Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions

Simultaneous confidence intervals for multinomial proportions are useful in many areas of science. Since 1964, approximate simultaneous 1 - α confidence intervals have been proposed for multinomial proportions. Although at each point in the parameter space, these confidence sets have asymptotic 1 - α coverage probability, the exact confidence coefficients of these simultaneous confidence intervals for a fixed sample size are unknown before. In this paper, we propose a procedure for calculating exact confidence coefficients for simultaneous confidence intervals of multinomial proportions for any fixed sample size. With this methodology, exact confidence coefficients can be clearly derived, and the point at which the infimum of the coverage probability occurs can be clearly identified.