Exact short confidence intervals from discrete data

Suppose that X is a discrete random variable whose possible values are {0, 1, 2,.... } and whose probability mass function belongs to a family indexed by the scalar parameter 0. This paper presents a new algorithm for finding a 1 - alpha confidence interval for 0 based on X which possesses the following three properties: (i) the infimum over 0 of the coverage probability is 1 - alpha; (ii) the confidence interval cannot be shortened without violating the coverage requirement; (iii) the lower and upper endpoints of the confidence intervals are increasing functions of the observed value x. This algorithm is applied to the particular case that X has a negative binomial distribution.