97 pages

Regression and degradation models in reliability theory and survival analysis ; Regresiniai ir degradaciniai modeliai patikimumo teorijoje ir išgyvenamumo analizėje

Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

VILNIUS UNIVERSITYInga Masiulaityte˙REGRESSION AND DEGRADATION MODELS IN RELIABILITYTHEORY AND SURVIVAL ANALYSISDoctoral dissertationPhysical sciences, mathematics (01P)Vilnius, 2010The scientific work was carried out in 2005–2009 at Vilnius University.Scientific supervisor:Prof. Dr. Vilijandas Bagdonaviˇcius (Vilnius University, Physical sciences, Mathe-matics - 01P)NotationF(t) Cumulative distribution function;f(t) Probability density function;S(t) Survival function;n Sample size;T Failure timeix Explanatory variable;λ(t) Hazard rate function;Λ(t) Cumulative hazard rate functionT ,...,T Failuretimesofn unitstestedin”hot”11 1n 11conditions;T ,...,T Failure times of n units tested in21 2n 22”warm” conditions;μ Mean failure time;2σ Variance;r Scale parameter;H Hypothesis;0α,β,ν,μ Parameters;K (t) Cumulative distribution function of re-jdundant system;k (t) Probability density function of redun-jdant system;(K (t),K (t)) Confidence interval;jjI Fisher information matrix;−1I Inverse of the Fisher information ma-trix;L(r,θ) Likelihood function;l(r,θ) Loglikelihood function;Z(t) Degradation process;z Critical level;0(0)T Moment of the non-traumatic failure;(k)Tt of the traumatic failure of thekth mode;(0)S Survival function of the non-traumaticfailure;3(k)S Survival function of the traumatic fail-ure of the kth mode;(k)˜λ (t|Z) Conditional failure rate;P Probability;C Censoring time of the ith unit .

Sujets

Informations

Publié par
Publié le 01 janvier 2010
Nombre de lectures 53
Poids de l'ouvrage 1 Mo

UNIVERSITYVILNIUS

IngaMasiulaityte˙

REGRESSIONTHEANODRDYEGRANDADSUARVTIONIVALMODEANALLSYSISINRELIABILITY

PhysicalDoscctoiences,ralmadissertatithematicson

0012Vilnius,

)P(01

The

tificscien

orkw

was

Scientificsupervisor:

sVilijandaDr.Prof.

icsmat

-

1P)0

diecarr

out

in

Bagdonaviˇcius

09–200520

(Vilnius

at

Vilnius

University,

.yersitUniv

Phlicasy

iences,sc

-heMat

ontitaNo

fF((tt))
)t(SnTix)t(λ)tΛ(T11,...,T1n1
T21,...,T2n2
µ2σrH0α,β,ν,µ
)t(Kj)t(kj(Kj(t),Kj(t))
I1−IL(r,θ)
lZ((rt,)θ)
z0(0)T)k(T)(0S

Cumulativedistributionfunction;
Probabilitydensityfunction;
Survivalfunction;
FSampleailuresiztimee;
HazaExplanardtorrateyvafunctriable;ion;
FCumailureulativtimesehaofzna1rdunitsratetestedfunctionin”hot”
ns;conditioFailuretimesofn2unitstestedin
ions;conditrm”a”wVMeanariafncae;iluretime;
HypScaleotheparasis;meter;
CParumulativameters;edistributionfunctionofre-
ystem;stdundanProbabilitydensityfunctionofredun-
system;tdanCoFishernfidencinfoeinrmattervional;matrix;
InverseoftheFisherinformationma-
;trixLoLikglikelihoelihoododffunctiounction;n;
Degradationprocess;
el;levCriticalMomentofthenon-traumaticfailure;
kthMomenmotde;ofthetraumaticfailureofthe
Survivalfunctionofthenon-traumatic
failure;

3

()kS

)k(˜λPCi

(t|Z)

Survivalfunctionofthetraumatic
ureofthekthmode;

ProbaConditbilitionay;lfailurerate;
Censoringtimeoftheith

of

4

the

ith

unit

.

fail-

Contents

1Acceleratedlifemodels
1.1Introduction................................
1.2GeneralizedSedyakin’smodel......................
1.2.1Definitionofthemodel......................
1.2.2GSmodelforstep-stresses....................
1.3Acceleratedfailuretimemodel......................
1.3.1Definitionofthemodelforconstantstresses..........
1.3.2Definitionofthemodelfortime-varyingstresses........
1.3.3Relationsbetweenthemeansandthequantiles........
1.4Proportionalhazardsmodel.......................
1.4.1Definitionofthemodelforconstantstresses..........
1.4.2Definitionofthemodelfortime-varyingstresses.......
1.5Wienerprocess..............................
1.6Wienerprocesswithdrift.........................
1.7Gammaprocess..............................
2Statisticalanalysisofredundantsystems
2.1Redundantsystemwithonemainandonestand-byunit.......
2.1.1Themodels............................
∗2.1.2Goodness-of-fittestforthehypothesisH...........
02.1.3Goodness-of-fittestforthehypothesisH...........
02.1.4Simulations:powerofthetests.................
2.2Redundantsystemwithonemainand(m−1)stand-byunits....
2.2.1Nonparametricestimation....................
2.2.2Parametricestimation......................
ˆ2.3AsymptoticdistributionofKandconfidenceintervalsforK(t)..
jj2.3.1Nonparametriccase........................
2.3.2Parametriccase..........................
3Failure-TimeDegradationModels
3.1FailureDegradationModelwithcovariates...............
3.2Estimationofmodelparameters.....................
3.2.1Thedata..............................
3.2.2Likelihoodfunctionconstruction.................
3.2.3Example1:Timescaledgammaprocess............

5

411418181820202122232324242425622627293437434448484860677680808182

A

tInroduciotn3.2.4Example2:Shockprocesses..
3.23.2.5.6EMxodifieampled3:loPglikathelihomooddels.........
3.3Estimationofreliabilitycharacteristics

dmethoaDelt

Bibliography

6

....

....

....

....

....

....

....

....

....

....

....

....

....

4............

79

....

84878889

29

Introduction

Towarranthighreliabilityofkeycomponentsofreliabilitysystems,stand-byunits
areused.Ifanycomponentfailsthenastand-byunitoperatesinsteadofthefailed
t.nneocompIfthestand-byunitsarefunctioninginthesame”hot”conditionsasthemain
Butunit”hotthen”usuallyredundancyafterswhasitchingdisadvtheantagesreliabilitbyecauseoftheanyofstand-bstandy-bunitsydounitsesnotfailscehaange.rlier
thanthemainonewiththeprobability0.5.
Ifthestand-byunitsarenotoperatinguntilthefailureofthemainunit(”cold”
reserving),itispossiblethatduringandaftercommutingthefailurerateincreases
becused:ausestathend-stbyand-bunitsyfunitunctioisnnot”wunderalormed”werenostressugh.thaSon”wthearm”mainoreservingne.Inissuchasometimescase
theunitandprobaitbilitisyalsoofptheossiblefailurethaofttheswitcstahingnd-bisyfluenunitt,isi.e.smallerswitcthahingnthatfromof”wthearm”mainto
”hot”conditionsdoesnotdoanydamagetounits.
Thedefinitionof”fluentswitching”asstatisticalhypothesisontheconditional
survivdistributialonregreoftssionhefamoiluredelsstimeuchofastheSedysystemakin’saafterndathecswceleraitchtedisgivfailuren.eWtimeellkno(AFT)wn
d.usearedelmoposed.GooAsdness-yomptotf-fiticteprostspfoertriesobtofainedproposedredundantesttstasystemstisticsareinreliabilitvestyigamoted.delsarepro-
Parametricandnon-parametricestimationproceduresforthereliabilityofsuch
systemsFailuresareofgivhigen.hlyPropreliableertiesounitfstheareprorapore.sedOneparwayameterofesobtatimatoiningrsaacreoomplemenbtained.tary
expreliabierimenlitytalinformfactoratios,nishencetodotoobtaacceleratinefailurdelifesquictestingkly.(ALAnotT),heri.e.watyoofuseobtahigherininglevelcom-of
plementaryreliabilityinformationistomeasuresomeparameterswhichcharacterize
theagingordegradationoftheproductintime.
StatisticalinferencefromALTispossibleiffailuretimeregressionmodelsrelating
failureinfluencingtimethedistributioreliabilitnyarwithewellexternaclhosen.explaStatnatoistryicavlainferiablesrence(covfromariates,failursterestimeses)-
timedegradisdattionributiondatanotwithcoonlyvariatwithesexneeternaldsevenbutamolsorewithcomplicatinternedalmoexpladelsnatorelarytingvariablesfailure
case(degrmoadadelstion,forweardegra)whicdatiohnexproplaincessthedistributiostateofnaunitresbeneeded,forettheoo.failures.Inthelast
Hence,theseconddirectionoftheworkismodellingandstatisticalestimationof

7

thereliabilityofsystemsorunitsinthecasewhenjointfailuretimeanddegradation
regressiondataareavailable.
Themodifiedmaximumlikelihoodmethodforestimationoffailureprocessand
degradationprocessparametersusingsimultaneousdegradationandmulti-modefail-
uretimeregressiondataisintroduced.
Estimatorsofvariousreliabilitycharacteristicsoftheunitsrelatedtotraumatic
andnon-traumaticfailuresaregiven.
Exampleswhenthedegradationprocessismodelledbytimescaledgammapro-
cess,pathprocesses,shockprocesseswiththenumberofshocksmodelledbynon-
homogenousPoissonprocessareconsidered.

yitualActTherearemanypublicationsonprobabilisticmodellingofredundantsystemsrelia-
bilitygiventhereliabilityofthesystemcomponents.Applyingoftheseresultsinreal
analysisofsystemreliabilityispossibleiftheprobabilitydistributionofthecompo-
nenreliabitsislityknoandwn.theSoapropveryertiesactuaoflthpreoblemestimatoisrstheeusingstimatioestimanotorsfthethereliaredundanbilittyofsystemthe
ts.nneocompMethodsofacceleratedlifetestinganddegradationprocessanalysisseparately
arewelldevelopedbutjointmodellingandstatisticalanalysisofsimultaneousfailure
time-degradationdatawithcovariatesisveryrecentresearchdirection.Thelast
internationalconferences”Mathematicalmethodsinreliability”(2005,2007,2009)
showincreasinginterestinthisdirection.

blemsproandmsAiThemainproblemsarethefollowing:
1.toformulatemathematicaldefinitionofstand-byunitfluentswitchingfrom
”warm”to”hot”conditions;
2.toconstructtestsforgeneral”fluentswitchinghypothesis”formulatedusing
Sedyakin’s”reliabilityprinciple”andforparticularfluentswitchinghypothesisfor-
mulatedusingacceleratedfailuretimemodel;
3.toinvestigateasymptoticpropertiesoftheteststatistics;
4.toconstructparametricandnonparametricestimatorsofthecumulativedistribu-
tionfunctionofredundantsystemusingreliabilitydataofcomponentstestedunder
es;stresstdifferen5.toinvestigateasymptoticpropertiesoftheparametricandnonparametricestima-
rs;to6.toconstructasymptoticconfidentialintervalsforcumulativedistributionfunction
system;tredundanof7.toinvestigatefinitesamplepropertiesoftheparametricandnonparametricesti-
matorsbysimulation;

8

8.toformulategeneralsimultaneousfailuretimeanddegradationregressiondata
dels;mo9.dattoionmoprodifycessmapaximrametumelikrselihousingodsimmethoultadfoneousrestimatdegraiondatiooffnaandiluremproculti-moessanddedegfailurera-
10time.toinvregressionestigatedatathestusingructurepredictoofmorsofdifieddegrlikadatelihoionodprofunctioncesses;whenthedegradation
prowithcesstheisnmoumbdeelrleodfbshoyckstimemoscaleddelledgambymanon-prohomogcess,enopausthPprooissocessnes,proscehocss.kprocesses

dsMethoCountingprocesstechniques,deltamethod,parametricandnon-parametricestima-
tionmethods,limittheoremsforthesequencesofrandomvariablesandstochastic
processes,numericandsimulationmethodswereused.

Novelty
Allresultsofthethesisarenew.

Statementspresentedforthedefence
1.Mathematicaldefinitionofstand-byunitfluentswitchingfrom”warm”to”hot”
conditionsisformulated.
2.Goodness-of-fittestforageneral”fluentswitchinghypothesis”basedonSedyakin’s
constructed.isprinciple3.Goodness-of-fittestfora”fluentswitchinghypothesis”basedonacceleratedfail-
uretimemodelisconstructed.
4.Asymptoticpropertiesofthetwoteststatisticsareinvestigated;
5.Parametricandnonparametricestimatorsofthecumulativedistributionfunction
ofredundantsystemusingreliabilitydataofcomponentstestedin”hot”and”warm”
conditionsareconstructed;
6.Asymptoticpropertiesoftheparametricandnonparametricestimatorsareinves-
ated;tig7.Asymptoticconfidenceintervalsforcumulativedistributionfunctionofredundant
nstructed.coreasystem8.Finitesamplepropertiesoftheparametricandnonparametricestimatorsarein-
vestigatedbysimulation.
9.Generalsimultaneousfailuretimeanddegradationregressiondatamodelsare
ulated.form10.Maximumlikelihoodmethodforestimationoffailureprocessanddegradation
processparametersusingsimultaneousdegradationandmulti-modefailuretimere-
gressiondataismodifiedusingpredictorsofdegradationprocesses.
11.Thestructureofmodifiedlikelihoodfunctionwhenthedegradationprocessis

9

modelledbytimescaledgammaprocess,pathprocessesandshockprocessesisinves-
ated.tig

Historyoftheproblem
Traditionallifedataanalysisinvolvesanalyzingtimes-to-failuredata(ofaproduct,
ofasystemorcomponent)obtainedundernormaloperatingconditionsinorderto
quantifythelifecharacteristicsoftheproduct,systemorcomponent.Failuresof
highlyreliableunitsarerare,forexample,thelifetimeofsemiconductorsisvery
long,andtotestdevicesunderusualconditionswouldrequirefartoomuchtesttime
andexcessivelylargesamplesize.Sootherinformationshouldbeusedinaddition
tofailure-timedata,whichcouldbecensored.
Onewayofobtainingacomplementaryreliabilityinformationistousehigher
levelofexperimentalfactors,stressesorcovariates(suchastemperature,voltageor
pressure)toincreasethenumberoffailuresand,hencetoobtainreliabilityinforma-
tionquickly.Thisprocedureprovidesthemethodsknowntodayastheaccelerated
lifetesting(ALT).Thesemethodsweredevelopedbymanyresearchers,seewidesur-
veysofthemodelsandmethodsinSingpurwalla(alsocanbefoundin[6],[26],[28],
[32],[37],[40]).
Thefirstpartofthisworkisthefirstattemptinthescientificliteraturetoapply
knownmodelsofALTtothestatisticalanalysisofredundantsystemswithstand-by
unitsin”warm”operatingconditions.Stand-byunitsoperatein”warm”conditions
whenthemainunitfunctionsandin”hot”conditionsaftertheirswitchonafterthe
failureofthemainunit,sothemodelsofALTcanbeapplied.Theproblemisthat
inALTthemomentsofstresslevelchangeisusuallyplannedandfixedbeforethe
experiment.Intheanalysisofredundantsystemsthemomentsofstresschangeare
randomandarerelatedwiththefailureofthemainunit.Sospecialgoodness-of-
fittestsfortheredundantsystemmodelsareneeded.Anotherparticularityisthe
following.InALTidenticunitsaretestedinvariousstressconditionsandinference
aboutthereliabilityoftheseunitsinusualstressconditionsisdone.Inthecaseof
redundantsystemscomponentsofthesystemfunctionindifferentstressconditions
andtheinferenceaboutthereliabilityofthesystem,notofthecomponentsmustbe
done.DifferentlyfromALTanotherwayofobtainingcomplementaryreliabilityinfor-
mationistomeasuresomeparameterswhichcharacterizetheagingorwearofthe
productintime.Inanalysisoflongevityofhighlyreliablecomplexindustrialor
biologicalsystems,thedegradationprocessesprovideanimportantadditionalinfor-
mationabouttheaging,degradationanddeteriorationofsystems,andfromthis
pointofviewthesedegradationdataarereallyaveryrichsourceofreliabilityinfor-
mationandoftenoffermanyadvantagesoverfailuretimedata.Degradationisthe
naturalresponseforsometests,anditisnaturalalsothatwithdegradationdatait
ispossibletomakeusefulreliabilityandstatisticalinference,evenwithnoobserved
failure.StatisticalinferencefromALTispossibleiffailuretimeregressionmodelsrelating

10

failuretimedistributionwithexternalexplanatoryvariables(covariates,stresses)
influencingthereliabilityarewellchosen.Statisticalinferencefromfailuretime-
degradationdatawithcovariatesneedsevenmorecomplicatedmodelsrelatingfailure
timedistributionnotonlywithexternalbutalsowithinternalexplanatoryvariables
(degradation,wear)whichexplainthestateofunitsbeforethefailures.Inthelast
casemodelsfordegradationprocessdistributionareneeded,too.
Themostappliedstochasticprocessesusedasdegradationmodelsaregeneralpath
models([7],[8],[27],[29],[30])andtimescaledstochasticprocesseswithstationary
andindependentincrementssuchasthegammaprocess([5],[21],[34]),compound
Poissonprocess([14],[18],[19],[20],[42],[43]),andWienerprocesswithdrift([11],
[12],[13],[44],[45],[46],[22],[23],[33]).Harlamov[17]discussesinversegamma-
processasawearmodel.Zacks[49]discussesgeneralcompoundrenewaldamage
s.cesseproIfjointfailuretimeanddegradationdataareavailablemodelsandestimation
methodsforanalysisofsuchdataareneeded.Excellentintroductionstofailure
time-degradationmodelsisgivenbySingpurwalla[38],YashinandManton[48].More
recentdevelopmentscanbefoundinBagdonaviˇciusandNikulin[3],Finkelstein[15],
Lehmann[24],Yashin[47].Methodsofestimationfromfailuretime-degradation
datamaybefoundinBagdonaviˇciusandNikulin[5],[6],Lehmann[22],[24],[25],
Bagdonaviˇciusetal[7],[8],LawlessandCrowder[21],Couallier[9].
Theseconddirectionofthisworkistoformulategeneralsimultaneousfailuretime
anddegradationregressiondatamodelsandgivemethodsofestimation.

ntioobaApprthSimTheularesultstion,St.ofPtheetersburgthesis,wRereussia,presen28tedJunea-t4thJulye,6200St.9;atPtheetersburgSecondWInterorkshonatiopnalon
Conference,ALT2008,Bordeaux,France,9–11June2008;attheMathematical
theMethocodsnferencinReseliaofLbilitithyuanianConferenceMa,thematicaGlasgolw,SocScotlaiety(nd,2001–48,20July09).2007;aswellasat

11

Principalpublications

Themainresultsofthethesisarepublishedinthefollowingpapers:

1.V.Bagdonaviˇcius,I.Masiulaityte˙,M.Nikulin,StatisticalAnalysisofRedundant
ReliaSystembilitwyithandoneQualitStand-byofyLife,Unit.(eds.InCMa.Hubthemater,Nic.Lalimnios,MethodsM.MesinbaSurvivh,alM.Nikulin)Analysis,,
ISTE&Wiley:London2008,p.179-189.

2.V.Bagdonaviˇcius,I.Masiulaityt˙e,M.Nikulin,Statisticalanalysisofredundant
systemswith’warm’stand-byunits.StochasticsAnInternationalJournalofProba-
bilityandStochasticProcesses,Volume80,Issue2and3,2008,p.115-128.

3.V.Bagdonavicˇius,I.Masiulaityt˙e,M.Nikulin,Statisticalanalysisofreliabilityof
aredundantsystemwithoneoperatingunitandonestand-byunitinwarmoperating
state.ProceedingofALT,2008.

4.V.Bagdonaviˇcius,I.Masiulaityt˙e,M.Nikulin,Asymptoticpropertiesofredundant
systemsreliabilityestimators.In:AdvancesinDegradationModellingApplications
toReliability,SurvivalAnalysis,andFinance.M.S.Limnios,N.Balakrishnan,N.;
Kahle,W.;Huber-Carol,C.(Editors).Birkhauser,2010,p.293-310ISBN:978-0-
8176-4923-4.

5.V.Bagdonaviˇcius,I.Masiulaityt˙e,M.Nikulin,Reliabilityestimationfromfailure-
degradationdatawithcovariates.In:AdvancesinDegradationModellingApplica-
tionstoReliability,SurvivalAnalysis,andFinance.M.S.Limnios,N.Balakrishnan,
N.;Kahle,W.;Huber-Carol,C.(Editors).Birkhauser,2010,p.275-292ISBN:978-0-
8176-4923-4.

12

Acknowledgments

Myfirst,andmostearnest,acknowledgementmustgotomyscientificsupervisor
ProfessorProfessorforV.hisBagconstadonavinˇtciusat.tenIwtion,ouldcarelikeandtobelief.expressMandeeypthanandksasinclsoeretoPgrratitofessoudertM.o
Nikulin,ProfessormeG.mKbersulldoofrff,thedoc.D.DepartmenKratpaofvickaitMathemae˙forticaltheirStasupptisticsortofduringVilniusthepUniverioersitdofy,
mydoctoralstudies.
andIoweunderstamylondingvingitwthankouldstohamveybpaeenrentsimpandossiblefriends.formetoWithoutfinishtheirthisewncoork.uragement

StructureoftheThesis

theThenotathesistioncoofnsisthetsfromfunctioinntroisductiopresenn,tedthreeinthechaptthesersis.Vandolumeconcoflusiowonsrk.isIn97additpages.ion,
IntheChapter1themainissuesandresultswhichotherauthorsanalyzedthat
themearepresented.
IntheChapter2redundantsystemwithonemainunitandm−1stand-by
eralunitsstopand-beraytingunitin””wfluenarm”tcoswitcnditionshingharypeaothesis”nalysed.basGedooondnessSedya-of-fitkin’stestsfprincipleorageandn-
forstructed.”fluentParaswitcmetrichingahndypnonpaothesis”rambaestedricoesntimatoaccelersratedandpropfailureertiestimeofmothedelestimatarecon-ors
ofthecumulativedistributionfunctionofredundantsystemusingreliabilitydata
ofconfidencomptoialnenintstervtestalsedforin”cumhot”ulatandive”diswtarm”ributioconnditiofunctionsanreofrepresendundated.ntsystemAsymptoticare
constructedandinvestigatedbysimulation.
dataInmothedelsChaptareerprese3gneneted.ralsimMaximultaumneouslikelihofailurodetimemethodandfordegestradaimatiotionnreofgressfailureion
processanddegradationprocessparametersisgiven.

13

C1hapter

Acceleratedlifemodels

1.1Introduction
Acceleratedlifemodelsrelatethelifetimedistributiontotheexplanatoryvariable
(stress,covariate,regressor).Thisdistributioncanbedefinedbythesurvival,cu-
mulativedistributionorprobabilitydensityfunctions.Nevertheless,thesenseof
acceleratedlifemodelsisbestseeniftheyareformulatedintermsofhazardrate
function.Supposeatfirstthattheexplanatoryvariableisadeterministictimefunction:
x(∙)=(x1(∙),...,xm(∙))T:[0,∞)→B∈Rm.
Ifx(∙)isconstantintime,weshallwritexinsteadofx(∙)inallformulas.
DenoteinformallybyTx(∙)thefailure-timeunderx(∙)andby
Sx(∙)(t)=P{Tx(∙)>t},Fx(∙)(t)=P{Tx(∙)≤t},fx(∙)(t)=−Sx(∙)(t),
thesurvival,cumulativedistributionandprobabilitydensityfunction,respectively.
Thehazard(rate)functionunderx(∙)is
1Sx(∙)(t)
λx(∙)(t)=h↓lim0hP{Tx(∙)∈[t,t+h)|Tx(∙)≥t}=−Sx(∙)(t).

Denotebyt
Λx(∙)(t)=λx(∙)(u)du=−ln{Sx(∙)(t)}
0thecumulativehazard(function)underx(∙).
Eachspecifiedacceleratedlifemodelrelatesthehazardrate(orotherfunction)
totheexplanatoryvariableinsomeparticularway.
IftheexplanatoryvariableisastochasticprocessX(t),t≥0,andTX(∙)isthe
failure-timeunderX(∙),thendenoteby
Sx(∙)(t)=P{TX(∙)>t|X(s)=x(s),0≤s≤t},

14

λx(∙)(t)=−Sx(∙)(t)/Sx(∙)(t),Λx(∙)(t)=−ln{Sx(∙)(t)}
theconditionalsurvival,hazardrateandcumulativehazardratefunctions.Inthis
casethedefinitionsofmodelsshouldbeunderstoodintermsoftheseconditional
functions.Tobeshortweshallusethewordthestressfortheexplanatoryvariableinthis
er.haptcSupposethatF1(t)andF2(t)arethec.d.f.ofunitsfunctionningunderaccelerated
andusualstresses,respectively.
Wesupposethatforallpositivetthefollowinginequalityisvalid:
F2(t)<F1(t),(1.1)
andwecanfindsomefunctiong(t)whichsatisfiestheequation:
F2(t)=F1(g(t)).(1.2)
Belowthedifferentexpressionoffunctiong(t)isgiven.Thec.d.f.F1andF2are
fromthesamedistributionfamily.
Example1.1.1.Exponentialdistribution
Supposetwocumulativedistributionfunctionofexponentialdistributionhave
form:wingfolloF1(t)=1−e−λ1t,F2(t)=1−e−λ2t.

Graph1.1.Cumulativedistributionfunction
Thefunctiong(t)whichsatisfied(1.2)hasthefollowingform:
g(t)=λ2t.(1.3)
λ1Example1.1.2.WeibulldistributionSupposetwocumulativedistribution
functionofweibulldistributionhavefollowingform:
tα1tα2
F1(t)=1−exp−β1,F1(t)=1−exp−β2.
15

Thefunctiong(t)Graphwhich1.2sat.isCfiedum(1ulativ.2)ehasthedistributfolloionwingfunctform:ion

/α1g(t)=β1αα11tα2/α1.(1.4)
2β2βIfα1=α2=1then(1.4)hasthefollowingform:g(t)=ββ21t.Ifα1=1andα2=2
then(1.4)hasthefollowingform:g(t)=β221t2.Ifα1=α2=2then(1.4)hasthe
followingform:g(t)=ββ21t.
Example1.1.3.LoglogisticdistributionSupposetwocumulativedistribution
functionofloglogisticdistributionhavefollowingform:
F1(t)=1tν1,F2(t)=1tν2.
1+µ11+µ2

Graph1.3.Cumulativedistributionfunction

16

.5)(1

Thefunctiong(t)whichsatisfied(1.2)hasthefollowingform:
tν2/ν1
g(t)=µ1µ.(1.5)
2Ifν1=ν2=1then(1.5)equationis
g(t)=µ1t.
µ2Intheexamplesthefunctiong(t)islinearfunction,whichingeneralcanbe
defined:g(t)=r∙t.(1.6)
Ifthefunctiong(t)hastheform(1.5)thenobtainF2(t)=F1(r∙t).
Thekthmomentsare:∞
α1(k)=tkdF1(t),(1.7)
0∞∞∞
α2(k)=tkdF2(t)=tkdF1(g(t))=tkdF1(r∙t),(1.8)
000andthecentralmomentsare:
∞µ1(k)=(t−α1(1))kdF1(t),(1.9)
0∞∞∞
µ2(k)=(t−α2(1))kdF2(t)=(t−α2(1))kdF1(g(t))=(t−α2(1))kdF1(r∙t).(1.10)
000Usingpropertyofthemomentµ(k)(r∙t)=rkµ(k)(t)weobtain:
α1(k)µ1(k)kg(t)k
α(k)=µ(k)=r=t.(1.11)
22Inthischapterwedescribesimplestacceleratedlifemodelsgiventheliterature.

17

1.2GeneralizedSedyakin’smodel
1.2.1Definitionofthemodel
Acceleratedlifemodelscouldbeatfirstformulatedforconstantexplanatoryvari-
ables.Nevertheless,beforeformulatingthem,letusconsiderageneralmethodfor
generalizingsuchmodelstothecaseoftime-varyingstresses.
In1966N.M.Sedyakin[36]formulatedthephysicalprincipleinreliability.The
ideaisthefollowing.Fortwoidenticalpopulationsofunitsfunctioningunderdifferent
stressesx1andx2,twomomentst1andt2areequivalentiftheprobabilitiesofsurvival
untilthesemomentsareequal:
P{Tx1>t1}=Sx1(t1)=Sx2(t2)=P{Tx2>t2}.
Ifaftertheseequivalentmomentstheunitsofbothgroupsareobservedunderthe
samethestressx2,i.e.thefirstpopulationisobservedunderthestep-thestress
x(τ)=x1,0≤τ<t1,
x2,τ≥t1,
andthesecondalltimeundertheconstantstressx2,thenforalls>0
λx(∙)(t1+s)=λx2(t2+s).
UsingtheideaofSedyakin,Bagdonaviˇcius[1]generalizedthemodeltothecaseof
anytime-varyingstressesbysupposingthatthehazardrateλx(∙)(t)atanymoment
tisafunctionofthevalueofthestressatthismomentandoftheprobabilityof
survivaluntilthismoment.Itisformalizedbythefollowingdefinition.
Definition1.2.1ThegeneralizedSedyakin’s(GS)modelholdsonasetofstresses
EifthereexistsapositiveonE×R+functiongsuchthatforallx(∙)∈E
λx(∙)(t)=gx(t),Sx(∙)(t).(1.12)
Equivalently,themodelcanbewrittenintheform
λx(∙)(t)=g1x(t),Λx(∙)(t).(1.13)
withg1(x,s)=g(x,exp{−s}).
1.2.2GSmodelforstep-stresses
Themostlyusedtime-varyingstressesinacceleratedlifetesting(ALT)arestep-
stresses:unitsareplacedontestataninitiallowstressandiftheydonotfailina
predeterminedtimet1,thestressisincreased.Iftheydonotfailinapredetermined
timet2>t1,thestressisincreasedoncemore,andsoon.Thus,step-stresseshave
rmofthex(u)=x2,t1≤u<t2,(1.14)
x1,0≤u<t1,
∙∙∙∙∙∙
xm,tm−1≤u<tm,
18

wherex1,∙∙∙,xmareconstantstresses.Ifm=1,thestep-stressiscalledsimple.
Setsofstep-stressesoftheform(1.14)willbeusuallydenotedbyEm.
Letusconsiderthemeaningoftherule(1.13)forthestep-stresses.
LetE1beasetofconstantstressesandE2beasetofsimplestep-stressesofthe
formx(u)=x1,0≤u<t1,(1.15)
x2,u≥t1,
wherex1,x2∈E1.
Proposition1.2.1(Bagdonaviˇcius[1]).IftheGSmodelholdsonE2thenthe
survivalfunctionandthehazardrateunderthestressx(∙)∈E2verifytheequalities
Sx1(t),0≤t<t1,
Sx(∙)(t)=Sx2(t−t1+t1∗),t≥t1,(1.16)
andαx1(t),0≤t<t1,
λx(∙)(t)=αx2(t−t1+t1∗),t≥t1,(1.17)
respectively;themomentt1∗isdeterminedbytheequalitySx1(t1)=Sx2(t1∗).
Corollary1.2.1UnderconditionsoftheProposition1.2.1foralls≥0
λx(∙)(t1+s)=λx2(t1∗+s).
ItisthemodelofSedyakin[36].
Intermsofcumulativedistributionfunctionsthemodelgraphicallycanbepre-
sentedbyfollowinginthefollowingfigures:

Graph1.4.Cumulativedistributionfunction:Fx2(t)–red,Fx1(t)–blue,Fx(∙)(t)–green

19

Graph1.5.Cumulativedistributionfunction:Fx1(t)–red,Fx2(t)–blue,Fx(∙)(t)–green
LetusconsiderasetEmofmoregeneralstepwisestressesoftheform(1.13).Set
t0=0.Weshallshowthattheruleoftime-shiftholdsandforthegeneralstep-stress.

Proposition1.2.2(Bagdonaviˇcius[4]).IftheGSmodelholdsonEmthenthe
survivalfunctionSx(∙)(t)verifiestheequalities:
Sx(∙)(t)=Sxi(t−ti−1+ti∗−1),ift∈[ti−1,ti),(i=1,2,...,m),(1.18)
wherewhereti∗verifytheequations
Sx1(t1)=Sx2(t1∗),...,Sxi(ti−ti−1+ti∗−1)=Sxi+1(ti∗),(i=1,...,m−1).(1.19)
N.M.Sedyakincalledhismodelthephysicalprincipleinreliabilitymeaningthat
thismodelisverywide.Nevertheless,thismodelandit’sgeneralizationcanbenot
appropriateinsituationsofperiodicandquickchangeofthestresslevelorwhen
switch-up’softhestressfromoneleveltotheanothercanimplyfailuresorshorten
e.lifthe

1.3Acceleratedfailuretimemodel

1.3.1Definitionofthemodelforconstantstresses
Supposethatunderdifferentconstantstressesthesurvivalfunctionsdifferonlyin
scale:foranyx∈E1
Sx(t)=G{r(x)t},(1.20)
wherethesurvivalfunctionGdoesnotdependonx.
ApplicabilityofthismodelinacceleratedlifetestingwasfirstnotedbyPieruschka
[35].ItisthemostsimpleandthemostlyusedmodelinFTRdataanalysisandALT.
UndertheAFTmodelthedistributionoftherandomvariable
R=r(x)Tx

20

doesnotdependonx∈E1andit’ssurvivalfunctionisG.Denotebym,σ2andtp
themean,thevarianceandthep-quantileofR,respectively.
TheAFTmodelimpliesthat
E(Tx)=m/r(x),Var(Tx)=σ2/r2(x),tx(p)=tp/r(x),
wheretx(p)isthep-quantileofTx.
Thecoefficientofvariation
E(Tx)m
Var(Tx)=σ
doesnotdependonx.
Thesurvivalfunctionsunderanyx1,x2∈E1arerelatedinthefollowingway:
Sx2(t)=Sx1{ρ(x1,x2)t},
whereρ(x1,x2)=r(x2)/r(x1).
Setε=ln{r(x)}+ln{Tx},a(x)=−ln{r(x)}.Then
ln{Tx}=a(x)+ε.
Thedistributionoftherandomvariableεdoesnotdependonx.Thelastequality
atthimpliesVar(lnTx)=Var(ε).
Thevarianceoftherandomvariableln{Tx}doesnotdependonx.
1.3.2Definitionofthemodelfortime-varyingstresses
Themodel(1.20)isgeneralizedtothecaseoftime-varyingstressesbysupposing
thattheGSmodelalsoholds,i.e.thehazardratesundertime-varyingstressesare
obtainedfromthehazardratesunderconstantstressesbytherule(1.13).
Proposition1.3.1(Bagdonaviˇcius[4]).TheGSmodelwiththesurvivalfunctions
(1.20)onE1holdsonE⊃E1ifthereexistapositiveonEfunctionrandapositive
on[0,∞)functionqsuchthatforallx(∙)∈E
λx(∙)(t)=r{x(t)}q{Sx(∙)(t))}.(1.21)
Proposition1.3.1suggeststhefollowingmodel.
Definition1.3.1Theacceleratedfailuretime(AFT)modelholdsonEifthere
existsapositiveonEfunctionrandapositiveon[0,∞)functionqsuchthatforall
x(∙)∈Etheformula(1.21)holds.
UndertheAFTmodelthehazardrateλx(∙)(t)atanymomenttisproportionalto
afunctionofthestressappliedatthismomentandtoafunctionoftheprobability
ofsurvivaluntiltunderx(∙).

21

Letusfindtheexpressionofthesurvivalfunctionundertime-varyingstresses.
Proposition2.6.(Bagdonaviˇcius[4]).Supposethattheintegral
xdv0q(v)(1.22)
convergesforallx≥0.
TheAFTmodelholdsonasetofstressesEifthereexistsasurvivalfunctionG
suchthatforallx(∙)∈E
tSx(∙)(t)=G0r{x(u)}du.(1.23)
IftheAFTmodelholdsonE2thenthesurvivalfunctionunderanystressx(∙)∈E2
oftheform(1.15)verifiestheequality
Sx(∙)(t)=Sx2(tS−x1t(1t),+t1∗),0τ≤≥τt<1,t1,(1.24)
ewhert1∗=r(x1)t1.(1.25)
)x(r21.3.3Relationsbetweenthemeansandthequantiles
Supposethatx(∙)isatime-varyingstress.Denotebytx(∙)(p)thep-quantileofthe
randomvariableTx(∙),andbyxτ=x(τ)1{t≥0}aconstantstressequaltothevalueof
time-varyingstressx(∙)atthemomentτ.
Proposition1.3.3(Bagdonaviˇcius[4]).SupposethattheAFTmodelholdsonE
andx(∙),xt∈Eforallt≥0.Then
tx(∙)(p)dτ=1.(1.26)
0txτ(p)
IfthemeansE(Tx(∙)),E(Txτ)existthen
Tx(∙)dτ
E0E(Txτ)=1.(1.27)
Themodel(1.27)isthemodelofMiner.
Corollary1.3.1Forthestressoftheform(1.14)theformula(1.27)impliesthe
qualityemE(Tk)
E(Txk)=1,(1.28)
=1k22

ewher0,Tx(∙)<tk−1,
tk−tk−1,Tx(∙)≥tk,
Tk=Tx(∙)−tk−1,tk−1≤Tx(∙)<tk,
isthelifeintheinterval[tk−1,tk)fortheunittestedunderthestressx(∙).
Theformula(1.26)impliesthatfortx(∙)(p)∈[tk−1,tk)thefollowingequalityholds:
k−1ti−ti−1+tx(∙)(p)−tk−1=1.(1.29)
i=1txi(p)txk(p)
Inthecasem=2,theformula(1.28)canbewrittenintheform
E(T1)E(T2)
E(Tx1)+E(Tx2)=1,(1.30)
andtheformula(1.29)canbewrittenintheform
t1+tx(∙)(p)−t1=1.(1.31)
tx1(p)tx2(p)
So)T(EE(Tx1)=1−E(1T2),(1.32)
E(Tx2)
andttx1(p)=tx(∙1)(p)−t1,iftx(∙)(p)≥t1.(1.33)
1−tx2(p)
Thus,iftheAFTmodelholdsonE2thenE(T1),E(T2)andE(Tx2)determineE(Tx1),
andtx(∙)(p)andtx2(p)determinetx1(p).

1.4Proportionalhazardsmodel
1.4.1Definitionofthemodelforconstantstresses
Insurvivalanalysisthemostlyusedmodeldescribingtheinfluenceofcovariateson
thelifetimedistributionistheproportionalhazards(PH)orCoxmodel,introduced
byD.Cox[10].
Supposethatunderdifferentconstantstressesx∈E1thehazardratesarepro-
portionaltoabaselinehazardrate:
λx(t)=r(x)λ(t).(1.34)
Forx∈E1thesurvivalfunctionshavetheform
Sx(t)=Sr(x)(t)=exp{−r(x)Λ(t)},(1.35)
wherett
S(t)=exp−λ(u)du,Λ(t)=λ(u)du=−lnS(t).
0023

1.4.2Definitionofthemodelfortime-varyingstresses
InthestatisticalliteraturethefollowingformalgeneralizationofthePHmodelto
thecaseoftime-varyingstressesisused.
Definition1.4.1Theproportionalhazards(PH)modelholdsonasetofstresses
Eifforallx(∙)∈E
λx(∙)(t)=r{x(t)}λ(t),(1.36)
thatimpliesndefinitioThistΛx(∙)(t)=r{x(u)}dΛ(u).(1.37)
0IntermsofsurvivalfunctionsthePHmodeliswritten:
tSx(∙)(t)=exp−r{x(u)}dΛ(u).(1.38)
0TheAFT,PHaareratherrestrictive.MoregiveninBagdonavicˇiusandNikulin
).002(2

1.5Wienerprocess
AWienerprocess(orBrownianmotion;notationWtorW)isatime-continuous
processwiththeproperties
0.=W1.02.Wt∼N(0,t)forallt≥0.Thatis,foreachttherandomvariableWtis
distributednormallywithmeanE(Wt)=0andvarianceVar(Wt)=E(Wt2)=t.
3.AllincrementsΔWt:=Wt+Δt−Wtonthenonoverlappingtimeintervalsarein-
dependent.Thatis,thedisplacementsWt2−Wt1andWt4−Wt3areindependent
forall0≤t1<t2≤t3<t4.
4.Wtdependscontinuouslyont.
Generallyfor0≤s<tthepropertyWt−Ws∼N(0,t−s)holds,inparticular
E(Wt−Ws)=0,Var(Wt−Ws)=E((Wt−Ws)2)=t−s.(1.39)
1.6Wienerprocesswithdrift
Astochasticprocess{W(t),t≥0}iscalledaWiener-processwithdriftifithasthe
erties:propwingfollo1.W(0)=0;

24

2.{W(t),t≥0}hasstationary,independentincrements,
3.EveryincrementW(t)−W(s)hasanormaldistributionwithexpectedvalue
µ(t−s)andvarianceσ2|t−s|.
Equivalently,{W(t),t≥0}isaWinerprocesswithdriftif
W(t)=µt+X(t),
where{X(t),t≥0}isaWienerprocesswithσ2=Var(X(1)).Theconstantµis
calleddriftparameter.

1.7Gammaprocess
Stoparchaametersticσpr,odecenosstedisabygZ(ammat)inprG(oνce(tss),1w/σith),tifheshapeparameterνandthescale
1.Z(0)=0;
2.vZ(ariat)hablessZ(indept1),Zenden(t2)t−Z(increment1),...,ts,Zi.(tme.)−forZ(tant−y1)0<aret1<indep...e<ndentmt;therandom
3.thedistributionofZ(t)−Z(s)isgammawithdensity
1pZ(t)−Z(s)(x)=Γ(ν(t)−ν(s)xν(t)−ν(s)−1σ−(ν(t)−ν(s))e−x/σ,x≥0.
Thegammaprocessisnon-decreasinganditsincrementsΔZ(t)=Z(t+Δt)−Z(t)
arefromthesamefamilyofgammadistributions.

25

2hapterC

Statisticalanalysisofredundant
systems

2.1Redundantsystemwithonemainandonestand-
uniybt

Consideraredundantsystemwithoneoperatingandonestand-byunit.Ifthemain
unitinsteadfailsofthethentmainheone.stand-bWyeunitsupp(ifoseitthaistnotcommfailedutingyet)isismocommmentaruteydandandtophereeratesare
irs.repanoIfreservingthe),stitand-bispyosunitsibleisthanottduringfunctioningandafteruntilcothemmfailurutingeothefthefailurmainerateunitincrea(”cold”ses
becfunctionauseofthethestmaand-binyeunitlementisnotpresen”wtedainrmed”thegraenoughph.[39].Theprobabilitydensity
thenIfustheuastllyaand-bfterycounitmmisutingfunctioningtheinreliabilittheyosamefthe”hostat”nd-conbyditionsunitdoasesthenotmacinhange.unit
But”hot”redundancyhasdisadvantagesbecausethestand-byunitfailsearlierthan
themainonewiththeprobability0.5.So”warm”reservingissometimesused[41]:
theprostandbabilitybyofunitthefaifunctionslureoftheunderlstandow-erbystressunitisthantsmallerhemathainnone.thatInofsucthehamacainseunitthe
anditisalsopossiblethatcommutingisfluent.Sothemainproblemistoverifythe
hypothesisthattheswitchonfrom”warm”to”hot”conditionsdoesnotdosome
units.togedamaLetusformulatethehypothesisstrictly.

26

2.1.1Themodels
Supposethatin”hot”conditionsthefailuretimesofthemainandthestand-byunits
areabsolutelycontinuousandhavethesamec.d.f.F1andtheprobabilitydensity
functionf1(seeGraph2.1.),thefailuretimeT2ofthestand-byelementhasthec.d.f.
F2andtheprobabilitydensityf2.Similarly,in”warm”conditionsthec.d.f.isF2
andthep.d.fisf2.

Graph2.1.Densityfunctionofthemainelement
ThefailuretimeofthesystemisT=max(T1,T2).
ybeDenotf2(y)(x)=fT2|T1=y(x)
theco(y)nditionalp.d.f.ofT2giventhatthemainunitfailsatthemomenty.Ifx≤y
thenf2(x)=f2(x).
Thec.d.f.ofthesystemfailuretimeTis
tF(t)=P(T1≤t,T2≤t)=P(T2≤t|T1=y)f1(y)dy=
0=f2(x)dx+f2(y)(x)dxf1(y)dy.(2.1)
tyt
y00Whenstand-byis”cold”thenf2(x)=0forx≤yandf2(y)(x)=f1(x−y)for
x>y,so
ttt
F(t)=f1(x−y)dxf1(y)dy=F1(t−y)dF1(y).
0y0

27

Graph2.2.Densityfunctionofthestand-byunitin”cold”reserving
Whenstand-byis”hot”thenf2(y)(x)=f2(x)=f1(x),
sotyt
0F(t)=f2(x)dx+f2(x)dxf1(y)dy=[F1(t)]2.
y0

Graph2.3.Densityfunctionofthestand-byunitin”hot”reserving
Inthecaseof”warm”reservingthefollowinghypothesisisassumed:
H0:f2(y)(x)=f1(x+g(y)−y),forallx≥y≥0,(2.2)
whereg(y)isthemomentwhichin”hot”conditionscorrespondstothemomentyin
”warm”conditionsinthesensethat
F1(g(y))=P(T1≤g(y))=P(T2≤y)=F2(y).
Wesupposethatthec.d.f.Fiarecontinuousandincreasingon(0,∞).Insuchacase
g(y)=F1−1(F2(y)).
28

Conditionally(givenT1=y)thehypothesiscorrespondstotheSediakin’smodel[36].
In[2]agoodness-of-fittestoflogrank-typeforSediakin’smodelusingexperiments
herewiththefixedswitcswitchhoffoffmomenmomentstsareisrpropandomosed.(Soseewealsoneed[6])a.Ingoothednessituas-of-fittiontestforconsideredthe
).(2.2delmoTheformula(2.1)impliesthatunderthehypothesisH0
tt
F(t)=0F2(y)+yf1(x+g(y)−y)dxf1(y)dy=
tt=0{F2(y)+F1(t+g(y)−y)−F1(g(y))}f1(y)dy=0F1(t+g(y)−y)dF1(y),
sotF(t)=F1(t+g(y)−y)dF1(y).(2.3)
0Inparticular,ifwesupposethatthedistributionoftheunitsfunctioningin”warm”
and”hot”conditionsdifferonlyinscale,i.e.
F2(t)=F1(rt),(2.4)
forsomer>0,theng(y)=ry.
InsuchacaseinsteadofthehypothesisH0narrowerhypothesis
H0∗:∃r>0:f2(y)(x)=f1(x+ry−y),forallx≥y≥0,(2.5)
istobeverified.Conditionally(givenT1=y)thehypothesiscorrespondstothe
acceleratedfailuretime(AFT)model[1],[31].In[4]agoodness-of-fittestforAFT
modelusingexperimentswith∗fixedswitchmomentsisproposed(seealso[6]).Sowe
needatestforthehypothesisH0.
2.1.2Goodness-of-fittestforthehypothesisH0∗
Supposethatthefollowingdataareavailable:
a)thefailuretimesT11,...,T1n1ofn1unitstestedin”hot”conditions;
b)thefailuretimesT21,...,T2n2ofn2unitstestedin”warm”conditions;
c)thefailuretimesT1,...,Tnofnredundantsystems(with”warm”stand-by
units).Thetestsarebasedonthedifferenceoftwoestimatorsofthec.d.f.F(t)of
thesystem.Thefirstistheempiricaldistributionfunctionobtainedfromthedata
T1,...,Tn:n
Fˆ(1)(t)=n11{Ti≤t}.(2.6)
=1iThesecondusesthedataT11,...,T1n1andT21,...,T2n2andisbasedonthe
formula(2.3):t
Fˆ(2)(t)=Fˆ1(t+gˆ(y)−y)dFˆ1(y),
029

where(ifwetestthehypothesisH0)
njgˆ(y)=Fˆ1−1(Fˆ2(y)),Fˆj(t)=n11{Tji≤t},Fˆ1−1(y)=inf{s:Fˆ1(s)≥t},(2.7)
j=1ior(ifwetestthehypothesisH0∗)
njgˆ(y)=rˆy,rˆ=µˆ1,µˆj=1Tji.(2.8)
µˆ2nji=1
Thetestsarebasedonthestatistic
∞X=√n(Fˆ(1)(t)−Fˆ(2)(t))dt.(2.9)
0InthefollowingsenseitisanalogoustotheStudent’st-testforcomparingthe
meansoftwopopulations.Indeed,themeanfailuretimeofthesystemwithc.d.f.F
is∞µ=[1−F(s)]ds,
0sothestatistic(2.9)isthenormeddifferenceoftwoestimators(thesecondbeing
nottheempiricalmean)ofthemeanµ.Student’st-testisbasedonthedifferenceof
empiricalmeansoftwopopulations.
ItwillbeshownthatinthecaseofbothhypothesisH0andH0∗thelimitdistri-
bution(asni/n→li∈(0,∞),n→∞)ofthestatisticXisnormalwithzeromean
andfinitevarianceσ2.
Letusfindtheasymptoticdistributionofthestatistic(2.9).
Theorem2.1.1Supposethatni/n→li∈(0,∞),n→∞andthedensitiesfi(x),
i=1,2arecontinuousandpositiveon(0,∞).ThenunderH0∗thestatistic(2.9)
convergesindistributiontothenormallawN(0,σ2),where
22rc1σ2=Var(Ti)+lVar(H(T1i))+lVar(T2i),(2.10)
21∞1H(x)=x[c+r−1−F1(x/r)−rF2(x)]+rE(1{T1i≤x/r}T1i)+rE(1{T2i≤x}T2i),
c=µ2y[1−F2(y)]dF1(y).
0Proof.Thelimitdistributionoftheempiricaldistributionfunctionsiswellknown:
√n(Fˆi−Fi)→DUi,√n(Fˆ(1)−F)→DU(2.11)
onD[0,∞),where→Dmeansweakconvergence,U1,U2andUareindependentGaus-
sianmartingaleswithUi(0)=U(0)=0andthecovariances
1cov(Ui(s1),Ui(s2))=liFi(s1∧s2)(1−Fi(s1∨s2)),
30

cov(U(s1),U(s2))=F(s1∧s2)(1−F(s1∨s2)).
TheGlivenko-Cantellitheoremstatesthattheempiricaldistributionfunctionscon-
vergeinprobabilitytwothec.d.f.uniformlyonR:
PPx∈supR|(Fˆi(x)−Fi(x)|→0,x∈supR|(Fˆ(x)−F(x)|→0.
UnderthehypothesisH0∗thedifferenceofthetwoestimatorsofthedistribution
functionFcanbewrittenasfollows:
ttFˆ(1)(t)−Fˆ(2)(t)=Fˆ(1)(t)−F(t)−Fˆ1(t+gˆ(y)−y)dFˆ1(y)+F1(t+g(y)−y)×
00t×dF1(y)=Fˆ(1)(t)−F(t)−[F1(t+gˆ(y)−y)−F1(t+g(y)−y)]dF1(y)−
0t−[(Fˆ1(t+gˆ(y)−y)−Fˆ1(t+g(y)−y))−(F1(t+gˆ(y)−y)−F1(t+g(y)−y))]dF1(y)−
0t−[Fˆ1(t+gˆ(y)−y)−Fˆ1(t+g(y)−y)](dFˆ1(y)−dF1(y))−
0t−[Fˆ1(t+g(y)−y)−F1(t+g(y)−y)]dF1(y)−
0t−[Fˆ1(t+g(y)−y)−F1(t+g(y)−y)](dFˆ1(y)−dF1(y))−
0t−F1(t+g(y)−y)(dFˆ1(y)−dF1(y)).
0Thestatistic(2.9)canbewritten
∞√X=n[Fˆ(1)(t)−F(t)]dt−
0t∞√−dtn[F1(t+gˆ(y)−y)−F1(t+g(y)−y)]dF1(y)−
√0∞0t
−dtn[Fˆ1(t+g(y)−y)−F1(t+g(y)−y)]dF1(y)−
0∞0t
−dtF1(t+g(y)−y)d{√n[Fˆ1(y)−F1(y)]}+oP(1);(2.12)
00hereoP(1)denotearandomvariablewhichconvergesinprobabilitytozero:oP(1)→P
0.Setσj2=Var(Tji),j=1,2.Theconvergence
∞∞√n(µˆj−µj)=−√n[Fˆj(y)−Fj(y)]dy→DYj=−Uj(y)dy∼N(0,σj2/li)
0031

.13)(2

impliestheconvergence
√1σ2(l+l)
n(rˆ−r)→DY=(Y1−rY2)∼N(0,1122).(2.13)
µ2l1l2µ2
Theformulas(2.11)-(2.13)imply
∞∞√n[Fˆ(1)(t)−F(t)]dt→DU(t)dt,
00∞t√D
dtn[F1(t+rˆy−y)−F1(t+ry−y)]dF1(y)→
00∞t∞∞
→DYdtyf1(t+ry−y)dF1(y)=YydF1(y)f1(t+ry−y)dt=
y000∞∞∞
=Yy[1−F1(ry)]dF1(y)=cY1−rcY2=−cU1(y)dy+rcU2(y)dy,
000t∞dt√n[Fˆ1(t+ry−y)−F1(t+ry−y)]dF1(y)→D
00∞t∞∞
→DdtU1(t+ry−y)dF1(y)=dF1(y)U1(u)du=
∞000ry
=U1(u)F1(u/r)du,
0t∞dtF1(t+ry−y)d{√n[Fˆ1(y)−F1(y)]}→D
00∞t
→DdtF1(t+ry−y)dU1(y)=
00∞t∞
=[F1(rt)U1(t)−U1(y)dF1(t+ry−y)]dt=F2(t)U1(t)dt−
000∞rt∞
−dtU1((v−t)/(r−1))dF1(v)=F2(t)U1(t)dt+
0t0∞v/r∞
+dF1(v)U1((v−t)/(r−1))dt=F2(t)U1(t)dt+
0t0∞v/r∞
+(1−r)dF1(v)U1(u))du=F2(t)U1(t)dt+
∞00∞0
+(r−1)[1−F1(ru)]U1(u))du=U1(y)[rF2(y)−r+1]dy.
00ainedobteWXD→V1+V2+V3,

32

where∞∞V1=U(y)dy,V2=h(y)U1(y)dy,
00∞h(y)=c+r−1−F1(y/r)−rF2(y),V3=−rcU2(y)dy.
0WeshallshowthatifG(x)=0tg(u)duthen
∞Var(G(Ti))=Var(U(y)dG(y)).(2.14)
0Indeed,setS(x)=1−F(x).
∞∞∞
Var(G(Ti))=−G2(x)dS(x)−(G(x)dS(x))2=2G(x)S(x)dG(x)−
000∞∞x
−(S(x)dG(x))2=2S(x)dG(x)dG(y)−
∞0x0∞0x
−2S(x)dG(x)S(y)dG(y)=2S(x)dG(x)F(y)dG(y)=
0∞x00∞0
=2EU(x)U(y)dG(x)dG(y)=Var(U(y)dG(y)).
000Analogousto(2.14)equalitiesaretruereplacingther.v.TibyTjiandtherandom
processesUbyUj,j=1,2.Theseequalitiesimplythatthevariancesoftherandom
variablesViare:
22rcVar(V1)=Var(Ti),Var(V3)=lVar(T2i)
22∞y1
Var(V2)=l10[1−F1(y)]h(y)dy0F1(z)h(z)dz=l1Var(H(T1i)),
wherexxH(x)=h(y)dy=x[c+r−1−F1(x/r)−rF2(x)]+ydF1(y/r)+
00x+rydF2(y)=x[c+r−1−F1(x/r)−rF2(x)]+
0+rE(1{T1i≤x/r}T1i)+rE(1{T2i≤x}T2i).
Theproofiscomplete.
Aconsistentestimatorofthevarianceσ2is
nn122n2
σˆ2=1(Ti−µˆ)2+n2[Hˆ(T1i)−H¯ˆ]2+cˆrˆ2n(T2i−µˆ2)2,
ni=1n1i=1n2i=1
33

whereµˆ=nTi,cˆ=µˆ2y[1−Fˆ2(y)]dFˆ1(y)=µˆ2n1T1i[1−Fˆ2(T1i)],
1n1∞1n1
0=1i=1iHˆ(x)=x[cˆ+rˆ−1−Fˆ1(x/rˆ)−rˆFˆ2(x)]+n1{T1i≤x/rˆ}T1i+n1{T2i≤x}T2i,
rˆn1rˆn2
1i=12i=1
n1Hˆ¯=1Hˆ(T1i).
n1=1iThestatistic√∞
X=n(Fˆ(1)(t)−Fˆ(2)(t))dt
0canbewritten∞
X=−√ntd(Fˆ(1)(t)−Fˆ(2)(t)).
0tthaeNotn1tFˆ(2)(t)=Fˆ1(t+rˆy−y)dFˆ1(y)=1Fˆ1(t+rˆT1i−T1i)1{T1i≤t}=
n10=1i1n1n11n1n1
=n21{T1j≤t+rˆT1i−T1i}1{T1i≤t}=n21{max(T1i,T1j−rˆT1i+T1i)≤t}.
1i=1j=11i=1j=1
So√√nn1n1
X=−nµˆ+n2max(T1i,T1j−rˆT1i+T1i).
1=1j=1iTheteststatistichastheform
X,=Tσˆwhereσˆ.ThedistributionofthestatisticTisapproximatedbythestandardnormal
on.distributiThetest.ThehypothesisH0∗isrejectedwiththeasymptoticsignificancevalueαif
|T|>zα/2,wherezα/2istheα/2-criticalvalueofthestandardnormaldistribution.
2.1.3Goodness-of-fittestforthehypothesisH0
AsinthecaseofthehypothesisH0∗thetestforthehypothesisH0isbasedonthe
statistic(2.9).Letusfindtheasymptoticdistributionofthisstatisticunderthe
hypothesisH0.

34

Theorem2.1.2Supposethatni/n→li∈(0,∞),n→∞andthedensitiesfi(x),
i=1,2arecontinuousandpositiveon(0,∞).ThenunderH0thestatistic(2.9)
convergesindistributiontothenormallawN(0,σ2),where
11σ2=Var(Ti)+l1Var(H(T1i))+l2Var(Q(T2i))
ewherH(x)=Q(x)−xF1(g−1(x))+g(x)[1−F2(x)]+E(1{g(T1i)≤x}g(T1i))+
+E(1{T2i≤x}g(T2i))−x,Q(x)=E{1{T1i≤x}[1−F2(T1i)]/f1(g(T1i))}.
Proof.SimilarlyasinTheorem2.1.1weobtain
∞∞√n[Fˆ(1)(t)−F(t)]dt→DU(t)dt,
00t∞dt√n[F1(t+gˆ(y)−y)−F1(t+g(y)−y)]dF1(y)→D
00∞→D−U1(g(y))−U2(y)f1(y)[1−F2(y)]dy,
0f1(g(y))
t∞√dtn[Fˆ1(t+g(y)−y)−F1(t+g(y)−y)]dF1(y)→D
00∞→DU1(u)F1(g−1(u))du,
0t∞dtF1(t+g(y)−y)d{√n[Fˆ1(y)−F1(y)]}→D
0∞0t∞
→DdtF1(t+g(y)−y)dU1(y)=F2(t)U1(t)dt−
∞00∞0∞
−0U1(y)[1−F2(y)]d(g(y)−y)=0U1(y)dy−0U1(y)[1−F2(y)]dg(y).
ainedobteWX→DV1+V2+V3,
where∞∞V1=0U(y)dy,V2=0h(y)U1(y)dy,
h(y)=f1(y)[1−F2(y)]−F1(g−1(y))−1+g(y)[1−F2(y)].
f1(g(y))
∞U2(y)
V3=−f1(g(y))[1−F2(y)]dF1(y).
0

35

ThevariancesoftherandomvariablesV1andV3are
1Var(V1)=Var(Ti),Var(V3)=lVar(Q(T2i));
2here∞1−F(y)
Q(x)=f(g(2y))dF1(y)=E{1{T1i≤x}[1−F2(T1i)]/f1(g(T1i))}.
10ThevarianceoftherandomvariableV2is
1Var(V2)=l1Var(H(T1i)),

whereH(x)=[1−F2(y)]dF1(y)−F1(g−1(y))dy−x+[1−F2(y)]dg(y)=
xxx
0f1(g(y))00
xx=Q(x)−xF1(g−1(x))+ydF1(g−1(y))−x+[1−F2(x)]g(x)+g(y)dF2(y)=
001−=Q(x)−xF1(g(x))+g(x)[1−F2(x)]−x+E(1{g(T1i)≤x}g(T1i))+E(1{T2i≤x}g(T2i)).
Theproofiscomplete.
Aconsistentestimatorofthevarianceσ2is
σˆ2=(Ti−ˆµ)2+2[Hˆ(T1i)−H¯ˆ]2+2[Qˆ(T2i)−Q¯ˆ]2,
1nnn1nn2
ni=1n1i=1n2i=1
wheren11Hˆ(x)=Qˆ(x)−xFˆ1(gˆ−1(x))+gˆ(x)[1−Fˆ2(x)]−x+1{gˆ(T1i)≤x}gˆ(T1i)+
n1=1i+1{T2i≤x}gˆ(T2i),Qˆ(x)=1{T1i≤x}[1−Fˆ2(T1i)]/fˆ1(gˆ(T1i)),
1n21n1
n2i=1n1i=1
gˆ−1(x)=Fˆ2−1(Fˆ1(x)),H¯ˆ=1Hˆ(T1i),Q¯ˆ=1Qˆ(T2i),
n1n2
n1i=1n2i=1
thedensityf1isestimatedbythekernelestimator
nfˆ1(x)=11Kx−X1i,
hhn=1i√√nn1n1
X=−nµˆ+2max(T1i,T1j−gˆ(T1i)+T1i).
n1i=1j=1
36

Theteststatistichastheform

X,=Tσˆwhereσˆ.ThedistributionofthestatisticTisapproximatedbythestandardnormal
on.distributiThetest.ThehypothesisH0isrejectedwiththeasymptoticsignificancevalueαif
|T|>zα/2,wherezα/2istheα/2-criticalvalueofthestandardnormaldistribution.
2.1.4Simulations:powerofthetests
Weinvestigatedthepoweroftheproposedgoodness-of-fittestswhenthedistribution
oftheunitsin”warm”and”hot”conditionsisexponential,Weibullandloglogistic.
LetusconsiderthefollowingalternativehypothesisH˜0∗:
f2(y)(x)=f1[x+F1−1(F2(y)+p(1−F2(y))−y],0<p<1.
Itmeansthatattheswitchingtimeythec.d.f.ofthestand-byunithasajumpof
sizep(1−F2(y)).
Setgp(y)=F1−1(F2(y)+p(1−F2(y)).
Underthealternativethec.d.f.ofthestand-bysystemis
ttF(t)=F1(t+gp(y)−y)dF1(y)=F1(t)−S1(t+gp(y)−y)dF1(y).(2.15)
00Example2.1.1.Simulatedexponentialdistribution:
T1j∼E(λ1),T2j∼E(λ2),λ2=rλ1.
Thec.d.f.ofTijforallt≥0isFi(t)=1−e−λit.
InthiscasethehypothesesH0andH0∗coincideandunderthesehypothesesthe
c.d.f.oftheredundantsystemis
F(t)=1−λ2+λ1e−λ1t+λ1e−(λ1+λ2)t.
λλ22UndertheexponentialdistributionandunderthealternativehypothesisH˜0∗we
haveλ21
gp(y)=λ1y−λ1ln(1−p),
andthec.d.f.oftheredundantsystemis
tF(t)=1−e−λ1t−λ1(1−p)e−λ1te−λ2ydy=
0

37

=1−e−λ1t−(1−p)λ1e−λ1t(1−e−λ2t).
λ2InthisexamplethetestforthehypothesisH0∗isconsidered,sothefollowingvalues
oftheparameterswereused:
λ1=1/100,λ2=1/300,r=1/3.
Thedistributionfunctionofaredundantsysteminthiscaseis
F(t)=1−4e−t/100+3e−(4/300)t.
ThehypothesisH0∗wastestedusingα=0.05asymptoticsignificancelevelandseveral
valuesofthesamplesizen,n=n1=n2.Thenumberofreplicationswas3000.
Table2.1.Significancelevelofthetest
SamplesizeSignificancelevel(%)
7.5505.30105.1020UndertheExponentialdistributionandunderthealternativehypothesisH˜0∗we
evha

F(t)=1−e−t/100−(1−p)3e−t/100(1−e−t/300).
Table2.2.Powerofthetest
Samplesize\Constant0.10.250.50.750.9
5058244579
1006123285100
200153988100100
Example2.1.2.Simulateddistribution:Weibull:
T1j∼W(α1,β1),T2j∼W(α2,β2),
Thec.d.f.ofTijforallt≥0isFi(t)=1−e−(t/βi)αi.
UnderthehypothesisH0thefunctiongisg(t)=β1(t/β2)α2/α1andthec.d.f.of
issystemtredundanthetF(t)=F1(t)−αα11yα1−1e−(y/β1)α1−[(t−y)/β1+(y/β2)α2/α1]α1dy.
β01ThehypothesisH0coincideswiththehypothesisH0∗ifα1=α2.Insuchacase
g(t)=rtandr=β1/β2.
UndertheWeibulldistributionandunderthealternativehypothesisH˜0∗wehave
gp(y)=β1[−ln(1−p)+(y/β2)α2]1/α1,
38

andthec.d.f.oftheredundantsystemis
F(t)=F1(t)−
t−αα11yα1−1e−(y/β1)α1e−[(t−y)/β1+(−ln(1−p)+(y/β2)α2)1/α1]α1dy.
β01InthisexamplethetestforthehypothesisH0∗isconsidered,sothefollowingvalues
oftheparameterswereused:
α1=α2=2,β1=100,β2=300.
Thedistributionfunctionofaredundantsysteminthiscaseis
tF(t)=1−e−t2/β12−22ye−1/β12(t2−34ty+913y2)dy.
β10ThehypothesisH0∗wastestedusingα=0.05asymptoticsignificancelevelandseveral
valuesofthesamplesizen,n=n1=n2.Thenumberofreplicationswas3000.
Table2.3.Significancelevelofthetest
SamplesizeSignificancelevel(%)
4.7505.30105.0020UnderthealternativehypothesisH˜0∗thec.d.f.oftheredundantsystemis
t22
F(t)=1−e−t2/β12−2ye−y2/β12e−(t−y)/β1+−log(1−p)+9yβ12dy.
2β10Table2.4.Powerofthetest
Samplesize\Constant0.10.250.50.750.9
50921307188
1001435457695
200255874100100
Example2.1.3.ContinuingExample2.1.2,insteadofthehypothesisH0∗we
consideredthehypothesisH0,takingdifferentvaluesofαi:
α1=1,α2=2,β1=100,β2=300.
SincetheGausserrorfunctionis
xerf(x)=√2e−t2dt
π039

andmalrelatiodistributnbionetwΦeeisnterhef(xGa)=uss2Φ(erroxr√2)−function1,theerfdistandtributiohecn.d.f.functionoftheofastandaredundardnornt-
issystemtF(t)=1−e−βt1−1e−βt1e−y2/(9β12)dy=
β01=1−e−t/β1[1+3√π(Φ(t√2/(3β1))−0.5)].
oftheThesahypmpleothessizeisn,H0wnas=tne1sted=n2.usingNum5pberercoenftreplicasignificancetionswlevasel3a00nd0.severalvalues
Table2.5.Significancelevelofthetest
SamplesizeSignificancelevel(%)
5.6502010005.15.2

Takingintoaccountthat
2yF1−1(u)=−β1ln(1−u),gp(y)=−β1ln(1−p)+9β1,
weobtainthatthec.d.f.oftheredundantsystemunderthealternativehypothesis
H0ist2
F(t)=1−e−t/β1−(1−p)e−t/β11e−9yβ12dy=
β01=1−e−t/β1[1+3(1−p)√π(Φ(t√2/(3β1))−0.5)].
Table2.6.Powerofthetest
Samplesize\Constant0.10.250.50.750.9
50711457993
20100024154533775610920101000
Example2.1.4.Simulateddistribution:loglogistic:
T1j∼L(α1,β1),T2j∼L(α2,β2),
Thec.d.f.ofTijforallt≥0isFi(t)=1−(1+(t/β1i)αi).
UnderthehypothesisH0thefunctiongisg(t)=β1(t/β2)α2/α1andthec.d.f.of
issystemtredundantheα1tyα1−1dy
F(t)=F1(t)−−β1α10(1+(y/β1)α1)2∗1+[(t−y)/β1+(y/β2)α2/α1]α1.
40

UndertheloglogisticdistributionandunderthealternativehypothesisH˜0∗wehave
gp(y)=β1p+(y/β2),
α21/α1
p1−andthec.d.f.oftheredundantsystemis
F(t)=F1(t)−
α1tyα1−11
β1α10(1+(y/β1)α1)21+t−y+p+(y/β2)α21/α1α1dy.
p−1β1InthisexamplethetestforthehypothesisH0∗isconsidered,sothefollowingvalues
oftheparameterswereused:
α1=α2=2,β1=100,β2=300.
Thedistributionfunctionofaredundantsysteminthiscaseis
12t1y
1β13β1β12
F(t)=1−1+βt22−β1201+t−2y2∗1+y22dy.
ThehypothesisH0∗istestedusing5percentsignificancelevelandseveralvaluesof
thesamplesizen,n=n1=n2.Numberofreplicationwas3000.
Table2.7.Significancelevelofthetest
SamplesizeSignificancelevel(%)
8.1506.30105.2020UnderthealternativehypothesisH0∗thec.d.f.oftheredundantsystemis
1+1F(t)=1−y2−
β1t1y2−β120y22t−yp+(y/β2)21/22dy.
1+β11+β1+1−p
Table2.8.Powerofthetest
Samplesize\Constant0.10.250.50.750.9
50144260100100
100215773100100
2002962100100100
41

Example2.1.5.ContinuingExample2.1.4.,insteadofthehypothesisH0∗we
consideredthehypothesisH0,takingdifferentvaluesofαi:
α1=1,α2=2,β1=100,β2=300.
Thedistributionfunctionofaredundantsysteminthiscaseis
11t11
11+β1+3β11+β1
F(t)=1−1+βt−β10t−yy2∗y2dy.
ThehypothesisH0istestedusing5percentsignificancelevelandseveralvaluesof
thesamplesizen,n=n1=n2.Numberofreplicationwas3000.
Table2.9.Significancelevelofthetest
SamplesizeSignificancelevel(%)
7.7502010005.46.5
UnderthealternativehypothesisH0thec.d.f.oftheredundantsystemis
11ty1
F(t)=1−1+βy1−β101+y21+tβ−y+p+(1y−/βp2)2dy.
1β1Table2.10.Powerofthetest
Samplesize\Constant0.10.250.50.750.9
50927396475
102000152341495279977110850

42

2.2Redundantsystemwithonemainand(m−1)
tsuniyand-bstLetusconsiderasystemofmunits:onemainunitandm−1stand-byunits.We
shallusenotationS(1,m−1)forsuchsystems.
DenotebyT1,F1andf1thefailuretime,thec.d.f.andtheprobabilitydensity
functionofthemainunit.Thefailuretimesofthestand-byunitsaredenotedby
T2,...,Tm.In”hot”conditionstheirdistributionfunctionsarealsoF1.In”warm”
conditionsthec.d.f.ofTiisF2andthep.d.fisf2,i=2,...,m.Ifastand-byunitis
switchedfrom”warm”to”hot”conditions,itsc.d.f.isdifferentfromF1andF2.
ThefailuretimeofthesystemS(1,m−1)isT(m)=T1∨T2∨∙∙∙∨Tm.As
T(m)=(T1∨T2∨∙∙∙∨Tm−1)∨Tm,wecanconsiderthissystemasasystemS(1,1)
withonemainelement(whichitselfisasystemS(1,m−2))andonestand-byelement.
DenotebyKjandkjthec.d.f.andthep.d.f.ofT(j),respectively,(j=2,...,m),
K1=F1,k1=f1.Thec.d.fKjcanbewrittenintermsofthec.d.fKj−1andF1:
tKj(t)=P(T(j)≤t)=P(T(j−1)≤t,Tj≤t)=P(Tj≤t|T(j−1)=y)dKj−1(y).
0.16)(2Wegeneralize(2.2)modellingtheconditionaldistributionP(Tj≤t|T(j−1)=y)and
definethefollowinghypothesis.
HypothesisH0:
f2(t)ift≤y,
fTj|T(j−1)=y(t)=f1(t+g(y)−y)ift>y;(2.17)
here(asinthecaseofthehypothesis(2.2))
g(y)=F1−1(F2(y)).
Theformulas(2.16)and(2.17)implytheequality
tKj(t)=F1(t+g(y)−y)dKj−1(y).(2.18)
0Sothecumulativedistributionfunctionofthesystemwithm−1stand-byunitsis
definedrecurrentlyusingformula(2.18)(j=2,...,m).
Inparticular,ifwesupposethatthehypothesisH0istrueandthedistributionof
unitsfunctioningin”warm”and”hot”conditionsdifferonlyinscale,i.e.
F2(t)=F1(rt),(2.19)
forallt≥0andsomer>0,theng(y)=ry.Sowedefinethefollowinghypothesis.
HypothesisH0∗:

fTj|T(j−1)=y(t)=ff2((tt)+ry−y)ififtt>≤yy.,
143

.17)(2

.20)(2

Underthemodel(2.20)thecumulativedistributionfunctionofthesystemisobtained
ulasformtrecurrenusingtKj(t)=F1(t+ry−y)dKj−1(y).(2.21)
0Ifswitchingfrom”warm”to”hot”conditionsdoesnotdamageunitsinthesystem
S(1,1)thenitisnaturalthatthisistrueforthesystemS(1,m−1),m>2.Soit
issufficienttousegoodness-of-fittestsforthehypothesesH0andH0∗whenonlyone
stand-byunitisused.SuchtestsweregiveninChapter2.1.
InwhatfollowswesupposethatoneofthehypothesisH0orH0∗isverifiedand
weshallconsidernonparametricandparametricestimationmethodsforredundant
systemsreliabilityestimationusingdatafromunitsreliabilitytrials.

2.2.1Nonparametricestimation
SupposethatthehypothesisH∗istrueandthefollowingdataareavailable:
a)completeorderedsample0T11,...,T1n1ofthefailuretimesofn1unitstestedin
ions;conditt””hob)theorderedfirstm2failuretimesT21,...,T2m2obtainedbytestingofn2units
uptothetimet1in”warm”conditions.
Thesecondsampleiscensoredbecausethetimetoobtaincompletedatain
”warm”conditionsmaybelong.
Setn1n2
N1(t)=1{T1i≤t},N2(t)=1{T2i≤t,t≤t1},
=1i=1iY1(t)=1{T1i≥t},Y2(t)=1{T2i≥t,t≤t1}.
n1n2
=1i=1iNotethattherandomvariablesT/randTcanbeinterpretedasorderstatistics
fromsamplesofsizen1andn2,resp1iectively,f2riomthepopulationhavingthec.d.fF2.
denoteewifSon1N˜1(t)=1{T1i/r≤t}=N1(rt),N˜2(t)=N2(t),
=1in1Y˜1(t)=1{T1i/r≥t}=Y1(rt),Y˜2(t)=Y2(t),
=1ithenthefollowingNelson-Aalentypeestimator(stilldependingonr)ofthecumula-
tivehazardfunctionΛ2=−lnS2canbeconsidered:
Λ˜2(t,r)=tdN˜1(u)+dN˜2(u)=tdN1(ru)+dN2(u).(2.22)
0Y˜1(u)+Y˜2(u)0Y1(ru)+Y2(u)

44

Takingintoconsiderationthefactthatthedifference
tM2(t)=N2(t)−Y2(u)dΛ2(u)
0isamartingaleon[0,t1]withrespecttothefiltrationgeneratedbythedata,and
EM2(t1)=0,theparameterrcanbeestimatedusingtheestimatingfunction
t1U˜(r)=N2(t1)−Y2(u)dΛ˜2(u,r)=(2.23)
0rt1Y2(v/r)dN1(v)t1Y2(u)dN2(u)
=N2(t1)−0Y1(v)+Y2(v/r)−0Y1(ru)+Y2(u).
U˜(r)isanon-increasingstepfunction,
t∞1U˜(0+)=N2(t1)−Y2(u)dN2(u)>0,U˜(+∞)=−n2dN1(v)<0,
0n1+Y2(u)0Y1(v)+n2
sotheparameterrisestimatedbythestatistic
rˆ=U˜−1(0)=sup{r:U˜(r)>0}.(2.24)
Thec.d.f.Kmoftheredundantsystemisestimatedusingthefollowingrecurrent
equations(j=2,...,m):
Kˆ1(t)=Fˆ1(t),
tKˆj(t)=Kˆj−1(t)−Sˆ1(t+rˆy−y)dKˆj−1(y)=
0=Kˆj−1(t)−Sˆ1(t+rˆT1i−T1i)(Kˆj−1(T1i)−Kˆj−1(T1,i−1));
i:T1i≤t
hereSˆ1=1−Fˆ1.Theestimatorofthemeanfailuretimeµofthesystemis
n1∞µˆ=tdKˆm(t)=T1i[Kˆm(T1i)−Kˆm(T1,i−1)].
0=1iThefollowingalternativeestimatorsofthec.d.f.Fimaybeconsidered.
TheestimatorsofthecumulativehazardsΛ1andΛ2are
∗∗tdN1(u)t/rˆdN2(u)
Λ1(t)=Λ2(t/rˆ,rˆ)=0Y1(u)+Y2(u/rˆ)+0Y1(rˆu)+Y2(u)=
=1+1,
T1i≤tY1(T1i)+Y2(T1i/rˆ)T2i≤t/rˆY1(rˆT2i)+Y2(T2i)
Λ2∗(t)=Λ1∗(rˆt).

45

The∗estima∗torsofthec.d.f.Si=1−Fiaretheproductintegralsoftheestimators
Λ1(t)andΛ2(t),so
F1∗(t)=1−π0≤s≤t(1−dΛˆ1(s))=
11
=1−T1i≤t1−Y1(T1i)+Y2(T1i/rˆ)T2i≤t/rˆ1−Y1(rˆT2i)+Y2(T2i),
F2∗(t)=F1∗(rˆt).
MixingallmomentsTandrˆTandorderingthem,weobtainthesequenceof
randomvariablesT1≤∙1∙i∙≤Tn1+2mj2.TheestimatorsF1∗(t)andF2∗(t)canbewritten:
∗1∗∗
F1(t)=1−Ti≤t1−Y1(Ti)+Y2(Ti/rˆ),F2(t)=F1(rˆt).
Thec.d.f.Kmoftheredundantsystemisestimatedusingthefollowingrecurrent
equations(j=2,...,m):
tKj∗(t)=Kj∗−1(t)−Sˆ1(t+rˆy−y)dKj∗−1(y)=
0=Kj∗−1(t)−S1∗(t+rˆTi−Ti)(Kj∗−1(Ti)−Kj∗−1(Ti−1));
i:Ti≤t
hereS∗1=1−F1∗.Theestimatorofthemeanfailuretimeµofthesystemis
∞n1+m2
µ∗=0tdKm∗(t)=Ti[Km∗(Ti)−Km∗(Ti−1)].
=1iThegraphsofthetrajectoriesoftheestimatorsofthec.d.f.F1andKm(m=2,3,4),
inthecaseofcompletesamplesanddifferentdistributionsarepresentedinGraph
2.4,2.5and2.6.Increasingthenumberofstand-byunitsincreasesthereliabilityof
system.tredundanthe

46

Graph2.4.estimatoGraphsrsFˆo1f,Kˆtheitra(Expjectoronenietsialofthedistributiononparan)metric

Graph2.5.Graphsofthetrajectoriesofthenonparametric
estimatorsFˆ1,Kˆi(Weibulldistribution)

Graph2.6.GestimatoraphsrsoFˆf1,theKˆitra(Loglojectorgisticiesofdistthenoribution)nparametric

47

2.2.2Parametricestimation
Supposethatinhotconditionsthec.d.f.F1(kt;θ)isabsolutelyTcTontinuousanddepends
onfinitedimensionalparameterθ∈Θ⊂R.Setγ=(r,θ).
Themaximumlikelihoodestimatorγ∗=(r∗,(θ∗)T)Toftheparameterγmaxi-
mizestheloglikelihoodfunction
n1m2
(γ)=lnf1(T1i;θ)+m2lnr+lnf1(rT2i;θ)+(n2−m2)lnS1(rt1;θ).(2.25)
=1i=1iUnderH0∗foranyt≥0andj≥2thec.d.f.Kj(t)isestimatedrecurrently:
tKˆj(t)=F1(t+r∗y−y;θ∗)dKˆj−1(y),Kˆ1(t)=F1(t;θ∗).(2.26)
02.3AsymptoticdistributionofKˆjandconfidence
intervalsforKj(t)
Weneedforanasymptoticdistributionoftheestimatorofthec.d.fKm(t)ofthe
redundantsystemtoconstructconfidentialintervalsforKm(t).Supposethat
ni=li+O(1),li∈(0,1),asn=n1+n2→∞.
nn2.3.1Nonparametriccase
Thelimitdistributionoftheempiricaldistributionfunctionsiswellknown:
√n(Fˆi−Fi)→DUi(2.27)
onD(Ai),whereD(Ai)isthespaceofcadlagfunctionswithsupremumnormmet-
ric,→Dmeansweakconvergence,A1=[0,∞),A2=[0,t1],U1,U2areindependent
GaussianmartingaleswithUi(0)=0andthecovariances
cov(Ui(u),Ui(v))=1Fi(u∧v)Si(u∨v).(2.28)
liUsing(2.27)weget
√n(Sˆi−Si)→D−Ui(2.29)
Letusfindtheasymptoticdistributionoftheestimatorrˆdefinedby(2.24).Denote
byr0∈(0,1)thetruevalueofr.UnderthehypothesisH0∗itistheratioofthe
meanfailuretimesµ1andµ2ofunitsfunctioningin”hot”and”warm”conditions,
.elyctiveresp

48

Lemma2.3.1Supposethatthec.d.f.F1isabsolutelycontinuouswithpositivep.d.f.
f1on(0,∞)andtheequalityF2(t)=F1(r0t)istrueforallt≥0.If
1r0t1
A=−uf1(u)dΛ1(u)−t1f1(r0t1)=0,(2.30)
r00then√W(r)
n(rˆ−r0)→dY=−0,(2.31)
Aewhert1W(r0)=−[U1(r0u)−U2(u)]dΛ2(u)−U1(r0t1)+U2(t1),(2.32)
0Proof.SetUˆ(r)=nnnU˜(r),Sˆi=1−Fˆi,whereU˜(r)isdefinedby(2.23).For
anyr>012
ttˆn1Y2(u)dN1(ru)1Y2(u)
U(r)=n1n2−Y1(ru)+Y2(u)+1−Y1(ru)+Y2(u)dN2(u)=
00t1Sˆ2(u−)dSˆ1(ru)t1Sˆ1(ru−)dSˆ2(u)
=:=−0nn1Sˆ1(ru−)+nn2Sˆ2(u−)0nn1Sˆ1(ru−)+nn2Sˆ2(u−)
=Zˆ2(u,r)dSˆ1(ru)−Zˆ1(u,r)dSˆ2(u),
t1t1
00Zˆ2(u,r)=n(1−n1Zˆ1(u,r)).
nn2Theconvergencesup|Sˆi(u)−Si(u)|→P0,sup|Sˆi(u−)−Si(u)|→P0impliesthat
u∈Aiu∈Ai
Uˆ(r)→PU(r),where
U(r)=S2(u)dS1(ru)−S1(ru)dS2(u)=:
t1t1
l1S1(ru)+l2S2(u)l1S1(ru)+l2S2(u)
00t1t1
=:Z2(u,r)dS1(ru)−Z1(u,r)dS2(u),(2.33)
001Z2(u,r)=l(1−l1Z1(u,r)).
2UsingtheequalityS1(r0u)√=S2(u),weobtainU(r0)=0.
DUsingtheconvergencen(Sˆi−Si)→−Uiandthefunctionaldeltamethodwe
ainobtˆ√n(Zˆ−Z)(u,r)=√nS1(ru)−S1(ru)→d
11l1Sˆ1(ru)+l2Sˆ2(u)l1S1(ru)+l2S2(u)
49

→dl2−U1(ru)S2(u)+U2(u)S21(ru)=:U1∗(u,r)(2.34)
(l1S1(ru)+l2S2(u))
and

ˆ√n(Zˆ2−Z2)(u,r)=√nS2(u)−S2(u)→d
l1Sˆ1(ru)+l2Sˆ2(u)l1S1(ru)+l2S2(u)
→dlU1(ru)S2(u)−U2(u)S1(ru)=−l1U∗(u,r)=:U∗(v,r)(2.35)
1(l1S1(ru)+l2S2(u))2l212
onD([0,t1]×[0,1]).Notethat
U1∗(u,r0)=l2−U1(r0u)S2(u)+U2(u)S1(r0u)=
(l1S1(r0u)+l2S2(u))2
=lU2(u)S2(u)−U1(r0u)S2(u)=lU2(u)−U1(r0u),
22(l1S2(u)+l2S2(u))2S2(u)
Z(u,r)=S1(r0u)=S2(u)≡1,
10l1S1(r0u)+l2S2(u)l1S2(u)+l2S2(u)
S2(ru)S2(u)
Z2(u,r0)=l1S1(r0u)+l2S2(u)=l1S2(u)+l2S2(u)≡1.
Bythefunctionaldeltamethodforstochasticintegrals(seeTheoremA.0.2)andusing
(2.34),(2.35),(2.29)wehave
√n(Uˆ(r)−U(r))=
√dt1t1t1t1
nZˆ2dSˆ1(ru)−Zˆ1dSˆ2(u)−Z2dS1(ru)+Z1dS2(u)→
0000t1→dW(r):=(U2∗(u,r)dS1(ru)−Z2(u,r)dU1(ru))−
0t1−(U1∗(u,r)dS2(u)−Z1(u,r)dU2(u))(2.36)
01].,[0onBythefunctionaldeltamethod(seeTheoremA.0.3)andusing(2.24),(2.36)we
get√√n(rˆ−r0)=n(U˜−1(r)−U˜−1(r0))=
=√nn1n2Uˆ−1(r)−n1n2Uˆ−1(r0)→d
nn→dY=−W(r0)=−W(r0);
U(r0)A
50

heret1W(r0)=−[U1(r0u)−U2(u)]dΛ2(u)−U1(r0t1)+U2(t1).
0yualiteqtheUsingrt1t1
U(r)=S2(v/r)dS1(v)−S1(ru)dS2(u),
l1S1(v)+l2S2(v/r)l1S1(ru)+l2S2(u)
00weobtainthederivative
tr1S2(t1)f1(rt1)l1vf2(v/r)S1(v)dS1(v)
U(r)=−t1l1S1(rt1)+l2S2(t1)+r2(l1S1(v)+l2S2(v/r))2+
0t1r0t1
+l2uf1(ru)S2(u)dS2(u),U(r0)=−t1f1(t1)−1vf1(v)dΛ1(v).
(l1S1(v)+l2S2(v/r))2r0
00Theproofiscomplete.
Remark2.3.1.Ifsamplesarecompleteandtf1(t)→0ast→∞then
∞∞1W(r0)=−[U1(r0u)−U2(u)]dΛ2(u),A=−ruf1(u)dΛ1(u),(2.37)
000and1WD√n(rˆ−r0)→Y=−∼N0,2.(2.38)
Al1l2A
1Proof.By(2.27)Ui(v)∼N(0,l1F1(v)S1(v))andS1(v)→0asv→∞.So
Ui(v)→P0asv→∞andtheformula(2.31)impliesthefirstformulain(2.36).The
conditiontf1(t)→0andtheformula(2.29)implythesecondformulain(2.36).
Usingtheequalitycov(Ui(s1),Ui(s2))=1Fi(s1∧s2)(1−Fi(s1∨s2)),i=1,2,
iweobtainthevariance
2∞V(W(r0))=E(U1(r0u)−U2(u))dΛ2(u)=
0∞∞=dΛ2(u)E((U1(r0u)−U2(u)(U1(r0v)−U2(v))dΛ2(v)=
0∞0∞
=dΛ2(u)(EU1(r0u)U1(r0v)+EU2(u)U2(v))dΛ2(v)=
00=2dΛ2(u)S1(r0u)F1(r0v)+S2(u)F2(v)dΛ2(v)=
∞u
0012

51

=2dΛ2(u)S2(u)F2(v)+S2(u)F2(v)dΛ2(v)=
∞u
0012
u∞2=S2(u)dΛ2(u)F2(v)dΛ2(v)=
2∞S2(u)dS2(u)uF2(v)dS2(v)
1200
==120S2(u)0S2(v)
u∞=2dS2(u)[1−S2(v)]dS2(v)=
1200S2(v)
2∞1
21021=−(lnS2(u)−S2(u)+1)dS2(u)=S2(u)=x=.
Theproofiscomplete.
Theorem2.3.2IfF1iscontinuouslydifferentiableon[0,∞)thenunderH0∗forany
t>0andanynaturalj≥2
√n(Kˆj(s)−Kj(s))→D
ssWj(s)=U1(s+r0y−y)dKj−1(y)+µ(j−1)(s)Y+F1(s+r0y−y)dWj−1(y)(2.39)
00onD[0,t],whereW1(s)=U1(s),µ(j−1)(s)=0syf1(s+r0y−y)dKj−1(y).
Proof.LetusprovethatunderconditionsofLemma2.3.1foranyt≥0
√n(Fˆ1(t+rˆy−y)−F1(t+r0y−y))→dU1(t+r0y−y)+yf1(t+r0y−y)Y(2.40)
onD[0,t].ItissufficienttoverifytheconditionsoftheTheoremA.0.4.
Fixε:0<ε<min(r0,r0t)andτ:t<τ<t+ε.Then
1)x=F1iscontinuouslydifferentiableon[0,τ];
2)ϕ(y,r)=t+ry−yiscontinuouson[0,τ]×Uε(r0),non-increasinginyand
ϕ(0,r0)=t<τ,ϕ(τ,r0)=t+r0τ−τ>−ε+r0τ>0;
3)Xn=Fˆ1∈D[0,τ]isasequenceofstochasticprocessessuchthat
√n(Xn−x)→DZ=U1
onD[0,τ],whereZ=U1isacontinuouson[0,τ]stochasticprocess;
4)rˆisasequenceofrandomvariablessuchthat
√n(rˆ−r0)→DY.
SoallconditionsoftheTheoremA.0.4areverified.Thistheoremimpliesthecon-
vergence(2.39)onD[0,τ]andconsequentlyonD[0,t]becausex(ϕ(y,r))=f1(t+
r0y−y),ϕr(y,r0)=y.

52

Weprovethetheorembyinduction.Ifj=2thenusing(2.26),(2.40),the
functionaldeltamethodforintegralsandtheestimatorof(2.18)
sKˆ2(s)=Fˆ1(s+rˆy−y)dKˆ1(y),Kˆ1(y)=Fˆ1(y)
0weobtain√
n(Kˆ2(s)−K2(s))=
√nFˆ1(s+rˆy−y)dFˆ1(y)−F1(s+r0y−y)dF1(y)→D
ss
00sss
→DU1(s+r0y−y)dF1(y)+yf1(s+r0y−y)YdF1(y)+F1(s+r0y−y)dU1(y)=
0s0s0
=U1(s+r0y−y)dF1(y)+µ(1)(s)Y+F1(s+r0y−y)dU1(y)=W2(s)(2.41)
00onD[0,t].So(2.39)istrueforj=2.
Ifj=3thenbythefunctionaldeltamethodforintegralsandusingtheestimator
sKˆ3(s)=Fˆ1(s+rˆy−y)dKˆ2(y),
0inbtaoew√n(Kˆ3(s)−K3(s))=
√ss
=nFˆ1(s+rˆy−y)dKˆ2(y)−F1(s+r0y−y)dK2(y)→D
00sss
→DU1(s+r0y−y)dK2(y)+yf1(s+r0y−y)YdK2(y)+F1(s+r0y−y)dW2(y)=
0s0s0
=U1(s+r0y−y)dK2(y)+µ(2)(s)Y+F1(s+r0y−y)dW2(y)=W3(s)(2.42)
00onD[0,t].So(2.39)istrueforj=3.
Supposingthat(2.38)istrueforj=landusingthefunctionaldeltamethodfor
integralsweobtaintheresultforj=l+1:
√n(Kˆl+1(s)−Kl+1(s))=
√ss
nFˆ1(s+rˆy−y)dKˆl(y)−F1(s+r0y−y)dKl(y)→D
s0s0s
→DU1(s+r0y−y)dKl(y)+yf1(s+r0y−y)YdKl(y)+F1(s+r0y−y)dWl(y)=
0s0s0
=U1(s+r0y−y)dKl(y)+µ(l)(s)Y+F1(s+r0y−y)dWl(y)(2.43)
00onD[0,t].
Theproofiscomplete.

53

√Theasymptoticvarianceofn(Kˆj(t)−Kj(t)),j≥2mightbeestimatedrecur-
rently,usingtheequation(2.39):thecovariances
Cov(Wj(s),Wj(t))=E(Wj(s),Wj(t))
canbewrittenintermsofthecovariances
E(Wj−1(u)Wj−1(v)),E(Wj−1(u)U1(v)),E(Wj−1(u)U2(v)),
E(U1(u)U1(v)),E(U2(u)U2(v)).
Notethatforj=2thesecovariancesare
1E(W1(u)W1(v))=E(W1(u)U1(v))=E(U1(u)U1(v))=l1F1(u∧v)S1(u∨v),
1E(W1(u)V2(v))=E(U1(u)U2(v))=0,E(U2(u)U2(v))=l2F2(u∧v)S2(u∨v).
√Letusfindtheasymptoticvarianceofn(Kˆ2(t)−K2(t))whichcoincideswiththe
varianceofW2(t).
Supposefirstthatsamplesarecomplete.Inthefollowingweskiptheindexinr0
.Using(2.37)weget
ttW2(t)=U1(t+ry−y)dF1(y)+µ(1)(t)Y+F1(t+ry−y)dU1(y)=
00tt=F2(t)U1(t)+U1(t+ry−y)dF1(y)−U1(y)dF1(t+ry−y)+
∞0∞0
+µ(t)U1(ry)dΛ2(y)−U2(y)dΛ2(y)=(V1+V2+V3+V4)(t).(2.44)
A0t0∞
µ(t)=µ(1)(t)=yf1(t+ry−y)dF1(y),A=−1uf1(u)dΛ1(u).(2.45)
r00TherandomvariableW2(t)haszeromean.Set
tν(t)=0F1(t+ry−y)dF1(y).
Takingintoaccountthat0<r<1andusingtheequality(2.27)foranyt≥0
weobtainthevariancesandthecovariancesoftherandomvariablesVi(t)multiplied
:lyb1l1EV12(t)=F22(t)F1(t)S1(t),
ttl1EV22(t)=l1EU1(t+ry−y)U1(t+rz−z)dF1(y)dF1(z)=
t0y0
=2S1(t+rz−z)dF1(z)F1(t+ry−y)dF1(y)=
0054

tt2
=2F1(z)F1(t+rz−z)dF1(z)−F1(t+rz−z)dF1(z)=
00t=2F1(z)F1(t+rz−z)dF1(z)−ν2(t),
0ttl1EV32(t)=l1EU1(y)U1(z)dF1(t+ry−y)dF1(t+rz−z)=
t0y0
=2F1(z)dF1(t+rz−z)S1(y)dF1(t+ry−y)=
t00t
=2F2(t)F1(z)dF1(t+rz−z)−2F1(z)F1(t+rz−z)dF1(t+rz−z)−
002t−F1(z)dF1(t+rz−z)=2F2(t)(F1(t)F2(t)−
0t−ν(t))−2F1(z)F1(t+rz−z)dF1(t+rz−z)−(F1(t)F2(t)−ν(t))2,
02l1EV42(t)=l1µ2(t)EY2=µ(t2),
Al2tl1EV1(t)V2(t)=l1F2(t)EU1(t+ry−y)U1(t)dF1(y)=
0t=S1(t)F2(t)F1(t+ry−y)dF1(y)=S1(t)F2(t)ν(t),
0tl1EV1(t)V3(t)=−l1F2(t)EU1(y)U1(t)dF1(t+ry−y)=
0t=−S1(t)F2(t)0F1(y)dF1(t+ry−y)=−S1(t)F2(t)(F1(t)F2(t)−ν(t)),
∞t/rl1EV1(t)V4(t)=l1F2(t)µ(t)[EU1(ry)U1(t)dΛ2(y)+EU1(ry)U1(t)dΛ2(y)]=
At/r0∞t/r=F2(t)µ(t)[S1(t)F1(ry)dΛ2(y)+F1(t)S1(ry)dΛ2(y)]=
At/r0∞t/r=−F2(t)µ(t)[S1(t)[1−1]dS2(y)+F1(t)dS2(y)]=
A0S2(y)t/r
=−F2(t)µ(t)S1(t)lnS1(t),
A

55

tt−l1EV2(t)V3(t)=l1EU1(t+ry−y)U1(z)dF1(y)dF1(t+rz−z)=
00tt+ry−y
=S1(t+ry−y)F1(z)dF1(t+rz−z)+
00t+F1(t+ry−y)S1(z)dF1(t+rz−z)dF1(y)=
t+ry−y
tt+ry−y
=F1(z)dF1(t+rz−z)dF1(y)−
00tt+ry−y
−F1(z)dF1(t+rz−z)F1(t+ry−y)dF1(y)+
00tt
+dF1(t+rz−z)F1(t+ry−y)dF1(y)−
0t+ry−y
tt
−F1(z)dF1(t+rz−z)F1(t+ry−y)dF1(y)=
0t+ry−y
tu
=−F1(z)dF1(t+rz−z)dF1((t−u)/(1−r))+
0trt+(F1(rt)−F1(t+(r−1)(t+ry−y)))F1(t+ry−y)dF1(y)−
0tt
−F1(z)dF1(t+rz−z)F1(t+ry−y)dF1(y)=
00rtt
=−dF1((t−u)/(1−r))F1(z)dF1(t+rz−z)−
tr0tt
−dF1((t−u)/(1−r))F1(z)dF1(t+rz−z)+
ztrt+(F1(rt)−F1(t+(r−1)(t+ry−y)))F1(t+ry−y)dF1(y)−
0tt−F1(t+ry−y)dF1(y)F1(z)dF1(t+rz−z)=
00tr=F1(t)F1(z)dF1(t+rz−z)+
0tt+F1((t−z)/(1−r))F1(z)dF1(t+rz−z)+F2(t)F1(t+ry−y)dF1(y)−
0trt−F1(t+(r−1)(t+ry−y))F1(t+ry−y)dF1(y)−
0tt−F1(t+ry−y)dF1(y)F1(z)dF1(t+rz−z)=
0056

tr=F1(t)0F1(z)dF1(t+rz−z)+
t+F1((t−y)/(1−r))F1(y)dF1(t+ry−y)+F2(t)ν(t)+
trt+F1(t+rz−z))F1(z)dF1((t−z)/(1−r))−ν(t)(F1(t)F2(t)−ν(t),
tr∞tl1EV2(t)V4(t)=l1µ(t)EU1(t+ry−y)U1(rz)dF1(y)dΛ2(z)=
A001)t(µt(t+ry−y)/r
=−AS1(t+ry−y)dF1(y)S2(y)−1dS2(y)+
00∞t+F1(t+ry−y)dF1(y)dS2(y)=
0(t+ry−y)/r
tA=−µ(t)S1(t+ry−y)[lnS1(t+ry−y)−S1(t+ry−y)+1]dF1(y)−
0t−F1(t+ry−y)S1(t+ry−y)dF1(y)=
0t=−µ(t)S1(t+ry−y)lnS1(t+ry−y)dF1(y),
A0µ(t)ty/r∞
Al1EV3(t)V4(t)=−l1EU1(y)dF1(t+ry−y)+U1(rz)dΛ2(z)=
/ry00/ryµ(t)t1
=AS1(y)dF1(t+ry−y)S2(z)−1dS2(z)+
00t∞t
A0+F1(y)dF1(t+ry−y)dS2(z)=µ(t)S1(y)lnS1(y)dF1(t+ry−y).
/ry0SothevarianceVar(W2(t))isdefinedbythefollowingformula:
tl1Var(W2(t))=−F1(t)F22(t)−4ν2(t)+F1(t+ry−y)[F1(t+ry−y)+2F1(y)]dF1(y)+
0

57

tµ2(t)
+2F1(t)ν(rt)+2F1(t+ry−y)F1((t−y)/(1−r))dF1(y)+2+
trtl2A
+2µ(t)ν(t)+[F1(t+ry−y)lnS1(y)−
A0−S1(t+ry−y)lnS1(t+ry−y)]dF1(y))].
Setn1Z1i=Fˆ1(t+(rˆ−1)T1i−),Fˆ1(t−)=11{T1i<t},Z2i=Fˆ1(t−T1i−),
n1i=11−ˆr
Z3i=Fˆ1(T1i−),Z4i=fˆ1(t+(rˆ−1)T1i−),µˆ(t)=1T1iZ4i,Z5i=fˆ1(T1i−).
n1T1i≤t
ThevarianceVar(W2(t))isestimatedusingthestatistic
2n1Vˆar(W2(t))=−Fˆ1(t)Fˆ22(t)−4φˆ12(t)+φˆ2(t)+nˆµ(t)+2µˆ(t)φˆ3(t);
nn2Aˆ2Aˆ
here

φˆ1(t)=Z1i,Aˆ=−1i5i,
11n1TZ
n1T1i≤trˆn1i=11−Z3i
1φˆ2(t)=Z1i[Z1i+2Z3i+2Fˆ1(t)1{T1i≤rˆt}+2Z2i1{T1i>rˆt}],
n1T1i≤t
1φˆ3(t)=[Z1i(1+ln(1−Z3i))−(1−Z1i)ln(1−Z1i)].
n1T1i≤t
Sothevarianceσ2Kˆ2oftheestimatorKˆ2(t)isestimatedby
σˆKˆ2=Vˆar(W2(t))=Vˆar(W2(t)).
211n1
nnn1Theasymptotic1−αconfidenceintervalforK2(t)is
Kˆ2(t)±σˆKˆ2z1−α/2.(2.46)
Alternativeasymptoticconfidenceintervaloftheform(K2(t),K2(t)),where
ˆ−1
2K(t)=1+1−K2(t)expσˆKˆ2z1−α/2,
Kˆ2(t)Kˆ2(t)(1−Kˆ2(t))
−1
ˆσˆz
K2(t)=1+1−K2(t)exp−Kˆ21−α/2,(2.47)
Kˆ2(t)Kˆ2(t)(1−Kˆ2(t))
58

considered.ebcanRemark2.3.2.Inthetcaseofcensoringtheexpressioninparenthesisoftheterm
1V4in(2.44)isreplacedby[U1(ru)−U2(u)dΛ2(u)+U1(rt1)+U2(t1),soonlyminor
0modificationsareneeded.
Example2.3.1.Exponentialdistribution:S1(t)=e−λt.Weinvestigatedfinite
sampleconfidenceleveloftheproposedasymptoticconfidenceintervals.Thefailure
timesT1jandT2jweresimulatedfromexponentialdistribution:
11T1j∼E(λ1),T2j∼E(λ2),λ1=100,λ2=300,t1=500.
Thenumberofreplicationswas2000.Forvariousvaluesofttheproportionsof
confidenceintervalrealizationscoveringthetruevalueofthedistributionalfunction
K2(t)aregivenbelow:
Table2.11.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K2(t)0.1140.3190.6670.8560.9410.977
Confidencelevel(%)94.591.991.890.490.789.2

Example2.3.2.Weibulldistribution:S1(t)=e−(t/β)α.ThefailuretimesT1j
andT2jweresimulatedfromexponentialdistribution:
T1j∼W(α1,β1),T2j∼W(α1,β1),

α1=α2=2,β1=100,β2=300,t1=500.
Thenumberofreplicationswas2000.Forvariousvaluesofttheproportionsof
confidenceintervalrealizationscoveringthetruevalueofthedistributionalfunction
K2(t)aregivenbelow:
Table2.12.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K2(t)0.0180.1940.8220.9920.9991.000
Confidencelevel(%)88.991.790.790.089.489.9

1T1jExamplandTe2j2.w3.ere3.simLoulaglotedgisticfromedistrxponenibutitiaonl:S1distribut(t)=ion:1+(t/β)α.Thefailuretimes
T1j∼L(α1,β1),T2j∼L(α1,β1),

α1=α2=2,β1=100,β2=300,t1=500.
59

Thenumberofreplicationswas2000.Forvariousvaluesofttheproportionsof
confidenceintervalrealizationscoveringthetruevalueofthedistributionalfunction
K2(t)aregivenbelow:
Table2.13.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K2(t)0.0160.1380.5170.7430.8510.905
Confidencelevel(%)92.191.791.389.290.591.5
2.3.2Parametriccase
Letusconsidertheparametricestimator(2.25).DenotebyIn(γ)=−E¨(γ)the
Fisherinformationmatrixandsupposethat1In(γ)→i(γ).Underclassicalassump-
tionsonthefamilyofdistributionsf1(t,θ)nthemaximumlikelihoodestimatorγ∗is
rmal:nollyoticaasympt√n(γ∗−γ)→dY=(Y1,Y2T)T∼Nk+1(0,i−1(γ)).
Y1isone-dimensional,Y2–k-dimensional.
Usingdeltamethodweobtain:
√n(Kˆ2(t)−K2(t))→DW2(t)=YTC2(t;γ),
wheret∂C2(t;γ)=(C21(t;γ),C22T(t;γ))T,C21(t;γ)=∂rF1(t+ry−y;θ)dF1(y;θ),
0t∂∂C22(t;γ)=0∂θF1(t+ry−y;θ)dF1(y;θ)+F1(t+ry−y;θ)d(∂θF1(y;θ)).
TherandomvariableW2(t)islinearfunctionofY.
then2jIf≥√n(Kˆj(t)−Kj(t))→DWj(t).
LetusprovebyrecurrencethattherandomvariableWj(t),j≥2,isalsolinear
Y:offunctionWj(t)=YTCj(t;γ),Cj(t;γ)∈(C[0,t])k+1.
Weshowedthatitistruefork=2.Byfunctionaldeltamethodforintegrals
(Theorem1.3.2)andusingtheassumptionthatthestatementistrueforWj−1we
ainobtt∂Wj(t)=YT(0∂γF1(t+ry−y;θ)dKj−1(y;γ)+F1(t+ry−y;θ)dCj−1(t;γ)).
rianceavtheSoVar(Wj(t))=Var(Cj(t;γ)TY)=CjT(t;γ)i−1(γ)Cj(t;γ)
60

.48)(2

isestimatedbynC2T(t;γˆ)I−1(γˆ)Cj(t;γˆ),andthevarianceσ2ˆKj(t)oftheestimatorKˆj(t)
ybtedestimaisσˆ2Kˆj(t)=CjT(t;γˆ)I−1(γˆ)Cj(t;γˆ).
ThematrixI(γˆ)maybereplacedby−¨(γˆ).
Theasymptotic1−αconfidenceintervalforKj(t)is
Kˆj(t)±σˆKˆj(t)z1−α/2,(2.48)
or,alternatively,(Kj(t),Kj(t)),where
−1
Kj(t)=1+1−Kˆj(t)expσˆKˆjz1−α/2,
Kˆj(t)Kˆj(t)(1−Kˆj(t))
−1
Kj(t)=1+1−Kˆj(t)exp−σˆKˆjz1−α/2.(2.49)
Kˆj(t)Kˆj(t)(1−Kˆj(t))
Example2.3.4.Exponentialdistribution:S1(t)=e−λt.

Letusconsiderthecaseofcompletesamples.By(2.25)theloglikelihoodfunction
rmfothehasl(r;θ)=lnf1(T1i;θ)+n2lnr+lnf1(rT2j;θ).
n1n2
=1j=1iInthecaseofexponentialdistribution
f1(t;λ)=λe−λt;lnf1(t;λ)=lnλ−λt,
son1n2
l(r;λ)=n1lnλ−λT1i+n2lnr+n2lnλ−λrT2j=
=1j=1in1n2
=nlnλ+n2lnr−λ(T1i+rT2j).
=1j=1iEquatingthescorefunctiontozeroweobtainthesystemofequations
˙r=∂l=n2−λT2j=0,˙λ=∂l=n−T1i−rT2j=0.
n2n1n2
∂rrj=1∂λλi=1j=1
Sotheestimatorsoftheparametersrandλare:
11T1ˆrˆ=T2;λ=rˆT2=T1.
61

Secondpartialderivativesare
∂r2=−r22;∂λ2=−λ2;∂λ∂r=−T2j,
∂2ln∂2ln∂2ln2
=1jsotheFisherinformationmatrixandtheinverseoftheFishermatrixare
I(r;λ)=nr22nλr;I−1(r;λ)=n1n2λˆrˆ−λˆ2n1.
n2n2nrˆ2λˆrˆ
λrλ2−n1n1
Ifn1andn2arelargethen−1thedistributionoftheestimator(ˆr,λˆ)isapproximated
bythenormalN((r,λ),I(r,λ)).
Takingintoconsiderationtheequality
Kj(t)=1+ir(−1)Cj−11+ir.
j−11j−1ii1−e−λ(1+ir)t
=0i=1itheweightsCj=(Cj1,Cj2)Tcanbecomputed:
j−1j−1j−1−λ(1+ir)t
Cj1(t;r,λ)=∂Kj(t)=1+1−11(−1)iCji−11−e+
∂ri=1irri=11+iri=01+ir
+(−1)Cj−1(1+ir)2,
j−1iie−λ(1+ir)t[iλt(1+ir)+i]−i
=0iCj2(t;r,λ)=∂Kj(t)=t1+1(−1)iCji−1e−λ(1+ir)t.
j−1j−1
∂λi=1iri=0
TheestimatorofvarianceoftheestimatorKˆj(t)is
σˆ2Kˆj(t)=CjT(t;rˆ,λˆ)I−1(rˆ,λˆ)Cj(t;rˆ,λˆ).
andtheasymptotic1−αconfidenceintervalforKj(t)hastheform(2.48)or,alter-
natively,(2.49).
Inthecasej=2theestimatorofthefunction
K2(t)=1−e−λt+1(e−λ(1+r)t−e−λt)=F1(t)−S1(t)F2(t);
rrisKˆ2(t)=Fˆ1(t)−Sˆ1(t)Fˆ2(t)=1−e−ˆλt+1(e−λˆ(1+rˆ)t−e−ˆλt).
rˆrˆSo

C21(t;r,λ)=S12(t)(F2(t)−rλtS2(t)),C22(t;r,λ)=(1+r)tS1(t)F2(t).
rr62

Soweobtained

Soweobtained
C2(t;r,λ)=(C21(t;r,λ),C22(t;r,λ))T=
=S12(t)(F2(t)−rλtS2(t)),(1+r)tS1(t)F2(t).
rr
VKˆ2(t)≈∂K2,∂K2I−1(λ,r)∂∂KrK22
λ∂r∂λTheestimatorofthevarianceoftheestimatorKˆ2(t)is
σˆ2Kˆ2(t)=C2T(t;rˆ,λˆ)I−1(rˆ,λˆ)C2(t;rˆ,λˆ)=
=Sˆ1ˆ(t)2lˆ1Fˆ2(t)−λˆrˆtSˆ2(t)+lˆ2(1−ˆλt)Fˆ2(t)−λˆrˆt;
ˆ222
nl1l2r
herelˆi=ni/n.
Wefoundbysimulationfinitesampleconfidencelevelsoftheintervalsobtained
usingasymptoticformulaswith1−α=0.9.ThefailuretimesT1jandT2jwere
simulatedfromexponentialdistributionwithfollowingparameters:
T1j∼E(λ1),T2j∼E(λ2),
11λ1=100,λ2=300.
Thenumberofreplicationswas2000.Forvariousvaluesofttheproportionsof
confidenceintervalrealizationscoveringthetruevalueofthedistributionalfunction
K2(t)aregivenbelow:
Table2.14.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K2(t)0.1140.3190.6670.8560.9410.977
Confidencelevel(%)89.989.488.990.290.091.5
Forvariousvaluesofttheproportionsofconfidenceintervalrealizationscovering
thetruevalueofthedistributionalfunctionK3(t)aregiveninTable2.15.
Table2.15.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K3(t)0.0180.0920.3090.4790.5730.617
Confidencelevel(%)88.691.892.890.889.990.3

63

here

Graphestimato2.7.rsGFˆr1,aphsKˆ2,oKˆf3,theKˆ4,tra(Expjectorieonensoftialthedistrparametribution)ic

Example2.3.5.Weibulldistribution:S1(t)=e−(t/µ)ν.Letusconsiderthe
caseofcompletesamples.By(2.25)theloglikelihoodfunctionhastheform
l(r;θ)=lnL(r,θ)=lnf1(T1i;θ)+n2lnr+lnf1(rT2j;θ);
n1n2
=1j=1ihereν−1tν
f1(t;ν,µ)=µνµte−(µ);
νtlnf1(t;ν,µ)=lnν−lnµ+(ν−1)(lnt−lnµ)−µ=
ν=lnν−νlnµ+(ν−1)lnt−t;
µlnf1(T1i;ν,µ)=lnν−νlnµ+(ν−1)lnT1i−T1i=
n1n1ν
µ=1i=1in1n1ν
=n1lnν−n1νlnµ+(ν−1)lnT1i−T1i;
µ=1i=1iSointhecaseofWeibulldistributiontheloglikelihoodfunctionis
n1n2
l(r;θ)=n(lnν−νlnµ)+νn2lnr+(ν−1)lnT1i+lnT2j−
=1j=1i1−µνT1νi+rνT2νj.
n1n2
=1j=1i

64

Equatingthescorefunctiontozerothefollowingsystemofequationsisobtained:
ν˙r=∂r=r−µνT2j=0;
∂ln2ννrν−1n2
=1j˙ν=∂ν=ν−nlnµ+n2lnr+lnT1i+lnT2j+
∂lnn1n2
=1j=1i+lnµT1νi+rνT2νj−1T1νilnT1i+rνlnrT2νj+
n1n2n1n2
ννµi=1j=1µi=1j=1
+rνT2νjlnT2j=n−nlnµ+n2lnr+lnT1i+lnT2j+
n2n1n2
j=1νi=1j=1
+1νlnµT1νi+rνT2νj−T1νilnT1i−rνlnrT2νj−
n1n2n1n2
µi=1j=1i=1j=1
n2−rνT2νjlnT2j=0;
=1j˙∂lnννn1ννn2ν
µ=∂µ=−µ+µν+1T1i+rT2j=0.
=1j=1iResolvingthissystemofequationsweobtainthattheestimatorsµˆandrˆareexplicit
functionsoftheestimatorνˆ:
TνˆTνˆ
n11/νˆn11/νˆ
i=11in2i=11i
11T2νˆj
µˆ=n,r=nn2.
=1jTheestimatorνsatisfiedtheequation:
n1n2n1νn2TνlnT
νˆi=1j=1i=1T1ij=1T2j
n+lnT1i+lnT2j−n1i=1nT11ilnT1i−n2j=1n22j2j.
Secondpartialderivativesoftheloglikelihoodfunctionare
¨r2=∂l=−n2ν−ν(ν−1)rTν;
2ν−2n2
∂r2r2µν2j
=1jnnrν=∂r∂ν=r−µνT2j−µνT2jlnT2j;
¨∂2ln2rν−1+νrν−1lnr−νrν−1lnµ2ννrν−12ν
=1j=1j

65

¨rµ=∂r∂µ=µν+1T2νj;
∂2lν2rν−1n2
=1jnn¨ν2=∂ν2=−ν2−µνlnµlnµT1i+rT2j−
∂2ln11νν2ν
=1j=1i−T1νilnT1i−rνlnrT2νj−rνT2νjlnT2j+
n1n2n2
i=1j=1j=1
+1νlnµT1νilnT1i+rνlnrT2νj+rνT2νjlnT2j−
n1n2n2
µi=1j=1j=1
−T1νiln2T1i−rνln2rT2νj−rνlnrT2νjlnT2j−
n1n2n2
i=1j=1j=1
−rνlnrT2νjlnT2j−rνT2νjln2T2j;
n2n2
=1j=1j¨µν=∂l=−n+µ−νµlnµT1νi+rνT2νj+
2ν+1ν+1n1n2
∂µ∂νµµ2(ν+1)
=1j=1innn+µν+1T1ilnT1i+rlnrT2j+rT2jlnT2j;
ν1νν2νν2ν
i=1j=1j=1
¨µ2=nν−ν(ν+1)T1νi+rνT2νj.
n1n2
µ2µν+2i=1j=1
νTherandomvariablesX1i=Tµ1ihavethestandardexponentialdistribution,i.e.
X1i∼E(1).Weobtain
/ν1T1i=µX1i,T1νi=µνX1i,
νµ1T1νilnT1i=µνX1i(lnµ+lnX1i)=µνlnµX1i+X1ilnX1i;
ννννµµ2T1νiln2T1i=µνln2µX1i+lnµX1ilnX1i+2X1iln2X1i;
ννTakingintoaccountthatX1i∼ε(1)wehave
∞EX1i=1,EX1ilnX1i=xlnxe−xdx=Γ(2),
0∞EX1νiln2X1i=xe−xln2xdx=Γ(2);
0

66

because∞∞
Γ(a)=xa−1e−xdx;Γ(k)(a)=xa−1e−xlnkxdx.
00SoET1νi=µν,
νET1νilnT1i=EµX1ilnX1i+µνlnµX1i=
νν=µ[Γ(2)+νlnµ];
νν2νET1νiln2T1i=µΓ(2)+2νΓ(2)lnµ+ν2ln2µ.
TherandomvariablesrT2jandT1ihavethesamedistribution,so
νµ1ET2νi=Eν(rT2i)ν=,
rr11T2νjlnT2j=ν(rT2j)ν[ln(rT2j)−lnr]=ν((rT2j)νln(rT2j)−lnr(rT2j)ν);
rrET2νjlnT2j=ν(Γ(2)+νlnµ)−µνlnr=νΓ(2)+νln.
1µνµνµ
rνrνrνET2νjln2T2j=1µΓ(2)+2νΓ(2)lnµ+ν2ln2µ.
2rrrνUsingtheobtainedmeanswecomputethesecondpartialderivativesoftheloglikeli-
hoodfunction:ν−2n2
rr2µν2j
E¨2=−En2ν+ν(ν−1)rTν=
=1j=−2+νn2=−2;
n2νν(ν−1)rν−2µνν2n2
rrµrνE¨rµ=Eµν+1T2j=µν+1r=rµ;
ν2rν−1n2ν2rν−1n2µνn2ν2
=1jErν=E−νT2j−νT2jlnT2j=
¨n2rν−1+νrν−1lnr−νrν−1lnµn2ννrν−1n2ν
rµj=1µj=1
=r−µνn2r−µννr[Γ(2)+νlnr]=
n2rν−1+νrν−1lnr−νrν−1lnµµννrν−1n2µνµ
µn2(1+νlnr−νlnµ)n2(Γ(2)+νlnr)n2n2
=r−r−r=−rΓ(2);
E¨µ2=E2−ν+2T1νi+rνT2νj=
nνν(ν+1)n1n2
µµi=1j=1
67

=µ2−µν+2n1µ+rn2r=µ2−µ2=−µ2;
nνν(ν+1)ννµνnνnν(ν+1)nν2
E¨µν=E−+ν+1T1νi+rνT2νj+
n1−νlnµn1n2
µµi=1j=1
ν+ν+1T1νilnT1i+rνlnrT2νj+rνT2νjlnT2j=
n1n2n2
µi=1j=1j=1
=−n+1−νlnµ(nµν)+
+1νµµ+ν+11[Γ(2)+νlnµ]+rνlnrn2+rν2[Γ(2)+νln]=
νnµνµνnµνµ
µνrνrr
=−n+n(1−νlnµ)+1n1[Γ(2)+νlnµ]+νn2lnr+n2[Γ(2)+νlnµ]=
rµµµ)(2Γn;=µE¨ν2=E−2−νlnµlnµ(T1νi+rνT2νj)−T1νilnT1i−
n1n1n2n1
νµi=1j=1i=1
rνlnrT2νj−rνT2νjlnT2j+
n2n2
=1j=1j1+νlnµ(T1νilnT1i+rνlnrT2νj+rνT2νjlnT2j)−
n1n2n2
µi=1j=1j=1
−T1νiln2T1i−rνln2rT2νj−rνlnrT2νjlnT2j−
n1n2n2
i=1j=1j=1
−rνlnrT2νjlnT2j−rνT2νjln2T2j=−2(1+Γ(2)).
n2n2n
ν=1j=1jThentheFisherinformationmatrixis
22nr22ν−nr2µνnr2Γ(2)
µµµrI(r;µ,ν)=−n2ν2nν22−nΓ(2).
nr2Γ(2)−µnΓ(2)νn2(1+Γ(2))
2222|I|=n2νnνn(1+Γ(2))+n2νnΓ(2)n2Γ(2)+n2νnΓ(2)n2Γ(2)−
r2µ2ν2rµµrrµµr
2222−n2Γ(2)nνn2Γ(2)−n2νn2νn(1+Γ(2))−n2νnΓ(2)nΓ(2)=
rµ2rrµrµν2r2µµ

68

n2ν2n2n22nν22n22ν2n2
=r2µ2(1+Γ(2))+µ2r2(Γ(2))+r2µ2(Γ(2))−
n22nν22n22ν2nn2n2ν22
−r2µ2(Γ(2))−r2µ2(1+Γ(2))−r2µ2(Γ(2))=
=12221+Γ(2)−[Γ(2)]2,
nnnν2
µrTheinverseoftheFisherinformationmatrixis
22nr2rµ0
1−n1rnµ2νµ2n[1+Γ(2)n]1−νµ2n2[Γ(2)]2µΓ(2)
µΓ(2)ν2
I=n1ν2n1nν2(1+Γ(2)−[Γ(2)]2)n(1+Γ(2)−[Γ(2)]2).
0n(1+Γ(2)−[Γ(2)]2)n(1+Γ(2)−[Γ(2)]2)
Thec.d.f.K2hastheform
tK2(t)=F1(t)−S1(t+ry−y)dF1(y)=
0=1−e−(µt)ν−νye−(t+rµy−y)ν−(µy)νdy,
tν−1
µµ0andthefunctionsC2iare
∂K2(t)ν2tyt+ry−y−(t+rµy−y)ν−(µy)ν
νν−1
C21(t)=∂r=µµµedy,
0ννC22(t)=∂K2(t)=−νte−(µt)+
µµµ∂µµν2tyν−1yνt+ry−yν−(t+ry−y)ν−(y)ν
+µ2µ1−µ−µedy,
0C23(t)=∂K2(t)=te−(µt)lnt−1ye−(t+rµy−y)−(µy)dy+
ννtν−1νν
∂νµµµ0µ
νtyν−1yνyt+ry−yνt+ry−y
+µµµlnµ+µlnµ×
0−(t+ry−y)ν−(y)ννtyν−1y−(t+ry−y)ν−(y)ν
×eµµdy−lneµµdy,
µµµ0evhaeWσˆ2Kˆ2(t)=C2T(t;rˆ,ˆµ,νˆ)I−1(rˆ,ˆµ,νˆ)C2(t;rˆ,ˆµ,νˆ),
C2(t;rˆ,ˆµ,νˆ)=(C21(t;rˆ,ˆµ,νˆ),C22(t;rˆ,ˆµ,νˆ),C23(t;rˆ,ˆµ,νˆ))T.
Theasymptotic1−αconfidenceintervalforKj(t)hastheform(2.48)or,alternatively,
(2.49)thenj=2.

69

Wefoundbysimulationfinitesampleconfidencelevelsoftheintervalsobtained
usingasymptoticformulaswith1−α=0.9.WesimulatingfailuretimesT1jandT2j
fromWeibulldistributionwithfollowingparameters:
T1j∼W(α1,β1),T2j∼W(α1,β1),
α1=α2=2,β1=100,β2=300.

Graph2.8.Graphsofthetrajectoriesoftheparametric
estimatorsFˆ1,Kˆ2(Weibulldistribution)
Thenumberofreplicationswas2000.Forvariousvaluesofttheproportionsof
confidenceintervalrealizationscoveringthetruevalueofthedistributionalfunction
K2(t)aregivenbelow:
Table2.16.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K2(t)0.0180.1940.8220.9920.9991.000
Confidencelevel(%)89.489.289.289.589.590.2
Example2.3.6.Loglogisticdistribution:S1(t)=1+(1t/µ)ν.Letusconsiderthe
caseofcompletesamples.By(2.25)theloglikelihoodfunctionhastheform
l(r;θ)=lnL(r,θ)=lnf1(T1i;θ)+n2lnr+lnf1(rT2j;θ).
n1n2
=1j=1iInthecaseofloglogisticdistribution
1−νtν+1µf1(t;µ,ν)=νtν2;
µνlnf1(t;µ,ν)=lnν+(ν−1)lnt−νlnµ−2ln1+µt,
70

lnf1(t;µ,ν)=n1lnν+(ν−1)lnT1i−νn1lnµ−2ln1+T1i,
n1n1n1ν
µtri=1i=1i=1ν
lnf1(rt;µ,ν)=lnν+(ν−1)(lnr+lnt)−νlnµ−2ln1+.
µSotheloglikelihoodfunctionhastheform
n1n2
l(r;µ,ν)=nlnν−νnlnµ+νn2lnr+(ν−1)lnT1i+lnT2j−
=1j=1i−2ln1+−2ln1+.
n1T1iνn2rT2iν
i=1µj=1µ
Partialderivativesare
n2rT2jν
∂rrrj=11+rTµ2j
˙r=∂l=νn2−2νµν=0;
n1T1iνn2rT2jν
∂µµµi=11+Tµ1iµj=11+rTµ2j
˙µ=∂l=−νn+2νµν+2νµν=0;
˙ν=∂ν=ν−nlnµ+n2lnr+lnT1i+lnT2j−
∂lnn1n2
=1j=1iνν
n1Tµ1ilogTµ1in2rTµ2jlogrTµ2j
i=11+Tµ1ij=11+rTµ2j
−2ν−2ν=0.
Xi=T1iYj=,rT2i.
Setνν
µµSecondpartialderivativesoftheloglikelihoodfunctionare
¨νn22νn2Yj2ν2n2Yj¨2ν2n2Yj
r2=−r2+r21+Yj−r2(1+Yj)2,rµ=rµ(1+Yj)2,
j=1i=1j=1
¨rν=n2−2Yj−2ν1YjlnYj,
n2n2
2rrj=11+Yjrνj=1(1+Yj)
¨nν2νn1Xin2Yi2ν2n1Xin2Yi
µ2=µ2−µ21+Xi+1+Yi−µ2(1+Xi)2+(1+Yi)2;,
i=1j=1i=1j=1
¨µν=−++++,
n2n1Xin2Yj2n1XilnXin2YilnYi
µµi=11+Xij=11+Yjµi=1(1+Xi)2j=1(1+Yi)2
71

nnν2ν2i=1(1+Xi)2j=1(1+Yi)2
¨ν2=−n−21Xiln2Xi+2Yiln2Yi.
TherandomvariablesXiandYjareidenticallydistributedwiththeprobability
densityfunctionofthestandardloglogisticdistribution:fXi(x)=fYj(x)=1/(1+x)2.
Itimpliesthatforanyk>−2anda∈(−1,k+1)
∞aag(a)=E(1+XiX)k=(1+xx)k+2dx=|y=1+1x;x=y1−1;dx=−ydy2|=
i01a1
=1−1ykdy=yk−a(1−y)ady=Γ(k−a+1)Γ(a+1).
0y0Γ(k+2)
XalnX∞xa
g(a)=E(1i+Xi)ik=(1+x)k+2lnxdx=
0=(Γ(k−a+1)Γ(a+1))a=−Γ(k−a+1)Γ(a+1)+Γ(k−a+1)Γ(a+1);
Γ(k+2)Γ(k+2)
Xaln2X∞xa
g(a)=E(1i+Xi)ki=(1+x)k+2ln2xdx=
0=Γ(k−a+1)Γ(a+1)−2Γ(k−a+1)Γ(a+1)+Γ(k−a+1)Γ(a+1).
)2+kΓ(Ifa=1andk=1,then
EXi=Γ(1)Γ(2)=1,EXilnXi=−Γ(1)+Γ(2);
1+XiΓ(3)21+Xi2
ifa=1andk=2,then

XiΓ(2)Γ(2)1
E(1+Xi)2=Γ(4)=6,
EXilnXi2=0,EXiln2X2i=Γ(2)−[Γ(2)]2,
(1+Xi)(1+Xi)3
ifa=2andk=2,then
EXi2=Γ(1)Γ(3)=1,EXi2lnXi=−2Γ(1)+Γ(3).
(1+Xi)2Γ(4)3(1+Xi)26
Nowweareabletocomputethemeansofthesecondpartialderivativesofthelog-
likelihoodfunction:
E¨r2=−ν2n2+22νn2−2ν22n2=−n2ν22,
rr2r63r

72

2νn2νn2ν2n22n22
E¨rµ==,−E¨rν=−−0=0;
r6µ3rµr2rr
nν2νn1+n22ν2n1+n2nνnνnν2nν2
¨Eµ2=µ2−µ22−µ26=µ2−µ2−3µ2=−3µ2,
nnnE¨µν=−++0=0,−E¨ν2=(3+2Γ(2)−2(Γ(2))2);
2ν3µµSotheFisherinformationmatrixis
n2ν22−n2ν20
nννn3r232rµ
n{3+2Γ(2)−2[Γ(2)]2}
I(r,λ)=−32rµ3µ20
002ν3andtheinverseoftheFisherinformationmatrixis
n1n2ν2n1ν20
3nr23rµ
2n1νn1ν2
ν3I−1=3rµ23µ20,
00n{3+2Γ(2)−2[Γ(2)]2}
2andtheestimatorofσˆKˆj(t)hastheform
1−T2σˆKˆj(t)=Cj(t;rˆ,ˆµ,νˆ)I(rˆ,ˆµ,νˆ)Cj(t;rˆ,ˆµ,νˆ).(2.50)
Theasymptotic1−αconfidenceintervalforKj(t)hastheform(2.48)or,alternatively,
..49)(2Takingintoaccountthat

1+1S1(t)=tν,
µtheMLestimatorofthereliabilityfunctionK2hastheform
tKˆ2(t)=Fˆ1(t)−Sˆ1(t+rˆy−y)dFˆ1(y)=
0t1νˆyνˆ−111
1+µˆ01+µˆ1+µˆy
=1−tνˆ−µˆµˆt+rˆy−yνˆνˆ2dy.
andthefunctionsC2iare
C21(t)=∂K2(t)=
r∂2tyνt+ry−yν−1yν−2t+ry−yν−2
=ν0µµ1+µ1+µdy,
73

C22(t)=∂K2(t)=−νt1+t+
νν−2
µµµµ∂yν−1y(ν−1)
tνµνµ(ν−1)
µ1+µ1+µµ1+µ1+µ
+02t+ry−yνyν2+2t+ry−yνyν2−
νµµ2νµµ
2y(ν−1)t+ry−yν2y(ν−1)yν
µ1+µ1+µµ1+µ1+µ
−2t+ry−yν2yν2−2t+ry−yνyν3dy=
1−ννtνtν−2tν2µy
µ1+µ1+µ
=−µµ1+µ−02t+ry−yνyν2×
1−µ−2µµ
yνyνt+ry−yν
1+µ1+µ
×t+ry−yνyνdy,
∂K2(t)tνtν−2t
C23(t)=∂ν=µ1+µlnµ−
lnνµµµtyν−1yν−1y
µ1+µ1+µµ1+µ1+µ
−0t+ry−yνyν2−t+ry−yνyν2+
νµµlnµνµµlnµ
yν−1t+ry−yνt+ry−yyν−1yνy
µ1+µ1+µµ1+µ1+µ
+t+ry−yν2yν2+2t+ry−yνyν3dy=
tttνν−2
=µ1+µlnµ−
1tyν−1yν−2t+ry−yν−1
−µµ1+µ1+µ×
0×1+y−νlny1−y1+y−
νν−1
µµµµt+ry−yνt+ry−yν−1t+ry−y
−νµ1+µlnµdy,
Sotheestimatorˆσ2Kˆ2(t)hastheform(2.50)andtheasymptotic1−αconfidence
intervalforK2(t)isoftheform(2.48)or,alternatively,(2.49)takingj=2.

74

Wefoundbysimulationfinitesampleconfidencelevelsoftheintervalsobtained
usingasymptoticformulaswith1−α=0.9.WesimulatingfailuretimesT1jandT2j
fromloglogisticdistributionwithfollowingparameters:
T1j∼L(α1,β1),T2j∼L(α1,β1),
α1=α2=2,β1=100,β2=300.

Graph2.9.Graphsofthetrajectoriesoftheparametric
estimatorsFˆ1,Kˆ2(Loglogisticdistribution)
Thenumberofreplicationswas2000.Forvariousvaluesofttheproportionsof
confidenceintervalrealizationscoveringthetruevalueofthedistributionalfunction
K2(t)aregivenbelow:
Table2.16.Confidencelevelforfinitesamples(n1=n2=100)
Time,t50100200300400500
K2(t)0.0160.1380.5170.7430.8510.905
Confidencelevel(%)89.088.890.489.689.590.5

75

3hapterC

Failure-TimeDegradationModels

3.1FailureDegradationModelwithcovariates
Forfailuretireliabilitmesy(pcosharasiblycteristicscensored),estimatioexplananthetoryvfolloariableswingdat(coavaweriates,aregoingstressesto)aandnalyze:the
valuesofsomeobservablequantitycharacterizingthedegradationofunits.
ortrWhenaumatwice.Aanalyzefailuretheiscalleddegranodation-tndaraumataticthenwhenthethefailuredegracandatiooccurnanottainsan-traumacriticaltic
tylevpeles:z0.relaOthertedfwithailuresproareductioncalleddetrafects,umatic.causedTbryaumatmecicfhanicalailuresdamagmayesbeorofbyfadifferentiguet
ofcomponents.
Supposethatunderfixedconstantcovariatethedegradationisstochasticprocess
Z(t),t≥0.
SupposethatthedegradationprocessZ(t)isnon-decreasingwithcadlagtrajec-
tories.(k)
WeDenotsueppboyseTthatthethemoramenndomtofvatheriablestraTumatic(1),∙∙fa∙,Tilure(s)ofartehekconditiothmonallyde,kinde=1p,∙∙∙enden,s.t
giventhedegradationZ.
Denotebyλ˜(k)(t|Z)=λ˜(k)(t|Z(s),0≤s≤t)theconditionalfailurerateofthe
traumaticfailureofthekthmodegiventhedegradation.
latedSupptooosebservthatedthisdegradatconditioionvnalalues,failureotherrate-tohastwonon-observadditivableedegcompradaonenttions:(oneagingre-)
anddegratodatpionossibleoftiresshocisksthecauswiengarofsuddenthetprraumaotectoticr.Thefailures.failurFoerraexateofmple,tireobservexploablesion
tiredependcompsononenthictsandknessonofinttheensitprotyeofctopr,ossibleonnoshon-cksmeasured(hittingdegakradaerb,tionnail,leveletc.).ofoSother
λ˜(k)(t|Z)=λ(k)(Z(t))+µ(k)(t).(3.1)
)k(oftheThekfthunctiomodenλon(z)degrachardation.acterizesThethefunctiondependencµ(k)ecofhartheraacterizesteofthetraumadepeticfndenceailuresof
theratSuppeofosetrathatumaticexternalfailurescovoarftheiatesk[2th5]modeinfluenceonotherdegratiredatioconmprateonent.andtraumatic
eventintensity.

76

Letx(t)=x1(t),...,xs(t)Tbeavectorofpossiblytimedependentcovariates.
Weassumeinwhatfollowsthatxiaredeterministicorrealizationsofboundedright
continuouswithfinitelefthandlimitsstochasticprocesses.
DenotebyZ(t|x)thedegradationlevelatthemomenttforunitsfunctioning
underthecovariatex.
Wesupposethatthecovariatesinfluencelocallythescaleofthetraumaticfailure
timedistributioncomponentrelatedtoaging(non-observabledegradation)andto
possibleshocks,i.e.theacceleratedfailuretime(AFT)[1]modelistrueforthis
component.Letusexplainitindetail.Denoteby
ttS1(k)(t|Z)=exp{−λ(k)[Z(u)]du},S2(k)(t)=exp{−µ(k)(u)du}
00thesurvivalfunctionscorrespondingtothefailureratesλ(k)(Z(u))andµ(k)(u).The
firstsurvivalfunctionisconditionalgiventhedegradation.
TheAFTmodeldefinesthefollowingrelationofthesecondsurvivalfunctionand
thecovariates:t
S2(k)(t|x)=S2(k)(eβkTx(s)ds);
0theparametersβkhavethesamedimensionasx.Thecovariatexmaybereplaced
bysomespecifiedfunctionϕ(x).
Settf(t,x,β)=eβTx(u)du,(3.2)
0anddenotebyg(t,x,β)theinverseoff(t,x,β)withrespecttothefirstargument.If
x=constthen
f(t,x,β)=eβTxt,g(t,x,β)=e−βTxt.
Thefunctionf(t,x,β)istimetransformationindependenceonx.Forunitsfunc-
tioningunderdifferentcovariatesx(1)andx(2)twomomentst1andt2,respectively,
areequivalentinthesenseofdegradationiftheyverifytheequalityf(t1,x(1),β)=
f(t2,x(2),β),i.e.weconsiderthefollowingmodelfordegradationprocessunderco-
:tesariavZ(t|x)=Z(f(t,x,β)).(3.3)
Thecovariateshavedoubleinfluenceonthedistributionofthefirsttraumaticfailure
component:viadegradationanddirectly.SowecombinetheAFTandthepropor-
tionalhazardsmodels:
tS1(k)(t|x,Z)=exp{−eβ˜kTx(u)λ(k)(Z(u|x)du}.
0ybeDenotS(k)(t|x,Z)=P(T(k)>t|x(u),Z(u|x),0≤u≤t),
λ˜(k)(t|x,Z)=−dlnS(k)(t|x,Z)
dt

77

.5)(3.6)(3.7)(3

theconditionaldistributionfunctionandthefailurerateofthetraumaticfailureof
thekthmodegiventhecovariatesandthedegradation.Soweconsiderthefollowing
del:mosP(T(1)>t,...,T(s)>t|x(u),Z(u|x),0≤u≤t)=S(k)(t|x,Z),(3.4)
=1ktS(k)(t|x,Z)=exp−λ˜(k)(t|x,Z)du=
0t=exp−eβ˜kTx(u)λ(k)(Z(u|x))du−H(k)(f(t,x,βk)),(3.5)
0whereλ˜(k)(t|x,Z)=eβ˜kTx(t)λ(k)(Z(t|x))+eβkTx(t)µ(k)(f(t,x,βk)),(3.6)
tH(k)(t)=µ(k)(u)du.(3.7)
0ybeDenotT(0)=inf{t:Z(t|x)≥z0}.(3.8)
andS(0)(t|x)=PT(0)>t|x(u),0≤u≤t=
=PZ(t|x)<z0|x(u),0≤u≤t(3.9)
thetimetonon-traumaticfailureanditssurvivalfunctionunderthecovariatex,
.elyctiverespThetimeoftheunitfailure
T=min(T(0),T(1),...,T(s))(3.10)
maybetraumaticornon-traumatic.
ybeDenotV=kifT=T(k),k=0,...,s,(3.11)
theindicatorofthefailuremode.Thefailuremode0isnon-traumatic.Othersare
.icumattraLetusconsiderreliabilitycharacteristicswhichareinterestingforapplications.
:eareThes1)Thesurvivalfunctionofthefailuretimeunderthecovariatex:
sS(t|x)=P(T>t|x)=ES(t|x,Z),S(t|x,Z)=1{Z(t|x)<z0}S(k)(t|x,Z).
=1k.12)(32)Meanfailuretimeunderthecovariatex:
(0)Tse(x)=E(T|x)=E(E(T|x,Z)),E(T|x,Z)=S(k)(t|x,Z)dt.(3.13)
=1k078

.10)(3

3)Theprobabilitythatunderthecovariatexthenon-traumaticfailureisobserved
intheinterval[0,t]:
P(0)(t|x)=EP(0)(t|x,Z),
sP(0)(t|x,Z)=1{Z(t|x)≥z0}S(k)(T(0)|x,Z).(3.14)
=1kInparticular,theprobabilityofobservednon-traumaticfailureunderthecovariate
xintheinterval[0,∞)isobtained.
4)Theprobabilitythatunderthecovariatexatraumaticfailureisobservedin
theinterval[0,t]:
P(tr)(t|x)=EP(tr)(t|x,Z),
sP(tr)(t|x,Z)=1−S(k)(t∧T(0)|x,Z).(3.15)
=1k5)Theprobabilitythatunderthecovariatexthetraumaticfailureofthekth
mode,k=1,...,s,isobservedintheinterval[0,t]:
P(k)(t|x)=EP(k)(t|x,Z),
t∧T(0)s
P(k)(t|x,Z)=S(l)(s|x,Z)λ(k)(s|x,Z)ds.(3.16)
=1l0Supposethatthecauseofsometraumaticfailuremodesareeliminated.Notethat
eliminationofafailuremodemayincreasethenumberoffailuresofothermodes.
Indeed,afailureofthelthmodeisnotobservedifitisprecededbyafailureof
thekthmodebutthisfailuremightbeobservedifthekthfailuremodewouldbe
d.eeliminatIfi1th,...,iqth(1≤i1<...<iq≤s)traumaticfailuremodesareeliminatedthen
thesurvivalfunctionS(t|x),themeane(x)andtheprobabilitiesP(0)(t|x),P(tr)(t|x),
andP(k)(t|x),(k=0,1,...,s)aremodifiedtakingl=i1,...,iqinsteadofls=1inthe
formulas(3.12)-(3.16).Soanexperimentusingunitswitheliminatedfailuremodesis
notneeded.Theestimatorsofsurvivalcharacteristicsofunitswitheliminatedfailure
modesisusefulforplanningpossiblewaysofreliabilityimprovement.
Supposethataunitdidnotfailtothemomentτandwehavesomeinformation
aboutitscovariableanddegradationprocesses(x(s),Z(s|x)|s≤τ).
LetGdenotetheσ-algebrageneratedbythepossessedinformationaboutthe
degradationprocessand
G¯τ=σ(G∪{T>τ}).
Theconditionalprobabilitiesoftheeventsconsideredintheprevioussectiongiven
theσ-algebraG¯τare:fort>τ.
S(t|x,τ,G)=EG{S(t|x,Z)},(3.17)
EG{S(τ|x,Z)}

79

P(k)(t|x,τ,G)=EG{P(k)(t|x,Z)}−EG{P(k)(τ|x,Z)},(3.18)
EG{S(τ|x,Z)}
P(tr)(t|τ,G)=EG{P(tr)(t|x,Z)}−EG{P(tr)(τ|x,Z)}.(3.19)
EG{S(τ|x,Z)}
Moreover,themeanresiduallifeoftheunitis
e(x,τ,G)=EG{(T−τ)1{T>τ}|x}.(3.20)
EG{S(τ|x,Z)}
IfG=σ(Z)then
EG{S(t|x,Z)})=E{S(t|x,Z)},(3.21)
(0)TssEG{(T−τ)1{T>τ}|x,Z}=S(l)(t|x,Z)dt−τ1{Z(τ|x)<z0}S(l)(τ|x,Z).
=1l=1lτ.22)(33.2Estimationofmodelparameters
dataThe1.2.3Supposethatnunits(i)areobserved.Theithunitistestedunderthevectorofex-
planatoryvariablesx,andatthemoments
0<ti1<ti2<....<timi
)i(theThevaluesmomenZtijstij=Zcorri(teijsp|xon)dtoofthethescadegleofradareationlfunlevctelaioning.resuppForosedexample,tobeinthemeasured.case
oftirewear,tijmeankilometersdonebythei-thtireuntilthej-thmeasurement.
Thevaluesofcovariatesaresupposedtobeobservedduringtheexperiment.The
mostoftentheyshouldbeconstantintimeorstep-functions.
DenotebyTi=min(Ti(0),...,Ti(s))thefailuretimeandVi-thefailuremode
indicator.Thedatamayberightcensored.DenotebyCithecensoringtimeofthe
ithunit,andset
C˜i=Ci∧timi,Xi=Ti∧C˜i,δi=1{Ti≤C˜i},δ˜i=1{Ti≤C˜i,Vi=0}.(3.23)
ybeDenotµi=j,ifXi∈(tij,ti,j+1],j=0,...,mi−1,(3.24)
mi,ifXi=timi,
theobservednumberofmeasurementsofthei-thunit.
Thedataaretherandomvectors
(Xi,δi,Vi,µi,Zi1,...,Ziµi,x(i)),i=1,...,n.(3.25)
Ifµi=0thenthedegradationvaluesarenotobserved.
80

3.2.2Likelihoodfunctionconstruction
Supposethatthefunctionsλ(k)(z)andµ(k)(t)isfromaclassoffunctions
λ(k)(z)=λ(k)(z,ηk),µ(k)(t)=µ(k)(t,γk),(3.26)
whereηk,γkarepossiblymulti-dimensionalparameters.Forexample,analysisof
tirefailuretimeandweardatashowsthattheintensitiesλ(k)(z)andµ(k)(t)typically
havetheform(z/η1k)η2kand(t/γ1k)γ2k.
SupposeatfirstthatdegradationprocessesZi(t)=Zi(t|x(i))ofallunitsare
continuouslyobservable.InthiscaseconditionallikelihoodLandloglikelihoodl
functionsgivendegradationfortheparameterscharacterizingtraumaticfailurescan
bewrittenasfollows:
L=1{Vi=k}eβ˜kTx(i)(Xi)λ(k)(Zi(Xi);ηk)+
ns
=1k=1i+eβkTx(i)(Xi)µ(k)(f(Xi,x(i),βk);γk)δ˜i×
sX
×exp−ieβ˜kTx(i)(u)λ(k)(Zi(u);ηk)du−H(k)(f(Xi,x(i),βk);γk),
0=1ksnl=1{Vi=k}lneβ˜kTx(i)(Xi)λ(k)(Zi(Xi);ηk)+
=1k=1i+eβkTx(i)(Xi)µ(k)(f(Xi,x(i),βk);γk)−
sX−ieβ˜kTx(i)(u)λ(k)(Zi(u);ηk)du−H(k)(f(Xi,x(i),βk);γk).(3.27)
0=1kIfcovariantsareabsentthen
l=1{Vi=k}lnλ(k)(Zi(Xi);ηk)+µ(k)(Xi;,γk)−
ns
=1k=1i−λ(k)(Zi(u);ηk)du−H(k)(Xi;γk).(3.28)
sXi
0=1kIfthevaluesofdegradationprocessesaremeasuredonlyatdiscretetimestijthenthe
conditionallikelihoodfunctionismodifiedreplacingZi(u)bytheirpredictorsZˆi(u)
obtainedfromdegradationdata.Theformofthesepredictorsdependsontheform
ofthedegradationprocesses.
Itwasmentionedinintroductionthatthemostappliedstochasticprocessesde-
scribingdegradationaregeneralpathmodelsandtimescaledstochasticprocesses
withstationaryandindependentincrementssuchasthegammaprocess,compound
PoissonprocessandWienerprocesswithdrift,thelastnotmonotone.
Letusfindthepredictorsforsomespecifieddegradationprocesses.

81

3.2.3Example1:Timescaledgammaprocess
Letmbearealtimefunction.ThedegradationprocessZistimescaled(bythescale
functionm(t))gammaprocessif
1)ithasindependentincrements,i.e.forany0<t1<∙∙∙<tktherandom
variablesZ(t1),Z(t2)−Z(t1),...,Z(tk)−Z(tk−1)areindependent;
2)foranyt>0therandomvariableZ(t)hasthegammadistribution;
3)foranyt≥0
E(Z(t))=m(t),Var(Z(t))=σ2m(t).
Thedefinitionimpliesthatforanyx>0thedensityofther.v.Z(tj)−Z(tj−1),
j=1,...,k,t0=0,is
1xΔ2mj−1x
Γσ2pZ(tj)−Z(tj−1)(x)=2Δmjσ2σe−σ2,(3.29)
σ∞Δmj=m(tj)−m(tj−1),Γ(a)=xa−1e−xdx.
0ThedensityofZ(tj)isofthesameform:Δmjmustbereplacedbym(tj)in(3.29).
Thedegradationanditscharacteristicsundercovariatexare
Z(t|x)=Z(f(t,β,x)),m(t|x)=E(Z(t|x))=m(f(t,β,x)),(3.30)
σ2(t|x)=Var(Z(t|x)=σ2m(t|x).
m(t|x)isthemeandegradationunderthecovariatex.
a)Parametricformofthemeandegradation
Theformofthemeandegradationm(t)maybesuggestedbytheformofobserved
degradationcurves.Insuchacasem(t)ischosenfromsomeparametricclassof
functions(powerorothertimefunctiondependingonafinite-dimensionalunknown
parameter):m(t)=m(t;ν),ν=(ν1,...,νq)T.Thedata
(Zij,µi),i=1,n,j=1,µi,(3.31)
areusedforestimationoftheparametersθ=(β,ν,σ2)T.
Foranyz>0thedensityoftheincrementΔZij=Zij−Zi,j−1hastheform
1zΔµijσ2(β,ν)−1−z
2σpΔZij(z;θ)=σ2ΓΔµij(β,ν)σ2eσ2,
whereΔµij(β,ν)=mf(ti,j,β,x(i));ν−mf(ti,j−1,β,x(i));ν.(3.32)

82

Thelikelihoodfunctionofthedegradationdata(3.31)is
Ld(θ)=pΔZijΔZij;θ,(3.33)
nµi
=1j=1iwherewesetpΔZijΔZij;θ=1,ifXi<ti1,i.e.whenatraumaticeventoccurs
earlierthenthefirstmeasurementofdegradation.
Denotebyθˆthemaximumlikelihoodestimator.Thenforanyxtheestimatorof
themeandegradationm(t|x)underthecovariatexis
mˆ(t|x)=mf(t,βˆ,x);νˆ.(3.34)
Inthecaseofthedata(3.25)degradationvaluesarenotmeasuredcontinuouslyand
theloglikelihoodfunction(3.27)cannotbeusedforestimationoftheparameters
βk,β˜k,ηkandγk.Formodificationoftheloglikelihood(3.27)weneedpredictorsof
Zi(t).Set
Z˜i(t)=E(Zi(t)|Zi(ti1),...,Zi(tiµi)).(3.35)
Forj=1,...,µiwehaveZ˜i(tij)=Zij.Fort∈(ti,j−1,tij),j=1,...,µi,the
conditionalmeans(3.35)are
Z˜i(t;θ)=Zi,j−1+Δmij(t;β,ν)ΔZij,
Δmij(β,ν)
Δmij(t;β,ν)=mf(t,β,x(i));ν−mf(ti,j−1,β,x(i));ν.(3.36)
Foranyt>ti,µi
Z˜i(t;θ)=ZiµiFχκ2i(2(z0−Ziµi))+Δmi,µi+1(t;β,ν)Fχκ2i+2(2(z0−Ziµi)),(3.37)
whereFχn2(x)isthec.d.f.ofthechisquaredistribution,κi=2Δmi,µi+1(t;β,ν).
Notethatforanyt>ti,µi
Z˜i(t;θ)→Ziµi+Δmi,µi+1(t;β,ν),asz0→∞.(3.38)
ThepredictorsZˆiofZiaredefinedas
Zˆi(t)=Z˜i(t;θˆ).(3.39)
b)Unknownformofmeandegradation
Ifthefunctionmiscompletelyunknownthennon-parametricestimatorofthisfunc-
tionisusedseekingpredictorsofthestochasticprocessesZi.
Apiecewise-linearapproximationoftheprocessZi(t)=Z(t|x(i))on[0,Xi]is
Zi∗(t)=Zi(ti,j−1)+t−ti,j−1(Zi(tij)−Zi(ti,j−1))1[ti,j−1,tij](t),(3.40)
µi+1
j=1tij−ti,j−1
83

ti0=0,ti,µi+1=Xi.Wedenotedbyg(t,x,β)theinverseoff(t,x,β)withrespect
tothefirstargument.Thedistributionofthestochasticprocess
Z(t)=Z(g(t,x,β)|x),(3.41)
)i(∗doproescessnotZdepandendcanonbex,usedsocothenstproructingcessesanZi(g(t,estimatoxr,βof))thearemeanapproximationsofthe
m(t)=EZ(t)=EZ(g(t,x,β)|x)).(3.42)
Theseapproximatingprocessesarecensoredatthepointsti∗(β)=g(Xi,x(i),β).Con-
sidertheorderedsequenceofdistinctmoments
t∗(1)(β)<∙∙∙<t(∗d)(β),d≤n.
Takethefollowingpseudo-estimator(dependingonβ)ofm(t):
n1m˜(t,β)=nZk∗(g(t,β,x(i))),t∈[0,t∗(1)(β)],
=1im˜(t,β)=m˜(t(∗j−1)(β),β)+
i:t∗(β)>t∗(β)Zi∗(g(t,β,x(i)))−Zi∗(g(t(∗j−1)(β),β,x(i)))
1+i(j−1),(3.43)
i:ti∗(β)>t(∗j−1)(β)
t∈(t(∗j−1)(β),t(∗j)(β)].Thelikelihoodfunctionfromdegradationdata(3.31)iswritten
intheform(3.33)putting
θ=(βT,σ2)T,Δmij(β)=m˜f(ti,j,β,x(i)),β−m˜f(ti,j−1,β,x(i)),β.(3.44)
Denotebyβˆ,σˆ2themaximumˆlikelihoodestimators.Thefunctionm(t)isestimated
bythestatisticmˆ(t)=m˜(t,β).
DefineZ˜i(t;θ)by(3.36)and(3.37)replacingΔmij(β,ν)byΔmij(β)givenin
..44)(3ThepredictorsofZi(t)areZˆi(t)=Z˜i(t;θˆ).

84

3.2.4Example2:Shockprocesses
Assumethatdegradationresultsfromshocks,eachofthemleadingtoanincrementof
degradation.LetTn,(n≥1)bethetimeofthenthshockandXnthenthincrement
ofthedegradationlevel.DenotebyN(t)thenumberofshocksintheinterval[0,t].
SetX0=0.Thedegradationprocessisgivenby
∞N(t)
Z(t)=1{Tn≤t}Xn=Xn.
=0n=1nKahleandWendt[20]modelTnasthemomentsoftransitionofthedoublystochastic
Poissonprocess,i.e.theysupposethatthedistributionofthenumberofshocksup
totimetisgivenby
kP{N(t)=k}=E(Yη(t))exp{−Yη(t)},
!kwhereη(t)isadeterministicfunctionandYisanonnegativerandomvariablewith
finiteexpectation.IfYisnon-random,Nisnon-homogenousPoissonprocess,in
particular,whenη(t)=λt,NishomogenousPoissonprocess.Othermodelsforη
maybeused,forexample,η(t)=tα,α>0.
AssumethatX1,,X2,∙∙∙areconditionallyindependentgiven{Tn}andassume
thattheprobabilitydensityfunctionsofXngiven{Tn}isg.
Letusconsiderthecasewhenthenumberofshocksismodelledbynon-homogenous
cess:prooissonPP{N(t1)=i1,N(t2)−N(t1)=i2,∙∙∙,N(tm)−N(tm−1)=im}=
η(t1)i1e−η(t1)[η(t2)−η(t1)]i2e−[η(t2)−η(t1)]∙∙∙[η(tm)−η(tm−1)]ime−[η(tm)−η(tm−1)].
i1!i2!im!
.45)(3ThedegradationprocessZ(t)isastochasticprocesswithindependentincrements
andforanyz≥0thedensityofther.v.Z(t)−Z(s),0≤s,t,is
∞kpZ(t)−Z(s)(z)=gk(z)[η(t)−η(s)]e−[η(t)−η(s)](3.46)
!k=1kwheregkistheconvolutionofkdensitiesg.Forexample,ifthesizesoftheshocks
XihaveexponentialdistributionE(ξ):g(u)=ξe−ξu,u≥0,then
gk(u;ξ)=ξue−ξu,u≥0,pZ(t)−Z(s)(z)=ξbe−ξz−b(ξbz),
kk−1∞k
(k−1)!k=0k!(k+1)!
whereb=η(t)−η(s).
Denotebya1=EX1anda2=EX12thefirsttwomomentsoftherandomvariable
X1.ThemomentsofZ(t)are
E(Z(t))=a1η(t),Var(Z(t))=a2η(t).

85

Thedegradationanditscharacteristicsundercovariatexare
Z(t|x)=Z(f(t,β,x)),m(t|x)=E(Z(t|x))=µ1η(f(t,β,x)),(3.47)
σ2(t|x)=Var(Z(t|x)=a2η(f(t,β,x)).
a)Parametricformofthemeandegradation
Supposethatgandηbelongtosomeparametricclassesg(t)=g(t,ξ),ξ=(ξ1,...,ξp)T.
andη(t)=η(t;ν),ν=(ν1,...,νq)T.Setθ=(βT,νT,ξT)T.Thelikelihoodfunctionof
thedegradationdata(3.31)isoftheform(3.33),whereforanyz>0,0≤s<t,the
densityoftheincrementZ(t)−Z(s)is
∞kpZ(t)−Z(s)(z;θ)=gk(z;ξ)[Δη(s,t;β,ν)]e−Δη(s,t;β,ν);(3.48)
!k=1khereΔη(s,t;β,ν)=ηf(t,β,x(i));ν−ηf(s,β,x(i));ν.(3.49)
Denotebyθˆthemaximumlikelihoodestimator.Thenforanyxtheestimatorofthe
meandegradationm(t|x)underthecovariatexhastheform(3.34).
Fort∈(ti,j−1,tij),j=1,...,µi,theconditionalmeans(3.35)are
Zji,1Z˜i(t;θ)=pΔZi,j(ΔZi,j;θ)ZzpZ(t)−Zi,j−1(z−Zi,j−1;θ)pZi,j−Z(t)(Zi,j−z;θ)dz.
i,j−1(3.50)
Foranyt>ti,µiz0
Z˜i(t;θ)=zpZ(t)−Zi,µi(z−Zi,µi;θ)dz.(3.51)
Zµi,iThepredictorsZˆiofZiaredefinedbytheformula(3.39).
Notethatasinthecaseofthegammaprocess(wesetσ2=a2/a1)
Var(Z(t))=σ2,(Z(s),Z(t))=Var(Z(s∧t)),
E(Z(t))
sointermsofthefirsttwomomentstheconsideredshockprocessandthegamma
processareofidenticalstructure.

b)Unknownformofmeandegradation
ThepredictorsofZi(t)aredefinedasZˆi(t)=Z˜i(t;θˆ),andZ˜i(t;θ)aredefinedby
(3.50)and(3.51)replacingΔη(s,t;β,ν)(givenin(3.49))by
Δη(s,t;β,ξ)=[m˜f(ti,j,β,x(i)),β−m˜f(ti,j−1,β,x(i)),β]/a1(ξ),
wherem˜(t;β)isthepseudoestimatorofthemeanm(t)=EZ(t)givenby(3.43).

86

3.2.5Example3:Pathmodels
SupposethatthedegradationprocessZ(t)isofthefollowingform:
Z(t)=ϕ(t,A,ν),(3.52)
whereϕisadeterministicfunctionandA=(A1,...,Ap)isafinitedimensional
randomvectorandνisafinitedimensionalnon-randomparameter.
Theformofthefunctionϕmaybesuggestedbytheformofindividualdegradation
curves.Thedegradationunderthecovariatexismodelledby
Z(t|x)=ϕ(f(t,x,β),A),m(t|x)=Eϕ(f(t,x,β),A).
Letusconsiderthefollowingtypicalexample:
Z(t)=(t/A)ν;(3.53)
hereAisapositiverandomvariablewithunknowncumulativedistributionfunction
F,νisapositiveparameter.Inparticularcaseν=1thismodelfitswellasthetire
wearmodel[29].
Thedegradationprocessundercovariatexis
Z(t|x)=Z(f(t,x,β))=(f(t,x,β)/A)ν.(3.54)
Eveninthecaseν=1itisnotnecessarylinear.
Supposethatnunitsareontest.Theithunitistestedunderexplanatoryvariable
x(i).DenotebyTi,Vithefailuretimesandthefailuremodes,respectively.Suppose
thatthedegradationvaluesZ(i)atthemomentsTiareobserved.Sothedatahas
rmofthe(Ti,Vi,Z(i),x(i)),i=1,...,n.(3.55)
Thecovariatesx(i)areobserveduntilthemomentsXi.
Takingintoaccountthattherandomvariables
lnAi=νf(Ti,x(i),β)−lnZ(i)
areindependentidenticallydistributedwiththemean,saym,whichdoesnotde-
pendonβandν,sotheseparametersareestimatedbythemethodofleastsquares,
umstheminimizingn(νlnf(Ti,x(i),β)−lnZ(i)−m)2,
=1iwhichgivesthesystemofequations
0nTix(i)eβTx(i)(u)du[νlnf(Ti,x(i),β)−lnZ(i)]
ni=1f(Ti,x(i),β)−

87

.56)(3

nTix(i)eβTx(i)(u)dun
−0f(Ti,x(i),β)[νlnf(Tj,x(i),β)−lnZ(j)]=0,
=1j=1innlnf(Ti,x(i),β)[νlnf(Ti,x(i),β)−lnZ(i)]−
=1inn−lnf(Ti,x(i),β)[νlnf(Tj,x(i),β)−lnZ(j)]=0.
=1j=1iIfx(i)areconstantthenthissystemis:
nβTx(i)[νβTx(i)+νlnTi−lnZ(i)]−βTx(i)[νβTx(i)+νlnTi−lnZ(i)]=0,
nnn
i=1i=1j=1
nx(i)[νβTx(i)+νlnTi−lnZ(i)]−x(i)[νβTx(i)+νlnTi−lnZ(i)]=0.
nnn
i=1i=1j=1
Denotebyβˆandνˆtheobtainedestimator.
ThepredictorsofZi(t)aredefinedas
νˆZˆi(t)=f(t,x(i),βˆ)Z(i).(3.56)
f(Ti,x(i),βˆ)
NotethatZˆi(Ti)=Z(i).Ifx(i)areconstantovertimethen
νˆZˆi(t)=TtZ(i).(3.57)
i3.2.6Modifiedloglikelihood
Themodifiedloglikelihoodfunctionfortheparametersβk,β˜k,ηkandγkfromthe
data(3.25)(and(3.55))isobtainedmodifyingtheloglikelihoodfunction(3.27):the
stochasticprocessesZiarereplacedbytheirpredictorsZˆiin(3.27)(inthecaseof
thedata(3.55)takeXi=Ti):
snl˜=1{Vi=k}lneβ˜kTx(i)(Xi)λ(k)(Zˆi(Xi);ηk)+
=1k=1i+eβkTx(i)(Xi)µ(k)(f(Xi,x(i),βk);γk)−
−eβ˜kTx(i)(u)λ(k)(Zˆi(u);ηk)du−H(k)(f(Xi,x(i),βk);γk).(3.58)
sXi
0=1kIfcovariantsareabsentthen
l˜=1{Vi=k}lnλ(k)(Zˆi(Xi);ηk)+µ(k)(Xi;,γk)−
ns
=1k=1i88

.57)(3

−λ(k)(Zˆi(u);ηk)du−H(k)(Xi;γk),(3.59)
sXi
0=1kwherethepredictorsZˆiaredefinedreplacingf(u,β,x(i))byuinallformulas.
Theloglikelihood(3.25)functioncanbemodifiedandtothecasewhenthetwo
functionsλ(k)(orthefunctionsH(k),butnotboth)arecompletelyunknown.Inthe
caseoflinearpathmodelssuchmodificationsandpropertiesofestimatorsaregiven
inBagdonaviˇciusetal([8]).
Investigatingthecaseofotherdegradationmodelsisasubjectforseparatework.

3.3Estimationofreliabilitycharacteristics
Letusconsiderestimationofreliabilitycharacteristics(3.12)-(3.22)whenthemean
degradationm(t)isofparametricform.Set
Zˆi(t|x)=Zˆig(f(t,βˆ,x),x(i),βˆ),(3.60)
whereZˆi(t)arethepredictorsofthediscreetlyobservedprocessesZi(t|x(i)),i=
1,...,n.Weconsideredconst(i)ructionofthepredictorsZˆi(t)intheprevioussection.
Inparticularcasewhenx,xareconstantovertime
Zˆi(t|x)=Zˆi(eβˆ(x−x(i))t).(3.61)
Thepredictorofthenon-traumaticfailureoftheithunitunderthecovariatexis
Tˆi(0)(x)=inf{t:Zˆi(t|x)≥z0}.(3.62)
Theformulas(3.12)-(3.16)implythefollowingestimators:
1)Theestimatorofthesurvivalfunctionofthefailuretimeunderthecovariate
x:1ns
Sˆ(t|x)=n1{Zˆi(t|x)<z0}Sˆ(k)(t|x,Zˆi),(3.63)
=1k=1iwheretSˆ(k)(t|x,Zˆi)=exp−eβ˜ˆkTx(u)λ(k)(Zˆi(u|x),ηˆk)du−H(k)(f(t,x,βˆk),γˆk).
0.64)(32)Theestimatorofthemeanfailuretimeunderthecovariatex:
nTˆi(0)(x)s
eˆ(x)=1Sˆ(k)(t|x,Zˆi)dt.(3.65)
ni=10k=1
3)Theestimatoroftheprobabilitythatunderthecovariatexthenon-traumatic
failureisobservedintheinterval[0,t]:
snPˆ(0)(t|x)=11{Zˆi(t|x)≥z0}Sˆ(k)(Tˆi(0)(x)|x,Zˆi).(3.66)
n=1k=1i89

4)Theestimatoroftheprobabilitythatunderthecovariatexatraumaticfailureis
observedintheinterval[0,t]:
Pˆ(tr)(t|x)=1−1nsSˆ(k)(t∧Tˆi(0)(x)|x,Zˆi).(3.67)
n=1k=1i5)oftheThekthestimamotorde,okf=the1,...proba,s,isbilityobservthatedinunderthetheinctervovalaria[0,tet]:xthetraumaticfailure

nt∧Tˆi(0)(x)s
Pˆ(k)(t|x)=1Sˆ(l)(s|x,Zˆi)λ(k)(s|x,Zˆi)ds.(3.68)
ni=10l=1
obtainedtakingl=i1,...,iqinsteadofl=1intheformulas(3.63)-(3.68).
Theestimatorsofsurvivalcharacteristicssofunitswitheliminatedfailuremodesare

90

nsusioonclC

Inthethesis,thefollowingresultswhichanalysedformulatedatthebeginningaims
ined:gaear1.Mathematicaldefinitionofstand-byunitfluentswitchingfrom”warm”to”hot”
conditionsisformulated;
2.Testsforgeneral”fluentswitchinghypothesis”formulatedusingSedyakin’s”re-
liabilityprinciple”andforparticularfluentswitchinghypothesisformulatedusing
acceleratedfailuretimemodelareconstructed;Asymptoticpropertiesofthetest
statisticsareinvestigated;
3.Parametricandnonparametricestimatorsofthecumulativedistributionfunc-
tionofredundantsystemusingreliabilitydataofcomponentstestedunderdifferent
constructed;aresstresse4.Asymptoticpropertiesoftheparametricandnonparametricestimatorsareinves-
ated;tig5.Asymptoticconfidenceintervalsforcumulativedistributionfunctionofredundant
systemareconstructed.Finitesamplepropertiesoftheparametricandnonparamet-
ricestimatorsareinvestigatedbysimulation;
6.Generalsimultaneousfailuretimeanddegradationregressiondatamodelsare
formulated.Maximumlikelihoodmethodforestimationoffailureprocessanddegra-
dationprocessparametersusingsimultaneousdegradationandmulti-modefailure
timeregressiondatausingpredictorsofdegradationprocessesismodified;
7.Thestructureofmodifiedlikelihoodfunctionwhenthedegradationprocessis
modelledbytimescaledgammaprocess,pathprocesses,shockprocesseswiththe
numberofshocksmodelledbynon-homogenousPoissonprocessisinvestigated.

91

AendixApp

dmethotalDe

TheoremA.0.1Let{an}beasequenceofrealnumbers,g=(g1,...,gq):Rp→Rq
beadifferentiablevector-function,and
Jg(x)=||∂gi(x)||q×p
x∂jbetheJacobimatrixofpartialderivativesofcoordinatefunctionsgi.If
an(X(n)−x)→DZasan→∞onRp,
thenan(g(X(n))−g(x))→DJg(x)Zasan→∞.(A.1)
TheoremA.0.2Supposethat
1){X1n∈D[0,τ]}and{X2n∈D[0,τ]}aresequencesofcatlagstochasticprocesses,
the2)secX1ond,Xb2eing∈Dof[0,bτ]oundearedcadlagvariationstoandchasticbproundeocdessesbyofabpoundeositivedconstantvariationMthe;second
beingboundedbyMsuchthat
(an(X1n−X1),an(X2n−X2))→D(Z1,Z2),
onD[0,τ]×D[0,τ];hereZ1,Z2∈D[0,τ].Then
....
anX1ndX2n−X1dX2→DZ1dX2+X1dZ2(A.2)
0000onD[0,τ].IfZ2isnotofboundedvariationthenthelastintegralisdefinedby
ttX1(u)dZ2(u)=X1(t)Z2(t)−X1(0)Z2(0)−Z2(u)dX1(u).
00

92

The1)orxem∈DA[0.0,τ.3]isSuppaosenondethatcreasingfunction,differentiableatthepoint
x−1(p)=inf{t:x(t)≥p}∈(0,τ),
wherep∈Risafixednumber.
2){X(n)∈D[0,τ]}isasequenceofnondecreasingstochasticprocessessuchthat
an(Xn−x)→DZ
on−1D[0,τ];hereZ∈D[0,τ]isanondecreasingprocess,continuousatthepoint
.)p(xThen1−an(Xn)−1(p)−x−1(p)D→−xZ((xx−1((pp)))).(A.3)
TheoremA.0.4Supposethat
1)xisacontinuouslydifferentiablefunctionon[0,τ];
2)ϕ=ϕ(t,θ):A×Bε(θ0)→R,Bε(θ0)⊂Rs,A=[0,τ0]or(0,τ0),isa
continuousnon-increasingintfunctionsuchthat0<ϕ(t,θ)<τfort∈A;
3){X(n)∈D[0,τ]}isasequenceofstochasticprocesses0suchthat
√n(Xn−x)→DZ
onD4)[0{,θˆτ(n],)}iswheraeseZisquencaceofrontinuousandomonvar[0,τiables]stosuchchasticthatprocess;
√n(θˆ(n)−θ0)→DY.
Then√n(Xn(ϕ(∙,θˆ0(n)))−x(ϕ(∙,θ0))→DZ(ϕ(∙,θ0))+x(ϕ(∙,θ0))ϕθ(∙,θ0)Y
onD(A).

93

Bibliography

[1]Bagdonavicˇius,V.Testingthehyphothesisoftheadditiveaccumulationofdam-
ages.Probab.TheoryanditsAppl.,23,1978,p.403–408.
[2]Bagdonavicˇius,V.andNikoulina,V.Agoodness-of-fittestforSedyakin’smodel.
RevueRoumainedeMath´ematiquesPuresetAppliqu´ees,42,1997,p.5–14.
[3]Bagdonavicius,V.andNikulin,M.AcceleratedLifeModels(BocaRaton:Chap-
manandHall/CRC),2002.
[4]Bagdonavicˇius,V.Acceleratedlifemodelswhenthestressisnotconstant.Ky-
bernetika,26,1990,p.289–295.
[5]Baexplanagdonatovirycˇvius,aV.riables,andLifetiNikulin,meDataM.S.AEnalysisstim,atio7,n200in1,p.degra85–datio103.nmodelswith
[6]BticaalgdoAnavnalysisicˇius,.CV.haandpman&Nikulin,Hall/M.CRAC:ccBelerocaatedRatolifen,mo200dels:2.ModelingandStatis-
[7]Badegragdodatnavionicˇius,andV.,failurBikeelist,imeA.,daandtaKawithzakmeviˇultiplecius,fV.ailureStatisticamoldes,analysLifetimeisoflinearData
Analysis,10,2004,p.65–81.
[8]BtimaagdotionnavifrocˇiusmV.,simBultikelis,Aaneous.,renewKazakeviˇciusal-failure-degra,V.,Nikuldatioin,nM.dataNon-parwithcompametricetinges-
risks.JournalofStatisticalPlanningandInference,137,2007,p.2191–2207.
[9]Couaelling.llieInrPrV.obSomeability,recentStatisticsresultsandonMojoindeltlingdegrainPdatioublicnaHendalthfailure(Eds.,timeNikulin,mod-
M.,Commenges,D.,Huber,C.),SpringerScience+BusinessMedia,2006,p.
9.–873[10]SoCoxcietD.y,BR.voRl.34egressionpp.197mo2,delsp.187and-2life20,tables,JournaloftheRoyalStatistical
[11]DoexpksumerimenKt.baA.sedandonHoaylandWienerA.Moprocdelsessforandvatheinriable-stressverseGaacceleratussianedlifedistributiotestingn,
Technometrics,34,1992,p.74–82.

94

[12]Doksum,K.A.andNormand,S.L.T.Gaussianmodelsfordegradationprocesses
-partI:Methodsfortheanalysisofbiomarkerdata,LifetimeDataAnalysis,1,
1995,p.131–144.
[13]Doksum,K.A.andNormand,S.T.Modelsfordegradationprocessesandevent
timesbasedongaussianprocesses.In:Jewell,N.P.etal.(eds),Lifetimedata:
Modelsinreliabilityandsurvivalanalysis,Kluweracademicpublishers,1996,p.
1.–985[14]Esary,J.D.,Marshall,A.W.andProshan,F.Shockmodelsandwearprocesses,
Theannalsofprobability,1,1973,p.627–649.
[15]Finkelstein,M.S.Ontheexponentialformulaofreliability,IEEETransactions
onReliability,53,2004,p.265–269.
[16]Greenwood,P.E.andNikulin,M.AGuidetoChi-SquaredTesting(NewYork:
JohnWileyandSons),1996.
[17]Harlamov,B.InverseGamma-ProcessasaModelofWear.In:Longevity,Ag-
ingandDegradationModelsinReliability,Health,MedicineandBiology,v.2,
(Eds.V.Antonov,C.Huber,M.Nikulin,V.Politschook),St.PetersburgState
PolytechnicalUniversity,SaintPetersburg,2004,p.180–190.
[18]Kahle,W.Simultaneousconfidenceregionsfortheparametersofdamagepro-
cesses.StatisticalPapers,35,1994,p.27–41.
[19]Kahle,W.andLehmann,A.Parameterestimationindamageprocesses:De-
pendentobservationsofdamageincrementsandfirstpassagetime,InAdvances
inStochasticModelsforReliability,QualityandSafety(Eds.,Kahle,W.,von
Collani,E.,Franz,F.,Jensen,U.).Birkhauser:Boston,1998,p.139–152,
[20]Kahle,W.andWendt,H.StatisticalAnalysisofSomeParametricDegradation
Models.In:Nikulin,M.,Commenges,D.,Huber,C.(eds),Probability,Statistics
andModellinginPublicHealth,SpringerScience+BusinessMedia,2006,p.266-
.79[21]LawlessJ.andCrowderM.Covariatesandrandomeffectsinagammaprocess
modelwithapplicationtodegradationandfailure,LifetimeDataAnalysis,10,
2004,p.213–227.
[22]Lehmann,A.AWienerprocessbasedmodelforfailureanddegradationdatain
dynamicenvironments,DresdnerSchriftenzurMathemat.Stochastik,4,2001,
.4035–p.[23]Lehmann,A.Onadegradation-failuremodelforrepairableitems,InSemipara-
metricModelsanditsApplicationsforReliability,SurvivalAnalysisandQual-
ityofLife(Eds.,Nikulin,M.,Balakrishnan,N.,Limnios,N.,Mesbah,M.),
Birkhauser:Boston,2004,p.65–79.

95

[24]Lehmann,A.Jointmodelsfordegradationandfailuretimedata,Proceedingsof
theInternationalWorkshopStatisticalModellingandInferenceinLifeSciences,
September1-4,Potsdam,2005,p.90–94.
[25]Lehmann,A.Degradation-Threshold-ShockModels.In:Probability,Statis-
ticsandModellinginPublicHealth,(Eds.M.Nikulin,D.Commenges,C.Huber).
Springer:NewYork,2006,p.286–298.
[26]Lawless,J.F.StatisticalModelsandMethodsforLifetimeData.NewYork:John
WileyandSons.2ndedition,2003.
[27]Lu,C.J.andMeeker,W.Q.Usingdegradationmeasurestoestimateatime-to-
failuredistribution,Technometrics,35,1993,p.161–174.
[28]Meeker,W.Q.andEscobar,L.A.Statisticalmethodsforreliabilitydata,Wiley,
NewYork,1998.
[29]Meeker,W.Q.,Escobar,L.A.andLu,C.J.Accelerateddegradationtest:
Modelingandanalysis,Technometrics,40,1998,p.89–99.
[30]Meeker,W.Q.,Escobar,L.A.StatisticalMethodsforReliabilityData.NewYork:
JohnWileyandSons,1998.
[31]Nelson,W.AcceleratedTesting:StatisticalModels,TestPlans,andDataAnal-
yses(NewYork:JohnWileyandSons),1990.
[32]Nelson,W.AcceleratedTesting:StatisticalModels,TestPlans,andDataAnal-
yses.NewYork:JohnWileyandSons.2ndedition,2004.
[33]Padgett,W.J.andTomlinson,M.A.Inferencefromaccelerateddegradation
andfailuredatabasedonGaussianprocessmodels,LifetimeDataAnalysis,10,
2004,p.191–206.
[34]PadgettW.J.andTomlinsonM.A.Accelerateddegradationmodelsforfail-
urebasedongeometricBrownianmotionandgammaprocesses,LifetimeData
Analysis,11,2005,p.511–527.
[35]Pieruschka,E.Relationbetweenlifitimedistributionandthestresslevelcausing
thefailures,1961.
[36]Sedyakin,N.M.Ononephysicalprincipleinreliabilitytheory.TechnicalCyber-
netics,3,1966,p.80–87.
[37]Singpurwalla,N.D.InferencefromAcceleratedLifeTestsWhenObservations
AreObtainedfromCensoredSamples,Technometrics,13,1971,p.161–170.
[38]Singpurwalla,N.D.Survivalindynamicenvironments,StatisticalScience,10,
1995,p.86–103.

96

[39]Veklerov,E.Reliabilityofredundantsystemswithunreliableswitches.IEEE
transactionsonreliability,36,1987,p.470-472.
[40]Viertl,R.StatisticalMethodsinAcceleratedLifeTesting.G¨ottingen:Vanden-
hoeckandRuprecht,1988.
[41]WassonCh.S.SystemAnalysis,Design,andDevelopment:Concepts,Principles,
andPractices.NewYork:Wiley,2005.
[42]Wendt,H.andKahle,W.OnaCumulativeDamageProcessandResultingFirst
PassageTimes.AppliedStochasticModelsinBusinessandIndustry,20,2004a,
.2617–p.[43]Wendt,H.,Kahle,W.OnParametricEstimationforaPosition-Dependent
MarkingofaDoublyStochasticPoissonProcess.In:ParametricandSemipara-
metricModelswithApplicationstoReliability,SurvivalAnalysis,andQuality
ofLife,(Eds.M.Nikulin,N.Balakrishnan,M.Mesbah,N.Limnios).Birkhauser:
Boston,2004b,p.473–486.
[44]WhitmoreG.A.EstimationdegradationbyaWienerdiffusionprocesssubject
tomeasurementerror,LifetimeDataAnalysis,1,1995,p.307–319.
[45]Whitmore,G.A.andSchenkelberg,F.Modellingaccelerateddegradationdata
usingWienerdiffusionwithatimescaletransformation,LifetimeDataAnalysis,
3,1997,p.27–45.
[46]WhitmoreG.A.,CrowderM.I.andLawlessJ.Failureinferencefromamarker
processbasedonabivariateWienermodel,LifetimeDataAnalysis,4,1998,p.
.2519–22[47]Yashin,A.I.SemiparametricModelsintheStudiesofAgingandLongevity.In:
ParametricandSemiparametricModelswithApplicationsforReliability,Sur-
vivalAnalysis,andQualityofLife,(Eds.M.Nikulin,N.Balakrishnan,N.Limnios,
M.Mesbah),Birkhauser:Boston,2004,p.149–166.
[48]Yashin,A.I.andManton,G.M.Effectsofunobservedandpartiallyobservedco-
variateprocessesonsystemfailure:Areviewamodelsandestimationstrategies,
StatisticalScience,12,1997,p.20–34.
[49]Zacks,Sh.FailureDistributionsAssociatedWithGeneralCompoundRenewal
DamageProcesses.In:Longevity,AgingandDegradationModelsinReliability,
PublicHealth,MedicineandBiology,v.2.(eds.V.Antonov,C.Huber,M.Nikulin,
V.Polischook),St.PetersburgStatePolytechnicalUniversity,St.Petersburg,Rus-
sia,2004,p.336–344.

97