Functional relations, random coefficients, and nonlinear regression with application to kinetic data
These notes are based on some lectures given by the author at the Institute of Mathematical Statistics, University of Copenhagen. The list of contents is as follows: Chapter 1: An example from physiology; Chapter 2: The general linear model; Chapter 3: The linear functional relation; Chapter 4: Random coefficient models; Chapter 5: Nonlinear regression; Chapter 6: Some examples of non-linear regression problems. An appendix gives basic concepts and results from linear algebra. The first chapter gives an example from physiology which motivates the problems studied in the later chapters. The second chapter discusses linear model theory when the observations are independent with the same variance and when the observations are correlated with known or unknown covariance matrix. The general linear model with estimated weights is studied in this chapter. The approach to the study of linear models throughout this chapter is geometric in nature. Let \((X_ i,Y_ i)\), 1\(\leq i\leq n\), be independent random variables distributed as bivariate normal with mean \((\xi_ i,\alpha +\beta \xi_ i)\) for 1\(\leq i\leq n\) and covariance matrix \(\sigma^ 2I\). This is not a linear model. The author studies the problems of estimation of \((\alpha,\beta,\xi_ 1,...,\xi_ n,\sigma^ 2)\) in Chapter 2. Asymptotic properties of the estimators and test statistics based on them are derived. Random coefficient models are discussed in Chapter 3. Here linear models with Gaussian errors are considered. Tests for the specification of the model and tests of hypotheses within the model are derived. Asymptotic properties of the estimator in a nonlinear regression model \(Y\simeq N_ n(f(\theta),\sigma^ 2I)\), \(\theta \in \Theta \subset R^ p\), are derived in Chapter 4. A likelihood ratio test for testing the hypothesis of the form \(H_ o:\theta =\theta(\beta)\), \(\beta \in B\), \(\theta\) (B)\(\subset \Theta\) is derived and its limiting distribution is obtained. An approximation to the distribution of the residual sum of squares in a nonlinear regression is derived in this chapter using the notion of normal radius curvature. Some examples illustrating these results are given in chapter 6. The notes are well written and can be used for a series of lectures on regression analysis at a graduate level.
general linear model with estimated weights
scientific article; zbMATH DE number 3864316