Analysis of Pham(Loglog) Reliability Model using Bayesian Approach

Transcription

1 Computer Scece Joural Volume, Issue 2, August 20 Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach Ashw Kumar Srvastava,, Vjay Kumar 2* Departmet of Computer Applcato, S K P G College, Bast, (U P), INDIA 2 Departmet of Mathematcs ad Statstcs, DDU Grorakhpur Uversty, Gorakhpur. ashw.skpg@gmal.com 2 vkgkp@redffmal.com Abstract: I ths paper, the two-parameter Pham(Loglog) model s cosdered to aalyze the software relablty data. The Markov Cha Mote Carlo (MCMC) method s used to compute the Bayes estmates of the model parameters. It has bee assumed that the parameters have gamma prors ad they are depedetly dstrbuted. Uder the above prors, Gbbs algorthm OpeBUGS has bee appled to geerate MCMC samples from the posteror desty fucto. Based o the geerated samples, the Bayes estmates ad hghest posteror desty credble tervals of the ukow parameters have bee computed. The maxmum lkelhood estmate ad assocated cofdece tervals have bee costructed to compare the performaces of the Bayes estmators wth the classcal estmators. Oe real software relablty data set has bee aalyzed to demostrate how the proposed method ca be used practce. Keywords: Software Relablty, Log log (Pham) model, Bayesa estmato, MLE, Markov Cha Mote Carlo(MCMC). Receved: Aprl 20, Revsed July 20, Publshed: August 20 * Correspodg Author: Vjay Kumar vkgkp@redffmal.com. Itroducto Recetly a two-parameter lfetme dstrbuto wth a Vtub-shaped hazard rate, called Loglog model or Pham model has bee troduced by Pham, []. As for the bathtubshaped, after the fat mortalty perod, the useful lfe of the system begs. Durg ts useful lfe, the system fals as a costat rate. Ths perod s the followed by a wear out perod durg whch the system starts slowly ad creases wth the o set of wear out. 79

2 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach For the Vtub-shaped, after the fat mortalty perod, the system starts to experece at a relatvely low creasg rate, but ths s ot costat, ad the creases wth falures due to agg. The Vtub-shaped hazard rate s defed as: If there exsts a chage pot x 0 such that the hazard rate h(x) s decreasg (0, x 0 ] ad slowly creasg, as a Vtubshaped, [x 0, ). Therefore, t s evdet that the loglog model wth Vtub-shaped ad Webull model wth bathtub-shaped falure rates are ot the same [2, 3]. The software qualty cotues to mprove as testg progresses. However, most real-lfe testg scearos, the software falure testy creases tally ad the decreases. There may be case where we ca observe Vtub-shaped falure rate fucto for software relablty data. 2. Related work A umber of models have bee advocated the lterature to deal wth o-mootoc especally the bathtub behavour of the hazard rate [4, 5]. The Loglog model has bee troduced ad studed by Pham []. The fereces to the loglog model are comparatvely meager perhaps because of the fact that the model has bee proposed oly recetly ad also because the form of the probablty desty fucto(pdf) s a bt complcated to provde closed-form fereces. The classcal fereces were cosdered by Pham[], ad obtaed pot estmates of the parameters usg maxmum lkelhood techque. Pham[6] summarzes some of the classcal results related to the model. The Pham model faled to draw the atteto of Bayesa aalysts geeral. We perhaps do ot fd ay referece that systematzes Bayesa fereces related to the model. The Bayesa fereces for some software relablty models have bee dscussed [7 0]. The Bayesa method of reasog s well kow. Its apparet advatages over the classcal paradgm are detaled by several authors [, 2]. I fact, the Bayesa method of reasog allows the corporato of formato about the ukow quattes a more formal ad structural maer. Sce the Bayesa aalysts cosder addtoal formato about the parameters the form of a pror dstrbuto, ths may add to the formato provded by the sample observatos ad thereby may result a moderate effect o the fal fereces. The techcal problem of evaluatg quattes requred for Bayesa ferece typcally reduces to the calculato of tegrals. As a result, oe requres the use of sophstcated Bayes computatoal techques for the same. We, however, stad for the use of sample-based procedures, especally the Markov cha Mote Carlo (MCMC) techques, as they provde the latter developmets routely avalable oce the samples from the posteror are obtaed (see, e.g., [ 3], etc.). I ths Paper, we preset Pham model for software relablty data. The correspodg hazard rate of the Pham model, called the Vtub-shaped hazard rate, ot oly cludes 80

3 Computer Scece Joural Volume, Issue 2, August 20 dstrbutos wth bathtub, creasg ad decreasg falure rates, but also provdes a broader class of mootoe falure rates. We llustrate the usefuless of the ew Vtubshaped hazard rate fucto by aalysg the real data set usg classcal as well as Bayesa methods. The Bayesa estmato of the model parameters s cosdered uder the depedet gamma prors usg Markov Cha Mote Carlo (MCMC) method. A procedure s developed to obta posteror sample usg MCMC smulato method by corporatg a module to OpeBUGS. The posteror aalyss s performed R software( A Programmg Laguage ad Computg Evromet), [4, 5]. 3. Model Aalyss 3. The cumulatve dstrbuto fucto(cdf) For > 0 ad, λ > 0 the two-parameter Pham model has the dstrbuto fucto; x F( x) = exp λ ; x > 0, > 0, λ > 0. (3.) Here ad λ are the shape ad scale parameters respectvely. The two-parameter Pham model wll be deoted by Pham(, λ). 3.2 The probablty desty fucto(pdf) The Pham probablty desty fucto s gve as follows (Pham 2002): x x f ( x) = l ( λ) x λ exp λ ; x > 0, > 0, λ > 0. (3.2) The R fuctos dloglog( ) ad ploglog( ), gve [6], ca be used for the computato of pdf ad cdf, respectvely. Some of the typcal Pham desty fuctos for dfferet values of ad for λ = are depcted Fgure. It s clear from the Fgure that the desty fucto of the Pham model ca take dfferet shapes. 3.3 Mode The mode ca be obtaed by solvg the o-lear equato x ( ) + x ( l λ) λ = 0 (3.3) 3.4 The Quatle fucto For a cotuous dstrbuto F(x), the p percetle (also referred to as fractle or quatle), x p, for a gve p, 0 < p <, s a umber such that P(X x p ) = F(x p ) = p (3.4) 8

4 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach The quatle fucto of Pham model ca be obtaed by solvg log ( log p) xp =. (3.5) log λ ( ) The computato of quatles ca be performed usg the R fucto qloglog( ), [6]. I partcular, for p=0.5 we get log ( log(0.5) ) Meda =. (3.6) log λ ( ) Fgure : Plots of the pdf of the Pham model for λ = ad dfferet values of 3.5 The radom devate geerato Let U be the uform (0,) radom varable ad F(.) a cdf for whch F - (.) exsts. The F - (u) s a draw from dstrbuto F(.). Therefore, the radom devate ca be geerated from Pham(, λ) by / x = log ( u ) 0.5 ; 0 < u <. (3.7) λ where u ~ U(0, ). The R fucto rloglog(), gve [6],geerates the radom devate from Pham(, λ). 3.6 The Relablty/ Survval fucto The relablty/survval fucto 82

5 Computer Scece Joural Volume, Issue 2, August 20 x R(x) = exp λ ; x > 0, > 0, λ > 0. (3.8) The R fucto sloglog( ), gve [6], computes the relablty/ survval fucto. 3.7 The hazard fucto The hazard rate fucto of Pham(, λ) s gve by x h (x) = l λ x λ (3.9) ( ) ad the assocated R fucto hloglog( ), gve [6]. Dfferetatg equato (3.9) w.r.t. x, we have x 2 h ( x) ( l ) x = λ λ ( ) + l λ.x (3.0) Settg h (x) = 0 ad after smplfcato, we obta the chage pot as x0 = l λ. (3.) It easly follows that the sg of h ( x) s determed by ( ) + l λ.x whch s egatve for all x x 0 ad postve for all x x 0. Therefore, h(x) s tally decreasg ad the creasg x. It s easy to see that whe the model F(.) s a IFR. The shape of the hazard fucto depeds upo both the parameters. Fgure 2 exhbts the dfferet hazard rate fuctos of Pham model. Fgure 2: Plots of the hazard fucto of the Log-log(Pham) model for λ= ad dfferet values of 3.8 Some Other Relablty Idcators The cumulatve hazard fucto H(x) defed as 83

6 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach ( ) l S( x ) H x = (3.2) that ca be obtaed wth the help of ploglog( ), [6], fucto by choosg argumets lower.tal = FALSE ad log.p = TRUE..e. - ploglog(x, alpha, lambda, lower.tal= FALSE, log.p = TRUE) Two other relevat fuctos useful relablty aalyss are falure rate average (fra) ad codtoal survval fucto(crf) The falure rate average of x s gve by H(x) x FRA(x) = = h(x) dx, x > 0, (3.3) x x 0 where, H(x) s the cumulatve hazard fucto. A aalyss for FRA(x) o x permts to obta the IFRA ad DFRA classes. The codtoal survval of X s defed by S (x + t) S(x t) =, t > 0, x > 0, S(x)= F(x), (3.4) S(x) where F( ) s the cdf of X. Smlarly to h(x) ad FRA(x), the dstrbuto of X belogs to the ew better tha used (NBU), expoetal, or ew worse tha used (NWU) classes, whe S(x t) < S(x), S(t x) = S(x), or S(x t) > S(x), respectvely. The R fuctos hra.loglog() ad crf.loglog(), gve [6], ca be used for the computato of falure rate average (fra) ad codtoal survval fucto(crf), respectvely. 4. Maxmum Lkelhood Estmato (MLE) ad Iformato Matrx I ths secto, we brefly dscuss the maxmum lkelhood estmators (MLE s) of the two-parameter Pham model ad dscuss ther asymptotc propertes to obta approxmate cofdece tervals based o MLE s [2]. Let x=(x,..., x ) be a radom sample of sze from Pham(, λ), the the lkelhood fucto L(, λ) ca be wrtte as =. (4.) x x ( λ ) = ( λ) λ λ L, l.x exp The log lkelhood fucto s x l (, λ ) = l + l ( l λ ) + ( ) l x + ( l λ ) x + λ. (4.2) = = = The frst dervatves of the log lkelhood fucto wth respect to ad λ are ad l (4.3) x (, λ ) = + l x + ( l λ) ( l x ).x x. λ.( l λ).( l x ) = = = ( ) λ λ l λ λ x l, λ = + x x λ, (4.4) = = 84

7 Computer Scece Joural Volume, Issue 2, August 20 respectvely. Settg equatos (4.3) ad (4.4) equal to zero, we ca obta the MLE of ad λ by solvg the followg smultaeous o-lear equatos: l λ. l x.x. λ = l x x ( ) = = x = l λ x. λ = 0. (4.5) We ca also obta the MLE's of ad λ by maxmzg equato (4.2) drectly wth respect to ad λ. We ext determe the cofdece tervals for parameter estmates ad λ. For the log- lkelhood fucto (4.2), we ca obta the Fsher formato matrx H as h h2 H = h2 h22 where, l l l h = E, h 2 2 = h2 = E, h22 = E. 2 λ λ The varace matrx, V, ca be obtaed as follows v v2 V = [ H] =. (4.6) v2 v22 The varaces of ad λ are Var() =v Var(λ) =v 22 Oe ca approxmately obta the (l-β)00% cofdece tervals for ad λ based o the ormal dstrbuto as ˆ Z v ˆ ˆ 22, ˆ Z v 22, ad β β Zβ v, Zβ v + λ λ + (4.7) respectvely, where v j s gve (4.6) ad Z β s ((l-β/2)00% of the stadard ormal dstrbuto. There are several other statstcal methods for estmatg model parameters such as the method of momets, the method of percetle ad the Bayesa method. Ufortuately, oe of these methods (excludg Bayesa) s approprate for small data sets [4, 5]. 5. Bayesa estmato usg Markov cha Mote Carlo method A Mote Carlo method s a algorthm that reles o repeated pseudo-radom samplg for computato, ad s therefore stochastc (as opposed to determstc). Mote Carlo methods are ofte used for smulato. The uo of Markov chas ad Mote Carlo methods s called MCMC. A Markov cha s a radom process wth a 85

8 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach fte state-space ad the Markov property, meag that the ext state depeds oly o the curret state, ot o the past. The revval of Bayesa ferece sce the 980s s due to MCMC algorthms ad creased computg power. The most prevalet MCMC algorthms may be the smplest: radom-walk Metropols ad Gbbs samplg. The qualty of the margal samples usually mproves wth the umber of teratos. I Bayesa ferece, the target dstrbuto of each Markov cha s usually a margal posteror dstrbuto, such as each parameter. Each Markov cha begs wth a tal value ad the algorthm terates, attemptg to maxmze the logarthm of the uormalzed jot posteror dstrbuto ad evetually arrvg at each target dstrbuto. Each terato s cosdered a state. The Gbbs algorthm starts by assumg some arbtrarly chose tal values for the cocered varates ad the geeratg the varate values from the varous full codtoals a cyclc order. That s, every tme a varate value s geerated from a full codtoal, t s flueced by the most recet values of all other codtog varables ad, after each cycle of terato, t s updated by samplg a ew value from ts full codtoal. The etre geeratg scheme s repeated uless the geeratg cha acheves a systematc patter of covergece. It ca be show that after a large umber of teratos the geerated varates ca be regarded as the radom samples from the correspodg posterors. Readers are referred to Gamerma [-3] for detals of the procedure ad the related covergece dagostc ssues. The most wdely used pece of software for appled Bayesa ferece s the OpeBUGS, Thomas [7]. The software offers a user-terface, based o dalogue boxes ad meu commads, through whch the model may the be aalyzed usg Markov Cha Mote Carlo techques. It s a fully extesble modular framework for costructg ad aalyzg Bayesa probablty models for the exstg probablty models, [8, 9]. As the Pham model s ot avalable OpeBUGS. Thus t requres corporato of a module to estmate parameters of Pham model. 6. Data Aalyss I ths secto we preset the aalyss of oe real data set for llustrato of the proposed methodology. The data represet the 86 tme-betwee-falures of a software (SYS2.DAT) [20]. The data set has bee modfed from tme-betwee-falures to tme to falures ad dvded all the values by 00 just for computatoal purpose. Frst we compute the maxmum lkelhood estmates. 6. Computato of MLE ad model valdato The Pham model s used to ft ths data set. We have started the teratve 86

9 Computer Scece Joural Volume, Issue 2, August 20 procedure by maxmzg the log-lkelhood fucto gve (4.2) drectly wth a tal guess for = 0.2 ad λ =.3, far away from the soluto. We have used optm() fucto R wth opto Newto-Raphso method. The teratve process stopped oly after 6 teratos. We obta ˆ = , ˆλ = ad the correspodg loglkelhood value = A estmate of varace-covarace matrx, usg (4.5), s gve by e e-06 Thus, usg (4.7) we ca costruct the approxmate 95% cofdece tervals for the parameters of Pham model based o MLE s. Table shows the MLE s wth ther stadard errors ad approxmate 95% cofdece tervals for ad λ. Table : Parameter MLE Std. Error 95% Cofdece Iterval alpha ( , ) lambda ( , ) Maxmum lkelhood estmate, stadard error ad 95% cofdece terval To study the goodess of ft of the Pham model, we compute the Kolmogorov- Smrov statstc betwee the emprcal dstrbuto fucto ad the ftted dstrbuto fucto whe the parameters are obtaed by method of maxmum lkelhood. For ths Fgure 3: The graph of emprcal dstrbuto fucto ad ftted dstrbuto fucto. we have used the R fucto ks.loglog() gve [6]. The result of K-S test s D = wth the correspodg p-value = , Therefore, the hgh p-value clearly 87

10 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach dcates that Pham model ca be used to aalyze ths data set, we also plot the emprcal dstrbuto fucto ad the ftted dstrbuto fucto Fgure 3. From above result ad Fgure 3, t s clear that the estmated Pham model provdes excellet ft to the gve data. Fgure 4: Quatle-Quatle(Q-Q) plot usg MLEs as estmate. The graphcal methods wdely used for checkg whether a ftted model s agreemet wth the data are Quatle-Quatle(Q-Q) ad Probablty-Probablty (P-P) plots model valdato. The correspodg R fuctos are qq.loglog( ) ad pp.loglog( ), [6]. Fgure 5: Probablty-Probablty(P-P) plot usg MLEs as estmate. 88

11 Computer Scece Joural Volume, Issue 2, August 20 The Q-Q plot shows the estmated versus the observed quatles. If the model fts the data well, the patter of pots o the Q-Q plot wll exhbt a 45-degree straght le. As ca be see from the straght le patter Fgure 4, the Pham model fts the data very well. Ths s also supported by the Probablty-Probablty plot Fgure Bayesa Aalyss uder Gamma Prors The Bayesa model s costructed by specfyg the pror dstrbutos for the model parameters alpha ad lambda, ad the multplyg wth the lkelhood fucto to obta the posteror dstrbuto fucto. Gve a set of data x=(x,..., x ), the lkelhood fucto L(, λ) s x = x = = L (, λ x) = ( l λ) x λ exp λ. (6.) Deote the pror dstrbuto of ad λ as p(, λ). The jot posteror s p, λ x L, λ x p, λ ( ) ( ) ( ) Let us cosder depedet gamma prors for the parameters ~ gamma(a, b) ad λ ~gamma(c, d) as ad a b a p( ) = exp( b ) ; > 0, (a, b) > 0 Γ(a) c d c p( λ ) = λ exp( d λ) ; λ > 0, (c, d) > 0. Γ(c) Combg the lkelhood fucto wth the pror va Bayes theorem yelds the posteror up to proportoalty as c + x + a = x = = p (, λ x) ( l λ) x λ exp b d λ λ. It may be observed that posteror s qute complcated ad o close form fereces appear possble. We, therefore, propose to cosder MCMC methods, amely the Gbbs sampler ad the Metropols algorthms, to smulate samples from the posteror so that sample-based fereces ca be easly draw. For Gbbs sampler mplemetato, oe eeds to obta the full codtoals for ad λ ad the same up to proportoalty ca be specfed as x + a = x = = p ( λ, x) x λ exp b λ, (6.2) 89

12 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach c + x = x p ( λ, x) ( l λ) λ exp d λ λ =. (6.3) It ca be see that (6.2) ad (6.3) are qute complcated ad do ot belog to ay wellkow famly of dstrbutos. These two full codtoals are otherwse also dffcult to smulate. Therefore, we ca coclude that the Gbbs sampler scheme s ot easy to mplemet drectly for the pror-lkelhood combatos that we have cosdered for defg the posteror. Therefore, we opt for stochastc smulato procedure, amely the Gbbs sampler OpeBUGS, a well-establshed ope source ad fully extesble software, to geerate samples from the posteror dstrbutos. The Bayesa aalyss of a probablty model ca be performed for the models defed OpeBUGS. As the Pham model s ot avalable OpeBUGS. Thus t requres corporato of a module to estmate parameters of Pham model. Recetly, a umber of probablty models have bee corporated OpeBUGS to facltate the Bayesa aalyss, [2]. The readers are referred to [8-0, 22] for mplemetato detals of some models. A module, dpham_t(alpha, lambda), s wrtte compoet Pascal for Pham model to perform full Bayesa aalyss OpeBUGS usg the method descrbed Thomas [7], Lu et al. [8] ad Thomas et al. [9]. The module code ca be obtaed from the author. The module dpham_t(alpha, lambda) eables to perform full Bayesa aalyss of Pham model to OpeBUGS usg the method descrbed [8]. Implemetato of Module - dpham_t(alpha, lambda) The developed module s mplemeted to obta the Bayes estmates of the Pham model usg MCMC method. The ma fucto of the module s to geerate MCMC sample from posteror dstrbuto for formatve set of prors,.e. Gamma prors. Model { for( : N ) { x[] ~ dpham_t(alpha, lambda) } # Pror dstrbutos of the Model parameters # Gamma pror for alpha alpha ~ dgamma(0.00, 0.00) # Gamma pror for lambda lambda ~ dgamma(0.00, 0.00) } Ital values 90

13 Computer Scece Joural Volume, Issue 2, August 20 # Cha lst(alpha=0.5,lambda=0.4) # Cha 2 lst(alpha=.0, lambda=.7) We ru the model to geerate two Markov Chas at the legth of 40,000 wth dfferet startg pots of the parameters. The covergece s motored usg trace ad ergodc mea plots, we fd that the Markov Cha coverge together after approxmately 2000 observatos. Therefore, bur of 5000 samples s more tha eough to erase the effect of startg pot(tal values). Fally, samples of sze 7000 are formed from the posteror by pckg up equally spaced every ffth outcome,.e. th=5, startg from 500.Ths s doe to mmze the auto correlato amog the geerated devates. Therefore, we have the posteror sample {,λ }, =,,7000 from cha ad { 2, λ 2 }, =,,7000 from cha 2. The cha s cosdered for covergece dagostcs plots. The vsual summary s based o posteror sample obtaed from cha 2 whereas the umercal summary s preseted for both the chas. Covergece dagostcs Sequetal realzato of the parameters alpha ad theta ca be observed Fgure 6. The Markov cha s most lkely to be samplg from the statoary dstrbuto ad s mxg well. Hstory(Trace) plot Fgure 6: Sequetal realzato of the parameters ad λ The plot looks lke a horzotal bad, wth o log upward or dowward treds, the we have evdece that the cha has coverged. Rug Mea (Ergodc mea) Plot 9

14 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach I order to study the covergece patter, we have plotted a tme seres (terato umber) graph of the rug mea for each parameter the cha. The mea of all sampled values up to ad cludg that at a gve terato gves the rug mea. I the Fgure 7 gve below, a systematc patter of covergece based o ergodc averages ca be see after a tal traset behavour of the cha. Autocorrelato Fgure 7: The Ergodc mea plots for ad λ The graph shows that the correlato s almost eglgble. We may coclude that the samples are depedet. Brooks-Gelma-Rub Fgure 8: The autocorrelato plots for ad λ The Brooks-Gelma-Rub plot uses the parallel chas wth dspersed tal values to test whether they all coverge to the same target dstrbuto. Falure could dcate the presece of a mult-mode posteror dstrbuto (dfferet chas coverge to dfferet local modes) or the eed to ru a loger cha (bur- s yet to be completed). 92

15 Computer Scece Joural Volume, Issue 2, August 20 Fgure 9: The BGR plots for ad λ From the Fgure 9, t s clear that covergece s acheved. Thus we ca obta the posteror summary statstcs. Numercal Summary I Table 2, we have cosdered varous quattes of terest ad ther umercal values based o MCMC sample of posteror characterstcs for Pham model uder Gamma prors. The umercal summary s based o fal posteror sample (MCMC output) of 7000 samples for alpha ad lambda. {, λ }, =,,7000 from cha ad { 2, λ 2 }, =,,7000 from cha 2. The Table 2, cotas the followg computed values :. Posteror Mea : Bayes estmate uder Squared Error Loss 2. The Mote Carlo(MC) error ad sample stadard devato (SD) 3. The fve pot summary : Mmum, Q, meda, Q 3 ad Maxmum 4. Posteror Meda : Bayes estmate uder Absolute Error loss 5. Posteror Mode : Bayes estmate uder Zero-Oe loss 6. Symmetrc Credble tervals: To compute the credble tervals of ad λ, order,, M ad λ,, λ M as (),, (M) ad λ (),, λ (M) respectvely. The the 00(-2γ)% symmetrc credble tervals of ad λ respectvely become ((Mγ), (M(- γ))) ad (λ(m γ), λ (M(- γ))) 7. Hghest probablty desty (HPD) terval: The HPD tervals are of the shortest legth of ay of the Bayesa credble tervals. Let () <... < M) be the ordered (). Cosder the followg 00(- γ)% credble tervals of ( (), ((- γ)m) ),... ( (γm) (M) ); the choose that terval whch has the shortest legth. Smlarly, we ca costruct a 00(- γ)% HPD credble terval of λ. 93

16 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach Table 2: Numercal summares based o MCMC sample of posteror characterstcs for Pham model uder Gamma prors The 95% symmetrc credble tervals ad HPD tervals are computed for γ = The Hghest probablty desty (HPD) tervals are computed the algorthm descrbed by Che ad Shao [23] uder the assumpto of umodal margal posteror dstrbuto We ca also estmate ay fucto G()(or G(λ)) of the parameters of terest (or λ) by smply cosderg {G( ),G(λ )}, =,,7000 from cha ad {G( 2 ), G λ 2 )}, =,,7000 from cha 2, as a MCMC sample of the desred parametrc fucto G()(or G(λ)). Vsual summary Box plots The boxes represet ter-quartle rages ad the sold black le at the cetre of each box s the mea; the arms of each box exted to cover the cetral 95 per cet of the dstrbuto - ther eds correspod, therefore, to the 2.5% ad 97.5% quatles. (Note that ths represetato dffers somewhat from the tradtoal.) 94

17 Computer Scece Joural Volume, Issue 2, August 20 Kerel desty estmates Fgure 0: The boxplots for alpha ad lambda. We plot of the probablty hstograms of the samples of ad λ geerated by MCMC method alog wth ther kerel desty estmates. The kerel desty estmates have bee draw usg R software, [4, 5], wth the assumpto of Gaussa kerel ad properly chose values of the badwdths. Fgure ad 2 provde the kerel desty estmate of ad λ. It ca be see that s symmetrc whereas λ shows postve skewess. The actual MCMC sample values are plotted alog the x-axs. Fgure : Hstogram ad kerel desty estmate of based o MCMC samples, vertcal les are draw at the correspodg MLE ad Bayes estmates. 95

18 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach Fgure 2: Kerel desty estmate ad hstogram of λ based o MCMC samples, vertcal les dcates the correspodg ML ad Bayes estmates. 7. Comparso wth MLE I ths secto our ma am s to demostrate the effectveess of proposed methodology. For the comparso wth MLE we have plotted three graphs. Fgure 3: The desty fuctos f(x; ˆ, λ ˆ ) usg MLEs ad Bayesa estmates, computed va MCMC samples uder gamma prors. 96

19 Computer Scece Joural Volume, Issue 2, August 20 I Fgure 3 the desty fuctos f(x; ˆ, λˆ ) usg MLEs ad Bayesa estmates, computed va MCMC samples uder gamma prors, are plotted. The Fgure 4 represets the Quatle-Quatle(QQ) plot of emprcal quatles ad theoretcal quatles computed from MLE ad Bayes estmates. Fgure 4: Quatle-Quatle(QQ) plot of emprcal quatles ad theoretcal quatles computed from MLE ad Bayes estmates. Fgure 5 represets the estmated relablty fucto(dashed le) usg Bayes estmate based o MCMC output uder depedet gamma prors for both the parameters ad the emprcal relablty fucto(sold le). Fgure 5: The estmated relablty fucto (dashed le) usg Bayes estmate uder gamma prors ad the emprcal relablty fucto (sold le). 97

20 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach It s clear from the above fgures, the MLEs ad the Bayes estmates wth respect to the gamma prors are qute close ad ft the data very well. Therefore, the MCMC method works qute well for Pham model uder gamma prors. The proposed method ca be easly mplemeted to other reasoable prors. 9. Cocluso A attempt has bee made to corporate Pham model for software relablty data. We have preseted the statstcal tools for emprcal modelg of the data geeral. These tools are developed R laguage ad evromet for model aalyss, model valdato ad estmato of parameters. The exact ad approxmate ML estmates of the parameters are obtaed. To check the valdty of the model, we have plotted a graph of emprcal dstrbuto fucto ad ftted dstrbuto fucto ad t s see that the estmated Pham model provdes excellet good ft to the gve data. We have also dscussed the Quatle-Quatle (Q-Q) plots ad the probablty probablty (P-P) model valdato. A module s developed OpeBUGS for Bayesa aalyss usg MCMC smulato method. Both umercal as well as vsual summary based o MCMC sample of posteror characterstcs uder gamma prors for Pham model has bee worked out. We have appled the proposed methodology ad tools developed o a real software relablty data set. We have show that Pham model ca be used for software relablty data sets. The paper successfully descrbes the scope of Markov cha Mote Carlo (MCMC) techque for the Loglog(Pham) model. Such procedures ca be developed for other smlar probablty models as well. The oly requremet s to develop a proper ad effcet algorthm OpeBUGS. 9. Ackowledgemet The authors are thakful to the edtor ad the referees for ther valuable suggestos, whch mproved the paper to a great extet. Refereces [] Pham, H. (2002). A Vtub-shaped hazard rate fucto wth applcatos to system safety, Iteratoal Joural of Relablty ad Applcatos, Vol. 3, No.l, -6. [2] Pham, H. (2000). Software Relablty, Sprger-Verlag. [3] Pham, H. (2003). Hadbook of Relablty Egeerg, Sprger. [4] Murthy, D.N.P., Xe, M., Jag, R. (2004). Webull Models, Wley, New Jersey. 98

21 Computer Scece Joural Volume, Issue 2, August 20 [5] Lawless, J. F., (2003). Statstcal Models ad Methods for Lfetme Data, 2 d ed., Joh Wley ad Sos, New York. [6] Pham, H. (2006). System Software Relablty, Sprger-Verlag: Lodo. [7] Cd, J. E. R. ad Achcar, J. A., (999). Bayesa ferece for ohomogeeous Posso processes software relablty models assumg omootoc testy fuctos, Computatoal Statstcs ad Data Aalyss, 32, [8] Srvastava, A.K. ad Kumar V. (20). Aalyss of software relablty data usg expoetal power model. Iteratoal Joural of Advaced Computer Scece ad Applcatos, Vol. 2(2), [9] Srvastava, A.K. ad Kumar V. (20). Markov Cha Mote Carlo methods for Bayesa ferece of the Che model, Iteratoal Joural of Computer Iformato Systems, Vol. 2 (2), [0] Srvastava, A.K. ad Kumar V. (20). Software relablty data aalyss wth Marshall-Olk Exteded Webull model usg MCMC method for o-formatve set of prors, Iteratoal Joural of Computer Applcatos, Vol. 8(4), [] Robert, C. P. ad Casella, G. (2004). Mote Carlo Statstcal Methods, 2 d ed., New York, Sprger-Verlag. [2] Che, M., Shao, Q. ad Ibrahm, J.G. (2000). Mote Carlo Methods Bayesa Computato, Sprger, NewYork. [3] Gelfad, A.E. ad Smth, A.F.M. (990). Samplg based approaches to calculatg margal destes, Jour. Amer. Statst. Assoc., 85, [4] Ihaka, R. ad Getlema, R.R. (996). R: A laguage for data aalyss ad graphcs, Joural of Computatoal ad Graphcal Statstcs, 5, [5] Veables, W. N., Smth, D. M. ad R Developmet Core Team (200). A Itroducto to R, R Foudato for Statstcal Computg, Vea, Austra. ISBN , [6] Kumar, V. ad Lgges, U. (20). relar : A package for some probablty dstrbutos. [7] Thomas, A. (200). OpeBUGS Developer Maual, Verso 3..2, [8] Lu, D.J., Adrew, A., Best, N. ad Spegelhalter, D. (2000). WBUGS - A Bayesa modelg framework: Cocepts, structure, ad extesblty, Statstcs ad Computg, 0, [9] Thomas, A., O Hara, B., Lgges, U. ad Sturtz, S. (2006). Makg BUGS Ope, R News, 6, 2-7, [20] Lyu, M.R., (996). Hadbook of Software Relablty Egeerg, IEEE Computer Socety Press, McGraw Hll, 996. [2] Kumar, V., Lgges, U. ad Thomas, A. (200). RelaBUGS User Maual : A subsystem OpeBUGS for some statstcal models, Verso.0, OpeBUGS 3.2., [22] Kumar, V. (200). Bayesa aalyss of expoetal exteso model, J. Nat. Acad. Math., Vol. 24,

22 Srvastava et al: Aalyss of Pham(Loglog) Relablty Model usg Bayesa Approach [23] Che, M. H. ad Shao, Q. M. (999). Mote Carlo estmato of Bayesa credble tervals ad HPD tervals, Joural of Computatoal ad Graphcal Statstcs, 8(),