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Joint Estimation of the Mean and Error Distribution in Generalized Linear Models

This article introduces a semiparametric extension of generalized linear models that is based on a full probability model, but does not require specification of an error distribution or variance function for the data. The approach involves treating the error distribution as an infinite-dimensional parameter, which is then estimated simultaneously with the mean-model parameters using a maximum empirical likelihood approach. The resulting estimators are shown to be consistent and jointly asymptotically normal in distribution. When interest lies only in inferences on the mean-model parameters, we show that maximizing out the error distribution leads to profile empirical log-likelihood ratio statistics that have asymptotic χ2 distributions under the null. Simulation studies demonstrate that the proposed method can be more accurate than existing methods that offer the same level of flexibility and generality, especially with smaller sample sizes. The theoretical and numerical results are complemented by a data analysis example. Supplementary materials for this article are available online.