Measurement Error
Models
WAYNE A. FULLER
在科学实验、工农业生产以及社会调研等领域中，对兴趣变量进行测量时，往往会受到多种因素的影响，导致一些偏差，如抽样误差、仪器误差、记录误差等等;另外，人们考察变量之间的关系时，往往只关心主要因素对兴趣变量的影响，其它影响不大的因素的效应将反映于兴趣变量取值的偏差中.文献中这种观测数据带有误差的问题通常称为“测量误差问题”，分析这些数据的统计模型通常称为“测量误差模型”或“ev(errors-in-variables)模型”.在实践中，利用各种ev模型(包括线性ev模型、非线性ev模型、半参数ev模型等)解决实际问题时，常常会遇到各种各样的复杂数据，例如删失数据、纵向数据、缺失数据以及时间序列数据等等.因此，研究各种复杂数据下的ev模型富有现实意义，此类问题目前已经成为统计学界研究的热点课题之一.

Fuller 的这本书正是这个领域的经典值得下载学习!

Contents
List of Examples xv
List of Principal Results xix
List of Figures xxiii
1. A Single Explanatory Variable 1
1.1. Introduction, 1
1.1.1.
1.1.2.
1.1.3. Identification, 9
1.2.1. Introduction and Estimators, 13
1.2.2. Sampling Properties of the Estimators, 15
1.2.3. Estimation of True x Values, 20
1.2.4. Model Checks, 25
1.3. Ratio of Measurement Variances Known, 30
1.3.1. Introduction, 30
1.3.2. Method of Moments Estimators, 30
1.3.3. Least Squares Estimation, 36
1.3.4. Tests of Hypotheses for the Slope, 44
1.4. Instrumental Variable Estimation, 50
1.5. Factor Analysis, 59
1.6. Other Methods and Models, 72
1.6.1. Distributional Knowledge, 72
Ordinary Least Squares and Measurement Error, 1
Estimation with Known Reliability Ratio, 5
1.2. Measurement Variance Known, 13
xi
xii CONTENTS
1.6.2. The Method of Grouping, 73
1.6.3. Measurement Error and Prediction, 74
1.6.4. Fixed Observed X, 79
Appendix 1 .A.
Appendix l.B.
Appendix l.C.
Appendix l.D.
Large Sample Approximations, 85
Moments of the Normal Distribution, 88
Central Limit Theorems for Sample Moments, 89
Notes on Notation, 95
2. Vector Explanatory Variables 100
2.1. Bounds for Coefficients, 100
2.2. The Model with an Error in the Equation, 103
2.2.1. Estimation of Slope Parameters, 103
2.2.2. Estimation of True Values, 113
2.2.3. Higher-Order Approximations for
Residuals and True Values, 118
2.3. The Model with No Error in the Equation, 124
2.3.1. The Functional Model, 124
2.3.2. The Structural Model, 139
2.3.3. Higher-Order Approximations for
Residuals and True Values, 140
2.4. Instrumental Variable Estimation, 148
2.5. Modifications to Improve Moment Properties, 163
2.5.1. An Error in the Equation, 164
2.5.2. No Error in the Equation, 173
2.5.3. Calibration, 177
Appendix 2.A. Language Evaluation Data, 18 1
3. Extensions of the Single Relation Model 185
3.1. Nonnormal Errors and Unequal Error Variances, 185
3.1.1. Introduction and Estimators, 186
3.1.2. Models with an Error in the Equation, 193
3.1.3. Reliability Ratios Known, 199
3.1.4. Error Variance Functionally Related to
Observations, 202
3.1.5. The Quadratic Model, 212
3.1.6. Maximum Likelihood Estimation for Known
Error Covariance Matrices. 217
CONTENTS xiii
3.2. Nonlinear Models with No Error in the Equation, 225
3.2.1. Introduction, 225
3.2.2. Models Linear in x, 226
3.2.3. Models Nonlinear in x, 229
3.2.4. Modifications of the Maximum Likelihood
Estimator, 247
3.3. The Nonlinear Model with an Error in the Equation, 261
3.3.1. The Structural Model, 261
3.3.2. General Explanatory Variables, 263
Measurement Error Correlated with True Value, 271
3.4.1. Introduction and Estimators, 271
3.4.2. Measurement Error Models for Multinomial
Random Variables, 272
3.4.
Appendix 3.A. Data for Examples, 281
4. Multivariate Models
4.1. The Classical Multivariate Model, 292
4.1.1. Maximum Likelihood Estimation, 292
4.1.2. Properties of Estimators, 303
Least Squares Estimation of the Parameters
of a Covariance Matrix, 321
4.2.1. Least Squares Estimation, 321
4.2.2. Relationships between Least Squares
and Maximum Likelihood, 333
4.2.3. Least Squares Estimation for the
Multivariate Functional Model, 338
4.2.
4.3. Factor Analysis, 350
4.3.1. Introduction and Model, 350
4.3.2. Maximum Likelihood Estimation, 353
4.3.3. Limiting Distribution of Factor Estimators, 360
Appendix 4.A. Matrix-Vector Operations, 382
Appendix 4.B.
Appendix 4.C.
Properties of Least Squares and Maximum
Likelihood Estimators, 396
Maximum Likelihood Estimation for
Singular Measurement Covariance, 404
Bibliography
Author Index
292
409
433
Subject Index 435