M-estimator Of Generalized Linear Model

In this paper,we apply M-estimator to Generalized Linear Model in order to estimate the unknow parameterβ₀.Using the law of large numbers and martingale central limits theorem, the consistency and asymptotic normality of the M-estimator are proved under some conditions. Further,in fixed design,we also prove the two properties:consistency and asymptotic normality. Last,numerical simulation is done.There are four parts in this paper:1.Introduction.In this part,we introduce some basic knowledge for this paper.First, we introduce the conceptions of generalized linear model and M-estimator,with the properties of M-estimator.Second,we introduce some important theorems we use in this paper,that is traverse theorem,martingale central limit theorem and slutsky theorem.2.M-estimator of generalized linear model.This is the core part of the whole paper.Based on maximum likelihood estimator of generalized linear model which was done before,we apply M-estimator to generalized linear model.Furthermore,we also prove consistency and asymptotic normality of the M-estimator.3.Fixed design of M-estimator.In this part,we combine fixed design and M-estimator of generalized linear model,and using the method of part two,we prove the two properties above.4.Numerical simulation.This is the last part of the whole paper.In this part,considering the fixed generalized linear model and true parameter,we prove the consistency of M-estimator through matlab.From the results,we can see M-estimator of generalize linear model is good.