Statistical Regression with Measurement Error

1. Introduction to Linear Measurement Error Models.

1.1 Preliminaries.

1.2 Elementary Properties of Measurement Error Models.

1.3 Maximum Likelihood Estimation in the Univariate Normal Measurement Error Model.

1.4 The ME Model with Correlated Errors.

1.5 The Equation Error Model.

1.6 The Berkson Model.

1.7 Maximum Likelihood Estimation of Transformed Data: Elimination of Nuisance Parameters.

1.8 Bibliographic Notes and Discussion.

1.9 Exercises.

2. Properties of Estimate and Predictors.

2.1 Asymptotic Properties of ME Model Parameter Estimates.

2.2 Asymptotic Properties of Equation Error Model Estimates.

2.3 Finite-Sample Properties.

2.4 Implications Regarding Confidence Regions.

2.5 Prediction and Calibration under Measurement Error Models.

2.6 Bibliographic Notes and Discussion.

2.7 Exercises.

2.8 Research Problems.

3. Comparing Model Assumptions and Modifying Least Squares.

3.1 Issues Facing Users of ME Models.

3.2 A Unified Approach to the Functional, Structural, and Ultrastructural Relationships.

3.3 Identifiability Assumptions and the Equation Error Model.

3.4 Generalized Least Squares.

3.5 Modified Least Squares.

3.6 Bibliographic Notes and Discussion.

3.7 Exercises.

4. Alternative Approaches to the Measurement Error Model.

4.1 Introduction and Overview.

4.2 Instrumental Variable Estimators.

4.3 Grouping Methods.

4.4 Methods Based on Ranks.

4.5 Methods of Higher-order Moments and Product Cumulants.

4.6 Bibliographic Notes and Discussion.

4.7 Exercises.

5. Linear Measurement Error Model with Vector Explanatory Variables.

5.1 Introduction.

5.2 Identifiability

5.3 The Equation Error Model.

5.4 Maximum Likelihood for the No-Equation-Error Model.

5.5 Alternative Approaches to Estimating the Parameters.

5.6 Asymptotic Properties of the Estimates.

5.7 Bibliographic Notes and Discussion.

5.8 Exercises.

5.9 Research Problem.

6. Polynomial Measurement Error Models.

6.1 Introduction.

6.2 The Nonlinear Structural Model.

6.3 Identifiability in Nonlinear ME Models.

6.4 Polynomial Model with Equation Error.

6.5 The Polynomial Functional Relationship without Equation Error.

6.6 Polynomial Berkson Model.

6.7 Bibliographic Notes and Discussion.

6.8 Exercises.

7. Robust Estimation in Measurement Error Models.

7.1 Introduction.

7.2 Robust Orthogonal Regression.

7.3 Robust Measurement Error Model Estimation via Robust Covariance Matrices.

7.4 Computational Methods for Robust Orthogonal Regression.

7.5 Bibliographic Notes and Discussion.

7.7 Exercises.

7.8 Research Problem.

8. Additional Topics.

8.1 Estimation of the True Variables.

8.2 Obtaining Identifiability Assumption Information.

8.3 Conclusions.

8.4 Relations to Other Latent Variables Models.

8.5 The factor analysis model.

8.6 Terminology.

8.7 Exercises.

Appendix A. Identification in Measurement Error Models.

A.1 Overview.

A.2 Structural Model.

A.3 Functional Model.

A.4 Identiability and Consistent Estimation.

Bibliography.

Author Index.

Subject Index.