Likelihood Analysis of a Binomial Sample Size Problem

The problem of estimating the binomial sample size N from k observed numbers of successes is examined from a likelihood point of view. The direct use of the likelihood function for inference about N is illustrated when p is known, and the problem of inference is considered when p is unknown, and has to be eliminated in some way from the likelihood. Different methods (Bayesian, integrated likelihood, conditional likelihood, profile likelihood) for eliminating the nuisance parameter are found to lead to very different likelihoods in N in an example. This occurs because of a strong ridge in the two-parameter likelihood in N and p. Integrating out the parameter p is found to be unsatisfactory, but reparameterization of the model shows that the inference about N is almost unaffected by the new nuisance parameter. The resulting likelihood in N corresponds closely to the profile likelihood in the original parameterization.