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Sampling and Estimation from Finite Populations

Wiley Series in Survey Methodology 1. Aufl.

von: Yves Tille
73,99 €
Verlag:	Wiley
Format:	PDF
Veröffentl.:	21.02.2020
ISBN/EAN:	9781119071266
Sprache:	englisch
Anzahl Seiten:	448

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Titelbeschreibung

A much-needed reference on survey sampling and its applications that presents the latest advances in the field Seeking to show that sampling theory is a living discipline with a very broad scope, this book examines the modern development of the theory of survey sampling and the foundations of survey sampling. It offers readers a critical approach to the subject and discusses putting theory into practice. It also explores the treatment of non-sampling errors featuring a range of topics from the problems of coverage to the treatment of non-response. In addition, the book includes real examples, applications, and a large set of exercises with solutions. Sampling and Estimation from Finite Populations begins with a look at the history of survey sampling. It then offers chapters on: population, sample, and estimation; simple and systematic designs; stratification; sampling with unequal probabilities; balanced sampling; cluster and two-stage sampling; and other topics on sampling, such as spatial sampling, coordination in repeated surveys, and multiple survey frames. The book also includes sections on: post-stratification and calibration on marginal totals; calibration estimation; estimation of complex parameters; variance estimation by linearization; and much more. <ul> <li>Provides an up-to-date review of the theory of sampling</li> <li>Discusses the foundation of inference in survey sampling, in particular, the model-based and design-based frameworks</li> <li>Reviews the problems of application of the theory into practice</li> <li>Also deals with the treatment of non sampling errors</li> </ul> Sampling and Estimation from Finite Populations is an excellent book for methodologists and researchers in survey agencies and advanced undergraduate and graduate students in social science, statistics, and survey courses.

Inhaltsverzeichnis

List of Figures xiii List of Tables xvii List of Algorithms xix Preface xxi Preface to the First French Edition xxiii Table of Notations xxv 1 A History of Ideas in Survey Sampling Theory 1 1.1 Introduction 1 1.2 Enumerative Statistics During the 19th Century 2 1.3 Controversy on the use of Partial Data 4 1.4 Development of a Survey Sampling Theory 5 1.5 The US Elections of 1936 6 1.6 The Statistical Theory of Survey Sampling 6 1.7 Modeling the Population 8 1.8 Attempt to a Synthesis 9 1.9 Auxiliary Information 9 1.10 Recent References and Development 10 2 Population, Sample, and Estimation 13 2.1 Population 13 2.2 Sample 14 2.3 Inclusion Probabilities 15 2.4 Parameter Estimation 17 2.5 Estimation of a Total 18 2.6 Estimation of a Mean 19 2.7 Variance of the Total Estimator 20 2.8 Sampling with Replacement 22 Exercises 24 3 Simple and Systematic Designs 27 3.1 Simple Random Sampling without Replacement with Fixed Sample Size 27 3.1.1 Sampling Design and Inclusion Probabilities 27 3.1.2 The Expansion Estimator and its Variance 28 3.1.3 Comment on the Variance–Covariance Matrix 31 3.2 Bernoulli Sampling 32 3.2.1 Sampling Design and Inclusion Probabilities 32 3.2.2 Estimation 34 3.3 Simple Random Sampling with Replacement 36 3.4 Comparison of the Designs with and Without Replacement 38 3.5 Sampling with Replacement and Retaining Distinct Units 38 3.5.1 Sample Size and Sampling Design 38 3.5.2 Inclusion Probabilities and Estimation 41 3.5.3 Comparison of the Estimators 44 3.6 Inverse Sampling with Replacement 45 3.7 Estimation of Other Functions of Interest 47 3.7.1 Estimation of a Count or a Proportion 47 3.7.2 Estimation of a Ratio 48 3.8 Determination of the Sample Size 50 3.9 Implementation of Simple Random Sampling Designs 51 3.9.1 Objectives and Principles 51 3.9.2 Bernoulli Sampling 51 3.9.3 Successive Drawing of the Units 52 3.9.4 Random Sorting Method 52 3.9.5 Selection–Rejection Method 53 3.9.6 The Reservoir Method 54 3.9.7 Implementation of Simple Random Sampling with Replacement 56 3.10 Systematic Sampling with Equal Probabilities 57 3.11 Entropy for Simple and Systematic Designs 58 3.11.1 Bernoulli Designs and Entropy 58 3.11.2 Entropy and Simple Random Sampling 60 3.11.3 General Remarks 61 Exercises 61 4 Stratification 65 4.1 Population and Strata 65 4.2 Sample, Inclusion Probabilities, and Estimation 66 4.3 Simple Stratified Designs 68 4.4 Stratified Design with Proportional Allocation 70 4.5 Optimal Stratified Design for the Total 71 4.6 Notes About Optimality in Stratification 74 4.7 Power Allocation 75 4.8 Optimality and Cost 76 4.9 Smallest Sample Size 76 4.10 Construction of the Strata 77 4.10.1 General Comments 77 4.10.2 Dividing a Quantitative Variable in Strata 77 4.11 Stratification Under Many Objectives 79 Exercises 80 5 Sampling with Unequal Probabilities 83 5.1 Auxiliary Variables and Inclusion Probabilities 83 5.2 Calculation of the Inclusion Probabilities 84 5.3 General Remarks 85 5.4 Sampling with Replacement with Unequal Inclusion Probabilities 86 5.5 Nonvalidity of the Generalization of the Successive Drawing without Replacement 88 5.6 Systematic Sampling with Unequal Probabilities 89 5.7 Deville’s Systematic Sampling 91 5.8 Poisson Sampling 92 5.9 Maximum Entropy Design 95 5.10 Rao–Sampford Rejective Procedure 98 5.11 Order Sampling 100 5.12 Splitting Method 101 5.12.1 General Principles 101 5.12.2 Minimum Support Design 103 5.12.3 Decomposition into Simple Random Sampling Designs 104 5.12.4 Pivotal Method 107 5.12.5 Brewer Method 109 5.13 Choice of Method 110 5.14 Variance Approximation 111 5.15 Variance Estimation 114 Exercises 115 6 Balanced Sampling 119 6.1 Introduction 119 6.2 Balanced Sampling: Definition 120 6.3 Balanced Sampling and Linear Programming 122 6.4 Balanced Sampling by Systematic Sampling 123 6.5 Methode of Deville, Grosbras, and Roth 124 6.6 Cube Method 125 6.6.1 Representation of a Sampling Design in the form of a Cube 125 6.6.2 Constraint Subspace 126 6.6.3 Representation of the Rounding Problem 127 6.6.4 Principle of the Cube Method 130 6.6.5 The Flight Phase 130 6.6.6 Landing Phase by Linear Programming 133 6.6.7 Choice of the Cost Function 134 6.6.8 Landing Phase by Relaxing Variables 135 6.6.9 Quality of Balancing 135 6.6.10 An Example 136 6.7 Variance Approximation 137 6.8 Variance Estimation 140 6.9 Special Cases of Balanced Sampling 141 6.10 Practical Aspects of Balanced Sampling 141 Exercise 142 7 Cluster and Two-stage Sampling 143 7.1 Cluster Sampling 143 7.1.1 Notation and Definitions 143 7.1.2 Cluster Sampling with Equal Probabilities 146 7.1.3 Sampling Proportional to Size 147 7.2 Two-stage Sampling 148 7.2.1 Population, Primary, and Secondary Units 149 7.2.2 The Expansion Estimator and its Variance 151 7.2.3 Sampling with Equal Probability 155 7.2.4 Self-weighting Two-stage Design 156 7.3 Multi-stage Designs 157 7.4 Selecting Primary Units with Replacement 158 7.5 Two-phase Designs 161 7.5.1 Design and Estimation 161 7.5.2 Variance and Variance Estimation 162 7.6 Intersection of Two Independent Samples 163 Exercises 165 8 Other Topics on Sampling 167 8.1 Spatial Sampling 167 8.1.1 The Problem 167 8.1.2 Generalized Random Tessellation Stratified Sampling 167 8.1.3 Using the Traveling Salesman Method 169 8.1.4 The Local Pivotal Method 169 8.1.5 The Local Cube Method 169 8.1.6 Measures of Spread 170 8.2 Coordination in Repeated Surveys 172 8.2.1 The Problem 172 8.2.2 Population, Sample, and Sample Design 173 8.2.3 Sample Coordination and Response Burden 174 8.2.4 Poisson Method with Permanent Random Numbers 175 8.2.5 Kish and Scott Method for Stratified Samples 176 8.2.6 The Cotton and Hesse Method 176 8.2.7 The Rivière Method 177 8.2.8 The Netherlands Method 178 8.2.9 The Swiss Method 178 8.2.10 Coordinating Unequal Probability Designs with Fixed Size 181 8.2.11 Remarks 181 8.3 Multiple Survey Frames 182 8.3.1 Introduction 182 8.3.2 Calculating Inclusion Probabilities 183 8.3.3 Using Inclusion Probability Sums 184 8.3.4 Using a Multiplicity Variable 185 8.3.5 Using a Weighted Multiplicity Variable 186 8.3.6 Remarks 187 8.4 Indirect Sampling 187 8.4.1 Introduction 187 8.4.2 Adaptive Sampling 188 8.4.3 Snowball Sampling 188 8.4.4 Indirect Sampling 189 8.4.5 The Generalized Weight Sharing Method 190 8.5 Capture–Recapture 191 9 Estimation with a Quantitative Auxiliary Variable 195 9.1 The Problem 195 9.2 Ratio Estimator 196 9.2.1 Motivation and Definition 196 9.2.2 Approximate Bias of the Ratio Estimator 197 9.2.3 Approximate Variance of the Ratio Estimator 198 9.2.4 Bias Ratio 199 9.2.5 Ratio and Stratified Designs 199 9.3 The Difference Estimator 201 9.4 Estimation by Regression 202 9.5 The Optimal Regression Estimator 204 9.6 Discussion of the Three Estimation Methods 205 Exercises 208 10 Post-Stratification and Calibration on Marginal Totals 209 10.1 Introduction 209 10.2 Post-Stratification 209 10.2.1 Notation and Definitions 209 10.2.2 Post-Stratified Estimator 211 10.3 The Post-Stratified Estimator in Simple Designs 212 10.3.1 Estimator 212 10.3.2 Conditioning in a Simple Design 213 10.3.3 Properties of the Estimator in a Simple Design 214 10.4 Estimation by Calibration on Marginal Totals 217 10.4.1 The Problem 217 10.4.2 Calibration on Marginal Totals 218 10.4.3 Calibration and Kullback–Leibler Divergence 220 10.4.4 Raking Ratio Estimation 221 10.5 Example 221 Exercises 224 11 Multiple Regression Estimation 225 11.1 Introduction 225 11.2 Multiple Regression Estimator 226 11.3 Alternative Forms of the Estimator 227 11.3.1 Homogeneous Linear Estimator 227 11.3.2 Projective Form 228 11.3.3 Cosmetic Form 228 11.4 Calibration of the Multiple Regression Estimator 229 11.5 Variance of the Multiple Regression Estimator 230 11.6 Choice of Weights 231 11.7 Special Cases 231 11.7.1 Ratio Estimator 231 11.7.2 Post-stratified Estimator 231 11.7.3 Regression Estimation with a Single Explanatory Variable 233 11.7.4 Optimal Regression Estimator 233 11.7.5 Conditional Estimation 235 11.8 Extension to Regression Estimation 236 Exercise 236 12 Calibration Estimation 237 12.1 Calibrated Methods 237 12.2 Distances and Calibration Functions 239 12.2.1 The Linear Method 239 12.2.2 The Raking Ratio Method 240 12.2.3 Pseudo Empirical Likelihood 242 12.2.4 Reverse Information 244 12.2.5 The Truncated Linear Method 245 12.2.6 General Pseudo-Distance 246 12.2.7 The Logistic Method 249 12.2.8 Deville Calibration Function 249 12.2.9 Roy and Vanheuverzwyn Method 251 12.3 Solving Calibration Equations 252 12.3.1 Solving by Newton’s Method 252 12.3.2 Bound Management 253 12.3.3 Improper Calibration Functions 254 12.3.4 Existence of a Solution 254 12.4 Calibrating on Households and Individuals 255 12.5 Generalized Calibration 256 12.5.1 Calibration Equations 256 12.5.2 Linear Calibration Functions 257 12.6 Calibration in Practice 258 12.7 An Example 259 Exercises 260 13 Model-Based approach 263 13.1 Model Approach 263 13.2 The Model 263 13.3 Homoscedastic Constant Model 267 13.4 Heteroscedastic Model 1 Without Intercept 267 13.5 Heteroscedastic Model 2 Without Intercept 269 13.6 Univariate Homoscedastic Linear Model 270 13.7 Stratified Population 271 13.8 Simplified Versions of the Optimal Estimator 273 13.9 Completed Heteroscedasticity Model 276 13.10 Discussion 277 13.11 An Approach that is Both Model- and Design-based 277 14 Estimation of Complex Parameters 281 14.1 Estimation of a Function of Totals 281 14.2 Variance Estimation 282 14.3 Covariance Estimation 283 14.4 Implicit Function Estimation 283 14.5 Cumulative Distribution Function and Quantiles 284 14.5.1 Cumulative Distribution Function Estimation 284 14.5.2 Quantile Estimation: Method 1 284 14.5.3 Quantile Estimation: Method 2 285 14.5.4 Quantile Estimation: Method 3 287 14.5.5 Quantile Estimation: Method 4 288 14.6 Cumulative Income, Lorenz Curve, and Quintile Share Ratio 288 14.6.1 Cumulative Income Estimation 288 14.6.2 Lorenz Curve Estimation 289 14.6.3 Quintile Share Ratio Estimation 289 14.7 Gini Index 290 14.8 An Example 291 15 Variance Estimation by Linearization 295 15.1 Introduction 295 15.2 Orders of Magnitude in Probability 295 15.3 Asymptotic Hypotheses 300 15.3.1 Linearizing a Function of Totals 301 15.3.2 Variance Estimation 303 15.4 Linearization of Functions of Interest 303 15.4.1 Linearization of a Ratio 303 15.4.2 Linearization of a Ratio Estimator 304 15.4.3 Linearization of a Geometric Mean 305 15.4.4 Linearization of a Variance 305 15.4.5 Linearization of a Covariance 306 15.4.6 Linearization of a Vector of Regression Coefficients 307 15.5 Linearization by Steps 308 15.5.1 Decomposition of Linearization by Steps 308 15.5.2 Linearization of a Regression Coefficient 308 15.5.3 Linearization of a Univariate Regression Estimator 309 15.5.4 Linearization of a Multiple Regression Estimator 309 15.6 Linearization of an Implicit Function of Interest 310 15.6.1 Estimating Equation and Implicit Function of Interest 310 15.6.2 Linearization of a Logistic Regression Coefficient 311 15.6.3 Linearization of a Calibration Equation Parameter 313 15.6.4 Linearization of a Calibrated Estimator 313 15.7 Influence Function Approach 314 15.7.1 Function of Interest, Functional 314 15.7.2 Definition 315 15.7.3 Linearization of a Total 316 15.7.4 Linearization of a Function of Totals 316 15.7.5 Linearization of Sums and Products 317 15.7.6 Linearization by Steps 318 15.7.7 Linearization of a Parameter Defined by an Implicit Function 318 15.7.8 Linearization of a Double Sum 319 15.8 Binder’s Cookbook Approach 321 15.9 Demnati and Rao Approach 322 15.10 Linearization by the Sample Indicator Variables 324 15.10.1 The Method 324 15.10.2 Linearization of a Quantile 326 15.10.3 Linearization of a Calibrated Estimator 327 15.10.4 Linearization of a Multiple Regression Estimator 328 15.10.5 Linearization of an Estimator of a Complex Function with Calibrated Weights 329 15.10.6 Linearization of the Gini Index 330 15.11 Discussion on Variance Estimation 331 Exercises 331 16 Treatment of Nonresponse 333 16.1 Sources of Error 333 16.2 Coverage Errors 334 16.3 Different Types of Nonresponse 334 16.4 Nonresponse Modeling 335 16.5 Treating Nonresponse by Reweighting 336 16.5.1 Nonresponse Coming from a Sample 336 16.5.2 Modeling the Nonresponse Mechanism 337 16.5.3 Direct Calibration of Nonresponse 339 16.5.4 Reweighting by Generalized Calibration 341 16.6 Imputation 342 16.6.1 General Principles 342 16.6.2 Imputing From an Existing Value 342 16.6.3 Imputation by Prediction 342 16.6.4 Link Between Regression Imputation and Reweighting 343 16.6.5 Random Imputation 345 16.7 Variance Estimation with Nonresponse 347 16.7.1 General Principles 347 16.7.2 Estimation by Direct Calibration 348 16.7.3 General Case 349 16.7.4 Variance for Maximum Likelihood Estimation 350 16.7.5 Variance for Estimation by Calibration 353 16.7.6 Variance of an Estimator Imputed by Regression 356 16.7.7 Other Variance Estimation Techniques 357 17 Summary Solutions to the Exercises 359 Bibliography 379 Author Index 405 Subject Index 411

Rezension

"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

Autorenportrait

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

Back cover copy

A much-needed reference on survey sampling and its applications that presents the latest advances in the field Seeking to show that sampling theory is a living discipline with a very broad scope, Sampling and Estimation from Finite Populations examines the modern development of the theory and the foundations of survey sampling. It offers readers a critical approach to the subject and discusses putting theory into practice. It also explores the treatment of non-sampling errors featuring a range of topics from the problems of coverage to the treatment of non-response. In addition, the book includes real examples, applications, and a large set of exercises with solutions. The book begins with a look at the history of survey sampling. It then offers chapters on: population, sample, and estimation; simple and systematic designs; stratification; sampling with unequal probabilities; balanced sampling; cluster and two-stage sampling; and other topics on sampling, such as spatial sampling, coordination in repeated surveys, and multiple survey frames. The book also includes sections on: post-stratification and calibration on marginal totals; calibration estimation; estimation of complex parameters; variance estimation by linearization; and much more. Sampling and Estimation from Finite Populations: <ul> <li>Provides an up-to-date review of the theory of sampling</li> <li>Discusses the foundation of inference in survey sampling, in particular, the model-based and design-based frameworks</li> <li>Reviews the problems of application of the theory into practice</li> <li>Deals with the treatment of non-sampling errors</li> </ul> Sampling and Estimation from Finite Populations is an excellent book for methodologists and researchers in survey agencies and advanced undergraduate and graduate students in social science, statistics, and survey courses.