The Generalized Least Absolute Deviation Estimation For The Generalized Linear Models

Generalized linear models (GLMs), which can model a large variety of data, have a wide area of application. The class of GLMs includes, as special cases, linear regression, analysis-of-variance models, log-linear models for the analysis of contingency tables, logistic models for binary data in the form of proportions and many others. Usually, the parameters in the generalized linear models are estimated by the method of maximum likelihood . But, in the literature, the nonrobustness of the maximum likelihood estimator forβhas been studied extensively. The quasi-likelihood estimator of the parameter of the generalized linear model shares the same non-robustness properties. In statistics, we should consider the robustness of statistical methods sometimes, that is, when there are small differences between the true model and the assumed model, there is no large influence on the performance of the statistical methods. We know, the least absolute deviation (LAD) method is a widely recognized superior robust method. The method has found many applications in econometrics and biomedical studies. In this paper, we develop generalized least absolute deviation criterion to estimate generalized linear models firstly. By the lack of differentiability of the criterion function and the nonlinear of the mean function, the standard approach to the demonstration of asymptotic normality, based on a Taylors series expansion of the objective function, is not directly applicable. Under suitable conditions, applied the method of empirical process and the result of stochastic equicontinuity, we proof that the estimator is consistent and asymptotically normal. Secondly, by analyzing the criterion function, we use two weighted generalized least absolute deviation criterions to estimate generalized linear models. Similarly, we give out the proof of the consistency and asymptotical normality of the estimators. At last, we conduct a simulation study for several special cases of GLMs.There are three parts in this paper: Section 1, Generalized least absolute deviation estimation; Section 2, The weighed generalized least absolute deviation estimation; Section 3, Simulation study.