11EmpiricaldistributionExampleSuppose that n=16 independent lifetimes (given in months)have been observed:31.739.257.587.788.394.265.0101.765.8105.870.0109.275.0110.075.2130.0Thesamplemeantofthelifetimesis16~81.64ti16=1NTNU-Trondheim莎Norwegian University ofScience and Technologywww.ntnu.edu
11 Empirical distribution Example Suppose that n = 16 independent lifetimes (given in months) have been observed: 31.7 39.2 57.5 87.7 88.3 94.2 65.0 101.7 65.8 105.8 70.0 109.2 75.0 110.0 75.2 130.0 The sample mean ݐ ̅ of the lifetimes is ݐ ̅ ൌ 1 ݐ 16 ଵ ୀଵ ൎ 81.64
12Empiricaldistribution1,00,80,6EF0,40,20,0020406080100120140TimetTheempiricaldistributionfunctionF,(t)forthedata in Example.NTNU-Trondheim?Norwegian University ofScience and Technologywww.ntnu.edu
12 Empirical distribution The empirical distribution function ܨሺݐሻ for the data in Example
13Kaplan-MeierestimatorThennumbered itemsareactivatedattimet=0>thecensoringtimeforitemi,C,isstochastic>the lifetime Ti,that would be observed if item i is not exposed to censoringItisonlypossibletorecordthesmallerof T,andC,foritemIY; = min[Ti, C)and theindicatorsifTi≤Ci(Failure)1ifT,>C(Censoring)0NTNU-TrondheimNorwegian University of梦ScienceandTechnologywww.ntnu.edu
13 Kaplan‐Meier estimator The ݊ numbered items are activated at time t =0 the censoring time for item ݅, ܥ is stochastic the lifetime ܶ, that would be observed if item ݅ is not exposed to censoring It is only possible to record the smaller of ܶ and ܥ for item I ܻ ൌ min ሼܶ, ܥሽ and the indicators ߜ ൌ ቊ 1 if ܶ ܥ ሺFailureሻ 0 if ܶ ܥ ሺCensoringሻ
14Kaplan-MeierestimatorKaplan and Meier suggested the following estimation procedure: Fix t > 0. Let t(i)<t(2)<..< t(n) denote the recorded functioning times, either until failure or to censoring, orderedaccording to size. Let Jt denote the set of all indices j where to)≤ t and t) represents afailuretime.Let n, denote the number of items functioning and in observation immediatelybefore timet(),j=1,2,..,n.Then theKaplan-MeierestimatorofR(t)isdefinedasfollows:R(t) =n-1njjeltThe Kaplan-Meier estimatorR (t)can be derived as a nonparametric maximum likelihoodestimator(MLE).NTNU-TrondheimNorwegianUniversityof莎ScienceandTechnologywww.ntnu.edu
14 Kaplan‐Meier estimator Kaplan and Meier suggested the following estimation procedure: Fix ݐ < 0. Let ݐሺଵሻ< ݐሺଶሻ < . < ݐሺሻ denote the recorded functioning times, either until failure or to censoring, ordered according to size. Let ܬ௧ denote the set of all indices j where ݐሺሻ ݐ and ݐሺሻ represents a failure time. Let ݊ denote the number of items functioning and in observation immediately before time ݐሺሻ, j = 1,2, . , n. Then the Kaplan‐Meier estimator of R(t) is defined as follows: ܴ ݐ ൌෑ݊ െ 1 ݊ ∈ The Kaplan‐Meier estimator ܴ ሺݐሻ can be derived as a nonparametric maximum likelihood estimator (MLE)
15Kaplan-MeierestimatorRankInverserankOrdered Failure andR(to)pjj(n-j+ 1)CensoringTimestu)Example011.000--475,PageSystem56433411631.70.938ReliabilityTheory,byRausandM21539.20.87531457.50.813141365.0*0.813H2O51265.80.74561170.00.677711075.0*0.677Censoringtimes89175.2*0.677are starred (*)19887.5*0.677788.3*1100.67711694.2*10.6775112101.7*0.677mi4134105.80.5083109.2*1140.508122Irondheim150.254110.0University of11130.0*160.254d Technologywww.ntnu.edu
15 Kaplan‐Meier estimator Example Page 475, System Reliability Theory, by Rausand M Censoring times are starred (*)