Title: On avoiding inconsistencies between confidence intervals and tests for parameters of some discrete distributions
Authors: Jan Klaschka - Institute of Computer Science of the Czech Academy of Sciences (Czech Republic) [presenting]
Jeno Reiczigel - University of Veterinary Medicine Budapest (Hungary)
Mans Thulin - Department of Statistics - Uppsala University (Sweden)
Abstract: A problem in testing and interval estimation of discrete distribution parameters is addressed: Some confidence interval construction methods, namely those by Sterne (also referred to as the probability based method) and Blaker (also called the combined tails method) suffer from inconsistency between the interval estimates and the corresponding tests. The test rejects, in some cases, hypothesis $\theta = \theta_0$ at significance level $\alpha$, though parameter value $\theta_0$ lies in the $1 - \alpha$ confidence interval. The problem stems from the fact that the set of those $\theta_0$ for which the hypothesis $\theta = \theta_0$ is not rejected, may not be connected, and the gaps have to be filled in order to obtain a confidence interval. A discrepancy then appears when $\theta_0$ lies in a gap. The proposed solution of the problem consists in a modification of the tests (so called $p$-value function unimodalization) that makes the (modified) tests and confidence intervals match perfectly. Programs implementing the test modification for the most frequented settings partly already are, and partly will be soon available on the web.