Small Area Estimation

327 Pages·2003·6.78 MB·English

by  J. N. K. Rao

Small Area Estimation
WILEY SERIES IN SURVEY METHODOLOGY
Established in part by WALTEAR . SHEWHAARNTD SAMUESL. WILKS
Editors: Robert M. Groves, Graham KaIton, J N. K. Rao, Norbert Schwarz,
Christopher Skinner
A complete list of the titles in this series appears at the end of this volume.
Small Area Estimation
J. N. K. RAO
Carleton University
@EZiciENCE
A JOHN WILEY & SONS, INC., PUBLICATION
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Library of Congress Cataloging-in-Publication Data:
Rao, J. N. K., 1937-
Small area estimation / J.N.K. Rao.
p. cm. -- (Wiley series in survey methodology)
lncludes bibliographical references and index.
lSBN 0-471-41374-7 (cloth)
1. Sampling (Statistics) 2. Estimation theoty. I. Title. 11. Series.
QA27b.b .R344 2003
5 19.5'2--dc2 I 2002033197
Printed in the United States of America.
10 9 8 7 6 5 4 3 2
To my mother, Sakuntalamma
and
to my wife, Neela
Contents
List of Figures xiii
List of Tables xv
Foreword xvii
Preface xxi
1 Introduction 1
1.1 What is a Small Area? . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Demand for Small Area Statistics . . . . . . . . . . . . . . . . . 3
1.3 Traditional Indirect Estimators . . . . . . . . . . . . . . . . . . 3
1.4 Small Area Models . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Model-Based Estimation . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Direct Domain Estimation 9
2
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Design-based Approach . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Estimation of Totals . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Design-unbiased Estimator . . . . . . . . . . . . . . . . 11
2.3.2 Generalized Regression Estimator . . . . . . . . . . . . 13
2.4 Domain Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Case of no Auxiliary Information . . . . . . . . . . . . . 15
2.4.2 GREG Estimation . . . . . . . . . . . . . . . . . . . . . 17
2.4.3 Domain-specific Auxiliary Information . . . . . . . . . . 17
2.5 Modified Direct Estimators . . . . . . . . . . . . . . . . . . . . 20
2.6 Design Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.7.1 Proof of YGR(X)= X . . . . . . . . . . . . . . . . . . . . 25
2.7.2 Derivation of Calibration Weights w; . . . . . . . . . . 25
2.7.3 Proof of Y = XTB when cj = vTxj . . . . . . . . . . . 25
vii
viii CONTENTS
3 Traditional Demographic Methods 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Symptomatic Accounting Techniques . . . . . . . . . . . . . . . 28
3.2.1 Vital Rates Method . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Composite Method . . . . . . . . . . . . . . . . . . . . . 30
3.2.3 Component Methods . . . . . . . . . . . . . . . . . . . . 30
3.2.4 Housing Unit Method . . . . . . . . . . . . . . . . . . . 30
3.3 Regression Symptomatic Procedures . . . . . . . . . . . . . . . 31
3.3.1 Ratio Correlation and Difference Correlation Methods . 31
3.3.2 Sample Regression Method . . . . . . . . . . . . . . . . 33
3.4 Dual-system Estimation of Total Population . . . . . . . . . . . 37
3.4.1 Dual-system Model . . . . . . . . . . . . . . . . . . . . . 37
3.4.2 Post-enumeration Surveys . . . . . . . . . . . . . . . . . 39
3.5 Derivation of Average MSEs . . . . . . . . . . . . . . . . . . . . 42
4 Indirect Domain Estimation 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Synthetic Estimation . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 No Auxiliary Information . . . . . . . . . . . . . . . . . 46
4.2.2 Auxiliary Information Available . . . . . . . . . . . . . . 46
4.2.3 Regression-adjusted Synthetic Estimator . . . . . . . . . 51
4.2.4 Estimation of MSE . . . . . . . . . . . . . . . . . . . . . 51
4.2.5 Structure Preserving Estimation . . . . . . . . . . . . . 5 3
4.3 Composite Estimation . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.1 Optimal Estimator . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Sample Size Dependent Estimators . . . . . . . . . . . . GO
4.4 James-Stein Method . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Common Weight . . . . . . . . . . . . . . . . . . . . . . 63
4.4.2 Equal Variances $, = . . . . . . . . . . . . . . . . . . 64
$J
4.4.3 Estimation of Component MSE . . . . . . . . . . . . . . 68
4.4.4 Unequal Variances 4% . . . . . . . . . . . . . . . . . . . 7 1
4.4.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Small Area Models 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Basic Area Level (Type A) Model . . . . . . . . . . . . . . . . 76
5.3 Basic Unit Level (Type B) Model . . . . . . . . . . . . . . . . . 78
5.4 Extensions: Type A Models . . . . . . . . . . . . . . . . . . . . 81
5.4.1 Multivariate Fay-Herriot Model . . . . . . . . . . . . . . 81
5.4.2 Model with Correlated Sampling Errors . . . . . . . . . 82
5.4.3 Time Series and Cross-sectional Models . . . . . . . . . 83
5.4.4 Spatial Models . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Extensions: Type B Models . . . . . . . . . . . . . . . . . . . . 87
5.5.1 Multivariate Nested Error Regression Model . . . . . . . 87
5.5.2 Random Error Variance Linear Model . . . . . . . . . . 88
CONTENTS ix
5.5.3 Two-fold Nested Error Regression Model . . . . . . . . 88
5.5.4 Two-level Model . . . . . . . . . . . . . . . . . . . . . . 89
5.5.5 General Linear Mixed Model . . . . . . . . . . . . . . . 90
5.6 Generalized Linear Mixed Models . . . . . . . . . . . . . . . . . 91
5.6.1 Logistic Regression Models . . . . . . . . . . . . . . . . 91
5.6.2 Models for Mortality and Disease Rates . . . . . . . . . 92
5.6.3 Exponential Family Models . . . . . . . . . . . . . . . . 93
5.6.4 Semi-parametric Models . . . . . . . . . . . . . . . . . . 93
6 Empirical Best Linear Unbiased Prediction: Theory 95
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 General Linear Mixed Model . . . . . . . . . . . . . . . . . . . 96
6.2.1 BLUP Estimator . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 MSE of BLUP . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.3 EBLUP Estimator . . . . . . . . . . . . . . . . . . . . . 99
6.2.4 ML and REML Estimators . . . . . . . . . . . . . . . .1 00
6.2.5 MSE of EBLUP . . . . . . . . . . . . . . . . . . . . . . 103
6.2.6 Estimation of MSE of EBLUP . . . . . . . . . . . . . .1 04
6.2.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Block Diagonal Covariance Structure . . . . . . . . . . . . . . .1 07
6.3.1 EBLUP Estimator . . . . . . . . . . . . . . . . . . . . . 107
6.3.2 Estimation of MSE . . . . . . . . . . . . . . . . . . . . . 108
6.3.3 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.4 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . 110
6.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.1 Derivation of BLUP . . . . . . . . . . . . . . . . . . . . 112
6.4.2 Equivalence of BLUP and Best Predictor E(mTvlATy) 113
6.4.3 Derivation of the Decomposition (6.2.26) . . . . . . . . 113
7 EBLUP: Basic Models 115
7.1 Basic Area Level Model . . . . . . . . . . . . . . . . . . . . . . 115
7.1.1 BLUP Estimator . . . . . . . . . . . . . . . . . . . . . . 116
7.1.2 Estimation of . . . . . . . . . . . . . . . . . . . . . . 118
0;
7.1.3 Relative Efficiency of Estimators of . . . . . . . . . . 1 20
0,”
7.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1.5 MSE Estimation . . . . . . . . . . . . . . . . . . . . . . 128
7.1.6 Conditional MSE . . . . . . . . . . . . . . . . . . . . . . 131
7.1.7 Mean Product Error of Two Estimators . . . . . . . . . 1 32
7.1.8 Estimation of Small Area Means . . . . . . . . . . . . . 133
7.1.9 Weighted Estimator . . . . . . . . . . . . . . . . . . . . 134
7.2 Basic Unit Level Model . . . . . . . . . . . . . . . . . . . . . . 134
7.2.1 BLUP Estimator . . . . . . . . . . . . . . . . . . . . . . 135
7.2.2 Estimation of 0,”a nd u: . . . . . . . . . . . . . . . . . .1 38
7.2.3 MSE of EBLUP . . . . . . . . . . . . . . . . . . . . . . 139
7.2.4 MSE Estimation . . . . . . . . . . . . . . . . . . . . . . 140
7.2.5 Non-negligible Sampling Rates . . . . . . . . . . . . . .1 41
X CONTENTS
7.2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2.7 Pseudo-EBLUP Estimation . . . . . . . . . . . . . . . .1 48
8 EBLUP: Extensions 153
8.1 Multivariate Fay-Herriot Model . . . . . . . . . . . . . . . . . .1 53
8.2 Correlated Sampling Errors . . . . . . . . . . . . . . . . . . . . 155
8.3 Time Series and Cross-sectional Models . . . . . . . . . . . . .1 58
8.3.1 Rao-Yu Model . . . . . . . . . . . . . . . . . . . . . . . 158
8.3.2 State Space Models . . . . . . . . . . . . . . . . . . . . 162
8.4 Spatial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.5 Multivariate Nested Error Regression Model . . . . . . . . . . . 1 69
8.6 Random Error Variances Linear Model . . . . . . . . . . . . . . 1 71
8.7 Two-fold Nested Error Regression Model . . . . . . . . . . . . . 1 72
8.8 Two-level Model . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9 Empirical Bayes (EB) Method 179
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2 Basic Area Level Model . . . . . . . . . . . . . . . . . . . . . . 180
9.2.1 EB Estimator . . . . . . . . . . . . . . . . . . . . . . . . 181
9.2.2 MSE Estimation . . . . . . . . . . . . . . . . . . . . . . 182
9.2.3 Approximation to Posterior Variance . . . . . . . . . . . 1 85
9.2.4 EB Confidence Intervals . . . . . . . . . . . . . . . . . .1 91
9.3 Linear Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . 194
9.3.1 EB Estimation . . . . . . . . . . . . . . . . . . . . . . . 194
9.3.2 MSE Estimation . . . . . . . . . . . . . . . . . . . . . . 195
9.3.3 Approximations to the Posterior Variance . . . . . . . . 1 96
9.4 Binary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.4.1 Case of no Covariates . . . . . . . . . . . . . . . . . . . 197
9.4.2 Models with Covariates . . . . . . . . . . . . . . . . . . 202
9.5 Disease Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9.5.1 Poisson-Gamma Model . . . . . . . . . . . . . . . . . .2 06
9.5.2 Log-normal Models . . . . . . . . . . . . . . . . . . . . . 208
9.5.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 209
9.6 Triple-goal Estimation . . . . . . . . . . . . . . . . . . . . . . . 211
9.6.1 Constrained EB . . . . . . . . . . . . . . . . . . . . . . 211
9.6.2 Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.6.3 Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.7 Empirical Linear Bayes . . . . . . . . . . . . . . . . . . . . . . 214
9.7.1 LB Estimation . . . . . . . . . . . . . . . . . . . . . . . 214
9.7.2 Posterior Linearity . . . . . . . . . . . . . . . . . . . . 217
9.8 Constrained LB . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.9 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.9.1 Proof of (9.2.11) . . . . . . . . . . . . . . . . . . . . . . 220
9.9.2 Proof of (9.2.30) . . . . . . . . . . . . . . . . . . . . . . 221
9.9.3 Proof of (9.6.6) . . . . . . . . . . . . . . . . . . . . . . . 221
9.9.4 Proof of (9.7.1) . . . . . . . . . . . . . . . . . . . . . . . 222