Sampling Designs Dependent on Sample Parameters of Auxiliary Variables

84 Pages·2015·1.463 MB·English

by  Janusz L. Wywiał (auth.)

SPRINGER BRIEFS IN STATISTICS
Janusz L. Wywiał
Sampling Designs
Dependent
on Sample
Parameters
of Auxiliary Variables
123
SpringerBriefs in Statistics
More information about this series at http://www.springer.com/series/8921
ł
Janusz L. Wywia
Sampling Designs Dependent
on Sample Parameters
of Auxiliary Variables
123
JanuszL. Wywiał
Department ofStatistics
Katowice University of Economics
Katowice
Poland
ISSN 2191-544X ISSN 2191-5458 (electronic)
SpringerBriefs inStatistics
ISBN978-3-662-47382-5 ISBN978-3-662-47383-2 (eBook)
DOI 10.1007/978-3-662-47383-2
LibraryofCongressControlNumber:2015942470
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For Eva and Peter
Preface
Wecanobservegradualincreaseintherangeofsurveysamplingmethodsandtheir
applications in the fields of economics, geology, and agriculture, among others.
These applications have been followed by new theoretical solutions that provide
bettersamplingdesignsorestimators.Recently,severalhighlyimportantproperties
of sampling designs have been discovered, and many new methods have been
published.
Thissmallmonographisthe result oftheauthor’s interest in survey designs.In
general, the book represents a synthesis of contributions on sampling designs that
are dependent on sample moments or the order statistics of auxiliary variables.
The book should be useful for students and statisticians whose work involves
survey sampling. The book can offer new inspirations for those looking for new
sampling designs dependent on auxiliary variables.
In some sense, this book has been prepared because of Professor Miodrag
Lovric who invited me to write some brief notes in the International Encyclopedia
of Statistical Science edited by him. This led to the proposal to write this book.
The Author is grateful to Reviewers for valuable comments.
This book is a result of the research supported by the grants: N N111 434137
from the Polish Ministry of Science and Higher Education and DEC-2012/07/
B/H/03073 from National Scientific Center.
Katowice, Poland Janusz L. Wywiał
October 2014
vii
Contents
1 Introduction and Basic Sampling Strategies. . . . . . . . . . . . . . . . . . 1
1.1 Aim and Outline of This Book . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Population and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Sampling Design and Sampling Scheme. . . . . . . . . . . . . . . . . . 4
1.4 Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Sampling Designs Dependent on Sample Moments
of Auxiliary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Sampling Design Proportional to Sample Mean. . . . . . . . . . . . . 15
2.2 Sampford’s Sampling Design. . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Sampling Design Proportional to Sample Variance. . . . . . . . . . . 17
2.4 Sampling Designs Proportional to the Generalized Variance . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Sampling Designs Based on Order Statistics
of Auxiliary Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Basic Properties of Order Statistics . . . . . . . . . . . . . . . . . . . . . 31
3.2 Sampling Design Proportional to Function
of One-Order Statistic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Sampling Design Proportional to Function
of Two-Order Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Sampling Design Proportional to Function
of Three-Order Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Simulation Analysis of the Efficiency of the Strategies . . . . . . . . . . 51
4.1 Description of the Simulation Experiments . . . . . . . . . . . . . . . . 51
4.2 Efficiency of Estimation Strategies Dependent
on Sample Moments or Order Statistics . . . . . . . . . . . . . . . . . . 52
ix
x Contents
4.3 Efficiency of Estimation Strategies Dependent
on the Sum of Order Statistics. . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Efficiency Estimation of Domain Mean. . . . . . . . . . . . . . . . . . . 66
4.5 Estimation of Quantiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chapter 1
Introduction and Basic Sampling Strategies
1.1 AimandOutlineofThisBook
Anestimationofthepopulationparametersinfiniteandfixedpopulationsistaken
into account. The estimation is focused on the population average of the variable
understudy.Thevaluesofthevariableunderstudyareobservedinrandomsamples
thatareselectedaccordingtopreassignedsamplingdesignsandsamplingschemes.
Weassumethatthevaluesofauxiliaryvariablesareobservedinthewholepopula-
tion. The population average is estimated by means of a sampling strategy, which
is the pairing of an estimator and a sampling design. Extensive reviews of sam-
pling designs or sampling schemes dependent on an auxiliary variable have been
previouslypresentedbyBrewerandHanif(1983),ChaudhuriandVos(1988),Singh
(2003),ChaudhuriandStenger(2005),Tillé(2006)andLovric(Ed),International
Encyclopedia(2011).
Ourconsiderationsarefocusedonparticularsamplingstrategiesthataredepen-
dentonsamplemomentsororderstatisticsofapositivelyvaluedauxiliaryvariable.
Themainpurposeofthisbriefmonographistopresentthebasicpropertiesofsuch
samplingstrategies.Moreover,theiraccuracywillbeanalyzed.
We start with basic notations and the presentation of ordinary sampling strate-
gies. In particular, the well-known Horvitz–Thompson statistic, as well as ratio or
regressionestimators,areconsidered.Theclassofstrategiesdependent onsample
momentsofauxiliaryvariablesarepresentedinChap.2.Thesamplingdesignspro-
portional to the sample mean or sample variance are defined. The last design is
generalized into a sampling design proportional to the determinant of the sample
variance–covariance matrix of the auxiliary variables. The new class of sampling
designsthatareproportionaltofunctionsoforderstatisticsoftheauxiliaryvariable
areintroducedinChap.3.Inparticular,thesamplingdesignproportionaltothevalue
ofthepositiveorderstatisticisconsidered.Theinclusionprobabilitiesofthesam-
plingdesignsdependentonsamplemomentsaswellasorderstatisticsarederived.
Several new estimators dependent on sample moments or order statistics are also
©TheAuthor(s)2015 1
J.L.Wywiał,SamplingDesignsDependentonSampleParameters
ofAuxiliaryVariables,SpringerBriefsinStatistics,
DOI10.1007/978-3-662-47383-2_1