Robustness in statistical forecasting

369 Pages·2013·2.45 MB·English

by  Kharin Yu

Yuriy Kharin
Robustness
in Statistical
Forecasting
Robustness in Statistical Forecasting

Yuriy Kharin
Robustness in Statistical
Forecasting
123
YuriyKharin
DepartmentofMathematicalModeling
andDataAnalysis
BelarusianStateUniversity
Minsk
Belarus
ISBN978-3-319-00839-4 ISBN978-3-319-00840-0(eBook)
DOI10.1007/978-3-319-00840-0
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013948622
MathematicsSubjectClassification(2010):62-02,62M20,62M10,62G35,62-07,62F35,62C20,62P20
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Preface
Statistical forecasting procedures are used to solve many applied problems in
engineering, economics, finance, medicine, environmental studies, etc. For the
majorityofthedevelopedstatistical forecastingprocedures,optimality(orasymp-
totic optimality as observation time increases) is proved w.r.t. the mean square
forecastriskundertheassumptionsofanunderlyinghypotheticalmodel.Inpractice,
however, the observed data usually deviates from hypothetical models: random
observation errors may be non-Gaussian, correlated, or inhomogeneous; the data
maybecontaminatedbyoutliers,levelshifts,ormissingvalues;trend,regression,
and autoregressionfunctionsdo not necessarily belong to the declared parametric
families, etc. Unfortunately, the forecasting procedures which have been proved
to be optimal under the hypothetical model often become unstable under even
small model distortions, resulting in forecast risks or mean square errors which
are significantly higher than the theoretical values obtained in the absence of
distortion.Thisnecessitatesthedevelopmentofrobuststatisticalalgorithms,which
are designed to retain most of their propertiesunder small deviationsfrom model
assumptions.
The available textbooks on the subjects of statistical forecasting and robust
statistical methodscan besplit intotwo distinctclusters.The first cluster includes
booksontheoreticalandappliedaspectsofstatisticalforecastingwherelittleorno
attentionispaidtorobustness.Thefocusofthesebooksisonvarioushypothetical
models, methods, and computer algorithms used in forecasting, as well as their
performanceintheabsenceofmodeldistortions.
Thesecondclusterincludesbooksonrobuststatisticswhicharededicatedtosuch
diversesubjectsasrobuststatisticalparameterestimation,robusthypothesistesting
in parametric (e.g., shift-scale) families of probability distributions, regression
analysis,discriminantanalysis,clusteranalysis,timeseriesanalysis,etc.However,
the topic of robustness in statistical forecasting remains barely touched upon,
and little or no information is provided on such important aspects of forecasting
as analysis of risk increments due to different types and levels of distortion,
estimation of critical distortion levels for the traditional forecasting procedures,
v
vi Preface
and development of robust forecasting procedures tailored to the distortion types
thatarecommonlyencounteredinapplications.
Thismonographisanattempttofillthedescribedgapintheliteraturebygoing
beyondthefundamentalsubjectsofrobuststatisticalestimationandrobuststatistical
hypothesistestingandpresentingasystematiccollectionofknownandnewresults
relatedtothefollowingtopicalproblems:
• Construction of mathematical models and descriptions of typical distortions in
appliedforecastingproblems;
• Quantitativeevaluationoftherobustnessoftraditionalforecastingprocedures;
• Evaluationofcriticaldistortionlevels;
• Construction of new robust forecasting proceduressatisfying the minimax-risk
criterion.
Solvingtheseproblemsanswersthefollowingquestions,whicharehighlyrelevant
toboththeoreticalandappliedaspectsofstatisticalforecasting:
• Whichdistortiontypescanbeaccommodatedbyforecastingprocedures?
• Whatarethemaximaldistortionlevelsallowingfor“safe”useofthetraditional
forecastingalgorithms?
• How can we estimate the effect of distortions on the mean square risk of
traditionalforecastingalgorithms?
• Whichrobustforecastingstatisticsarethemostsuitableunderdifferenttypesof
distortions?
The monograph is organized into ten chapters. Chapter 1 serves as a general
introductiontothesubjectofstatisticalforecasting,presentingitshistoryandsome
of the possible applications. Chapter 2 describes the decision-theoretic approach
to forecasting, which is different from the general statistical approach used in
Chaps.3–10.Chapter3presentsmathematicalmodelsofthetimeseriescommonly
used in statistical forecasting. Chapter 4 classifies types of model distortions and
defines metrics for optimality and robustness in statistical forecasting. Chapter 5
presentsmethodsforoptimalparametricandnonparametrictime series regression
forecasting. In Chap.6, robustness of these methods is evaluated, and robust
forecasting statistics are constructed. A similar treatment of the ARIMA.p;d;q/
autoregressiveintegratedmovingaveragetimeseriesmodelispresentedinChap.7.
Chapter8presentsananalysisofoptimalityandrobustnessofforecastingbasedon
autoregressivetimeseriesmodelsundermissingvalues.Robustnessofmultivariate
timeseriesforecastingbasedonsystemsofsimultaneousequationsisinvestigated
inChaps.9,and10discussesforecastingofdiscretetimeseries.
The interdependence of the chapters is illustrated in Fig.1 below; solid lines
representprerequisites,anddashedlinesindicateweakerrelations.
Presentation of the material within the chapters follows the pattern “model !
method!algorithm!computationresultsbasedonsimulatedorreal-worlddata.”
The theoretical results are illustrated by computer experiments using real-world
statisticaldata(eightinstances)andsimulateddata(teninstances).
Preface vii
Fig.1 Theinterdependenceofthechapters
Therobustforecastingalgorithmsdescribedinthemonographhavebeenimple-
mentedascomputerprograms;however,thedevelopedsoftwarepackagecannotbe
madeavailabletothereader(inparticular,onlytheRussian-languageuserinterface
hasbeendesigned).Theauthorhasintentionallydeferredfromconnectingthemate-
rialpresentedinthisbooktoanyspecificsoftwarepackageorframework.Instead,
the methods and algorithms are presented in the most general form, allowing the
reader to implement them within a statistical package of their choice (e.g., S-
PLUSorR). Thisalso ensuresthatsoftware developerscan easily incorporatethe
developedmethodsintotheirproducts.
viii Preface
The book is primarily intended for mathematicians, statisticians, and software
developersinappliedmathematics,computerscience,dataanalysis,econometrics,
financialengineering,andbiometrics.Itcanalsoberecommendedasatextbookfor
aone-semestercourseforadvancedundergraduateandpostgraduatestudentsofthe
mentioneddisciplines.
The author would like to thank Serguei Ayvazyan, Christophe Croux, Peter
Filzmoser, Ursula Gather, Peter Huber, William Mallios, Helmut Rieder, and
Peter Rousseeuw for the discussions that aroused his interest in robust statistics.
Theauthorisgratefultotheanonymousreviewersfortheirattentioninanalyzingthe
manuscript,andtheresultingcommentsandproposalsonimprovingthebook.The
authoralsothankshiscolleaguesfromtheResearchInstituteforAppliedProblems
of Mathematics and Informatics, as well as the Department of Mathematical
ModelingandDataAnalysisoftheBelarusianStateUniversity.Theauthorthanks
hisPh.D.studentsAlexanderHuryn,AndreyKostevich,VladimirMalugin,Maksim
Pashkevich, Andrey Piatlistkiy, Valery Voloshko, and Andrey Yarmola for their
contributions to the joint research in the field of robust statistical methods and
algorithms. The author would like to give special thanks to Nikolai Kalosha for
hisassistanceineditingtheEnglishversionofthemanuscript.Toconcludethelist
ofpeoplewhomadeasignificantimpactonthebook,theauthormustmentionthe
wisecounselandhelpfuladviceprovidedbyDr.NielsPeterThomasandVeronika
RosteckofSpringer.
Minsk,Belarus YuriyKharin
April2013
Contents
1 Introduction................................................................. 1
References.................................................................... 4
2 ADecision-TheoreticApproachtoForecasting......................... 7
2.1 TheMathematicalModelofDecisionMaking ..................... 7
2.2 Minimax,Admissible,andBayesianFamilies
ofDecisionRules..................................................... 8
2.3 TheBayesianForecastDensity...................................... 13
2.4 ForecastingDiscreteStatesbyDiscriminantAnalysis ............. 17
2.4.1 TheMathematicalModel................................... 17
2.4.2 CompletePriorKnowledgeoff(cid:2) ;p0.(cid:2)/g................. 18
i i
2.4.3 PriorUncertainty ........................................... 23
References.................................................................... 29
3 TimeSeriesModelsofStatisticalForecasting........................... 31
3.1 RegressionModelsofTimeSeries .................................. 31
3.2 StationaryTimeSeriesModels...................................... 33
3.3 ARIMA.p;d;q/TimeSeriesModel................................ 35
3.4 NonlinearTimeSeries ............................................... 39
3.4.1 AGeneralNonlinearModel................................ 39
3.4.2 BilinearModelBL.p;q;P;Q/............................ 40
3.4.3 Functional-CoefficientAutoregression
ModelFAR.p;d/........................................... 40
3.4.4 GeneralizedExponentialAutoregression
ModelEXPAR.p;d/....................................... 41
3.4.5 ThresholdAutoregressionModelTAR.k/ ................ 41
3.4.6 ARCH.p/Model........................................... 42
3.4.7 GARCH.p;q/Model ...................................... 42
3.5 MultivariateTimeSeriesModels.................................... 42
3.5.1 MultivariateStationaryTimeSeries....................... 42
3.5.2 VectorAutoregressionModelVAR.p/.................... 43
3.5.3 VectorMovingAverageModelVMA.q/.................. 45
ix