Confidence Regions for the Multinomial Parameter With Small Sample Size

Abstract

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 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 n 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.

Keywords:  :

• Confidence region
• Multinomial distribution
• Small sample