Likelihood, Quasi-Likelihood and Pseudolikelihood: Some Comparisons

SUMMARY There is considerable interest in the fitting of models jointly to the mean and dispersion of a response. For the mean parameter, the Wedderburn estimating equations are widely accepted. However, there is some controversy about estimating the dispersion parameters. Finite sampling properties of several dispersion estimators are investigated for three models by simulation. We compare the maximum extended quasi-likelihood estimator, the maximum pseudolikelihood estimator and the maximum likelihood estimator, if it exists. Of these estimators, the maximum extended quasi-likelihood estimator is usually superior in minimizing the mean-squared error.