Orthogonality of the mean and error distribution in generalized linear models

TL;DR

This paper proves that in generalized linear models, the mean parameter is orthogonal to the error distribution, ensuring asymptotic efficiency of the mean estimator regardless of the error distribution's specification, and highlights the benefits of nonparametric error estimation.

Contribution

It establishes the orthogonality between mean and error distribution in GLMs, extending known results and enabling efficient inference with nonparametric error estimation.

Findings

Mean-model parameter is orthogonal to error distribution in GLMs.

Maximum likelihood estimator of the mean is asymptotically efficient regardless of error distribution knowledge.

Asymptotic independence of mean and error distribution estimators.

Abstract

We show that the mean-model parameter is always orthogonal to the error distribution in generalized linear models. Thus, the maximum likelihood estimator of the mean-model parameter will be asymptotically efficient regardless of whether the error distribution is known completely, known up to a finite vector of parameters, or left completely unspecified, in which case the likelihood is taken to be an appropriate semiparametric likelihood. Moreover, the maximum likelihood estimator of the mean-model parameter will be asymptotically independent of the maximum likelihood estimator of the error distribution. This generalizes some well-known results for the special cases of normal, gamma and multinomial regression models, and, perhaps more interestingly, suggests that asymptotically efficient estimation and inferences can always be obtained if the error distribution is nonparametrically estimated along with the mean. In contrast, estimation and inferences using misspecified error distributions or variance functions are generally not efficient.