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, 2015. 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 e-mail: (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: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
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: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
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) 0.01 0.1 0.5 1.2 0 0 0.039 5.998 0.117 3 0.143 5.994 0.144 0.010 0.000 0.001 0.500 0.007 1. 0.008 0.027 0.082 0.028 0.082 02 0.994 (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: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
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) 4.275 1.516 0.969 0.973 0.017 0.104 0.106 4.667 1.282 0.974 0.973 0.103 5.703 03 0.991 0.991 0.140 0.007 0.097 6.552 0.533 0.996 0.127 0.089 0.088 7.003 0.426 0.123 0.087 0.086 (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) 4.958 1.516 0.964 0.967 0.017 0.103 5.366 1.282 0.979 0.973 0.144 0.101 0.102 6.398 03 0.992 0.992 0.137 0.007 0.098 0.095 7.244 0.533 0.128 0.090 0.093 7.696 0.426 0.127 0.089 0.090 (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: 2078-0958 (Prit); ISSN: 2078-0966 (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) 0.5 5.298 0.486 02 0.110 0.076 0.078 5.297 0.751 0.997 02 0.128 0.005 0.092 0.091 5.297 0.914 0.995 03 0.135 0.006 0.093 0.095 5.305 1.591 04 03 0.008 0.103 5.349 2.507 22 29 0.157 0.011 0.109 0.109 5.322 3.130 12 12 0.163 0.111 0.113 (The upper clum represets mea value f each estimati; the lwer clum represets stadard deviati f each estimati)
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) 0.5 5.990 0.486 0.116 0.081 0.082 5.987 0.751 0.997 0.130 0.005 0.092 0.093 5.994 0.914 02 0.137 0.006 0.096 0.097 5.996 1.591 06 0.145 0.008 0.098 35 2.507 22 21 0.151 0.011 0.105 0.104 52 3.130 33 28 0.160 0.111 0.111 (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: 2078-0958 (Prit); ISSN: 2078-0966 (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 =10000. 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 =10000. 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.
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. 135 149. [2] W. Q. Meeker ad Y. Hg, Reliability meets big data: Opprtuities ad challeges, Quality Egieerig, vl. 26,. 1, 2014, pp. 102-116. [3] W. Nels, Accelerated Testig: Statistical Mdel, Test Plas, ad Data Aalyzes, New Yrk: Jh Wiley, 1990. [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. 611-621. [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, 1998. [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. 2768-2785. [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. 89-101. [9] H. White, "Maximum likelihd estimati f misspecified mdels," Ecmetrica, vl. 50, 1982, pp. 1-25. [10] F. G. Pascual, "Maximum likelihd estimati uder misspecified lgrmal ad Weibull distributis," Cmmuicatis i Statistics Simulati ad Cmputati, vl. 34,. 3, 5, pp. 503-524. ISSN: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)