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

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

• Mathematics
• Reliability
• Statistical hypothesis testing

Informations

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

Extrait

UNIVERSITYVILNIUS

IngaMasiulaityte˙

REGRESSIONTHEANODRDYEGRANDADSUARVTIONIVALMODEANALLSYSISINRELIABILITY

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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

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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............

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411418181820202122232324242425622627293437434448484860677680808182

A

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

dmethoaDelt

Bibliography

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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.Parametricandnonparametricestimatorsofthecumulative