinomial distribution with vector of probabilities $(p_1, p_2, \ldots, p_k)$, from which I draw a size-$N$ sample $(c_1, c_2, \ldots, c_k), \sum\limits_{s=1}^k c_s = N$. Assume for simplicity that $c_j \geq 1$ for all $j$. Next, I sort the sample in descending order by count to obtain $(c_{(1)}, c_{(2)}, \ldots, c_{(k)})$ where $c_{(\ell)} \geq c_{(\ell+1)}$.

Are there any research papers or other resources that describe the distribution over $c_{(i)}$, the $i$th largest count in a multinomial sample? For example, the distribution over $c_{(1)}$ is the distribution over the largest count and $c_{(k)}$ is the distribution over the smallest.

Or, asked a different way, what is the distribution describing the probability that the $j$th class's count is the $i$th largest in the sample? I've been googling around for a while but haven't really found anything relevant.

• Perhaps this helps? repository.tudelft.nl/islandora/object/… Jul 1, 2017 at 0:22
• Thank you! Yes, I found this in my own research as well. It seems that statisticians have studied this from a slightly different angle for some time, and I have to figure out how to apply the past research to this problem. Jul 1, 2017 at 4:18

The distribution of $$i$$-th largest entry in multinomial sample is described in «Exact distributions of multinomial order statistics» by Anton Ogay. The formulations of his results, however, are too long to be reposted there.