Confidence regions for the multinomial parameter with small sample size

Consider the observation of n iid realizations of an experiment with d ≥ 2 possible outcomes, which corresponds to a single observation of a multinomial distribution Md(n,p) where p is an unknown discrete distribution on {1, . . . , d}. In many applications, the construction of a confidence region for p when n is small is crucial. This concrete challenging problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing non-asymptotic regions with controlled coverage are limited to the binomial case d = 2. In the present work, we propose a new method valid for any d ≥ 2. This method provides confidence regions with controlled coverage and small volume, and consists of the inversion of the “covering collection” associated with level-sets of the likelihood. The behavior when d/n tends to infinity remains an interesting open problem beyond the scope of this work.