三参数威布尔分布参数估计方法比较-严晓东

18320059()JOURNALOFNINGBOUNIVERSITY(NSEE)Vol.18No.3Sept.2005:100125132(2005)03203012051,2,1,1(1.,315211;2.,315000):2004-10-15.:(2002417).:(1967-),,,,.:5(),7,5,.:;;;;;:O213.2;TB114.3:AComparisonoftheParametersEstimationMethodsfor32ParameterWeibullDistributionYANXiao2dong1,MAXiang2,ZHENGRong2yue1,WULiang1(1.FacultyofArchitectural,CivilEngieeringandEnvironment,NingboUniversity,Ningbo315211,China;2.NingboRealestateCo.,Ltd.,Ningbo315000,China)Abstract:TheWeibulldistributionisoneofthemostpopularmodelsinthefieldofreliability.Since1939,manyparameterestimationmethodshavebeenproposed,includingtheprobabilityweightedmomentmethod,themaxi2mumlikelihoodmethod,bilinearregressionmethod,correlationcoefficientmethodandGreyestimationmethod.Themethodsmentionedabovearedisscussed,programsandexamplesaregiven.Keywords:32parameterWeibulldistribution;themaximumlikelihoodmethod;bilinearregressionmethod;probabilityweightedmoments;correlationcoefficientmethod;greyestimationmethodCLCnumber:O213.2;TB114.3Documentcode:AW.Weibull[1],[2].,.,,;,,,,.,,[3].,3,,,,,,.,5(),,.1F(x)=1-exp{-[(x-)/]},(x)(1),,0,0.f(x)=(/)[(x-)/]-1exp{-[(x-)/]},(2)r(x)=(/)[(x-)/]-1,(3)=0,.,1,(3),;=1,,;1,,.22.1.,.=-,(4)(2),[4],:lnL=nln-nln(-)+(-1)ni=1ln(xi-)-1(-)ni=1(xi-r),(5)n,xi(i=1,2,,n),.Weibull:9lnL9=n--(-1)ni=11xi--(-)(-1)ni=1(xi-)+(-)ni=1(xi-)(-1)=0,9lnL9=n-nln(-)+ni=1ln(xi-)+ln(-)(-)ni=1(xi-)-ni=1[(xi-)ln(xi-)](-)=0,9lnL9=-n-+(-)(+1)ni=1(xi-)=0.(6),,,[5].(6).Newton2Raphson.Newton2Raphson,,,0,0,0,0+=;0+=;0+=,0,0,0,,(6).,,,,,,,0+0+0+,,,,.,,..2.2[6](1),=x01/,2:lnln11-F(x)=ln(x-)-lnx0,(7)(ln11-F(x))1/=x-.(8)203()2005(7)(8),,^=[ln(x-^)]lnln11-F(x)[ln(x-^)]2-[ln(x-^)]2-[ln(x-^)]lnln11-F(x)[ln(x-^)]2-[ln(x-^)]2,(9)^=x-[x2-€x2]ln11-F(x)1/^xln11-F(x)1-^-€xln11-F(x)1-^,(10)x^0=exp^[ln(x-^)]-lnln11-F(x).(11)(9)(10),(11),^=f1(^),^=f2(^),x0=f3(^,^).(12)(13)(14),,.(12)(13),.(14),x^0.2.3(PWM)[7]M1,0,k=1+k+(1+1)(1+k)1+1,(15),k=0,1,3.M1,0,1M1,0,2M1,0,3=4(M1,0,3M1,0,0-M21,0,1)4M1,0,3+M1,0,0-4M1,0,1,=M1,0,0-ln(M1,0,0-2M1,0,1M1,0,1-2M1,0,3)/ln2,=ln2lnM1,0,0-2M1,0,12(M1,0,1-2M1,0,3).(16)Ms1,0,0=1nni=1xi,Ms1,0,1=1nni=1xi(1-i-0.35n),Ms1,0,3=1nni=1xi(1-i-0.35n)3.(17)2.4[8](1)1-F(x)=exp-x-,(18)ln[-ln(1-F(x))]=ln(x-)-ln,(19):Y=ln[-ln(1-F(x))],X=ln(x-),A=-ln,B=.(20)Y=A+BX.(21)XY,(xi,F(xi)),(20)(Xi,Yi),ABrA^=Y-B^X,B^=LXY/LXX,r=LXY/LXXLYY,(22)X=1nni=1Xi,Y=1nni=1Yi,LXX=ni=1X21-n…X2,LYY=ni=1Y21-n…Y2,LXY=ni=1XiYi-nXY.ABr,A()B()r(),,r|r|,d|r()|/d=0,,(22)ABr,(20).2.5[9]R(x)=1-F(x),(1)lnln1R(x)=lnx-,(23)x=e1lnln1R(x)+,3033:ti=-lnln1R(xi),xti=e-1ti+.=c,1/=a,=b,x=ce-ati+b.GM(1,1).GM(1,1).3,,[10]:1n=5,(:MPa)381,395,408,423,4312n=20,(:)350,380,400,430,450,470,480,500,520,540,550,570,600,610,630,650,670,730,770,8403n=31,30926,34554,36381,38423,40103,40501,42200,44392,46092,46125,46175,48025,48025,48055,48055,48055,48055,48056,51675,52344,52345,52345,52345,52379,55997,56202,57709,57709,57709,57709,634964n=71,3956.42,4004.18,4091.61,4355.05,4355.40,4376.01,4391.79,4487.68,4487.68,4736.67,4736.67,4939.85,4963.62,5220.19,5353.41,5372.72,5418.04,5444.11,5603.17,5698.10,5746.17,5843.52,6175.14,6197.41,6249.69,6279.76,6279.76,6572.74,6740.48,6887.65,7183.09,7209,7209,7209,7209,7366.4,7581.64,7581.64,7581.64,7645.59,8246,8599.7,8713.97,8936.34,9044.22,9197.45,9511.73,9754.47,9967.45,10136.31,10172.88,10172.88,10308.04,10395,10609.23,10609.23,10788.97,10879.97,10971.75,11594.41,11990.59,12237.31,12400.31,12400.31,12550.01,13198.73,13947.78,15557.12,17646.12,19848.23,23199.07.1Tab.1Resultsofparameterestimationandverification1(n=5)K-S-13.7914-642.65861080.73880.2566000.263360.8470955.2291133.1607284.24960.2105820.0426450.9950695.2341133.1500284.21980.2106000.0426450.99507017.6007417.8990.21350.2094700.0449310.99468312.6499258.4210158.35130.1987480.0052130.9999842(n=20)K-S2.1248330.2207264.54400.0729850.0223310.9991701.9940316.4807280.82420.0669120.0207070.9991081.9950316.5078280.79840.0670120.0206990.9991004.5241608.88700.00011200.0873910.0571630.9944052.0493305.3332288.73390.0598730.0099540.9999503(n=31)K-S5.959645489.54856271.11150.1139070.0789750.9869067.000653073.7953-1290.99820.1137310.0790190.9869046.999653070.3289-1291.75620.1137400.0790200.9869076.790351782.53550.04100.1146800.0791800.9868805.915542713.37058821.18590.1028770.0223230.999749403()20054(n=71)K-S1.28705208.145343578.95260.0609950.0358310.9874011.13864946.13583821.50410.0582580.0391930.9969301.14164943.26493821.23590.0545910.0391980.9970002.74969426.40580.59890.1254560.0956010.9875061.03724150.34444368.22770.0568870.0525170.991255451,K-S.1,,5,.:(1),,;3,,,3,;3,3,.(2),,,.(3),;,,.10,5.5.(4),5.(5),5,.:[1]WeibullW.Astatisticaldistributionfunctionofwideapplicability[J].JournalofAppliedMicroelectron,Re2liab.,1951,28(4):613617.[2]HallinanAJ.AreviewoftheWeibulldistribution[J].JournalofQualityTechnology,1993,25(2):8593.[3]JingLing,JwoPan.Anewmethodforselectionofpop2ulationdistributionandparameterestimation[J].Relia2bilityEngineeringandSystemSafety,1998,60:247255.[4].Weibull[J].,2000,19(3):8894.[5].[J].,1997,16(4):5357.[6].[J].,1999,5:27.[7]GreenwoodAJ