Test and data analysisIn order to obtain information about a particular life distribution F(t) for an item(component/system),itisoftennecessarytocarryoutatestwherenidenticalitemsareactivated in order to record their lifetimes. The data set thus obtained is said to becomplete.Often we have to be satisfied with incomplete data sets.This may be because it isimpractical ortoo expensive to wait until all theitemshavefailed.Insuch situationsthedatasetissaidtobecensored.NTNU-TrondheimNorwegian UniversityofScienceand Technologywww.ntnu.edu
6 Test and data analysis In order to obtain information about a particular life distribution ܨሺݐሻ for an item (component/system), it is often necessary to carry out a test where ݊ identical items are activated in order to record their lifetimes. The data set thus obtained is said to be complete. Often we have to be satisfied with incomplete data sets. This may be because it is impractical or too expensive to wait until all the items have failed. In such situations the data set is said to be censored
Types of censoringCensoringof typelAll items are activated at timet=o andfollowed until failure or until timeto when theexperiment isterminated.CensoringoftypellAll items areactivated at timet =o, and the resultingdata set contains afixed numberrofobserved lifetimesandfurthermorewanttoterminatethetestasfastaspossibleCensoringof type IllAll items are activated at time t =O, and the test terminates at the time that occursfirst,toortherthfailure.CensoringoftypeIVn numbered identical itemsare activated at differentgivenpoint(s)in time.Censoringtimeisstochastic.NTNU-TrondheimNorwegian University of莎Science and Technologywww.ntnu.edu
7 Types of censoring • Censoring of type I All items are activated at time t =0 and followed until failure or until time ݐ when the experiment is terminated. • Censoring of type II All items are activated at time t =0, and the resulting data set contains a fixed number r of observed lifetimes and furthermore want to terminate the test as fast as possible • Censoring of type III All items are activated at time t =0, and the test terminates at the time that occurs first, ݐ or the rth failure. • Censoring of type IV n numbered identical items are activated at different given point(s) in time. Censoring time is stochastic
8Empiricaldistributiond(t)ationoftestingd(n(a)d(t)(b)NTNU-TrondheimNorwegian University ofScience and Technologywww.ntnu.edu
8 Empirical distribution
0NonparametricmethodsWedo not makeanyassumptionsabout the lifedistribution F(t)T;-potential lifecyleti-the observed value of T;ThesamplemeanSm1-1Thesamplevariancet?(t)2tPNTNU- Trondheim?NorwegianLniversityofScience and Technologywww.ntnu.edu
9 Nonparametric methods We do not make any assumptions about the life distribution F(t) ܶ െpotential lifecyle ݐ െthe observed value of ܶ The sample mean ݐ ̅ ൌ 1 ݐ ݊ ୀଵ The sample variance ݏଶ ൌ 1 ݊െ1ሺݐ െ ݐ̅ሻଶൌ 1 ݊െ1 ሺ ݐଶ ୀଵ െ 1 ݐ ሺ݊ ୀଵ ሻଶሻ ୀଵ
10EmpiricaldistributionLetF(t)denotethe lifedistributionfora certaintypeof items.Theempirical distributionfunctionisdefinedasNumberoflifetimes≤tFn(t)=nIf we assume that there are no ties in the data set, the empirical distribution function maybewrittenfort<t(1)Fn(t) :for t() ≤t<t(i+1);i = 1,2, .,(n - 1)n1fort(n) ≤tNTNU -Trondheim福NorwegianUniversityofScienceand Technologywww.ntnu.edu
10 Empirical distribution Let ܨሺݐሻ denote the life distribution for a certain type of items. The empirical distribution function is defined as ൌ ݐ ܨ Number of lifetimes ݐ ݊ If we assume that there are no ties in the data set, the empirical distribution function may be written ൌ ݐ ܨ 0 for ݐ ൏ ݐሺଵሻ ݅ ݊ for ݐሺሻ ݐ൏ݐ ାଵ ; ݅ ൌ 1, 2, . , ሺ݊ െ 1ሻ 1 for ݐሺሻ ݐ t