Sampling and Estimation from Finite Populations

YVES TILLÉ, PhD, is a Professor at the University of Neuchâtel (Université de Neuchâtel) in Neuchâtel, Switzerland.

• List of Figures xiiiList of Tables xviiList of Algorithms xixPreface xxiPreface to the First French Edition xxiiiTable of Notations xxv1 A History of Ideas in Survey Sampling Theory 11.1 Introduction 11.2 Enumerative Statistics During the 19th Century 21.3 Controversy on the use of Partial Data 41.4 Development of a Survey Sampling Theory 51.5 The US Elections of 1936 61.6 The Statistical Theory of Survey Sampling 61.7 Modeling the Population 81.8 Attempt to a Synthesis 91.9 Auxiliary Information 91.10 Recent References and Development 102 Population, Sample, and Estimation 132.1 Population 132.2 Sample 142.3 Inclusion Probabilities 152.4 Parameter Estimation 172.5 Estimation of a Total 182.6 Estimation of a Mean 192.7 Variance of the Total Estimator 202.8 Sampling with Replacement 22Exercises 243 Simple and Systematic Designs 273.1 Simple Random Sampling without Replacement with Fixed Sample Size 273.1.1 Sampling Design and Inclusion Probabilities 273.1.2 The Expansion Estimator and its Variance 283.1.3 Comment on the Variance–Covariance Matrix 313.2 Bernoulli Sampling 323.2.1 Sampling Design and Inclusion Probabilities 323.2.2 Estimation 343.3 Simple Random Sampling with Replacement 363.4 Comparison of the Designs with and Without Replacement 383.5 Sampling with Replacement and Retaining Distinct Units 383.5.1 Sample Size and Sampling Design 383.5.2 Inclusion Probabilities and Estimation 413.5.3 Comparison of the Estimators 443.6 Inverse Sampling with Replacement 453.7 Estimation of Other Functions of Interest 473.7.1 Estimation of a Count or a Proportion 473.7.2 Estimation of a Ratio 483.8 Determination of the Sample Size 503.9 Implementation of Simple Random Sampling Designs 513.9.1 Objectives and Principles 513.9.2 Bernoulli Sampling 513.9.3 Successive Drawing of the Units 523.9.4 Random Sorting Method 523.9.5 Selection–Rejection Method 533.9.6 The Reservoir Method 543.9.7 Implementation of Simple Random Sampling with Replacement 563.10 Systematic Sampling with Equal Probabilities 573.11 Entropy for Simple and Systematic Designs 583.11.1 Bernoulli Designs and Entropy 583.11.2 Entropy and Simple Random Sampling 603.11.3 General Remarks 61Exercises 614 Stratification 654.1 Population and Strata 654.2 Sample, Inclusion Probabilities, and Estimation 664.3 Simple Stratified Designs 684.4 Stratified Design with Proportional Allocation 704.5 Optimal Stratified Design for the Total 714.6 Notes About Optimality in Stratification 744.7 Power Allocation 754.8 Optimality and Cost 764.9 Smallest Sample Size 764.10 Construction of the Strata 774.10.1 General Comments 774.10.2 Dividing a Quantitative Variable in Strata 774.11 Stratification Under Many Objectives 79Exercises 805 Sampling with Unequal Probabilities 835.1 Auxiliary Variables and Inclusion Probabilities 835.2 Calculation of the Inclusion Probabilities 845.3 General Remarks 855.4 Sampling with Replacement with Unequal Inclusion Probabilities 865.5 Nonvalidity of the Generalization of the Successive Drawing without Replacement 885.6 Systematic Sampling with Unequal Probabilities 895.7 Deville’s Systematic Sampling 915.8 Poisson Sampling 925.9 Maximum Entropy Design 955.10 Rao–Sampford Rejective Procedure 985.11 Order Sampling 1005.12 Splitting Method 1015.12.1 General Principles 1015.12.2 Minimum Support Design 1035.12.3 Decomposition into Simple Random Sampling Designs 1045.12.4 Pivotal Method 1075.12.5 Brewer Method 1095.13 Choice of Method 1105.14 Variance Approximation 1115.15 Variance Estimation 114Exercises 1156 Balanced Sampling 1196.1 Introduction 1196.2 Balanced Sampling: Definition 1206.3 Balanced Sampling and Linear Programming 1226.4 Balanced Sampling by Systematic Sampling 1236.5 Methode of Deville, Grosbras, and Roth 1246.6 Cube Method 1256.6.1 Representation of a Sampling Design in the form of a Cube 1256.6.2 Constraint Subspace 1266.6.3 Representation of the Rounding Problem 1276.6.4 Principle of the Cube Method 1306.6.5 The Flight Phase 1306.6.6 Landing Phase by Linear Programming 1336.6.7 Choice of the Cost Function 1346.6.8 Landing Phase by Relaxing Variables 1356.6.9 Quality of Balancing 1356.6.10 An Example 1366.7 Variance Approximation 1376.8 Variance Estimation 1406.9 Special Cases of Balanced Sampling 1416.10 Practical Aspects of Balanced Sampling 141Exercise 1427 Cluster and Two-stage Sampling 1437.1 Cluster Sampling 1437.1.1 Notation and Definitions 1437.1.2 Cluster Sampling with Equal Probabilities 1467.1.3 Sampling Proportional to Size 1477.2 Two-stage Sampling 1487.2.1 Population, Primary, and Secondary Units 1497.2.2 The Expansion Estimator and its Variance 1517.2.3 Sampling with Equal Probability 1557.2.4 Self-weighting Two-stage Design 1567.3 Multi-stage Designs 1577.4 Selecting Primary Units with Replacement 1587.5 Two-phase Designs 1617.5.1 Design and Estimation 1617.5.2 Variance and Variance Estimation 1627.6 Intersection of Two Independent Samples 163Exercises 1658 Other Topics on Sampling 1678.1 Spatial Sampling 1678.1.1 The Problem 1678.1.2 Generalized Random Tessellation Stratified Sampling 1678.1.3 Using the Traveling Salesman Method 1698.1.4 The Local Pivotal Method 1698.1.5 The Local Cube Method 1698.1.6 Measures of Spread 1708.2 Coordination in Repeated Surveys 1728.2.1 The Problem 1728.2.2 Population, Sample, and Sample Design 1738.2.3 Sample Coordination and Response Burden 1748.2.4 Poisson Method with Permanent Random Numbers 1758.2.5 Kish and Scott Method for Stratified Samples 1768.2.6 The Cotton and Hesse Method 1768.2.7 The Rivière Method 1778.2.8 The Netherlands Method 1788.2.9 The Swiss Method 1788.2.10 Coordinating Unequal Probability Designs with Fixed Size 1818.2.11 Remarks 1818.3 Multiple Survey Frames 1828.3.1 Introduction 1828.3.2 Calculating Inclusion Probabilities 1838.3.3 Using Inclusion Probability Sums 1848.3.4 Using a Multiplicity Variable 1858.3.5 Using a Weighted Multiplicity Variable 1868.3.6 Remarks 1878.4 Indirect Sampling 1878.4.1 Introduction 1878.4.2 Adaptive Sampling 1888.4.3 Snowball Sampling 1888.4.4 Indirect Sampling 1898.4.5 The Generalized Weight Sharing Method 1908.5 Capture–Recapture 1919 Estimation with a Quantitative Auxiliary Variable 1959.1 The Problem 1959.2 Ratio Estimator 1969.2.1 Motivation and Definition 1969.2.2 Approximate Bias of the Ratio Estimator 1979.2.3 Approximate Variance of the Ratio Estimator 1989.2.4 Bias Ratio 1999.2.5 Ratio and Stratified Designs 1999.3 The Difference Estimator 2019.4 Estimation by Regression 2029.5 The Optimal Regression Estimator 2049.6 Discussion of the Three Estimation Methods 205Exercises 20810 Post-Stratification and Calibration on Marginal Totals 20910.1 Introduction 20910.2 Post-Stratification 20910.2.1 Notation and Definitions 20910.2.2 Post-Stratified Estimator 21110.3 The Post-Stratified Estimator in Simple Designs 21210.3.1 Estimator 21210.3.2 Conditioning in a Simple Design 21310.3.3 Properties of the Estimator in a Simple Design 21410.4 Estimation by Calibration on Marginal Totals 21710.4.1 The Problem 21710.4.2 Calibration on Marginal Totals 21810.4.3 Calibration and Kullback–Leibler Divergence 22010.4.4 Raking Ratio Estimation 22110.5 Example 221Exercises 22411 Multiple Regression Estimation 22511.1 Introduction 22511.2 Multiple Regression Estimator 22611.3 Alternative Forms of the Estimator 22711.3.1 Homogeneous Linear Estimator 22711.3.2 Projective Form 22811.3.3 Cosmetic Form 22811.4 Calibration of the Multiple Regression Estimator 22911.5 Variance of the Multiple Regression Estimator 23011.6 Choice of Weights 23111.7 Special Cases 23111.7.1 Ratio Estimator 23111.7.2 Post-stratified Estimator 23111.7.3 Regression Estimation with a Single Explanatory Variable 23311.7.4 Optimal Regression Estimator 23311.7.5 Conditional Estimation 23511.8 Extension to Regression Estimation 236Exercise 23612 Calibration Estimation 23712.1 Calibrated Methods 23712.2 Distances and Calibration Functions 23912.2.1 The Linear Method 23912.2.2 The Raking Ratio Method 24012.2.3 Pseudo Empirical Likelihood 24212.2.4 Reverse Information 24412.2.5 The Truncated Linear Method 24512.2.6 General Pseudo-Distance 24612.2.7 The Logistic Method 24912.2.8 Deville Calibration Function 24912.2.9 Roy and Vanheuverzwyn Method 25112.3 Solving Calibration Equations 25212.3.1 Solving by Newton’s Method 25212.3.2 Bound Management 25312.3.3 Improper Calibration Functions 25412.3.4 Existence of a Solution 25412.4 Calibrating on Households and Individuals 25512.5 Generalized Calibration 25612.5.1 Calibration Equations 25612.5.2 Linear Calibration Functions 25712.6 Calibration in Practice 25812.7 An Example 259Exercises 26013 Model-Based approach 26313.1 Model Approach 26313.2 The Model 26313.3 Homoscedastic Constant Model 26713.4 Heteroscedastic Model 1 Without Intercept 26713.5 Heteroscedastic Model 2 Without Intercept 26913.6 Univariate Homoscedastic Linear Model 27013.7 Stratified Population 27113.8 Simplified Versions of the Optimal Estimator 27313.9 Completed Heteroscedasticity Model 27613.10 Discussion 27713.11 An Approach that is Both Model- and Design-based 27714 Estimation of Complex Parameters 28114.1 Estimation of a Function of Totals 28114.2 Variance Estimation 28214.3 Covariance Estimation 28314.4 Implicit Function Estimation 28314.5 Cumulative Distribution Function and Quantiles 28414.5.1 Cumulative Distribution Function Estimation 28414.5.2 Quantile Estimation: Method 1 28414.5.3 Quantile Estimation: Method 2 28514.5.4 Quantile Estimation: Method 3 28714.5.5 Quantile Estimation: Method 4 28814.6 Cumulative Income, Lorenz Curve, and Quintile Share Ratio 28814.6.1 Cumulative Income Estimation 28814.6.2 Lorenz Curve Estimation 28914.6.3 Quintile Share Ratio Estimation 28914.7 Gini Index 29014.8 An Example 29115 Variance Estimation by Linearization 29515.1 Introduction 29515.2 Orders of Magnitude in Probability 29515.3 Asymptotic Hypotheses 30015.3.1 Linearizing a Function of Totals 30115.3.2 Variance Estimation 30315.4 Linearization of Functions of Interest 30315.4.1 Linearization of a Ratio 30315.4.2 Linearization of a Ratio Estimator 30415.4.3 Linearization of a Geometric Mean 30515.4.4 Linearization of a Variance 30515.4.5 Linearization of a Covariance 30615.4.6 Linearization of a Vector of Regression Coefficients 30715.5 Linearization by Steps 30815.5.1 Decomposition of Linearization by Steps 30815.5.2 Linearization of a Regression Coefficient 30815.5.3 Linearization of a Univariate Regression Estimator 30915.5.4 Linearization of a Multiple Regression Estimator 30915.6 Linearization of an Implicit Function of Interest 31015.6.1 Estimating Equation and Implicit Function of Interest 31015.6.2 Linearization of a Logistic Regression Coefficient 31115.6.3 Linearization of a Calibration Equation Parameter 31315.6.4 Linearization of a Calibrated Estimator 31315.7 Influence Function Approach 31415.7.1 Function of Interest, Functional 31415.7.2 Definition 31515.7.3 Linearization of a Total 31615.7.4 Linearization of a Function of Totals 31615.7.5 Linearization of Sums and Products 31715.7.6 Linearization by Steps 31815.7.7 Linearization of a Parameter Defined by an Implicit Function 31815.7.8 Linearization of a Double Sum 31915.8 Binder’s Cookbook Approach 32115.9 Demnati and Rao Approach 32215.10 Linearization by the Sample Indicator Variables 32415.10.1 The Method 32415.10.2 Linearization of a Quantile 32615.10.3 Linearization of a Calibrated Estimator 32715.10.4 Linearization of a Multiple Regression Estimator 32815.10.5 Linearization of an Estimator of a Complex Function with Calibrated Weights 32915.10.6 Linearization of the Gini Index 33015.11 Discussion on Variance Estimation 331Exercises 33116 Treatment of Nonresponse 33316.1 Sources of Error 33316.2 Coverage Errors 33416.3 Different Types of Nonresponse 33416.4 Nonresponse Modeling 33516.5 Treating Nonresponse by Reweighting 33616.5.1 Nonresponse Coming from a Sample 33616.5.2 Modeling the Nonresponse Mechanism 33716.5.3 Direct Calibration of Nonresponse 33916.5.4 Reweighting by Generalized Calibration 34116.6 Imputation 34216.6.1 General Principles 34216.6.2 Imputing From an Existing Value 34216.6.3 Imputation by Prediction 34216.6.4 Link Between Regression Imputation and Reweighting 34316.6.5 Random Imputation 34516.7 Variance Estimation with Nonresponse 34716.7.1 General Principles 34716.7.2 Estimation by Direct Calibration 34816.7.3 General Case 34916.7.4 Variance for Maximum Likelihood Estimation 35016.7.5 Variance for Estimation by Calibration 35316.7.6 Variance of an Estimator Imputed by Regression 35616.7.7 Other Variance Estimation Techniques 35717 Summary Solutions to the Exercises 359Bibliography 379Author Index 405Subject Index 411

"A task for the current, and future, generation is the research and development of methods for integrating data from multiple sources by explicitly addressing the different measurement errors. Those who read this book and address its challenges will be well placed to deal with the research opportunities ahead—both foreseen and yet to be identified."—Carl M. O'Brien, Lowestoft Laboratory, International Statistical Review (2020) doi:10.1111/insr.12420