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.


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    $\begingroup$ 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 $p_i$? $\endgroup$
    – whuber
    Jan 4, 2012 at 15:46
  • $\begingroup$ Thanks @whuber. I forgot the notion of simultaneous confidence intervals already exists. I've simplified my question statement significantly. $\endgroup$
    – DavidR
    Jan 4, 2012 at 20:21

1 Answer 1


Glaz and Sison (Journal of Statistical Planning and Inference, 1999) contains formulae for the Sison and Glaz confidence intervals for the MLE, which simulation showed perform quite well, and also some parametric bootstrap confidence intervals, also for the MLEs. I won't try to reproduce the math here, since there's rather a lot of it and it's in the paper anyway.

  • $\begingroup$ @whuber = thanks, I appreciate it. I'd missed the exclamation point when I tried to fix it up. $\endgroup$
    – jbowman
    Jan 4, 2012 at 23:32
  • $\begingroup$ The link is broken again $\endgroup$
    – uLoop
    May 15, 2019 at 19:20
  • $\begingroup$ @uLoop - eventually all links break; that's why we give the original reference as well. I'll see if I can dig up an alternative, my first try didn't turn up anything. $\endgroup$
    – jbowman
    May 15, 2019 at 19:39

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