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1、Bernoulli15(1),2009,4068DOI:10.3150/08-BEJ141Maximumlikelihoodestimationofalog-concavedensityanditsdistributionfunction:BasicpropertiesanduniformconsistencyLUTZDÜMBGEN1andKASPARRUFIBACH21InstituteofMathematicalStatisticsandActuarialScience,UniversityofBern,Sidlerstra
2、sse5,CH-3012Bern,Switzerland.E-mail:duembgen@stat.unibe.ch2AbteilungBiostatistik,InstitutfürSozial-undPräventivmedizin,UniversitätZürich,Hirschengraben84,CH-8001Zürich,Switzerland.E-mail:kaspar.rufibach@ifspm.uzh.chWestudynonparametricmaximumlikelihoodestimationofalog
3、-concaveprobabilitydensityanditsdis-tributionandhazardfunction.Somegeneralpropertiesoftheseestimatorsarederivedfromtwocharac-terizations.Itisshownthattherateofconvergencewithrespecttosupremumnormonacompactintervalforthedensityandhazardrateestimatorisatleast(log(n)/n)
4、1/3andtypically(log(n)/n)2/5,whereasthedifferencebetweentheempiricalandestimateddistributionfunctionvanisheswithrateop(n−1/2)undercertainregularityassumptions.Keywords:adaptivity;bracketing;exponentialinequality;gapproblem;hazardfunction;methodofcaricatures;shapecons
5、traints1.IntroductionTwocommonapproachestononparametricdensityestimationaresmoothingmethodsandqual-itativeconstraints.Theformerapproachincludes,amongothers,kerneldensityestimators,esti-matorsbasedondiscretewaveletsorotherseriesexpansionsandestimatorsbasedonroughnessp
6、enalization.GoodstartingpointsforthevastliteratureinthisfieldareSilverman(1982,1986)andDonohoetal.(1996).Acommonfeatureofallofthesemethodsisthattheyinvolvecertaintuningparameters,forexample,theorderofakernelandthebandwidth.Aproperchoiceoftheseparametersisfarfromtrivia
7、lsinceoptimalvaluesdependonunknownpropertiesoftheunder-lyingdensityf.Thesecondapproachavoidssuchproblemsbyimposingqualitativepropertiesonf,forexample,monotonicityorconvexityoncertainintervalsintheunivariatecase.Suchassumptionsareoftenplausibleorevenjustifiedrigorously
8、inspecificapplications.DensityestimationundershapeconstraintswasfirstconsideredbyGrenander(1956),whofoundthatthenonparametricmaximuml
