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Linear regression / (Gross, Jürgen.)

Bibliographical information (record 270806)

Linear regression /
Author:	Gross, Jürgen. Search in Online Databases
Publisher:	Springer,
ISBN:	3540401784
Edition:	c2003.
Classification:	QA278.2

Detailed notes

- Includes bibliographical references (p. [381]-387) and index.

- 1 Fundamentals -- 1.1 Linear Models -- 1.1.1 Application of Linear Models -- 1.1.2 Types of Linear Models -- 1.1.3 Proceeding with Linear Models -- 1.1.4 A Preliminary Example -- 1.2 Decision Theory and Point Estimation -- 1.2.1 Decision Rule -- 1.2.2 Non-operational Decision Rule -- 1.2.3 Loss and Risk -- 1.2.4 Choosing a Decision Rule -- 1.2.5 Admissibility -- 1.2.6 Squared Error Loss -- 1.2.7 Matrix Valued Squared Error Loss -- 1.2.8 Alternative Loss Functions -- 1.3 Problems -- 2 The Linear Regression Model -- 2.1 Assumptions -- 2.2 Ordinary Least Squares Estimation -- 2.2.1 The Principle of Least Squares -- 2.2.2 Coefficient of Determination R2 -- 2.2.3 Predictive Loss -- 2.2.4 Least Squares Variance Estimator -- 2.2.5 Properties of the Ordinary Least Squares Estimator -- 2.2.6 Properties Under Normality -- 2.3 Optimality of Least Squares Estimation -- 2.3.1 Linear Unbiased Estimation -- 2.3.2 Gauss-Markov Theorem -- 2.3.3 Normality Assumption -- 2.3.4 Admissibility -- 2.4 Unreliability of Least Squares Estimation -- 2.4.1 Estimation of the Covariance Matrix -- 2.4.2 Unbiased Versus Biased Estimation -- 2.4.3 Collinearity -- 2.4.4 Consistency -- 2.4.5 Biased Estimation -- 2.5 Inadmissibility of the Ordinary Least Squares Estimator -- 2.5.1 The Reparameterized Regression Model -- 2.5.2 Risk Comparison of Least Squares and Stein Estimator -- 2.5.3 An Example for Stein Estimation -- 2.5.4 Admissibility -- 2.6 Problems -- 3 Alternative Estimators -- 3.1 Restricted Least Squares Estimation -- 3.1.1 The Principle of Restricted Least Squares -- 3.1.2 The Parameter Space -- 3.1.3 Properties of Restricted Least Squares Estimator -- 3.1.4 Risk Comparison of Restricted and Ordinary Least Squares Estimator -- 3.1.5 Pretest Estimation -- 3.2 Other Types of Restriction -- 3.2.1 Stochastic Linear Restrictions -- 3.2.2 Inequality Restrictions -- 3.2.3 Elliptical Restrictions -- 3.3 Principal Components Estimator -- 3.3.1 Preliminary Considerations -- 3.3.2 Properties of the Principal Components Estimator -- 3.3.3 Drawbacks of the Principal Components Estimator -- 3.3.4 The Marquardt Estimator -- 3.4 Ridge Estimator -- 3.4.1 Preliminary Considerations -- 3.4.2 Properties of the Linear Ridge Estimator -- 3.4.3 The Choice of the Ridge Parameter -- 3.4.4 Standardization -- 3.4.5 Ridge and Restricted Least Squares Estimator -- 3.4.6 Ridge and Principal Components Estimator -- 3.4.7 Jackknife Modified Ridge Estimator -- 3.4.8 Iteration Estimator -- 3.4.9 An Example for Ridge Estimation -- 3.5 Shrinkage Estimator -- 3.5.1 Preliminary Considerations -- 3.5.2 Risk Comparison to Ordinary Least Squares -- 3.5.3 The Choice of the Shrinkage Parameter -- 3.5.4 Direction Modified Shrinkage Estimators -- 3.6 General Ridge Estimator -- 3.6.1 A Class of Estimators -- 3.6.2 Risk Comparison of General Ridge and Ordinary Least Squares Estimator -- 3.7 Linear Minimax Estimator -- 3.7.1 Preliminary Considerations -- 3.7.2 Inequality Restrictions -- 3.7.3 Linear Minimax Solutions -- 3.7.4 Alternative Approaches -- 3.7.5 Admissibility -- 3.8 Linear Bayes Estimator -- 3.8.1 Preliminary Considerations -- 3.8.2 Characterization of Linear Bayes Estimators -- 3.8.3 Non-Operational Bayes Solutions -- 3.8.4 A-priori Assumptions -- 3.9 Robust Estimator -- 3.9.1 Preliminary Considerations -- 3.9.2 Weighted Least Squares Estimation -- 3.9.3 The 11 Estimator -- 3.9.4 M Estimator -- 3.9.5 Robust Ridge Estimator -- 3.10 Problems -- 4 Linear Admissibility -- 4.1 Preliminary Considerations -- 4.2 Linear Admissibility in the Non-Restricted Model -- 4.2.1 Linear Admissibility in the Simple Mean Shift Model -- 4.2.2 Characterization of Linearly Admissible Estimators -- 4.2.3 Ordinary Least Squares and Linearly Admissible Estimator -- 4.2.4 Linear Transforms of Ordinary Least Squares Estimator -- 4.2.5 Linear Admissibility of Known Estimators -- 4.2.6 Shrinkage Property and Linear Admissibility -- 4.2.7 Convex Combination of Estimators -- 4.2.8 Linear Bayes Estimator -- 4.3 Linear Admissibility Under Linear Restrictions -- 4.3.1 The Assumption of a Full Rank Restriction Matrix -- 4.3.2 Restricted Estimator -- 4.3.3 Characterization of Linearly Admissible Estimators -- 4.4 Linear Admissibility Under Elliptical Restrictions -- 4.4.1 Characterization of Linearly Admissible Estimators -- 4.4.2 Linear Admissibility of Certain Linear Estimators -- 4.4.3 Admissible Improvements Over Ordinary Least Squares -- 4.5 Problems -- 5 The Covariance Matrix of the Error Vector -- 5.1 Estimation of the Error Variance -- 5.1.1 The Sample Variance -- 5.1.2 Nonnegative Unbiased Estimation -- 5.1.3 Optimality of the Least Squares Variance Estimator -- 5.1.4 Non-Admissibility of the Least Squares Variance Estimator -- 5.2 Non-Scalar Covariance Matrix -- 5.2.1 Preliminary Considerations -- 5.2.2 The Transformed Model -- 5.2.3 Two-Stage Estimation -- 5.3 Occurrence of Non-Scalar Covariance Matrices -- 5.3.1 Seemingly Unrelated Regression -- 5.3.2 Heteroscedastic Errors -- 5.3.3 Equicorrelated Errors -- 5.3.4 Autocorrelated Errors -- 5.4 Singular Covariance Matrices -- 5.5 Equality of Ordinary and Generalized Least Squares -- 5.6 Problems -- 6 Regression Diagnostics -- 6.1 Selecting Independent Variables -- 6.1.1 Mallows' Cp -- 6.1.2 Stepwise Regression -- 6.1.3 Alternative Criteria -- 6.2 Assessing Goodness of Fit -- 6.3 Diagnosing Collinearity -- 6.3.1 Variance Inflation Factors -- 6.3.2 Scaled Condition Indexes -- 6.4 Inspecting Residuals -- 6.4.1 Normal Quantile Plot -- 6.4.2 Residuals Versus Fitted Values Plot -- 6.4.3 Further Residual Plots -- 6.5 Finding Influential Observations -- 6.5.1 Leverage -- 6.5.2 Influential Observations -- 6.5.3 Collinearity-Influential Observations -- 6.6 Testing Model Assumptions -- 6.6.1 Preliminary Considerations -- 6.6.2 Testing for Heteroscedasticity -- 6.6.3 Testing for Autocorrelation -- 6.6.4 Testing for Non-Normality -- 6.6.5 Testing for Non-Linearity -- 6.6.6 Testing for Outlier -- 6.7 Problems -- A.1 Preliminaries -- A.1.1 Matrices and Vectors -- A.1.2 Elementary Operations -- A.1.3 Rank of a Matrix -- A.1.4 Subspaces and Matrices -- A.1.5 Partitioned Matrices -- A.1.6 Kronecker Product -- A.1.7 Moore-Penrose Inverse -- A.2 Common Pitfalls -- A.3 Square Matrices -- A.3.1 Specific Square Matrices -- A.3.2 Trace and Determinant -- A.3.3 Eigenvalue and Eigenvector -- A.3.4 Vector and Matrix Norm -- A.3.5 Definiteness -- A.4 Symmetric Matrix -- A.4.1 Eigenvalues -- A.4.2 Spectral Decomposition -- A.4.3 Rayleigh Ratio -- A.4.4 Definiteness -- A.5 Lowner Partial Ordering -- B.1 Expectation and Covariance -- B.2 Multivariate Normal Distribution -- B.3 X2 Distribution -- B.4 F Distribution -- C.1 Problem and Goal -- C.2 The Data -- C.3 The Choice of Variables -- C.3.1 The Full Model -- C.3.2 Stepwise Regression -- C.3.3 Collinearity Diagnostics -- C.4 Further Diagnostics -- C.4.1 Residuals -- C.4.2 Influential Observations -- C.5 Prediction.