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Topics in Small Area Estimation with Applications to the National Resources Inventory Junyuan Wang Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Topics in small area estimation with applications to the National Resources Inventory Junyuan Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Wang, Junyuan, "Topics in small area estimation with applications to the National Resources Inventory " (2000). Retrospective Theses and Dissertations. 12372. https://lib.dr.iastate.edu/rtd/12372 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter ^ce, while others may t>e from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author dkJ not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate ttie deletkxi. 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UMI Number 9990499 UMI^ UMI Microfomi9990499 Copyright 2001 by Bell & Howell information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 II Graduate College Iowa State University This is to certify that the Doctoral dissertation of Junyuan Wang has met the dissertation requirements of Iowa State University Signature was redacted for privacy. Co-major Professor Signature was redacted for privacy. Signature was redacted for privacy. For the Major Program Signature was redacted for privacy. For the Graduate College Ill To my parents IV TABLE OF CONTENTS 1 OVERVIEW OF SMALL AREA ESTIMATION 1 1.1 Introduction 1 1.2 Random effects model 2 1.2.1 EBLUP -Approach 4 1.2.2 Empirical Bayes approach 9 1.2.3 Hierarchical Bayes approach 11 1.3 Nested error regression model 1-5 1.4 Some e.xtensions and recent development IS 1.4.1 Multi-level model IS 1.4.2 Models with discrete measurements IS 1.4.3 Time series model 19 1.5 Dissertation organization 20 2 AN EXAMPLE OF SMALL AREA ESTIMATION IN THE NA­ TIONAL RESOURCES INVENTORY 21 2.1 The U.S. N'ational Resources Inventory 21 2.2 Small area estimation 22 2.2.1 Small area estimation model for urban change 22 2.2.2 Results and model checks 28 3 MSE OF EBLUP WHEN SAMPLING ERROR VARIANCES ARE ESTIMATED 33 3.1 Introduction 3.2 MSE of EBLUP under alternative states of knowledge 3 3.3 Simulation study 5 3.4 Improved estimator for M5£'(y, — y,) 5 4 ESTIMATION UNDER A RESTRICTION .... 4.1 Introduction 4.2 Best linear unbiased estimator under a restriction . 4.3 The MSE of the modified estimator 4.4 Simulation study 5 THE EFFECTS OF NONNORMAL ERRORS . •5.1 Introduction •5.2 .Approximation to bieis for non-normal sampling errors •5.3 Simulation study 98 6 SUMMARY I BIBLIOGRAPHY 1 ACKNOWLEDGEMENTS I VI LIST OF TABLES Table 2.1 Summary statistics for mean model 2S Table 2.2 Summary statistics for variance model 31 Table 3.1 Simulation study of MSE(yi — j/,) [d = o.m = 36) 60 Table 3.2 Simulation study of MSE{yi — yi) {d = 9.m = 36) 61 Table 3.3 Simulation study of MSE[yi — y.) {d = 9.m = 99) 62 Table 3.4 Simulation study of MSE{yi — yi) (d = 14. m = 99) 63 Table 3.5 Simulation study of MSE{yi — yi) {d = 14. m = 225) 64 Table 4.1 Simulation study of MSE{yf^ — j/,) (d = 5. m = 36) SS Table 4.2 Simulation study of MSE[yf'' — yi) {d = 9. m = 36) 89 Table 4.3 Simulation study of MSEiyf'^ — j/,) {d = 9.m — 99) 90 Table 4.4 Simulation study of MSE(y'^ — yi) (d = 14. m = 99) 91 Table 4.5 Simulation study of MSE{yf^ — y,) (d = 14. m = 225) 92 Table 5.1 Comparison of Monte Carlo Bicis and MSE of y — y. (1000 samples) 103 VII LIST OF FIGURES Figure 2.1 Mean Model 29 Figure 2.2 Variance Model 30 Figure .3.1 Variance Model for Colorado (Grouped) 34 Figure 3.2 Variance Model for Colorado (Individual HUCCO) 35 Figure 5.1 Monte Carlo bias of y, when (m.d) = (36.5) and e, are centered ,\3 distribution with variance cr^,. The plots are for = 1. 1.5, and 2. respectively 104 Figure 5.2 Monte Carlo bias of y, when {m.d) = (99,9) and e, are centered \3 distribution with variance cr^,. The plots are for = 1, 1.5. and 2, respectively 105 Figure 5.3 Monte Carlo bias of y,- when (m. d) = (225. 14) and e; are centered distribution with variance cr^,. The plots are for = I. 1.5. and 2, respectively 106 Figure 5.4 Monte Carlo MSE ratio when {m. d) = (36,5) and e, are centered distribution with variance <7^,. The plots are for cr^, = 1, 1-5. and 2, respectively 107 Figure 5.5 Monte Carlo MSE ratio when (m. d) = (99,9) and e, are centered \'§ distribution with variance cr^,. The plots are for cr^i = 1, 1-5, and 2, respectively 108 VIII Figure 0.6 Monte Carlo MSE ratio when {m.d) = (225.14) and e, are centered \3 distribution with variance cr^,. The plots are for = 1. 1.5. and 2. respectively 109 1 1 OVERVIEW OF SMALL AREA ESTIMATION 1.1 Introduction Sample surveys are a more cost-effective way of obtaining information than complete enumerations or censuses for most purposes. The surveys are usually designed to ensure that reliable estimators of totals and means for the population, pre-specified domains of interest, or major subpopulations can be derived from the survey data. There are also many situations in which it is desirable to derive reliable estimators for additional domains of interest, especially small geographical areas or small subpopulations. from existing survey data. The terms "small area" and "local area" are commonly used to denote a small geographical area, such as a county, a municipality or a census division and the term "small domain" is used to denote a small subpopulation. such as a specific age-sex-race group of people. In this discussion, these terms will not be differentiated and only "small area" will be used. For example, suppose there is a state-wide survey about the average household in­ come. Many cities in that state want to know the average household income for individ­ ual cities without conducting their own survey. .Naturally, they turn to the state-wide survey. However, sample sizes for these cities (small arezis) are typically small due to the size of cities. Therefore, the usual direct survey estimators of the average household income for such small areas, based on data only from the sample units in the area, are likely to yield unacceptably large standard errors (compared to the interesting statistic, the average household income) due to the small sample size in the area. This makes 2 it necessary to "borrow strength" from data related to the city of interest using some form of model-dependent estimator to find more accurate estimates for the given area or. simultaneously, for several areas. The potential sources for the related data sets can be divided into three groups: • .A.dministrative data known at the small area level or sample element level • Data measured for the same characteristics in other 'similar' areas • Data meaisured for the same characteristics in the same area in a previous sample or census. Classical survey sampling practitioners and Bayesians frequently disagree on the philosophical basis for estimation. But small area estimation is one area where these groups of statistician have a consensus on the need for model-dependent estimation. The idea is to relate similar small areas via supplementary data (e.g. census and admin­ istrative data) through explicit or implicit models. Ghosh and Rao (1994) provide an excellent review of many of the models found in the current literature and evaluate them in the light of practical considerations.
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