The Neyman-Scott Phenomenon in Generalized Linear Models and Overdispersed Exponential Families

In fixed effects balanced one-way analysis of variance models with homoscedastic normal errors, the maximum likelihood estimator (MLE) of the error variance is inconsistent as the cell-size remains fixed but the number of cells grows to infinity. This is the famous Neymnn-Scott phenomenon. The present paper shows that the Neyman-Scott phenomenon continues to hold for estimating the scale parameter in the canonical version of generalized linear models when the number of nuisance parameters grows to infinity. A similar result holds for overdispersed exponential faruily of distributions. It is also pointed out how the conditional MLE in such cases does not suffer from the inconsistency problem. The relationship between the conditional score function and the corrected score function in general mixture models is also pointed out. The Neyman-Scott phenomenon is also shown to hold for the two-parameter exponential family typically used for modelling overdispersion.