Linear Models and Regression
Lecture notes for the two-semester course "Linear Models and Regression I-II"
Reiter
- Introduction (Definition of linear models, examples)
- Estimation of Parameters (Vector and matrix representation, estimation of \(\theta\), estimation of \(\sigma\), Gauss-Markov theorem and standard errors, performance of the estimators in case of misspecified models, parametrizations for categorical covariates)
- Tests and Confidence Regions (Multivariate normal distributions, the joint distribution of \(\hat{\theta}\) and \(\hat{\sigma}\), student confidence intervalle and tests, F confidence regions and tests, further simultaneous confidence regions, non-central F distributions and approximation errors, calibration, random effects, data-scientific interpretation of F tests)
- Regression diagnostics (Leverage, an application of the Central Limit Theorem, residual analyses, transformations)
- Nonparametric Regression (Spline regression, local polynomials, regularization)
- General Considerations about Estimation (Means and quantiles als optimal predictors, loss and risk functions, maximum likelihood estimation, application to regression problems)
- Logistic Regression and Related Models (Logistic regression, general asymptotic considerations, methods for multicategorical response, Poisson regression, supplements)
- Bootstrap procedures (Bootstrap for simple samples, bootstrap for regression)
- Empirical Likelihood Procedures
- Appendix: Various Auxiliary Results (QR decomposition, Iteratively reweighted least squares, mean vectors and covariance matrices, B-splines, weak convergence of distributions, Central Limit Theorem, couplings and Mallows distances, an inequality for sums of independent random vectors)
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