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Sampling and Estimation from Finite Populations

Sampling and Estimation from Finite Populations

Autorzy
Wydawnictwo John Wiley & Sons Inc
Data wydania 2020
Liczba stron 448
Forma publikacji książka w twardej oprawie
Poziom zaawansowania Dla profesjonalistów, specjalistów i badaczy naukowych
Język angielski
ISBN 9780470682050
Kategorie Prawdopodobieństwo i statystyka
404.25 PLN (z VAT)
$90.93 / €86.67 / £75.24 /
Produkt na zamówienie
Dostawa 5-6 tygodni
Ilość
Do schowka

Opis książki

This comprehensive text takes a critical look at the modern development of the theory of survey sampling as well as the foundations of survey sampling, and explains how to put this theory into practice. The treatment of non-sampling errors is featured, and a range of other topics, from the problems of coverage to the treatment of non-response, is explored. Real examples, applications, and a large set of exercises are also provided. "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

Sampling and Estimation from Finite Populations

Spis treści

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 Riviere 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

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