Ranked Set Sampling

This is the first book on the concept and applications of ranked set sampling. It provides a comprehensive review of the literature, and it includes many new results and novel applications. Scientists and researchers on this subject will find a balanced presentation of theory and applications. The mathematical rigor of the theoretical foundations makes it beneficial to researchers. The detailed description of various methods illustrated by real or simulated data makes it useful for scientists and practitioners in application areas such as agriculture, forestry, sociology, ecological and environmental science, and medical studies. It can serve as a reference book and as a textbook for a short course at the graduate level. Zehua Chen is Associate Professor of Statistics at the National University of Singapore. Zhidong Bai is Professor of Statistics at the National University of Singapore; he is a Fellow of the Institute of Mathematical Statistics. Bimal Sinha is the Presidential Research Professor at University of Maryland Baltimore County; he is a Fellow of the Institute of Mathematical Statistics and the American Statistical Association. "This comprehensive and up-to-date monograph covers, systematically and in simple language, the theory and applications of ranked set sampling (RSS), an improved technique related to traditional simple random sampling (SRS). Strong emphasis is placed on theoretical developments in RSS. In the meanwhile, the practical orientation and broad coverage will appeal to researchers and scientists working in sampling techniques, experimental designs, nonparametric statistics, and related fields...I would highly recommend this well-written and reasonably priced book." Techometrics, February 2005 " Ranked Set Sampling: Theory and Applications represents a major achievement, providing an up-to-date account of major research in ranked set sampling....this book would be a good addition to the library of anyone involved in statistical, environmental, and ecological research." Journal of the American Statistical Association, September 2005 "The book is well laid out with concepts well explained...Those who are recently getting interested in the topic will find it an excellent start." Biometrics, December 2005

Bimal K. Sinha is the Presidential Research Professor at University of Maryland Baltimore County; he is a Fellow of the Institute of Mathematical Statistics and the American Statistical Association.

1 Introduction

1

(10)

1.1 What is ranked set sampling?

1

(6)

1.2 A historical note

7

(1)

1.3 Scope of the rest of the monograph

8

(3)

2 Balanced Ranked Set Sampling I: Nonparametric

11

(42)

2.1 Ranking mechanisms

12

(2)

2.2 Estimation of means using ranked set sample

14

(4)

2.3 Estimation of smooth-function-of-means using ranked set sample

18

(2)

2.4 Estimation of variance using an RSS sample

20

(4)

2.4.1 Naive moment estimates

20

(2)

2.4.2 Minimum variance unbiased non-negative estimate

22

(2)

2.5 Tests and confidence intervals for population mean

24

(5)

2.5.1 Asymptotic pivotal method

25

(1)

2.5.2 Choice of consistent estimates of σxRSS

25

(4)

2.5.3 Comparison of power between teats based on RSS and SRS

29

(1)

2.6 Estimation of quantiles

29

(6)

2.6.1 Ranked set sample quantiles and their properties

29

(3)

2.6.2 Inference procedures for population quantiles based on ranked set sample

32

(3)

2.6.3 The relative efficiency of RSS quantile estimate with respect to SRS quantile estimate

35

(1)

2.7 Estimation of density function with ranked set sample

35

(9)

2.7.1 RSS density estimate and its properties

36

(4)

2.7.2 The relative efficiency of the RSS density estimate with respect to its SRS counterpart

40

(4)

2.8 M-estimates with RBB data

44

(4)

2.8.1 The RSB M-estimates and their asymptotic properties

44

(2)

2.8.2 The relative efficiency of the RSS M-estimates

46

(2)

2.9 Appendix: Technical details for Section 2.4

48

(2)

2.10 Bibliographic notes

50

(3)

3 Balanced Ranked Set Sampling II: Parametric

53

(20)

3.1 The Fisher information of ranked set samples

54

(9)

3.1.1 The Fisher information of ranked set samples when ranking is perfect

54

(4)

3.1.2 The Fisher information of ranked set samples when ranking is imperfect

58

(5)

3.2 The maximum likelihood estimate and its asymptotic relative efficiency

63

(3)

3.3 The best linear unbiased estimation for location-scale families

66

(22)

3.4 The regularity conditions and the existence of Fisher information matrix of ranked set samples

88

3.5 Bibliographic notes

71

(2)

4 Unbalanced Ranked Set Sampling and Optimal Designs

73

(30)

4.1 General structure of unbalanced RSB

74

(1)

4.2 Nonparametric analysis of unbalanced ranked-set data

75

(9)

4.2.1 Asymptotic properties of unbalanced ranked-set sample quantiles

75

(1)

4.2.2 Nonparametric estimation of quantiles and distribution function

76

(2)

4.2.3 Confidence interval and hypothesis testing

78

(1)

4.2.4 Estimation of statistical functionals

79

(3)

4.2.5 Inference by bootstrapping

82

(1)

4.2.6 Simulation results

82

(2)

4.3 Parametric analysis of unbalanced ranked-set data

84

(2)

4.4 Optimal design for location-scale families

86

(5)

4.4.1 Optimal RBS for MLE

86

(5)

4.4.2 Optimal DBB for BLUE

91

(1)

4.5 Optimal design for estimation of quantiles

91

(4)

4.5.1 Optimal RSS scheme for the estimation of a single quantile

91

(3)

4.5.2 Optimal RSS scheme for the simultaneous estimation of several quantiles

94

(1)

4.6 The computation of optimal designs

95

(1)

4.7 Asymptotic relative efficiency of the optimal schemes

95

(3)

4.7.1 Asymptotic relative efficiency of the optimal schemes for parametric location-scale families

96

(1)

4.7.2 Relative efficiency of the optimal schemes for the estimation of quantiles

97

(1)

4.8 Other design methodologies

98

(2)

4.8.1 Bayesian design

99

(1)

4.8.2 Adaptive design

99

(1)

4.9 Bibliographic notes

100

(3)

5 Tests with Ranked Set Sampling

103

(40)

5.1 Sign tee with DSB

103

(12)

5.1.1 Distributional properties of S+RSS

104

(6)

5.1.2 Decision ruls of the sign test and confidence intervals for median

110

(1)

5.1.3 Effect of imperfect ranking on RSS sign test

110

(3)

5.1.4 Comparison of efficiency and power with respect to S+SRS

113

(2)

5.2 Mann-Whitney-Wilcoxon test with RSB

115

(9)

5.2.1 Distributional properties of URSS

117

(5)

5.2.2 Decision rules of the RSS Mann-Whitney-Wilcoxon test

122

(1)

5.2.3 The estimate and confidence interval of Δ

123

(1)

5.2.4 Effect of imperfect ranking on RBB Mann-Whitney-Wilcoxon

124

(1)

5.3 Wilcoxon signed rank test with RSS

124

(9)

5.3.1 Distributional properties of W+RSS

125

(5)

5.3.2 Decision rules of the Wilcoxon signed rank test

130

(1)

5.3.3 Estimation of Θ

131

(2)

5.3.4 Effect of imperfect ranking on RSB Wilcoxon signed rank test

133

(1)

5.4 Optimal design for distribution-free tests

133

(9)

5.4.1 Optimal design for sign test

133

(2)

5.4.2 Optimal design for Wilcoxon signed rank tests

135

(7)

5.5 Bibliographic notes

142

(1)

6 Ranked Set Sampling with Concomitant Variables

143

(32)

6.1 Multi-layer ranked set sampling

144

(7)

6.1.1 Motivation and definition

144

(2)

6.1.2 Consistency of multi-layer ranked set sampling

146

(1)

6.1.3 Comparison between multi-layer RSS and marginal RSS by simulation studies

147

(3)

6.1.4 Issues on the choice of concomitant variables

150

(1)

6.2 Adaptive ranked set sampling

151

(4)

6.2.1 The best ranking mechanism

151

(1)

6.2.2 Adaptive ranked set sampling procedure

152

(1)

6.2.3 A simulation study

153

(2)

6.3 Regression analysis based on RSS with concomitant variables

155

(6)

6.3.1 Estimation of regression coefficients with RSS

156

(1)

6.3.2 Regression estimate of the mean of Y with RSS

157

(1)

6.3.3 Comparison of the RSS regression estimate and the ordinary RSS estimate

158

(2)

6.3.4 How does a specific ranking mechanism affect the efficiency of the regression estimate?

160

(1)

6.4 Optimal RSS schemes for regression analysis

161

(9)

6.4.1 Asymptotically optimal criteria

162

(2)

6.4.2 Asymptotically optimal schemes for simple linear regression

164

(2)

6.4.3 Applications

166

(4)

6.5 Technical details

170

(4)

6.6 Bibliographic notes

174

(1)

7 Ranked Set Sampling as Data Reduction Tools

175

(16)

7.1 Remedian: a motivation

175

(2)

7.2 Repeated ranked-set procedure for a single quantile

177

(3)

7.3 Repeated ranked-set procedure for multiple quantiles

180

(4)

7.4 Balanced repeated ranked-set procedure

184

(2)

7.5 Procedure for multivariate populations

186

(5)

8 Case Studies

191

(22)

8.1 Plantations of cinchona

191

(3)

8.2 Cost effective gasoline sampling

194

(1)

8.3 Estimation of weights of browse and herbage

195

(3)

8.4 Estimation of shrub phytomass in Appalachian oak forests

198

(3)

8.5 Single family homes sales price data

201

(1)

8.6 Tree data

202

(11)

References

213

(10)

Index

223

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.