The Research Of Generalized Linear Model Parameter Estimation Problems

Generalized Linear Model is an important model in statistical analysis, and it is a natural generalization of the classical linear model. The dependent variable’s distribution in generalized linear model only needs to obey exponential family distribution, that makes the generalized linear model can fit on a wide range of variables, it includes both continuous variables and discrete variables, include large skewed variables and symmetrical variables. At the same time, Generalized Linear Model changes the non-linear relationship between the dependent variables and independent variables into the linear relationship by the connected functions, which can handle the complex non-linear relationship between the dependent variables and independent variables, thus has great advantage. By now, studies of Generalized Linear Model mainly focus on its application and its parameter estimation problem, and the analysis of the model in practice can’t be separated from the application of parameter estimation. So parameter estimation problem is particularly important, it not only provides convenience for the analysis of the model in theory, but also provides theoretical basis for the prediction and parameter estimation etc in practice.This paper mainly studies the parameter estimation problem of generalized linear models. First introduces the common parameter estimation method of the generalized linear model-maximum likelihood estimation (ML), and gives the detailed calculation steps of the algorithm. Then, on the basis of the maximum likelihood estimation, other parameter estimation methods are proposed in Generalized Linear Model, namely parameter estimation methods in classic linear model are extended to generalized linear models when there is a collinear relationship in independent variables. Gives principal component estimation, ridge estimation, and Liu estimation in generalized linear model, respectively proves that the estimations are better than maximum likelihood estimation in the sense of Mean Square Error (MSE) when they satisfy some conditions. Uses the data of the practical examples to do the compared experiment, respectively lists two typical models of Poisson model and Logistic model in the generalized linear model. Respectively by fitting and comparing the numerical size of the error value shows the superiority of the estimation methods in the generalized linear model.Finally, presents two new estimation methods, ridge type principal component estimation and Liu type principal component estimation of the generalized linear model, and mainly studies their properties. Deduces conditions what ridge parameter and the number of selected principal components meet when ridge type principal estimation of the generalized linear model is better than the maximum likelihood estimation, principal comppnent estimation, and ridge estimation of the generalized linear model. Deduces conditions what ridge parameter and the number of selected principal components meet when Liu type principal estimation of the generalized linear model is better than the maximum likelihood estimation, principal component estimation, and Liu estimation of the generalized linear model. Through practical calculation examples, the advantages of the two methods are verified that they are better than previously proposed estimation methods. So these methods can be widely used to other generalized linear models, and provide new methods to generalized linear model parameter estimation.