Tolerance intervals with improved coverage probabilities for binomial and poisson variables

The construction of tolerance intervals (TIs) for discrete variables, such as binomial and Poisson variables, has been critical in industrial applications in various sectors, including manufacturing and pharmaceuticals. Inaccurate estimation of coverage probabilities leads to improper construction of tolerance intervals and may lead to serious financial losses for the manufacturers. This article proposes procedures to compute the exact minimum and average coverage probabilities of the tolerance intervals for Poisson and binomial variables. These procedures are illustrated with examples and real data applications. Based on these procedures, improved tolerance intervals are proposed that can ensure that the true minimum or average coverage probabilities are very close to the nominal levels.