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1、yQuantileRegressionRogerKoenkerDepartmentofEconomicsUniversityofIllinois,Urbana-ChampaignChampaign,IL61820USAVersion:November10,2000.Thismaterialconstitutesverypreliminaryversionsofseveralchaptersfromaproposedmonographonquantileregression.Portionsofthepresentversionwerepreparedforaminicours
2、eo eredbyRogerKoenkerinFebruary1997atCamposdosJord~ao,SaoPaulo,Brazil.Commentsonanyaspectofthechapterswouldbemostwelcomeandcouldbesenttorkoenker@uiuc.eduortothepostaladdressprovidedabove.12Contents3CONTENTSiPrefaceFrancisGaltoninafamouspassagedefendingthecharmsofstatistics"againstitsmanyde
3、tractors,chidedhisstatisticalcolleagues[who]limitedtheirinquiriestoAverages,anddonotseemtorevelinmorecomprehensiveviews.Theirsoulsseemasdulltothecharmofvarietyasthatofanativeofoneofour
atEnglishcounties,whoseretrospectofSwitzerlandwasthat,ifthemountainscouldbethrownintoitslakes,twonuisances
4、wouldbegotridofatonce.[NaturalInheritance,p.62]Itisthefundamentaltaskofstatisticstobringorderoutofthediversity,attimestheapparentchaos,ofscienti cobservation.Andthistaskisoftenverye ectivelyaccomplishedbyexploringhowaveragesofcertainvariablesdependuponthevaluesofotherconditioning"variables
5、.Themethodofleastsquareswhichpervadessta-tisticsisadmirablysuitedforthispurpose.Andyet,likeGalton,onemayquestionwhethertheexclusivefocusonconditionalmeanrelationsamongvariables,ignoressomecharmofvariety"inmattersstatistical.Asresidentsofoneofthe
attestAmericancounties,ourrecollectionsofSwi
6、tzer-landanditsattractivenuisancesarequitedi erentfromtheonesdescribedbyGalton.NotonlytheSwisslandscape,butmanyofitsdistinguishedstatisticianshaveinrecentyearsmadeusmoreawareofthecharmsandperilsofthediversityofobservations,andtheconsequencesoftoo-blindlylimitingourinquirytoaverages.Quantile
7、regressiono erstheopportunityforamorecompleteviewofthestatisti-callandscapeandtherelationshipsamongstochasticvariables.Thesimpleexpedientofreplacingthefamiliarnotionsofsortingandrankingobservationsinthemostel-ementaryone-samplecontextbyoptimizationenable
