A Study on Estimation of Lifetime Distribution with Covariates Under Misspecification

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1 Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA A Study Estimati f Lifetime Distributi with Cvariates Uder Misspecificati Masahir Ykyama, Member, IAENG Abstract I these days, the lie mitrig ifrmati which icludes usage histry, system cditis, ad evirmetal cditis is reprted. O statistical mdelig, these variables frm the lie mitrig are primary cadidates fr cvariates which affect the failure mechaism. There is sme literature mdelig by the cumulative expsure mdel fr a prducts lifetime distributi with cvariate effects. Sme existig literatures require a already kw parametric baselie distributi f the cumulative expsure. Hwever such kwledge may be difficult t acquire i advace i sme cases. Whe a icrrect baselie distributi is assumed, it is called misspecificati. A previus study prpsed the strategy which use a likelihd fucti uder a lg-rmal distributi t estimate parameters which represet cvariate effects whe the truly uderlyig baselie distributi is either a Weibull distributi r a lg-rmal distributi. I this time, my paper wides the rage f applicati f the strategy usig the likelihd fucti uder a lg-rmal distributi t estimate parameters f cvariate effects. O that accut, the simulati study ad the discussi fr the bias f estimati are shw. Idex Terms Lg-rmal distributi, Gamma distributi, Birbaum-Sauders distributi, Cumulative expsure, Misspecificati O I. INTRODUCTION NLINE mitrig usig the Iteret has becme a cmm tl fr keepig prducts reliably. This ifrmati icludes usage histry ad evirmetal cditis. I the statistical mdelig f failure mechaisms, the variables ctaied i this ifrmati are primary cadidates fr cvariates that affect a failure mechaism. Ifrmati cvariates ca be used t imprve a reliability aalysis. Fr example, i the aalysis f the lifetime distributi f a truck egie, the lifetime is usually estimated the basis f the distace traveled (mileage) befre catastrphic failure. Hwever, ther variables bserved by each truck, such as a average slpe f the rutes traveled ad a average weight f the lads carried, affect the lad the egie. Thus, the lad the egie ca differ eve thugh the mileage is the same. Therefre, the mileage is t always sufficiet ifrmati fr ptimizig ispecti ad maiteace times. Mauscript received July 10, 2015; revised July 29, This wrk was supprted by a Grat-i-Aid fr Yug Scietists (B) (N. 15K16301) frm the Japa Sciety fr the Prmti f Sciece. Masahir Ykyama is a assistat prfessr at the Departmet f Idustrial ad Systems Egieerig at Chu Uiversity, Japa (masa1034@htmail.cm). II. PREVIOUS STUDIES AND MY PROPOSAL A. Previus Studies I [1] ad [2], the utilizati f dyamic cvariate t predict field-failure is preseted. They used the cumulative expsure (CE) mdel t describe a effect f a dyamic cvariate the failure-time distributi. Referece [3] ad [4] described the CE mdel i the ctext f life tests t icrprate time-varyig cvariates it failure-time mdels. The CE mdel is als kw as the cumulative damage (CD) mdel, i [5]. Besides, referece [6] called the CE mdel a prprtial quatities (PQ) mdel r a scale accelerated failure-time (SAFT) mdel. Furthermre, it is remarked that the CE mdel ca be csidered as a limit f the step-stress mdel whe the legth f each step iterval ges t zer, i [4]. Give the etire cvariate histry, the cumulative expsure u by time t defied as tr u[ t; β, z( t)] exp[ β z( s)] ds. (1) t 0 Here, β is a vectr f cvariate parameters (accelerati cefficiets) ad z(t) is a cvariate vectr acquired at ctiuus time pit t, ad tr represets traspsed. The value f the cvariates are either discrete r ctiuus. Equati (1) represets a liear trasfrmati f t t quatity u, which icrprates dyamic cvariate ifrmati f idividual user s prducts with the cvetial lifetime aalysis. Let T be the failure time f the uit ad β * be the true value f β. The uit fails at time T whe the amut f cumulative expsure reaches a radm threshld U(β * ). Thus, the relatiship betwee the cumulative expsure U(β * ) ad the failure time T is * * U ( β ) exp[ β tr z( s)] ds. (2) T 0 The cumulative expsure threshld U(β * ) has the baselie cumulative distributi fucti (cdf) F 0(U) with distributi parameter θ 0. Equati (2) represets a accumulati f damage. Thus, i this mdel, if the prduct is heavily used r used i a harsh evirmet, the prduct fails ser. Fr this apprach based the CE mdel, the baselie distributi F 0(U) is eeded t idetify i advace. I [1], a Weibull distributi is used fr the CE f actual prducts ad it is said that egieerig kwledge based labratry tests, previus experiece with similar failure mdes, r kwledge ISSN: (Prit); ISSN: (Olie)

2 Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA f the physics f failure ca als prvide useful ifrmati t idetify the baselie distributi. As a special case, the baselie distributi is the same as the distributi f the prduct failure time T, whe it is used uder a cstat cvariates. Hwever csumer prducts are used uder varius evirmet ad the CE value U(β * ) cat be bserved. There is a research abut a statistical test t idetify the baselie distributi i the ctext f accelerated failure time mdels, i [7]. O the ther had, sme study fcus the estimati bias f β uder misspecificati fr the baselie distributi F 0(U). [8] prpsed a strategy ad its validity t use a likelihd fucti uder a lg-rmal distributi t estimate the cvariate effects parameter β whe the uderlyig distributi fr F 0(U) is assumed a Weibull distributi r a lg-rmal distributi. With this apprach, a mdel ca be built that icrprates cvariates eve if a Weibull baselie distributi is misspecified as a lg-rmal e. B. My Prpsal Fr the estimati f β uder misspecificati, this research wide the scpe f the type f distributi frm [8]. I this research, tw-parameter gamma distributi ad Birbaum-Sauders (Fatigue Life) distributi are csidered. This paper prpses a strategy ad its validity t use a likelihd fucti uder a lg-rmal distributi t estimate the cvariate effects parameter β whe the uderlyig distributi fr F 0(U) which is assumed a gamma distributi ad a Birbaum-Sauders (Fatigue Life) distributi. With this apprach, a mdel ca be built that icrprates cvariates eve if a gamma ad Birbaum-Sauders baselie distributi is misspecified as a lg-rmal e. This study makes the lifetime predicti fr idividual user s prducts usig the bserved cvariates easy t use. The gamma distributi is cmmly used i reliability aalysis fr cases where partial failures ca exist, i.e., whe a give umber f partial failures must ccur befre a item fails (e.g., redudat systems). Fr example, the gamma distributi ca describe the time fr a electrical cmpet t fail. The geeral frmula fr the prbability desity fucti f the gamma distributi withut referece t the lcati parameter is ( f ( x) G ) x m1 exp( ( m) x ), m, 0, where m is the shape parameter, η is the scale parameter, ad Γ is the gamma fucti. The Birbaum-Sauders distributi is als cmmly kw as the fatigue life distributi. The assumpti f the Birbaum-Sauders distributi is csistet with a determiistic mdel frm materials physics kw as Mier's rule. Mier s rule is e f the mst widely used cumulative damage mdels fr failures caused by fatigue. Whe the physics f failure suggests Mier's rule applies, the Birbaum-Sauders mdel is a reasable chice fr a lifetime distributi mdel. The frmula fr the prbability desity fucti ad cumulative distributi fucti f the Birbaum-Sauders distributi withut referece t the lcati parameter are x ( ) x x x f BS x, m, 0, 2mx m x ad ( ) x F BS x, m, 0, m where m is the shape parameter, η is the scale parameter, φ is the prbability desity fucti f the stadard rmal distributi, ad Φ is the cumulative distributi fucti f the stadard rmal distributi. III. ANALYTICAL STUDY USING LIKELIHOOD FUNCTION A. Cvariate Here we deal with the case i which a target cvariate is bserved at time t j (j=1,,j) fr a discrete iterval (Fig. 1). The cvariate value is assumed t be cstat betwee each pair f t j. Fig. 1. Cvariate bserved at discrete itervals (J=4) The symbls here used are defied as fllws. T Lifetime t j Cvariate acquisiti pits ( j=1,,j ) J Number f cvariate acquisiti pits up t failure q Cvariate type ( q= 1,,Q, Q: ttal umber f cvariates) z q(t j ) Value f cvariate acquired at t j The liear trasfrmati f lifetime T t quatity U(β) is illustrated i Fig. 2. Fig. 2. Trasfrmati t U(β) Let i (i=1,,) be the prduct umber f each user, T i be the lifetime f each prduct, z(t ij ) be the value f the bserved cvariate at t ij fr each prduct, ad U i (β * ) be the amut f trasfrmati fr each prduct. ISSN: (Prit); ISSN: (Olie)

3 Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA B. Likelihd Fucti Let μ be the mea ad σ be the stadard deviati f the lg-trasfrmed lg-rmal distributi. The likelihd fucti fr a lg-rmal distributi is give by lg LLN (β,, ) β tr z (ti, Bi ) lg( 2 ) The lg-likelihd fucti fr a gamma distributi is give by β tr i, Bi z (t ) (m 1) lg(u i (β)) U i (β) m lg( ) lg( (m)). lg LBS (β, m, ) β tr z (ti, B ) lg i lg( 2mU i (β)) lg( 2 ) 1 2m 2 (: Iterval is less tha Ji ly fr last bservati.) Here, exp J i 1 U i (β* ) exp 1* z1 (tij ) 2* z 2 (tij ) j 1 The lg-likelihd fucti fr a Birbaum-Sauders distributi is give by (: Iterval is icreasig by 1.) (ti, J i ti, J i 1 ) J i (lg(u i (β)) ) 2 lg(u i (β)). 2 2 lg LG (β, m, ) (3). Ti The values f Ti were determied frm the Ui (β*) ad z(tij) geerated as described abve. The iterval betwee each pair f cvariate bservatis was determied as fllws. ti1 1, (ti2 ti1 ) 2,, (ti, J i 1 ti, J i 2 ) J i 1 2. IV. SIMULATION STUDY AND RESULTS A. Simulati Settig I this simulati study, β was estimated frm cvariate z(tij) ad lifetime Ti usig MLE uder assumig U(β*) as a lg-rmal distributi. These data were geerated as explaied belw. (1). Ui (β*) The values f Ui (β*) were geerated i accrdace with a lg-rmal distributi, a gamma distributi r a Birbaum-Sauders distributi. I secti Ⅳ-B, the results f estimati uder misspecificati is shw. O the ther had, i secti Ⅳ-C, the results f estimati uder misspecificati fr a gamma distributi is shw. Furthermre, i secti Ⅳ-D, the results f estimati uder misspecificati fr a Birbaum-Sauders distributi is shw. * 1 z1 (ti, J i ) 2* z 2 (ti, J i ) (ti, J i ti, J i 1 ). Therefre, lifetime Ti fr each prduct was btaied frm the summati f t i, J i. B. Simulati Results (Lg-rmal Distributi) First we shw that U(β*) ca be crrectly estimated usig MLE assumig a lg-rmal distributi fr U(β*) geerated i accrdace with a lg-rmal distributi. Table I shws the estimati result f β. The μ ad σ represet the f the mea value ad the stadard deviati value f the lg-trasfrmed lg-rmal distributi, respectively. The estimati frm simulati data was repeated 0 times. The results shw i the upper rws are the average estimated values, ad thse shw i the lwer rws are the stadard deviatis. These results shw that the value f β was crrectly estimated by MLE. TABLE I Estimatis btaied fr Ui (β*) whe Ui (β*) was geerated usig a lg-rmal distributi (β1*=β2* =0, =10000,. f repetitis=0) (The upper clum represets mea value f each estimati; the lwer clum represets stadard deviati f each estimati) (2). z(tij) As was explaied abve fr Fig. 1, the values f z(tij) was geerated as cstat durig each bservati. We assumed tw kids f cvariate; cvariates z1(tij) ad z2(tij) were idepedetly geerated fr each prduct fr each bserved time pit as fllwig a rmal distributi N(,0.1), that is takig psitive values. Thus, i ur simulati, it is assumed that each prduct is used uder a rmal cditi that time-varyig cvariates have tred fr time. Here, true values f parameter vectr β*=(β1*, β2*) were set as 0 fr bth. ISSN: (Prit); ISSN: (Olie)

4 Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA C. Simulati Results (Gamma Distributi) Here we shw that β ca be apprximately estimated usig MLE assumig a lg-rmal distributi fr Ui (β*) geerated i accrdace with a gamma distributi. That is, i this simulati study, β is estimated whe a gamma distributi is misspecified as a lg-rmal distributi. The true value f β was set t 0, i.e., β1*=β2* =0. The umber f samples was set as 10000, the shape parameter m was set as,,,, ad the lcati parameter η was set as,. The estimati f β was repeated 0 times. Table II ad Table III shw the mea value ad the stadard deviatis f estimated values. TABLE II Estimatis btaied fr Ui (β*) whe Ui (β*) was geerated usig a gamma distributi (β1*=β2* =0, =10000,. f repetitis=0) (The upper clum represets mea value f each estimati; the lwer clum represets stadard deviati f each estimati) TABLE III Estimatis btaied fr Ui (β*) whe Ui (β*) was geerated usig a gamma distributi (β1*=β2* =0, =10000,. f repetitis=0) (The upper clum represets mea value f each estimati; the lwer clum represets stadard deviati f each estimati) The results Table II ad Table III fr differet distributi assumptis shw that parameter β ca be apprximately btaied. Next, it is ivestigated that the variati i the estimated results i each value f shape parameter m. Fig. 3 shws a bx plt (fr η=, =10000) f estimati result β1 i Table III. ISSN: (Prit); ISSN: (Olie) Fig. 3. Bx plt f estimati result β1 fr the case f Table II (η=100, β1*=β2* =0, =10000,. f repetitis=0) These result shws that whe the value f the shape parameter m f the gamma distributi was large, the bias f the resultig estimates f β was small. D. Simulati Results (Birbaum-Sauders Distributi) Here we shw that β ca be apprximately estimated usig MLE assumig a lg-rmal distributi fr Ui (β*) geerated i accrdace with a Birbaum-Sauders distributi. That is, i this simulati study, β is estimated whe a Birbaum-Sauders distributi is misspecified as a lg-rmal distributi. The true value f β was set t 0, i.e., β1*=β2* =0. The umber f samples was set as 10000, the shape parameter m was set as 0.5,,,,, ad the lcati parameter η was set as,. The estimati f β was repeated 0 times. Table IV ad Table V shw the mea value ad the stadard deviatis f estimated values. TABLE IV Estimatis btaied fr Ui (β*) whe Ui (β*) was geerated usig a Birbaum-Sauders distributi (β1*=β2* =0, =10000,. f repetitis=0) (The upper clum represets mea value f each estimati; the lwer clum represets stadard deviati f each estimati)

5 Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA TABLE V Estimatis btaied fr Ui (β*) whe Ui (β*) was geerated usig a Birbaum-Sauders distributi (β1*=β2* =0, =10000,. f repetitis=0) (The upper clum represets mea value f each estimati; the lwer clum represets stadard deviati f each estimati) The results i Table IV ad Table V shw that parameter β ca be apprximately btaied i the case f misspecificati fr the Birbaum-Sauders distributi. Next, it is ivestigated that the variati i the estimated results i each value f shape parameter m. Fig. 4 shws a bx plt (fr η=, =10000) f estimati result β1 i Table V. Fig. 4. Bx plt f estimati result β1 fr the case f Table V (η=100, β1*=β2* =0, =10000,. f repetitis=0) These result shws that whe the value f the shape parameter m f the Birbaum-Sauders distributi was small, the bias f the resultig estimates f β was small. Hwever, there is a strage value bserved fr the mea value f estimati f β i the case f (m=6, η=) at the Table IV. The cause is that there is may samples that the value f the cvariate has t chaged eve ce befre failure pit i this case, as Fig. 5. Thereby, bias has appeared the estimati result i this case. Uder the rmal cditi, the cvariates are assumed t have chagig i time. The validati fr the ifluece due t the bserved cvariates which has t chaged eve ce befre failure pit t estimati bias will be with future challeges. ISSN: (Prit); ISSN: (Olie) Number f cvariate acquisiti pits up t failure Fig. 5. Number f acquisiti pits f cvariate ifrmati up t failure i the case f a certai e estimati uder a Birbaum-Sauders distributi (m=6, η=, =10000). I this simulati study, the cvariate value is assumed t be cstat betwee each acquisiti pit. V. CONCLUSION AND FUTURE WORK I this secti, the strategy uder misspecificati fr a baselie distributi is prpsed. This paper ivestigates the asympttic bias f the maximum likelihd estimatr fr a cvariate parameter β uder misspecificati. The results i [9] ad [10] als discussed the asympttic bias ad the asympttic distributi f the MLE whe the assumed mdel is icrrect. Besides, they called this icrrect MLE quasi-mle (QMLE). Frm [9], it is shw that a bias betwee MLE ad QMLE is idepedet fr a sample size. I the case f misspecificati that the true mdel is a gamma distributi r a Birbaum-Sauders distributi ad the icrrect mdel is a lg-rmal distributi, the results f umerical simulati shw that cvariate parameter β ca be apprximately estimated by maximum likelihd estimati assumig a lg-rmal distributi fr the baselie distributi f cumulative expsure. Frm the results f umerical simulati fr a gamma distributi, it is shw that the value f a lcati parameter η has ifluece fr the estimati bias. O the ther had, it is shw that a scale parameter m affects the bias f estimati f cvariate parameters β uder misspecificati. Frm Table II ad Table III, whe a shape parameter takes m 4, it is cfirmed that the rate f the bias fr the true value (=0) is fall uder 0.5% i the case f = Furthermre, frm the results f umerical simulati fr a Birbaum-Sauders distributi, it is shw that the value f a lcati parameter η has ifluece fr the estimati bias. O the ther had, it is shw that a scale parameter m affect the bias f estimati f cvariate parameters β uder misspecificati. Frm the result Table IV ad Table V, whe a shape parameter takes m 2, it is cfirmed that the rate f the bias fr the true value (=0) is fall uder 0.5% i the case f = Thereby, whe the truly uderlyig baselie distributi is either a gamma distributi r a Birbaum-Sauders distributi, these results prvide a strategy t use a likelihd fucti uder a lg-rmal distributi t estimate cvariate effect parameters β. Frm the results f umerical simulati, it seems that, if the value f the estimated σ is small, the bias f β is small.

6 Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA Future wrk icludes fidig ways t imprve reliability by usig lie cditi mitrig. It will be exteded this study t the case f time-varyig cvariates havig tred fr time. REFERENCES [1] Y. Hg ad W. Q. Meeker, Field-failure predictis based failure-time data with dyamic cvariate ifrmati, Techmetrics, vl. 55,. 2, 2013, pp [2] W. Q. Meeker ad Y. Hg, Reliability meets big data: Opprtuities ad challeges, Quality Egieerig, vl. 26,. 1, 2014, pp [3] W. Nels, Accelerated Testig: Statistical Mdel, Test Plas, ad Data Aalyzes, New Yrk: Jh Wiley, [4] W. Nels, Predicti f field reliability f uits, each uder differig dyamic stresses, frm accelerated test data, Hadbk f Statistics, vl. 20, 1, pp [5] V. Bagdavicius ad M. Nikuli, Accelerated Life Mdels: Mdelig ad Statistical Aalysis. Bca Rat, FL: Chapma & Hall/CRC, 1. [6] W. Q. Meeker ad L. A. Escbar, Statistical Methds fr Reliability Data, New Yrk: Jh Wiley, [7] V. Bagdavicius, R. Levuliee ad M. Nikuli, "Chi-squared gdess-f-fit tests fr parametric accelerated failure time mdels," Cmmuicatis i Statistics-Thery ad Methds, vl. 42,. 15, 2013, pp [8] M. Ykyama, W. Yamamt ad K. Suzuki, "A Study Estimati f Lifetime Distributi with Cvariates Usig Olie Mitrig," Ttal Quality Sciece, vl. 1,. 2, 2015, pp [9] H. White, "Maximum likelihd estimati f misspecified mdels," Ecmetrica, vl. 50, 1982, pp [10] F. G. Pascual, "Maximum likelihd estimati uder misspecified lgrmal ad Weibull distributis," Cmmuicatis i Statistics Simulati ad Cmputati, vl. 34,. 3, 5, pp ISSN: (Prit); ISSN: (Olie)