Weighted least-squares and quasilikelihood estimation for categorical data under singular models

Abstract

Analysis of categorical variables by generalized linear models having singular covariance matrices is considered. A weighted least-squares estimator is proposed, and is shown to be “asymptotically best linear unbiased” under general sampling schemes. This estimator is also shown to be equivalent to estimators obtained from two other weighted least-squares approaches. Finally a “quasilikelihood” estimator is proposed for special covariance structures, which include product multinomial sampling and Dirichlet-multinomial models for two-stage cluster sampling. This is obtained directly without having to take explicit account of the sampling restrictions on the parameters. As a corollary, the “asymptotically best linear unbiased estimator” is shown to be “best asymptotically normal” for product multinomial sampling. Large-sample tests of goodness of fit and of hypotheses on the model parameters, and examples of applications of the results, are also presented.