| The profile likelihood is commonly used in cases where the maximum likelihood estimator for a shape or dispersion parameter depends on knowledge of the mean. We demonstrate that, in stratified models with many mean parameters, the maximum profile likelihood estimator for a common shape parameter can be severely biased or even inconsistent when the sample size per stratum is low. We note a 'marginal' likelihood function that eliminates these problematic mean parameters, but is usually intractable or even impossible to calculate in practice. We discuss approximations to this marginal likelihood - notably the modified profile likelihood of Barndorff-Nielsen [5], the adjusted profile likelihood of Cox & Reid [16], and quasi-likelihood variants - and demonstrate that estimators based on these functions have better bias properties than those based on the full likelihood. We apply these estimators to a stratified negative binomial model and achieve accurate estimates for the negative binomial dispersion parameter k in a simulation experiment. Finally, we provide an application of our methods to fishery data. |
