Elements of Copula Modeling with R

274 Pages·2018·65.43 MB·English

by  Marius Hofert

UseR!
Marius Hofert · Ivan Kojadinovic 
Martin Mächler · Jun Yan
Elements
of Copula
Modeling
with R
Use R!
Serieseditors
RobertGentleman KurtHornik GiovanniParmigiani
Moreinformationaboutthisseriesathttp://www.springer.com/series/6991
Marius Hofert • Ivan Kojadinovic (cid:129)
¨
Martin Machler (cid:129) Jun Yan
Elements of Copula
Modeling with R
123
MariusHofert IvanKojadinovic
DepartmentofStatisticsandActuarial LaboratoryofMathematicsandits
Science Applications
UniversityofWaterloo UniversityofPauandPaysdel’Adour
Waterloo,Ontario,Canada Pau,France
MartinMa¨chler JunYan
SeminarforStatistics DepartmentofStatistics
ETHZurich UniversityofConnecticut
Zurich,Switzerland Storrs,Connecticut,USA
ISSN2197-5736 ISSN2197-5744 (electronic)
UseR!
ISBN978-3-319-89634-2 ISBN978-3-319-89635-9 (eBook)
https://doi.org/10.1007/978-3-319-89635-9
LibraryofCongressControlNumber:2018940269
MathematicsSubjectClassification(2010):62H05,65C10,62H12,62H15,62P05,62P12,65C60
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Preface
Theaimofthisbookistoshowhowsomeofthemainstepsinvolvedinthestatistical
modelingofcontinuousmultivariatedistributionsusingcopulascanbecarriedout
intheRstatisticalenvironment(RCoreTeam,2017)withtheRpackagecopula.
The R package copula originally emerged from the authors’ research interests;
it has been available on the Comprehensive R Archive Network (CRAN) since
2005 and has been under constant development ever since. Judging from users’
feedback,thepackageisappliedinareassuchashydrology,environmentalsciences,
quantitativeriskmanagement,insurance,andfinance.
Comparedto existingmonographsoncopulas,the originalityofthisbookis to
illustrate how theoretical concepts can be applied in practice with the R package
copula. To this end, numerousstand-alone examples are provided. This is both
of pedagogical and of practical interest as the R source code and reproduced
outputssuchasfiguresnotonlyallowonetobetterunderstandthesubtletiesofthe
theoreticalnotionsbutalso enableonetosolvereal-worldproblemsinthevarious
fieldsofinterest.
Thebooktargets(possiblyfuture)statisticiansand(financial,hydrological,and
other) engineers alike who would like to understand how theoretical notions and
practical computations around copula modeling can be applied in R without an
overwhelmingamountofmathematics.Readersarenonethelessexpectedtohavea
basicknowledgeof(multivariate)probabilityandstatistics,inparticularofrandom
vectors,simulationalgorithms,estimationmethods,andstatisticaltests.
Althoughthebookaddressesmostofthepracticalissuesarisingwhenmodeling
multivariatedatawithcopulas,itonlycoversamodestpartofthefield.Inparticular,
important aspects such as dynamic copula models, vine copulas, and copula
modelingfor censoredor discontinuousdata are notdealtwith. Thisis partly due
toaselectionbias:ConceptsthatarenotimplementedintheRpackagecopulaat
the time ofwritingare notpresentedin the book.Manyof these morespecialized
notionsare,however,availablein other R packages.For an up-to-datedescription
of the functionalities of other packages dealing with copulas, see, for example,
the CRAN View “Distributions” available at https://cran.r-project.org/web/views/
Distributions.html.
vii
viii Preface
AstheRstatisticalenvironment,andmostfreesoftware,theRpackagecopula
comeswithnowarranty.Itisdistributedwiththehopethatitcanbeusefultoothers.
Waterloo,ON,Canada MariusHofert
Pau,France IvanKojadinovic
Zürich,Switzerland MartinMächler
Storrs,CT,USA JunYan
November2017
Reference
RCoreTeam.(2017).R:ALanguageandEnvironmentforStatisticalComputing.RFoundation
forStatisticalComputing,Vienna,Austria.https://www.R-project.org.
Contents
1 Introduction .................................................................. 1
1.1 AMotivatingExample.................................................. 1
1.2 ProbabilityandQuantileTransformations............................. 3
1.3 Copulas.................................................................. 5
1.4 StructureandPhilosophyoftheBook ................................. 6
1.5 AdditionalReading ..................................................... 7
References..................................................................... 8
2 Copulas........................................................................ 9
2.1 DefinitionandCharacterization........................................ 9
2.2 TheFréchet–HoeffdingBounds........................................ 18
2.3 Sklar’sTheorem......................................................... 22
2.4 TheInvariancePrinciple................................................ 35
2.5 SurvivalCopulasandCopulaSymmetries............................. 40
2.6 MeasuresofAssociation................................................ 45
2.6.1 FallaciesRelatedtotheCorrelationCoefficient............... 46
2.6.2 RankCorrelationMeasures .................................... 51
2.6.3 TailDependenceCoefficients.................................. 58
2.7 RosenblattTransformandConditionalSampling ..................... 68
References..................................................................... 77
3 ClassesandFamilies......................................................... 81
3.1 EllipticalDistributionsandCopulas ................................... 81
3.1.1 EllipticalDistributions.......................................... 82
3.1.2 EllipticalCopulas............................................... 85
3.2 ArchimedeanCopulas .................................................. 97
3.3 Extreme-ValueCopulas................................................. 112
3.4 SelectedCopulaTransformationsandConstructions ................. 117
3.4.1 RotatedCopulas ................................................ 117
3.4.2 Khoudraji’sDevice ............................................. 120
3.4.3 MixturesofCopulas............................................ 127
References..................................................................... 130
ix
x Contents
4 Estimation .................................................................... 133
4.1 EstimationUnderaParametricAssumptionontheCopula........... 133
4.1.1 ParametricallyEstimatedMargins............................. 134
4.1.2 NonparametricallyEstimatedMargins ........................ 139
4.1.3 EstimatorsofEllipticalCopulaParameters ................... 148
4.1.4 OtherSemi-parametricEstimators............................. 152
4.1.5 EstimationofCopulaModelswithPartlyFixed
Parameters....................................................... 153
4.2 NonparametricEstimationoftheCopula.............................. 157
4.2.1 TheEmpiricalCopula .......................................... 158
4.2.2 UnderExtreme-ValueDependence............................ 161
References..................................................................... 163
5 GraphicalDiagnostics,Tests,andModelSelection ...................... 167
5.1 BasicGraphicalDiagnostics............................................ 168
5.2 HypothesisTests ........................................................ 173
5.2.1 TestsofIndependence.......................................... 173
5.2.2 TestsofExchangeability ....................................... 176
5.2.3 ATestofRadialSymmetry .................................... 178
5.2.4 TestsofExtreme-ValueDependence .......................... 179
5.2.5 Goodness-of-FitTests .......................................... 181
5.2.6 AMixtureofGraphicalandFormalGoodness-of-Fit
Tests............................................................. 188
5.3 ModelSelection......................................................... 191
References..................................................................... 195
6 Ties,TimeSeries,andRegression.......................................... 197
6.1 Ties ...................................................................... 198
6.2 SelectedCopulaTestsandModelsforTimeSeries ................... 216
6.2.1 TestsofStationarity............................................. 216
6.2.2 TestsofSerialIndependence................................... 226
6.2.3 Models for Multivariate Time Series Based on
ConditionalCopulas............................................ 230
6.3 Regression............................................................... 238
References..................................................................... 252
A RandPackageVersions..................................................... 255
Index............................................................................... 259