Samp Algorithms

Contents......Page 8
Preface......Page 6
1.2 Representativeness......Page 13
1.3.2 Conditional Poisson Sampling......Page 14
1.3.4 Balanced Sampling......Page 15
1.5 Aim of This Book......Page 16
1.6 Outline of This Book......Page 17
2.2 Population and Variable of Interest......Page 19
2.3.1 Sample Without Replacement......Page 20
2.4 Support......Page 21
2.5 Convex Hull, Interior, and Subspaces Spanned by a Support......Page 24
2.7 Reduction of a Sampling Design With Replacement......Page 26
2.8 Expectation and Variance of a Random Sample......Page 27
2.9 Inclusion Probabilities......Page 29
2.10 Computation of the Inclusion Probabilities......Page 30
2.11 Characteristic Function of a Sampling Design......Page 31
2.13 Observed Data and Consistency......Page 32
2.14 Statistic and Estimation......Page 33
2.15 Sufficient Statistic......Page 34
2.16.2 Variance of the Hansen-Hurwitz Estimator......Page 38
2.16.3 Variance Estimation......Page 39
2.17.3 Variance Estimation......Page 40
2.18 More on Estimation in Sampling With Replacement......Page 41
3.2 Sampling Algorithms......Page 42
3.4 Martingale Algorithms......Page 43
3.5 Sequential Algorithms......Page 44
3.6 Draw by Draw Algorithms......Page 46
3.7 Eliminatory Algorithms......Page 48
3.8 Rejective Algorithms......Page 49
4.2 Definition of Simple Random Sampling......Page 51
4.3.1 Sampling Design......Page 53
4.3.3 Sequential Sampling Procedure for BERN......Page 54
4.4.1 Sampling Design......Page 55
4.4.2 Estimation......Page 56
4.4.3 Draw by Draw Procedure for SRSWOR......Page 57
4.4.5 Sample Reservoir Method......Page 58
4.4.6 Random Sorting Method for SRSWOR......Page 60
4.5.1 Sampling Design......Page 61
4.5.2 Estimation......Page 62
4.6.1 Sampling Design......Page 63
4.6.2 Distribution of n[r(S)]......Page 65
4.6.3 Estimation......Page 67
4.6.4 Draw by Draw Procedure for SRSWR......Page 70
4.7 Links Between the Simple Sampling Designs......Page 71
5.1 Introduction......Page 73
5.2.1 Minimum Kullback-Leibler Divergence......Page 74
5.2.2 Exponential Designs (EXP)......Page 75
5.3.1 Sampling Design......Page 77
5.3.2 Estimation......Page 79
5.4.1 Sampling Design......Page 80
5.4.2 Estimation......Page 82
5.4.3 Sequential Procedure for Multinomial Design......Page 84
5.5.1 Sampling Design......Page 86
5.5.2 Distribution of n(S)......Page 87
5.5.3 Estimation......Page 88
5.6.1 Sampling Design......Page 89
5.6.2 Inclusion Probabilities......Page 90
5.6.3 Computation of λ from Predetermined Inclusion Probabilities......Page 91
5.6.4 Joint Inclusion Probabilities......Page 94
5.6.5 Joint Inclusion Probabilities: Deville\'s Technique......Page 95
5.6.6 Computation of α(λ, S[sub(n)])......Page 97
5.6.7 Poisson Rejective Procedure for CPS......Page 99
5.6.8 Rejective Procedure with Multinomial Design for CPS......Page 100
5.6.9 Sequential Procedure for CPS......Page 101
5.6.10 Alternate Method for Computing π from λ......Page 102
5.6.11 Draw by Draw Procedure for CPS......Page 103
5.8 Links Between Exponential Designs and Simple Designs......Page 105
5.9 Exponential Procedures in Brewer and Hanif......Page 106
6.2.1 A General Technique of Splitting into Two Vectors......Page 109
6.2.2 Splitting Based on the Choice of π[sup(a)](t)......Page 111
6.2.4 Minimum Support Design......Page 112
6.2.5 Splitting into Simple Random Sampling......Page 114
6.2.6 The Pivotal Method......Page 116
6.2.8 Generalized Sunter Method......Page 118
6.3.1 A General Method of Splitting into M Vectors......Page 121
6.3.2 Brewer\'s Method......Page 122
6.3.3 Eliminatory Method......Page 124
6.3.4 Tillé\'s Elimination Procedure......Page 125
6.3.5 Generalized Midzuno Method......Page 127
6.3.6 Chao\'s Method......Page 129
6.4 Splitting Procedures in Brewer and Hanif......Page 130
7 More on Unequal Probability Designs......Page 132
7.1 Ordered Systematic Sampling......Page 133
7.2 Random Systematic Sampling......Page 136
7.3 Deville\'s Systematic Sampling......Page 137
7.4.1 The Sampford Sampling Design......Page 139
7.4.2 Fundamental Results......Page 140
7.4.3 Technicalities for the Computation of the Joint Inclusion Probabilities......Page 143
7.4.4 Implementation of the Sampford Design......Page 144
7.5.1 Approximation of the Variance......Page 146
7.5.2 Estimators Based on Approximations......Page 149
7.6 Comparisons of Methods with Unequal Probabilities......Page 151
7.7 Choice of a Method of Sampling with Unequal Probabilities......Page 154
8.2.1 Definition of Balanced Sampling......Page 155
8.2.2 Particular Cases of Balanced Sampling......Page 156
8.2.3 The Rounding Problem......Page 157
8.3.2 Yates\'s Procedure......Page 158
8.4.1 Constraint Subspace and Cube of Samples......Page 159
8.4.2 Geometrical Representation of the Rounding Problem......Page 161
8.5 Enumerative Algorithm......Page 163
8.6.1 The Flight Phase......Page 167
8.6.2 Fast Implementation of the Flight Phase......Page 169
8.6.3 The Landing Phase......Page 171
8.6.4 Quality of Balancing......Page 172
8.7.1 Simple Random Sampling......Page 174
8.7.2 Stratification......Page 175
8.7.3 Unequal Probability Sampling with Fixed Sample Size......Page 176
8.8.1 Construction of an Approximation......Page 177
8.8.2 Application of the Variance Approximation to Stratification......Page 179
8.9.1 Construction of an Estimator of Variance......Page 181
8.9.2 Application to Stratification of the Estimators of Variance......Page 182
8.10 Recent Developments in Balanced Sampling......Page 184
9.1 Implementation of the Cube Method......Page 185
9.3 The Sample of Municipalities......Page 186
9.4 Quality of Balancing......Page 188
Appendix: Population of Municipalities in the Canton of Ticino (Switzerland)......Page 189
List of Tables......Page 195
List of Figures......Page 197
List of Algorithms......Page 198
List of Abbreviations......Page 200
Table of Notations......Page 202
References......Page 205
F......Page 215
S......Page 216
Y......Page 217
D......Page 218
P......Page 219
S......Page 220
V......Page 221