Confidence regions for the multinomial parameter with small sample size

Djalil Chafai (1) , Didier Concordet (1)

1 UPTE - Physiopathologie et Toxicologie Expérimentales

Consider the observation of n iid realizations of an experiment with d>1 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 in Biology, Medicine, Physics, and Engineering, 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 build from asymptotic statistics do not have good coverage for small n. In the binomial case (d=2), Clopper and Pearson provided a nice way to construct non-asymptotic confidence regions. We show how to generalize their approach to any d, by using the concept of covering collections. We also propose an attractive new alternative method which provides small confidence regions of controlled coverage. It corresponds to a special covering collection based on level sets of the multinomial distribution. We compare the performance of our new method to various other methods, including a method of Wald based on the Central Limit Theorem, a method based on concentration of measure and deviation probabilities, and a Bayesian method based on Dirichlet-Jeffrey priors.

Mots clés: Confidence regions; small samples; multinomial distribution

Domaines: Calcul [stat.CO]

Djalil Chafai, Didier Concordet. Confidence regions for the multinomial parameter with small sample size. 2008. ⟨hal-00278790v1⟩