产品市场生存函数的一种非参数极大似然估计法

200388:100026788(2003)08200872041,2,1(1.,410083;2.,410083):,,L,,L:;;;;:F062.4:AAnImprovedLikelihoodMethodonEstimatingMarket2Life2FunctionofProductsSUYue2liang,GAOYang(SchoolofBusiness,CentralSouthUniversity,Changsha410083,China)Abstract:Inthispaper,consideringtheforceofmarketandcensoreddata,wemakeastudyonanim2provedlikelihoodmethodonestimatingproduct2life2function,andweapplyittopractice.Thestudynotonlycanovercomethedifficultiesanddrawbacksintheexistingmethodsofestimatingmarket2lifeofproduct,butalsoprovideanewmethodandprocedurefordecisionmakers.Keywords:market2life2dataofproducts;market2life2functionofproducts;likelihoodmethod;im2provedmethod:2002204215:(70172015);(02ZRN2014):(1969-),,,,,:,,L(1943-),,,,,,:,1,L,,,,L,L,(censoredtime),,L,,L2,T,ZTZ2.1S(t),tZ:S(t)=P(t)=P(Tt)(2.1)TF(t):©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.S(t)=1-P(t)=1-F(t)(2.2),S(t)t,t=0,S(t)=1;t=,S()=0:01,0Z,F(t),:limtt+0S(t)=1-limtt+0F(t)=1-F(t0)=S(t0),S(t)Z:S(a)-S(b)=P(atFb);S(a-0)-S(a)=P(t=a)S(t),S(t),,,Z1,a,1,b,Z,t:Sd(t)=t(2.3)Sd(t)S(t)Z,(2.3),(2.3)S(t)ZS(t),Z2.2Tf(t):f(t)=lim$t0P((t,t+$t))$t(2.4)f(t)Z2Z2,:fd(t)=t(2.5),(5)f(t).2.3Th(t),:h(t)=lim$t01$tP(t(t,t+$t)),h(t)t,ZF(t)f(t):h(t)=f(t)1-F(t)(2.6)3,:hd(t)=tt=tt(2.7)Z2.41)(2.2)(2.6):8820038©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.h(t)=f(t)S(t)(2.8)2),:f(t)=ddt[1-S(t)]=-S(t)(2.9)3)(2.9)(2.8):h(t)=-S(t)S(t)=-ddt[logeS(t)](2.10)4)(2.4)0tS(0)=1:-t0h(x)dx=logeS(t)S(t)=exp-t0h(x)dx(2.11)5)(2.8)(2.11):f(t)=h(t)exp{-logeS(t)}(2.12):,Z,T,ZZ3:,L,,LKaplanMeier(PL),L:,Lt1,t2,,tnnZnti1Ftt2FFtinZ(2.2)(2.3),tii:Sd(tii)=n-in=1-in(3.1)n-itiiZtii,iZ:ti2=ti3=ti,Sd(ti2)=Sd(ti3)=Sd(ti4)=n-4nSd(t),ZZ,,ZKa2planMeier(PL)Zp1;p2;pkk-1,kk:Sd(k)=p1p2pk(3.2):Sd(t)=Sd(t-1)pt(3.3)PLZ,n,ti1Fti2FFtin,:Sd(t)=7trFtn-rn-r+1(3.4),,;,,Z(3.4)988©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,,,,(3.4):,ZZ41998Z,1996100Z100Z,100199660,40Z,20ZZ2,60Z,220,20;215,5Z22Z,2,1Z,Sd(2)=20100=0.2,2201ZS(2)Z211Z,S(2)=P(11)=P(12)P(1)(4.1)PL:Sd(2)=(12)(1)(4.2),240220,,(3.2)2040Z401,22051,,(3.2)240+5100+20ZPL:Sd(2)=204040+5100+200.187Z,2k=0.8,Sd(2)=0.1870.8=0.1496Z5,,,L,,L:[1]ELISATLEE.[M].:,1998.9-300.[2]HalperinM.Maximumlikelihoodestimationintruncatedsamples[J].AnnalsofMathematicalStatistics,1952,23(3):226-238.[3],.[M].:,1985.1-190.[4].[J].,2000,17(1):66-69.[5]CharnesA,CooperWW,RhodesE.MeasuringtheefficiencyofDMU[J].EuropeanJournalofOperationalRe2search,1978,2(8):429-444.[6].[J].,1992,12(1):1-12.[7]LeontiefW.Input2outputEconomics[M].NY:OxfordUniversityPress,1966.[8]BalachandraR.AcomparisonofR&Dprojectterminationfactorsinfourindustrialnation[J].IEEETransactiononEngineeringManagement,1996,43(1):89-96.0920038©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.