These notes on regression give an introduction to some of the techniques that I have found useful when working with various data sets in collaboration with Dr. S. Keiding (Copenhagen) and Dr. J.W.L. Robinson (Lausanne). The notes are based on some lectures given at the Institute of Mathematical Statistics, University of Copenhigen, 1978-81, for graduate students, and assumes a familiarity with statistical theory corresponding to the book by C.R. Rao: "Linear Statistical Inference and its Applications". Wiley, New York (1973) . The mathematical tools needed for the algebraic treatment of the models are some knowledge of finite dimensional vector spaces with an inner product and the notion of orthogonal projection. For the analytic treatment I need characteristic functions and weak convergence as the main tools. The most important statistical concepts are the general linear model for Gaussian variables and the general methods of maximum likelihood estimation as well as the likelihood ratio test. All these topics are presented in the above mentioned book by Rao and the reader is referred to that for details. For convenience a short appendix is added where the fundamental concepts from linear algebra are discussed
CONTENT
List of Contents -- 1. An example from physiology -- 1.1 The experiment and the data -- 1.2 The model -- 2. The general linear model -- 2.1 Independent observations with the same variance -- 2.2 Correlated observations with known covariance matrix -- 2.3 Correlated observations with unknown covariance matrix -- 2.4 The general linear model with estimated weights -- 3. The linear functional relation -- 3.1 Introduction -- 3.2 The model and the estimates -- 3.3 Comments on the model -- 3.4 Asymptotic theory -- 3.5 Asymptotic properties of the estimators -- 3.6 Asymptotic properties of the test statistics -- 3.7 Comments on the asymptotic results -- 4. Random coefficient models -- 4.1 Introduction -- 4.2 The balanced case -- 4.3 Test for the specification of the model -- 4.4 Test of hypotheses within the model -- 4.5 An analysis of the unbalanced case -- 5. Non linear regression -- 5.1 The model -- 5.2 Estimation -- 5.3 Hypothesis testing -- 5.4 Curvature -- 6. Some examples of non linear regression problems -- 7. References -- 8. Index -- Examples -- 9. Appendix
