Sampling Strategies for Natural Resources and the Environment

480 Pages·2007·1.968 MB·English

by  Gregoire, Timothy G., Valentine, Harry T

Chapman & Hall/CRC
Applied Environmental Statistics
S S
ampling trategieS
n r
for atural eSourceS
e
and the nvironment
C3707_FM.indd 1 6/8/07 3:19:05 PM
CHAPMAN & HALL/CRC
APPLIED ENVIRONMENTAL STATISTICS
Series Editor
Richard Smith, Ph.D.
Published Titles
Timothy G. Gregoire and Harry T. Valentine, Sampling Strategies for Natural
Resources and the Environment
Steven P. Millard and Nagaraj K. Neerchal, Environmental Statistics
with S Plus
Michael E. Ginevan and Douglas E. Splitstone, Statistical Tools for
Environmental Quality
Forthcoming Titles
Daniel Mandallaz, Sampling Techniques for Forest Inventory
Thomas C. Edwards and Richard R. Cutler, Analysis of Ecological
Data Using R
Bryan F. J. Manly, Statistics for Environmental Science and
Management, 2nd Edition
Song S. Qian, Environmental and Ecological Statistics with R
C3707_FM.indd 2 6/8/07 3:19:05 PM
Chapman & Hall/CRC
Applied Environmental Statistics
S S
ampling trategieS
n r
for atural eSourceS
e
and the nvironment
timothY g. gregoire
harrY t. valentine
C3707_FM.indd 3 6/8/07 3:19:05 PM
CRC Press
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Contents
Preface xvii
1 Introduction 1
1.1 Theneedforsamplingstrategies 1
1.2 Amedleyofsamplingscenarios 3
1.2.1 Samplingforbiomassestimation 3
1.2.2 Samplingtoestimatecomposition 3
1.2.3 Samplingforcoverinformation 4
1.2.4 HowmanytreesinSweden? 4
1.3 Probabilitysample 5
1.3.1 Selectionprobabilityofasample 5
1.3.2 Inclusionprobability 7
1.3.3 Samplingframe 8
1.4 Inference 9
1.5 Populationdescriptiveparameters 10
1.5.1 Discretepopulations 10
1.5.2 Continuouspopulations 12
1.6 Historicalnote 13
1.7 Termstoremember 14
1.8 Appendix 14
1.8.1 Alternativeexpressionsfor2 14
y
2 SamplingDistributionofanEstimator 15
2.1 Distributionofvalues 15
2.2 Estimation 18
2.2.1 Consistency 19
2.2.2 Expectedvalue 21
2.2.3 Bias 24
2.2.4 Variance 25
2.2.5 Standarderror 27
2.2.6 Meansquareerror 27
2.3 Intervalestimation 28
2.4 Theroleofsimulatedsampling 32
2.5 Otherconsiderations 32
2.6 Termstoremember 33
2.7 Exercises 33
vii
viii CONTENTS
2.8 Appendix 34
2.8.1 Derivationoftherelationshipbetweenmeansquareerror,
variance,andsquaredbias 34
3 SamplingDesignsforDiscretePopulations 35
3.1 Introduction 35
3.2 Equalprobabilitydesigns 35
3.2.1 Simplerandomsampling 35
Simplerandomsamplingwithoutreplacement 36
SelectingaSRSwoRsample 42
Simplerandomsamplingwithreplacement 44
SelectingaSRSwRsample 48
3.2.2 Systematicsampling 49
Circularsystematicsampling 56
3.2.3 Bernoullisampling 57
3.3 Unequalprobabilitydesigns 61
3.3.1 Listsampling 62
Manipulatingselectionprobabilitiestoincreaseprecision 63
3.3.2 Poissonsampling 68
3.3.3 Unequalprobabilitysystematicsampling 71
3.3.4 Rao,Hartley,Cochransamplingstrategy 74
3.4 Termstoremember 77
3.5 Exercises 77
3.6 Appendix 79
3.6.1 Factorialandcombinatorialnotation 79
3.6.2 Derivationoftheinclusionprobabilityofanelementfor
SRSwoR. 79
3.6.3 ProofofunbiasednessofO asanestimatorof 80
y y
3.6.4 DerivationofV ŒNyNin(3.6)followingSRSwoR 81
3.6.5 Proofofunbiasednessofs2 asestimatorof2 following
y y
SRSwoR 83
3.6.6 Derivationoftheinclusionprobabilityofanelementfor
SRSwR 84
3.6.7 Derivation of EŒn and VŒn following Bernoulli
sampling 84
3.6.8 DerivationoftheexpectedvalueandvarianceofO 85
yp
3.6.9 ProofoftheunbiasednessofvOŒO asanestimatorof
yp
VŒO  89
yp
3.6.10 Varianceofestimatedproportions 89
3.6.11 Nearlyexactconfidenceintervalsforproportions 90
3.6.12 Expectedsamplesize 91
4 SamplingDesignsforContinuousPopulations 93
4.1 Introduction 93
4.2 CrudeMonteCarlo 94
CONTENTS ix
4.2.1 Definitions 95
4.2.2 Selection 97
4.2.3 Estimation 98
4.2.4 Antitheticvariates 102
4.2.5 Systematicselection 104
4.3 Importancesampling 106
4.3.1 Proxyfunction 107
4.3.2 Selectionbytheinverse-transformmethod 108
4.3.3 Selectionbytheacceptance-rejectionmethod 109
4.3.4 Estimation 110
4.4 Controlvariateestimation 114
4.5 Samplingintwoorthreedimensions 116
4.5.1 Selectionofasamplepoint 117
4.5.2 Estimation 118
4.5.3 Systematicselection 119
4.5.4 Threedimensions 120
4.6 Generalnotation 121
4.7 Termstoremember 121
4.8 Exercisesandprojects 122
4.9 Appendix 123
4.9.1 Proofoftheunbiasednessof O asanestimatorof  124
 
4.9.2 DerivationofV O 125
4.9.3 TheHorvitz-Thompsonestimatorof anditsvariance 126

5 StratifiedSamplingDesigns 127
5.1 Introduction 127
5.2 Rationaleforstratifiedsampling 127
5.3 Estimationwithstratifiedsampling 129
5.3.1 Notation 129
5.3.2 HTestimation 130
5.3.3 Moregeneralestimation 134
5.3.4 Stratifiedrandomsampling 137
5.4 Sampleallocationamongstrata 140
5.4.1 Equalallocation 140
5.4.2 Proportionalallocation 141
5.4.3 x-Proportionalallocation 143
5.4.4 Optimalallocation,equalsamplingcosts 143
5.4.5 Optimalallocation,unequalsamplingcosts 145
5.4.6 Powerallocation 145
5.4.7 Allocationformultiresourcesurveys 146
5.4.8 Comparisonofallocationrules 147
5.5 Incorrectassignmentofpopulationelementsintostrata 148
5.6 Doublesamplingforstratification 149
5.7 Poststratification 152
5.7.1 Preliminarydetailsaboutpoststratification 152