# Simultaneous confidence intervals for multinomial parameters, for small samples, many classes?

Suppose I have a sample from a multinomial distribution $\text{Multi}(p_1,\ldots,p_k)$, where $p_1+\dots+p_k=1$. I'd like simultaneous confidence intervals for the $p_i$'s. That is, for any $\alpha\in(0,1)$, I want to define $\ell_i$'s and $u_i$'s (random variables depending on the random sample) such that $$P(p_1\in [\ell_1,u_1],\ldots,p_n\in[\ell_n,u_n])\ge 1-\alpha,$$ and obviously the closer the left hand side is to $1-\alpha$, the better. The special consideration is that the number of classes won't be too small (e.g. 10-100 classes), so I'm wondering if there's something significantly better than a Bonferroni approach.

Thanks!

• It's a reasonable question but the sense of 'confidence band' needs to be made more precise. What properties do you need it to have? E.g., should there be $1-\alpha$ confidence that every $p_i$ lies within $[l_i,u_i]$? Or maybe that at least one $p_i$ lies in its interval? Or perhaps should each $[l_i,u_i]$ be a $1-\alpha$ CI for $p_i$? Or maybe you would prefer the bounding box for a $1-\alpha$ simultaneous CI for all the $p_i$? – whuber Jan 4 '12 at 15:46
• Thanks @whuber. I forgot the notion of simultaneous confidence intervals already exists. I've simplified my question statement significantly. – DavidR Jan 4 '12 at 20:21