Large-Sample Theory Of The Maximum Likelihood Estimate In Generalized Linear Models

We address in this thesis some important topics in the large-sample theory forgeneralized linear models (GLM). They are weak consistency, strong consistencyand asymptotic normality of the maximum likelihood estimate (MLE) of the para-meters. It is well-known that generalized linear models are the further developmentof classical linear models. Including many other models that have been found usefulin statistical analysis, generalized linear models are applied in a wide sense. Basedon the asymptotic theory of the maximum likelihood estimate for generalized linearmodels introduced by Fahrmeir.L & Kaufmann.H (1985), we improve and generalizetheir theory about the consistency of the MLE. Then, we develop their models inthe way that we consider the case that the regressors are stochastic matrixes andhave different distributions. In this case the large-sample theory of the MLE ofthe parameters are studied under the assumption of a natural link function and anon-natural link function, respectively. Since the improvement in theory and thenewly set-up asymptotic properties are more realistic, our study on them has morepractical value.This thesis consists of five parts as follows:In Chapter 1, we introduce first the background of generalized linear modelsand the results about the asymptotic properties of the MLE of Fahrmeir.L & Kauf-mann.H (1985). Following, we introduce in general the results that we obtain in thisthesis.In Chapter 2, we review first the conditions about the consistency of the MLEsuggested by Fahrmeir.L & Kaufmann.H (1985). Then, by an example, we showthat the conditions could be improved. Finally, we raise a more reasonable sufficientcondition for the weak consistency of the MLE.In Chapter 3, we recall the conditions about the strong consistency of the MLE in the case of the regressors with a compact range also suggested by Fahrmeir.L &Kaufmann.H (1985). Following this, we advance the theory to a large degree byshowing that the condition with natural link function remains to be sufficient evenin the case of non-natural link function.In Chapter 4, we consider the further case that the regressors are stochasticmatrixes and have different distributions. The weak and strong consistency andasymptotic normality of the MLE are studied under the assumption of a naturallink function. Specially, the case of regressors with a compact range is considered.In Chapter 5, in case that the regressors are stochastic and have different dis-tributions and the observations of the responses may have difibrent dimensionality,the asymptotic properties of the MLE are studied, under the assumption of a non-natural link function. What is more, two useful cases are considered. They includethe case of regressors with a compact range and the case of responses with a boundedrange.