Theory & Methods: 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 θ . This paper presents a new algorithm for finding a 1 − α confidence interval for θ based on X which possesses the following three properties: (i) the infimum over θ of the coverage probability is 1 − α ; (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.