Combining quasi and empirical likelihoods in generalized linear models with missing responses

By only specifying the conditional mean and variance functions of the response variable given covariates, the quasi-likelihood can produce valid semiparametric inference for regression parameter in generalized linear models (GLMs). However, in many studies, auxiliary information is available as moment restrictions of the marginal distribution of the response variable and covariates. We propose the combined quasi and empirical likelihood (CQEL) to incorporate such auxiliary information to improve the efficiency of parameter estimation of the quasi-likelihood in GLMs with missing responses. We show that, when assuming responses are missing at random (MAR), the CQEL estimator achieves better efficiency than the maximum quasi-likelihood (MQL) estimator due to utilization of the auxiliary information. When there is no auxiliary information, we show that the CQEL estimator of the mean response is more efficient than the existing imputation estimators. Based on the asymptotic property of the CQEL estimator, we also develop Wilks’ type tests and corresponding confidence regions for the regression parameter and mean response. The merits of the CQEL are further illustrated through simulation studies.