16Kaplan-Meierestimator1,00,8iieiardean0,60,40,20,0801001201400204060TimetR(t)t=1.000031.7≤I<565656 53531.739.2=0.938ST39.257.5=0.875MT<57.565.8=0.813≤t<HEHEUEND70.0= 0.74565.8≤I<h70.0105.8= 0.677》t<3mi4105.8110.0=0.508≤T<NTNU-Trondheim5110.0=0.254≤1ONorwegian University ofScienceand Technologywww.ntnu.edu
16 Kaplan‐Meier estimator
17Nelson's estimatorThefailureratefunctionwasdefinedasdf(t)z(t) =InR(t)R(t)dtThecumulativefailureratefunctionisz(t) =z(u)du = -lnR(t)Anatural estimator of thecumulativefailure rateZ(t)is thendeductedfromtheKaplan-Meierestimator,R(t),aspj = -In 2(t) = -lnR(t) = -ln(1)+)niinjeJtCR*(t) = e-2(t)NTNU-Trondheim福Norwegian Lniversity ofScience and Technologywww.ntnu.edu
17 Nelson’s estimator The failure rate function was defined as ݖ ݐ ൌ ݂ሺݐሻ ܴሺݐሻ ൌ െ ݀݀ݐlnܴሺݐሻ The cumulative failure rate function is ܼ ݐ ൌනݖ ݑ݀ ݑ ൌ െlnܴሺݐሻ ௧ A natural estimator of the cumulative failure rate ܼሺݐሻ is then deducted from the Kaplan‐ Meier estimator, ܴ ሺݐሻ, as ܼ መ ݐ ൌ െlnܴ ̂ െlnෑ ൌ ݐ ୀ ൌ െlnෑ ݊ െ 1 ݊ ୀ ൌ െ ln 1 െ 1݊ ∈ ൌ ሺ 1݊ 12݊ଶ ⋯ሻ ∈ ൎ 1݊ ∈ ି݁ൌ ݐ ∗ܴ ሺ௧ሻ
18Nelson's estimatorKaplan-TimetoNelsonMeierR(tO)Nelson Estimate 2(t)jFailureR*(to)V=0.00001.0001.000151131.7=0.06250.9390.938122t+t39.2=0.12920.8790.875Nelson Estimator for the3357.5++=0.20060.8180.813CensoredDataSet in4Example,Compared with55++++++#the65.8=0.28390.7530.745Kaplan-MeierEstimator.6670.00.6771+1+.+1=0.37480.6877891011121313105.81+果=0.62480.5350.508141515110.0#+.+++2=1.12480.3200.254 r16www.ntnu.edu
18 Nelson’s estimator Nelson Estimator for the Censored Data Set in Example, Compared with the Kaplan‐Meier Estimator
19NelsonplotZ(t)Z(t)Z(t)A(c)(b)(a)TimetTimetTimetEstimated cumulativefailurerateZ(t)indicating(a)increasingfailure rate(IFR),(b)decreasingfailurerate(DFR),and (c)bathtub-shapedfailurerate.NTNU-Trondheim?NorwegianUniversityofScienceandTechnologywww.ntnu.edu
19 Nelson plot Estimated cumulative failure rate ܼ መ ݐ indicating (a) increasing failure rate (IFR), (b) decreasing failure rate (DFR), and (c) bathtub‐shaped failure rate
20Total time of test (TTT)Thetotaltimeontestattimet,TTT(t)isdefinedasTTT(t) =To) +(n-i)tThetotaltimeontestT(t)denotesthetotal observed lifetimeofthenitems.Weassumethat all the n items are put into operation at time t =O and that the observation isterminated at timet.In the time interval (O, tl, a number, i, of the items have failed.Thetotal functioningtimeoftheseiitems isZj=oTo.Theremainingn-iitems survivethetimeinterval (o,t).Thetotalfunctioningtimeofthesen-i items isthus (n-i)t.Ifweplotthepointsi TTT(T()'TTT(T(n))WeobtaintheTTTplotofthedatasetNTNU-TrondheimNorwegian University ofScienceand Technologywww.ntnu.edu
20 Total time of test (TTT) The total time on test at time t, ܶܶܶ ݐ is defined as ܶܶܶ ݐ ൌܶሺሻ ݊െ݅ ݐ ୀଵ The total time on test ܶ ݐ denotes the total observed lifetime of the ݊ items. We assume that all the ݊ items are put into operation at time t = 0 and that the observation is terminated at time ݐ .In the time interval (0, ݐ ,[a number, ݅, of the items have failed. The total functioning time of these ݅ items is ∑ ܶሺሻ ୀ . The remaining ݊ െ ݅ items survive the time interval (0, ݐ .[The total functioning time of these ݊ െ ݅ items is thus ሺ݊ െ ݅ሻݐ. If we plot the points ሺ ݅ ݊ , ܶܶܶሺܶ ሻ ܶܶܶሺܶ ሻሻ We obtain the TTT plot of the data set