**Diploma Thesis** submitted to the

Department of Mathematics

Faculty of Sciences, University of Fribourg Switzerland

in partial fulfilment of the requirements for the attainment of the

Diploma in Mathematics

by Michael Beer

**Academic supervisor:** Prof. Dr André Antille

**Submission date:** November 2001

**Keywords:** logistic regression, consistency, asymptotic normality

**Abstract:** In a first section of this thesis, the logistic regression model as well as the maximum likelihood procedure for the estimation of its parameters are introduced in detail. Furthermore, several ideas on the motivation of the model and the interpretation of its parameters are outlined. At the end, a short paragraph illuminates a few historical aspects in relation to the development of this model.

The second part examines the consistency of the maximum likelihood estimator for the model parameters. It is shown that, if certain hypotheses are satisfied, this estimator converges in probability to the true parameter values as the sample size increases. As a peculiarity of the procedure presented in this thesis, the number of explanatory variables is allowed to grow simultaneously with the number of observations. Conditions on the admissible interdependence of both of these characteristics are given in order to ensure the consistency of the estimator. Finally, these conditions are compared to the assumptions of an existing consistency theorem.

As another important property, the asymptotic normality of the estimator being discussed is proved in the third section. Picking up the thread of the previous part, this aspect is established in a similar context. In contrast to the proof of consistency, however, this section is based on the «classical» approach where the number of independent variables is held constant.

To conclude, a short example attempts to illustrate the application of the logistic regression model. Proceeding from a survey of health enhancing physical activity carried out by the Swiss Federal Offices of Sports and Statistics, an estimation is made what effects the linguistic background of a Swiss inhabitant has on his or her daily or weekly physical activity. The results are presented along with the Mathematica code used for the calculations.

Diploma thesis titled “Asymptotic Properties of the Maximum Likelihood Estimator in Dichotomous Logistic Regression Models”

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