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 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.

Coalescence of low-viscosity fluids in air

“…al. proposed in [6] a confidence region with guaranteed confidence level and a small volume. However, such confidence region is difficult to visualize for category number κ greater than 2.…”

Simultaneous confidence intervals for comparisons of several multinomial samples

“…A Bonferroni-correction of individual intervals would be needed to obtain simultaneous confidence intervals. Assuming one multinomial sample, Hayter (2014) describes the efficient computation for multinomial probabilities, and Chafai and Concordet (2009) and Wang (2000) consider simultaneous confidence intervals for the corresponding vector of multinomial proportions. Recent methods by Fay and Proschan (2015) could be used to combine confidence intervals constructed for each multinomial sample in order to obtain confidence intervals for comparisons between groups, e.g., differences or ratios of proportions.…”

“…Since then, numerous authors have considered simultaneous confidence intervals for proportions or pairwise comparisons of proportions in a single multinomial sample (e.g. Glaz and Sison, 1999;Piegorsch and Richwine, 2001;Hou et al, 2003;Wang, 2000;Chafai and Concordet, 2009). To our knowledge, simultaneous confidence intervals for the comparison of multiple odds between multiple multinomial samples have not been considered any further, although there is room for improvement compared to the seminal methods of Gold (1963) and Goodman (1964): The test statistics related to comparisons of multiple logits of multinomial proportions asymptotically follow a multivariate normal distribution (e.g., Agresti, 2013) and multiple multinomial samples can be considered as a special case for the application of multivariate generalized linear models (e.g.…”

Exact Confidence Intervals for Linear Combinations of Multinomial Probabilities

“…When the desired inference is simultaneous confidence intervals on the multinomial parameters instead of linear combinations of the parameters, large and small sample methods exist. For large samples see Gold (1963); Quesenberry and Hurst (1964); Goodman (1965); Fitzpatrick and Scott (1987); Sison and Glaz (1995), and for small samples Chafai and Concordet (2009) developed a confidence region around multinomial parameters. Simultaneous inference on the multinomial parameter space does not efficiently or directly translate to the linear combinations of those parameters, and therefore the work of this paper fills a gap by providing an exact confidence interval on linear combinations of multinomial probabilities.…”