Confidence Regions for the Multinomial Parameter With Small Sample Size

Consider the observation of it iid realizations of an experiment with d >= 2 possible outcomes, which corresponds to a single observation of a multinomial distribution M(d)(n, p) where p is an unknown discrete distribution on {1,...,d}. In many applications, the construction of a confidence region for p when it is small is crucial. This challenging concrete 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 nonasymptotic regions with controlled coverage are limited to the binomial case d = 2. Here we propose a new method valid for any d >= 2 that provides confidence regions with controlled coverage and small Volume. The method involves 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.