Weibull分布寿命数据的参数估计

华中科技大学硕士学位论文Weibull分布寿命数据的参数估计姓名：雷刚申请学位级别：硕士专业：概率论与数理统计指导教师：刘次华20060401IWeibullWeibullWeibullMLEBayesWeibullWeibullGammaWeibullWeibullMLEBayesBayesBayesWeibullMLEBayesIIAbstractTopeoplewhoareengineersorworkinbiomedicalfieldvariousdatawithlifetimesurvivetimeorfailuretimehavebeenconcernedAndmanystatisticalanalysismethodshavewidelyusedinthereliabilityofproductaswellasindiseasesresearchInthispaperweconcernedtheparameterestimationinthereliability-testwhentheproductslifedistributionisWeibulldistributionInchapteronetheproblem’sbackgroundisintroducedincludingthecharacteristicsoflifetimedataandanalysismethodsInchaptertwowediscussedtheparameterestimationofWeibulldistributionundercompletesampleIncludingmomentestimationmaximumlikelihoodestimation(MLE)BayesianestimationgraphmethodNumericalmethodhasbeenusedtosolvethemomentestimationforalongtimeInthispapertakingadvantageofsomepropertiesofGammafunctionwegettheexplicitsolution.SomesimulationresultsarealsogiveninthispaperInchapterthree,wediscusstheproblemundercensoreddatainthreecircumstances.TotheType-censoreddatawetakeitconcernedfromtheExtremedistribution.BecausetheparametersinExtremedistributionarelocation&scaleparameters,whileinWeibulldistributionareshape&scaleparameters.Therefore,fromthefirstwecandiscusstheproblemeasier.Fromthispoint,makinguseofsomepropertiesoftheextremevaluedistributionandthefixed-pointtheorem,wealsogettheexplicitsolution.Andanewalgorithmisgiven,throughsomesimulationresults,weseethatthisalgorithmconvergesfastanddoesnotdependonanyconditions.IIIInchapterfour,weintroducethezero-failuredata.Becausemanytraditionalmethodscannotbeuseddirectlyinthiscase,sosomeimprovementshavemade.Wegetthequasi-momentestimation,modifiedMLE.Wepaymoreattentiontotheapplicationofpartitiondistributioncurvemethodandleastsquareestimation.Inthisproblemthemostimportantworkishowtogivethefailureprobabilityreasonably.WediscusstheBayesianestimatorsandhierarchicalBayesianestimators.Atlast,someunsolvedproblemshavebeenlisted.KeyWords:LifetimedataWeibulldistributionExtremedistributionMLEMomentestimationBayesianestimationZero-failuredata......_____..111.11.1.1n12,,,nXXXn12,,,nxxxn1n0t=rrr(1)(2)()rxxx≤≤≤20tr(1)(2)()0rxxxt≤≤≤≤2312,,,nXXX()Ftmin{,}iiiYXL=1,2,,in=iLiiXLiiL0min{,}rxtr0t12,,,nLLL{}iL{}iXzerofailuredataMartzWaller1979198919891.1.23Weibull1.1.3WeibullWeibull1939WeibullWeibullNelson[1],1972WeibullWhittemoreAltschuler[2],1976Peto[3],1972WeibullWeibull1.2WeibullMLEBayesWeibullGammaMLEBayesWeibull4WeibullWeibullMLEBayesMMLEipipBayesBayes522.1Weibull2.1.1WeibullWeibull()1exp{()},0xFxxβη=−−2.10β0ηβη(,)XWeiηβ∼1()()()exp{()}xxfxβββηηη−=⋅−2.21β=2.1.2WeibullWeibull()1exp{()},xFxxβγγη−=−−2.30β0η0γ≥γ(,;)XWeiηβγ∼1()()()exp{()}xxfxβββγγηηη−−−=⋅−2.40γ=Weibull1β=0γ=1β=2.1.3Weibull()1exp()tubFte−=−−2.51(,,)exp{},tubtuftubetbb−−=−−∞∞2.66()uu−∞∞(0)bb(,)TGumub∼(,)XWeiβη∼lnTX=Tlnuη=1bβ−=ub2.2Weibull2.2,βη2.6,ubWeibull,2.2.1X),(βηWei12,,nXXX1222111(1)21(1)niiniiEXXnEXXnηβηβ===Γ+==Γ+=∑∑2.7[24][41]Gamma)(⋅Γ2.2.1.1Gamma)(⋅Γ∫+∞−−=Τ01)(dxexsxs0sGamma1)(sΓ0s2)1(+Γs=s)(sΓsn+1!)1(nn=+Γ3)(sΓ)(sΓ=ss)1(+Γ01−s7)(sΓ-10)(sΓ0)(sΓ-2-1)(sΓ0)(sΓs=0-1-24xxxππsin)1()(=−ΓΓ10x4x,1+nxn10−nx)()()2)(1()1()1()11()(nxnxxxxxxx−Γ−−−==−Γ−=−+Γ=Γ)()2)(1()1()2)(1()3(1)2()1()1(xnxxnxxxxxxxx−−−++−Γ==−−+−Γ=+−+−Γ=+−Γ=−Γxnxnxnxxxnnnnππππsin)1()1()sin()1()1()()1()1()(−−=−−=++−Γ−Γ−=−ΓΓ=xππsinx2.82.2.1.2WeibullX),(βηWeiX2.1X),(βηWeiα)1(βαηαα+Γ=EXk≠βαk(1)EXαααηβ−−=Γ−X),(βηWei})(exp{1)(βηxxF−−=∫∫∞+∞+−−==00)()1()(βηαααxedxxdFxEXβη)(xz=βη1zx=dzezEXz−∞+∫=∴0βαααη=)1(0βαηηαβαα+Γ=−∞+∫dzezz8EXα−)(sΓs=0-1-22.1βα=(1)EXEXαβββηββη==Γ+=.2.2X),(βηWeiαk≠βαk222222)()()()(αααααααααπβ−−−−−⋅=EXEXEXEXEXEXEXEX2.9(1)EXαααηβ=Γ+2.102.1α)1(βαηαα+Γ=EXk≠βα)1(βαηαα−Γ=−−EX2.8αEXα−EX)1(βαηα+Γ=)1(βαηα−Γ−=)1()(βαβαβα−ΓΓ=sinαπβαπβααβαπβαπ−=EXEXsin2.111)21(22βαηαα+Γ=EXk≠βα)21(22βαηαα−Γ=−−EX2.8ααβαπβαπ2222sin−=EXEX2.12θθθcossin22sin=)sin1(sin42sin222θθθ−=2.112.124)(4222222=−ααβπαEXEX2222)(ααβπα−EXEX))(1(2222ααβπα−−EXEX9222222)()()()(αααααααααπβ−−−−−⋅=EXEXEXEXEXEXEXEX2.10(1)EXαααηβ=Γ+2.21=α21222122)()()()(−−−−⋅−⋅⋅⋅=EXEXEXEXEXEXEXEXπβ1(1)EXηβ=Γ+.2.12.2k≠βα)1(βαηαα−Γ=−−EX)21(22βαηαα−Γ=−−EXk≠βαk≠βα2kβα2kβαk112βα2βααβηβα=0.0010.050.12βα2.2.1.3ηβX),(βηWeinXXX,,21n∑==niiXnEX11αα∑=−−=niiXnEX11αα2.92.10βη)()2()()2(ˆ22αααααπβAAAA−=2.13ˆ()(1)Bααηαβ=Γ+2.1410α∑==niiXnB11)(αα∑∑==−=niniiiXnXnA1111)(ααα1α=1βη)1()2()1()2(ˆ22AAAA−=πβ1ˆ(1)(1)Bηβ=Γ+)1(B,)2(),1(AA2α=β2.1ββη=EX∑==niiXnEX11ββ∑==niiXn11ββη.ηβη2.132.142.3X),(βηWeinXXX,,21nαβα22.132.14βˆηˆβηnXXX,,,21nαiiXY=αiXnYYY,,,21∑=niiXn11→pEX∑=niiYn11→pEY∑=niiXn11α→pαEX)(+∞→nα−=iiXZ∑=niiZn11→pEZ∑=−niiXn11α→pα−EX)(+∞→n2.132.14ββ→pˆηη→pˆ)(+∞→n11βˆηˆβη2.2.1.4Montle-Carlonαββˆηηˆ10.00052.01.990433029.721220.0012.02.023773030.280930.012.02.010853030.125540.052.02.013503030.041950.12.02.022223030.210260.52.02.000893029.935171.02.02.030603030.056581.32.01.885773029.915091.52.01.478293028.1766101.72.01.201883025.2909110.00050.80.798513030.1713120.0010.80.807893029.5409130.010.80.789453029.8863140.050.80.80913302