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1、StatisticsandProbabilityLetters82(2012)1710–1717ContentslistsavailableatSciVerseScienceDirectStatisticsandProbabilityLettersjournalhomepage:www.elsevier.com/locate/staproQuadraticapproximationfornonconvexpenalizedestimationswithadivergingnumberofparametersSanginLee,Yongd
2、aiKim,SunghoonKwon∗SeoulNationalUniversity,RepublicofKoreaarticleinfoabstractArticlehistory:Weproposeanapproximatedpenalizedestimator(APE)thatcoversvariousstatisticalReceived8August2011modelsandnonconvexpenaltiesincludingthesmoothlyclippedabsolutedeviation(SCAD)Receivedi
3、nrevisedform14May2012penalty(FanandLi,2001)asaspecialcase.TheAPEachievestheoraclepropertywithaAccepted16May2012divergingnumberofparameterswhichextendstheresultsofKwonetal.(2011).SeveralAvailableonline26May2012numericalstudiesconfirmthetheoreticalresults.©2012ElsevierB.V.
4、Allrightsreserved.Keywords:NonconvexpenaltySCADSmoothlyclippedabsolutedeviationOraclepropertyQuadraticapproximation1.IntroductionVariableselectionisoneofthefundamentalissuesforsparsestatisticalmodeling,andpenalizedestimationshavebeenstudiedaseffectivetoolsforthis.Theleas
5、tabsoluteshrinkageandselectionoperator(LASSO)(Tibshirani,1996)andsmoothlyclippedabsolutedeviation(SCAD)penalties(FanandLi,2001)aretworepresentativeexamples.Tibshirani(1996)gaveaniceoverviewontheLASSOasasimultaneousprocedureofvariableselectionandparameterestimation.Howeve
6、r,theLASSOtendstoselectmorevariablesthannecessaryanditbecomesamaindeficiencyoftheLASSO.FanandLi(2001)pointedoutthattheSCADisanimprovedversionoftheLASSOsinceitachievestheoracleproperty:theasymptoticequivalencebetweenthepenalizedestimatorandtheidealnon-penalizedestimatorob
7、tainedwithtruesignalvariablesonly.Foryears,theoraclepropertyhasbeenstudiedintensivelyundervariousstatisticalmodelsandpenalties:theSCAD(FanandLi,2001;FanandPeng,2004),adaptiveLASSO(Zou,2006),bridge(Huangetal.,2008),adaptiveelasticnet(ZouandZhang,2009)andminimaxconcavepena
8、lty(MCP)(Zhang,2010).Inthispaper,weconsiderthequadraticapproximationframeworkstudiedbyKwonetal.(2011).T
