Abstract
Abstract The maximum negative binomial distribution is the distribution of the smallest number of independent Bernoulli trials needed in order to observe at least c successes and c failures. This distribution arises in a design for a medical experiment. We describe the moments and modes for this distribution. When c is large there are normal and half-normal approximate distributions. If the Bernoulli parameter is extremely close to either zero or one then a gamma approximate distribution is demonstrated. Estimates of the Bernoulli parameter are described using the EM algorithm and a Bayesian prior distribution.
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