Taylor quasi-likelihood for limited generalized linear models.

It is a major research topic of limited generalized linear models, namely, generalized linear models with limited dependent variables. The models are developed in many research fields. However, quasi-likelihood estimation of the models is an unresolved issue, due to including limited dependent variables. We propose a novel quasi-likelihood, called Taylor quasi-likelihood, to handle with the unified estimation problem of the limited models. It is based on Taylor expansion of distribution function or likelihood function. We also extend the likelihood to a generalized version and an adaptive version and propose a distributed procedure to obtain the likelihood estimator. In low-dimensional setting, we give selection criteria for the proposed method and make arguments for the consistency and asymptotic normality of the estimator. In high-dimensional setting, we discuss feature selection and oracle properties of the proposed method. Simulation results confirm the advantages of the proposed method.