Exact interval estimation for the linear combination of binomial proportions

The weighted sum of binomial proportions and the interaction effect are two important cases of the linear combination of binomial proportions. Existing confidence intervals for these two parameters are approximate. We apply the $h$-function method to a given approximate interval and obtain an exact interval. The process is repeated multiple times until the final-improved interval (exact) cannot be shortened. In particular, for the weighted sum of two proportions, we derive two final-improved intervals based on the (approximate) adjusted score and fiducial intervals. After comparing several currently used intervals, we recommend these two final-improved intervals for practice. For the weighted sum of three proportions and the interaction effect, the final-improved interval based on the adjusted score interval should be used. Three real datasets are used to detail how the approximate intervals are improved.