Reliability Engineering and System Safety

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1 Relablty Engneerng and System Safety 111 (213) Contents lsts avalable at ScVerse ScenceDrect Relablty Engneerng and System Safety journal homepage: A new modfed Webull dstrbuton Saad J. Almalk n Jngsong Yuan School of Mathematcs Unversty of Manchester Manchester M13 9PL UK artcle nfo Artcle hstory: Receved 2 February 212 Receved n revsed form 29 October 212 Accepted 3 October 212 Avalable onlne 22 November 212 Keywords: Webull dstrbuton Addtve Webull Modfed Webull Mamum lkelhood estmaton abstract We ntroduce a new lfetme dstrbuton by consderng a seral system wth one component followng a Webull dstrbuton and another followng a modfed Webull dstrbuton. We study ts mathematcal propertes ncludng moments and order statstcs. The estmaton of parameters by mamum lkelhood s dscussed. We demonstrate that the proposed dstrbuton fts two well-known data sets better than other modfed Webull dstrbutons ncludng the latest beta modfed Webull dstrbuton. The model can be smplfed by fng one of the parameters and t stll provdes a better ft than estng models. & 212 Elsever Ltd. All rghts reserved. 1. Introducton The Webull dstrbuton [28] has been used n many dfferent felds wth many applcatons see for eample [18]. The hazard functon of the Webull dstrbuton can only be ncreasng decreasng or constant. Thus t cannot be used to model lfetme data wth a bathtub shaped hazard functon such as human mortalty and machne lfe cycles. For many years researchers have been developng varous etensons and modfed forms of the Webull dstrbuton wth number of parameters rangng from 2 to 5. The two-parameter fleble Webull etenson of Bebbngton et al. [5] has a hazard functon that can be ncreasng decreasng or bathtub shaped. Zhang and e [31] studed the characterstcs and applcaton of the truncated Webull dstrbuton whch has a bathtub shaped hazard functon. A threeparameter model called eponentated Webull dstrbuton was ntroduced by Mudholkar and Srvastave [17]. Another three-parameter model s by Marshall and Olkn [15] and called etended Webull dstrbuton. e et al. [3] proposed a threeparameter modfed Webull etenson wth a bathtub shaped hazard functon. The modfed Webull () dstrbuton of La et al. [13] multples the Webull cumulatve hazard functon a b by e l whch was later generalzed to eponentated form by Carrasco et al. [6] FðÞ¼ð1 e ab e l Þ y Z: ð1þ Recent studes of the modfed Webull nclude [112627]. n Correspondng author. Tel.: þ E-mal address: saad.al-malk-2@postgrad.manchester.ac.uk (S.J. Almalk). Among the four-parameter dstrbutons the addtve Webull dstrbuton () of e and La [29] wth cumulatve dstrbuton functon (CDF) FðÞ¼1 e ay b g Z has a bathtub-shaped hazard functon consstng of two Webull hazards one ncreasng (y41) and one decreasng (ogo1). The modfed Webull dstrbuton of Sarhan and Zandn (SZ) [21] can be derved from the addtve Webull dstrbuton by settng y ¼ 1. A four-parameter beta Webull dstrbuton was proposed by Famoye et al. [1]. Cordero et al. [8] ntroduced another fourparameter called the Kumaraswamy Webull dstrbuton. Fve-parameter modfed Webull dstrbutons nclude Phan s modfed Webull [2] the beta modfed Webull (B) ntroduced by Slva et al. [24] and further studed by Nadarajah et al. [19]. The latest eamples nclude the beta generalzed Webull dstrbuton by Sngla et al. [25] eponentated generalzed lnear eponental dstrbuton by Sarhan et al. [22] and the generalzed Gomprtz dstrbuton by El-Gohary et al. [9]. We propose a new lfetme dstrbuton based on the Webull and the modfed Webull () dstrbutons by combnng them n a seral system. The hazard functon of the new dstrbuton s the sum of a Webull hazard functon and a modfed Webull hazard functon. Secton 2 gves defnton motvaton and usefulness of ths model and lsts ts sub-models. Secton 3 consders propertes of the new dstrbuton such as hazard moments and order statstcs. Secton 4 dscusses estmaton of the parameters. Two real data sets are analyzed n Secton 5 and the results are compared wth estng dstrbutons. Secton 6 concludes the paper /$ - see front matter & 212 Elsever Ltd. All rghts reserved.

2 f() S.J. Almalk J. Yuan / Relablty Engneerng and System Safety 111 (213) The model 2.1. Defnton We defne a new modfed Webull dstrbuton () by the followng CDF: FðÞ¼1 e ay b g e l Z ð2þ where a b y g and l are non-negatve wth y and g beng shape parameters and a and b beng scale parameters and l acceleraton parameter. The probablty densty functon (PDF) s f ðþ¼ðay y 1 þbðgþlþ g 1 e l Þe ay b g e l 4: ð3þ It can be rewrtten as f ðþ¼½h W ð; ayþþh ð; bglþšs W ðayþs ð; bglþ where S W h W S and h are survval and hazard functons of the Webull and modfed Webull dstrbutons respectvely. Ths functon can ehbt dfferent behavor dependng on the values of the parameters when chosen to be postve as shown n Fg Sub models Ths dstrbuton ncludes sub models that are wdely used n survval analyss. Table 1 shows a lst of models that can be derved from the dstrbuton α=1.15β=.5γ=5θ=.75λ=2 α=.5β=5γ=1.25θ=5λ=.5 α=2β=.75γ=15θ=1.2λ=.5 α=5β=4γ=5θ=2.5λ=.15 α=4β=.5γ=1.5θ=.4λ=.75 ð4þ 2.3. Motvaton and nterpretaton The survval functon of the new dstrbuton s gven by SðÞ¼e ay b g e l Z ð5þ and the hazard functon s hðþ¼ay y 1 þbðgþlþ g 1 e l 4 ð6þ whch can be nterpreted as that of a seral system wth two ndependent components one of whch follows the Webull dstrbuton wth parameters a and y and the other follows the modfed Webull dstrbuton of La et al. [13] wth parameters b g and l. Therefore the dstrbuton can be used when there are two types of falure e.g. a normal type and a premature type. The purpose of the Webull component s to provde a decreasng hazard functon when requred as n the addtve Webull [29] by choosng yo1. (It wll be ncreasng when y41.) The modfed Webull component has ether an ncreasng or a bathtub shaped hazard functon. Together they provde a bathtub shaped hazard (unless both hazards are ncreasng) wth more fleblty than the addtve Webull. The fleblty s useful when there s a second peak n the dstrbuton as shown n Secton Propertes of the model 3.1. The hazard functon The hazard functon can have many dfferent shapes ncludng bathtub as shown n Fg. 2. We can deduce from (6) that t s ncreasng f y gz1 decreasng f y go1 and l ¼ and bathtub shaped otherwse. It s desrable for a bathtub shaped hazard functon to have a long useful lfe perod [12] wth relatvely constant falure rate n the mddle. A few dstrbutons have ths property so does the as shown n Fg The moments Fg. 1. Probablty densty functons of the. It s customary to derve the moments when a new dstrbuton s proposed. Usng the Taylor epanson of e twce the rth non-central moment of the s m r ¼ r dfðþ ¼ 8 r de ay b g e l 6 Table 1 The sub-models of the. h() 4 Model a b g y l S() Reference Addtve Webull e ay b g e and La [29] Modfed Webull e bg e l La et al. [13] S Z modfed Webull 1 e a bg Sarhan and Zandn [21] Lnear falure rate 2 1 e a b2 Ban [4] Etreme-value 1 e e l Ban [4] Webull e a y Webull [28] Raylegh 2 e a2 Ban [4] Eponental 1 e a Ban [4] 2 α = 1.2β= 1.5γ=.5θ= 3λ=.75 α =.5β= 5γ= 1.25θ= 5λ=.5 α = 2β=.75γ= 15θ= 1.2λ=.5 α = 5β=.7γ=.5θ=.15λ= α =.5β=.7γ= 5θ= 2.5λ=.15 α = 4β=.5γ= 1.5θ=.4λ= Fg. 2. Hazard functons of the.

3 166 S.J. Almalk J. Yuan / Relablty Engneerng and System Safety 111 (213) ¼ ¼ 1 ¼ r y n ¼ m ¼ 1 r e ay b g e l d 1 1 n ¼ m ¼ ð bþ n ðlnþ m nm ð bþ n ðlnþ m nm r ng þ m e ay d a ðng þ m þ rþ=y G ngþmþr y for r ¼ where GðÞ s the gamma functon Order statstcs It wll also be useful to derve the pdf of the rth order statstc ðrþ of a random sample 1 y n drawn from the wth parameters a b y g and l. From Balakrshnan and Nagaraja [3] the pdf of ðrþ s gven by f r:n ðþ¼ FðÞ ð1 FðÞÞ n r f ðþ ð7þ Bðrn r þ1þ where BðÞ s the beta functon. Usng FðÞ¼1 e HðÞ f ðþ¼hðþe HðÞ ð8þ where h() s the hazard functon (6) and HðÞ¼a y þb g e l s the cumulatve hazard then ð1 FðÞÞ n r ¼ e ðn rþhðþ and FðÞ ¼ð1 e HðÞ Þ ¼ h() ¼ ð 1Þ e HðÞ : α= 3β= 5e 6γ=.1θ=.5λ=.15 α=.25β= 5e 4γ=.1θ=.5λ=.9 α=.11β= 2.5e 7γ= 1θ= 1λ=.12 α=.25β= 5e 7γ= 1θ= 1λ=.12 ð9þ ð1þ Substtutng (9) and (1) nto (7) we get 1 f r:n ðþ¼ Brn r ð þ1þ ¼ ¼ n n 1 ¼ ¼ ð 1Þ ðn þ þ 1 rþhðþ hðþe ¼ n n 1 ð 1Þ ðn þ þ 1 rþhðþ hðþe ð 1Þ ðay y 1 þbðgþlþ g 1 e l Þe ðn þ þ 1 rþðay þ b g e l Þ ¼ n n 1 ð 1Þ ðnþþ1 rþ f ð; a b yglþ ¼ where f ð; a b yglþ s the PDF of the wth parameters a ¼ðnþþ1 rþa b ¼ðnþþ1 rþb y g and l. Usng (7) the kth non-central moment of the rth order statstc ðrþ s m ðr:nþ ¼ nk n 1 1 k y 1 ¼ j ¼ ¼ ð 1Þ þ b ðlþ j ðgþjþ gþjþk ðnþþ1 rþ ðg þ j þ kþ=y G þ 1 aðg þ j þ kþ=y y 4. Parameter estmaton Gven a random sample 1 y n from the wth parameters ðabyglþ the usual method of estmaton s by mamum lkelhood [7]. Other possble approaches nclude Bayesan estmaton usng Lndley appromaton [14] or MCMC [2627]. The log-lkelhood functon s gven by L ¼ n a n lnðbðgþl Þ g 1 y n b g el : e l þay y 1 Þ : ð11þ Settng the frst partal dervatves of wth respect to a b y g and l to zero the lkelhood equatons are Fg. 3. Hazard functons of the wth long useful lfe perod. n n n y y 1 hð ; abgylþ ðgþl Þ g 1 e l hð ; abgylþ a y 1 n n ð1þy lnð ÞÞ hð ; abgylþ a n y ¼ g el ¼ y lnð Þ¼ ð12þ ð13þ ð14þ Table 2 MLEs of parameters and correspondng standard errors n brackets for the Aarset data. Model ^a ^b ^g ^y ^l.71 7: (.31) (1: ) (3.62) (.128) (.184) ).23 ða ¼ y ¼ ) (.27) (.113) (4: ) 1: (l ¼ ) (5: ) (.36) (.12) (1.33) SZ.13 8: (y ¼ 1 l ¼ ) (2: ) (4: ) (1.14)

4 S.J. Almalk J. Yuan / Relablty Engneerng and System Safety 111 (213) n n g 1 e l ððgþl Þ lnð Þþ1Þ n g hð ; abgylþ el lnð Þ¼ ð15þ ð1þgþl Þ g el n g þ 1 hð ; abgylþ e l ¼ : ð16þ Table 3 Log-lkelhood K S statstc the correspondng P-values AIC and BIC values of models ftted to Aarst data for comparson wth beta modfed Webull (Slva et al. [24]). Model Log-lk K S P-value AIC BIC SZ B The mamum lkelhood estmates can be obtaned by solvng the non-lnear equatons numercally for a b y g and l. Ths can be done usng R Matlab and Mathcad among other packages. The relatvely large number of parameters can cause problems especally when the sample sze s not large. A good set of ntal values s essental. We have also obtaned all the second partal dervatves of the log-lkelhood functon for the constructon of the Fsher nformaton matr so that standard errors of the parameter estmates can be obtaned n the usual way. These are n the Append. 5. Applcatons In ths secton we provde results of fttng the to two well-known data sets and compare ts goodness-of-ft wth other modfed Webull dstrbutons usng Kolmogorov Smrnov (K S) statstc as well as Akake nformaton crteron (AIC) [2] and Bayesan nformaton crteron (BIC) [23] values. 1 S() Nonpar SZ B Scaled TTT Transform Emprcal SZ B Kaplan Meer SZ B h().15 f() Fg. 4. For Aarst data: (a) hazard functon (b) TTT-transform plot (c) pdf and (d) survval functon usng plus sub models and beta modfed Webull. Table 4 MLEs of parameters and correspondng standard errors n brackets for the Meeker and Escobar data. Model ^a ^b ^g ^y ^l.24 5: (.19) (8: ) (1.29) (.15) (.24).18 :454 7: (a ¼ y ¼ ) (.18) (.22) (2: ) 1: (l ¼ ) (7: ) (.18) (.197) (.974) SZ 2: : (y ¼ 1 l ¼ ) (9: ) (1: ) (1.314)

5 168 S.J. Almalk J. Yuan / Relablty Engneerng and System Safety 111 (213) Aarset data 5.2. Meeker and Escobar data The data represent the lfetmes of 5 devces [1]. Many authors have analysed ths data set ncludng Mudholkar and Srvastava [17] e and La [29] La et al. [13] Sarhan and Zandn [21] and Slva et al. [24]. It s known to have a bathtub-shaped hazard functon (Fg. 3a) as ndcated by the scaled TTT-Transform plot (Fg. 3b). Table 2 gves ML estmates of parameters of the and sub-models wth standard errors n brackets and goodness of ft statstcs are n Table 3. We fnd that the dstrbuton wth the same number of parameters provdes a better ft than the beta modfed Weull dstrbuton (B) whch was the best n Slva et al. [24]. It has the largest lkelhood and the smallest K S AIC and BIC values among those consdered n ths paper. It s clear n Fg. 3c that the fts the left and rght peaks n the hstogram better and ts survval functon follows the Kaplan Meer estmate more closely (Fg. 3d). Table 5 Log-lkelhood K S statstc the correspondng P-values AIC and BIC values of models ftted to Meeker and Escobar data. Model Log-lk K S P-value AIC BIC SZ B The data are falure and runnng tmes of a sample of 3 devces (Meeker and Escobar [16 p. 383]). Two types of falures were observed for ths data. It was shown by Nadarajah et al. [19] to be best ft by the beta modfed Webull dstrbuton. The data have a bathtub shaped hazard functon (Fg. 4a and b). Agan the dstrbuton (Table 4) provdes a better ft than the B as can be seen from Table 5 (Fg. 5). 6. Sub-model of the wth c ¼ 1 To smplfy the statstcal nference t s always a good dea to reduce the number of parameters of any dstrbuton and nvestgate how that affects the ablty of the reduced model to ft the data. In ths secton we reduce the number of parameters from fve to four by settng g ¼ 1. We test the reduced model H : g ¼ 1Þ aganst the orgnal model H a : ga1þ. For each data set Table 6 shows ML estmates of the four parameter the loglkelhood value under H lkelhood rato statstc (LRT) wth P- value n brackets AIC K S statstc wth P-value n brackets. The lkelhood rato statstcs aganst the full model wth fve parameters are 1.31 (P-value¼.252) and 2.45 (P-value¼.118) respectvely on 1 d.f. Therefore we can choose the reduced model wth four parameters. The lkelhood and AIC value also ponts to ths model when the modfed beta dstrbuton s ncluded n the comparson. Fg. 6 shows the reduced model s nearly as good as the full model for both data sets. 1 h() Nonpar SZ B Scaled TTT Transform Emprcal f() SZ B S() Kaplan Meer SZ B Fg. 5. For Meeker and Escobar data: (a) hazard functon (b) TTT-transform plot (c) pdf and (d) survval functon usng plus sub models and beta modfed Webull.

6 S.J. Almalk J. Yuan / Relablty Engneerng and System Safety 111 (213) Table 6 Results of fttng wth g ¼ 1 to both data sets. Data ^a ^b ^y ^l Log-lk AIC LRT (P-value) K S (P-value) Aarset.92 2: (.39) (2:1 1 8 ) (.14) (.11) (.252) (.63) Meeker.17 3: (.13) (4:2 1 8 ) (.141) ð4: 1 3 ) (.118) (.44) S() S() Kaplan Meer Kaplan Meer f().4 f() Fg. 6. (a and b) Ftted pdf and survval functons for Aarst data and (c and d) those for Meeker and Escobar data fve parameters (sold lnes) vs four parameters (dotted lnes). 7. Conclusons A new dstrbuton based on Webull and modfed Webull dstrbutons has been proposed and ts propertes studed. The dea s to combne two components n a seral system so that the hazard functon s ether ncreasng or more mportantly bathtub shaped. Usng a modfed Webull component the dstrbuton has fleblty to model the second peak n a dstrbuton. We have shown that the new modfed Webull dstrbuton fts certan well-known data sets better than estng modfcatons of the Webull dstrbuton. Reducng the number of parameters to four by fng one of the parameters stll provdes a better ft than estng models. Future work ncludes MCMC methods wth censored data regresson problems wth covarates and parameter reducton. Acknowledgments We would lke to thank the referees for ther comments and suggestons whch mproved the presentaton of the paper. The frst author wshes to thank the Saud Araba Culture Bureau n the UK and the Taf Unversty for ther fnancal support. Append A The log-lkelhood functon of the ðabyglþ can be wrtten as LðWÞ¼ n ½lnðhð ; WÞÞ a y bg el Š ð17þ where hð ; WÞ s the hazard rate functon (6) of the and W ¼ ðabyglþ s the vector of parameters. The second partal dervatves are as follows: L aa ¼ n 2 h a ð ; WÞ hð ; WÞ

7 17 S.J. Almalk J. Yuan / Relablty Engneerng and System Safety 111 (213) L ab ¼ n L ay ¼ n L ag ¼ n L al ¼ n L bb ¼ n L by ¼ n L bg ¼ n L bl ¼ n L yy ¼ n L yg ¼ n L yl ¼ n L gg ¼ n L gl ¼ n h a ð ; WÞh b ð ; WÞ ðhð ; WÞÞ 2 hð ; WÞh ay ð ; WÞ h a ð ; WÞh y ð ; WÞ 2 y lnð Þ hð ; WÞ h a ð ; WÞh g ð ; WÞ ðhð ; WÞÞ 2 h a ð ; WÞh l ð ; WÞ ðhð ; WÞÞ 2 2 h bg ð ; WÞ hð ; WÞ h b ð ; WÞh y ð ; WÞ ðhð ; WÞÞ 2 hð ; WÞh bg ð ; WÞ h b ð ; WÞh g ð ; WÞ g ðhð ; WÞÞ 2 el lnð Þ hð ; WÞh bl ð ; WÞ h b ð ; WÞh l ð ; fþ g þ 1 ðhð ; WÞÞ 2 e l 2 hð ; WÞh yy ð ; WÞ h y ð ; WÞ a y ðhð ; WÞÞ 2 ln 2 ð Þ h y ð ; WÞh g ð ; WÞ ðhð ; WÞÞ 2 h y ð ; WÞh l ð ; WÞ ðhð ; WÞÞ 2 hð ; WÞh gg ð ; WÞ ðh g ð ; WÞÞ 2 b g ðhð ; WÞÞ 2 el ln 2 ð Þ hð ; WÞh gl ð ; WÞ h g ð ; WÞh l ð ; WÞ b g þ 1 ðhð ; WÞÞ 2 e l ln 2 ð Þ L ll ¼ d hð ; WÞh ll ð ; WÞ ðh l ð ; WÞÞ 2 b g þ 2 ðhð ; WÞÞ 2 where h ay ð ; WÞ¼ y 1 ð1þy lnð ÞÞ h bg ð ; WÞ¼ g 1 ð1þðgþl Þ lnð ÞÞe l h bl ð ; WÞ¼ g ð1þgþl Þ lnð ÞÞe l h yy ð ; WÞ¼a y 1 ð2þy lnð ÞÞ lnð Þ h gg ð ; WÞ¼b g 1 ð2þðgþl Þ lnð ÞÞ lnð Þe l h gl ð ; WÞ¼b g ð1þð1þgþl Þ lnð ÞÞe l h ll ð ; WÞ¼b g þ 1 ð2þgþl Þ lnð ÞÞe l h a ð ; WÞ¼y y 1 h b ð ; WÞ¼ g 1 ðgþl Þ lnð Þe l h y ð ; WÞ¼ah ay ð ; WÞ e l h g ð ; WÞ¼bh bg ð ; WÞ h l ð ; WÞ¼bh bl ð ; WÞ: References [1] Aarset MV. How to dentfy bathtub hazard rate. IEEE Transactons on Relablty 1987;36(1):16 8. [2] Akake H. A new look at the statstcal model dentfcaton. IEEE Transactons on Automatc Control 1974;AC-19: [3] Balakrshnan AN Nagaraja HN. A frst course n order statstcs. New York: Wley-Interscence; [4] Ban LJ. Analyss for the lnear falure-rate lfe-testng dstrbuton. Technometrcs 1974;16(4): [5] Bebbngton MS La CD Ztks R. A fleble Webull etenson. Relablty Engneerng & System Safety 27;92(6): [6] Carrasco M Ortega EM Cordero GM. A generalzed modfed Webull dstrbuton for lfetme modelng. Computatonal Statstcs and Data Analyss 28;53(2): [7] Fsher RA. On the mathematcal foundaton of theoretcal statstcs. Phlosophcal Transactons of the Royal Socety A 1922;222: [8] Cordero GM Ortega EM Nadarajah S. The Kumaraswamy Webull dstrbuton wth applcaton to falure data. Journal of the Frankln Insttute 21;347: [9] El-Gohary A Alshamran A Al-Otab A. The generalzed Gompertz dstrbuton. Appled Mathematcal Modelng 211;37(1):1324. [1] Famoye F Lee C Olumolade O. The beta-webull dstrbuton. Journal of Statstcal Theory and Applcatons 25;4(2): [11] Jang H e M Tang LC. On MLEs of the parameters of a modfed Webull dstrbuton for progressvely type-2 censored samples. Journal of Appled Statstcal Scence 21;37(4): [12] Kuo W Zuo MJ. Optmal relablty modelng: prncples and applcatons. Wley; 22. [13] La CD e M Murthy DNP. A modfed Webull dstrbuton. IEEE Transactons on Relablty 23;52(1):33 7. [14] Lndley DV. Appromate Bayesan method. Trabajos Estadst 198;31(1): [15] Marshall AW Olkn I. A new method for addng a parameter to a famly of dstrbutons wth applcaton to the eponental and Webull famles. Bometrka 1997;84(3): [16] Meeker WQ Escobar LA. Statstcal methods for relablty data vol. 78. New York: John Wley; [17] Mudholkar GS Srvastava DK. Eponentated Webull famly for analysng bathtub falure rate data. IEEE Transactons on Relablty 1993;42(2): [18] Murthy DNP e M Jang R. Webull models vol New York: Wley; 23. [19] Nadarajah S Cordero GM Ortega EMM. General results for the betamodfed Webull dstrbuton. Journal of Statstcal Computaton and Smulaton 211;81(1): [2] Phan KK. A new modfed Webull dstrbuton functon. Communcatons of the Amercan Ceramc Socety 1987;7(8): [21] Sarhan AM Zandn M. Modfed Webull dstrbuton. Appled Scences 29;11: [22] Sarhan AM Ahmad AEBA Alasbah IA. Eponentated generalzed lnear eponental dstrbuton. Appled Mathematcal Modelng accepted. [23] Schwarz G. Estmatng the dmenson of a model. Annals of Statstcs 1978;6: [24] Slva GO Ortega EM Cordero GM. The beta modfed Webull dstrbuton. Lfetme Data Analyss 21;16:49 3. [25] Sngla N Jan K Kumar Sharma S. The beta generalzed Webull dstrbuton: propertes and applcatons. Relablty Engneerng & System Safety 212;12:5 15. [26] Solman AA Abd-Ellah AH Abou-Elheggag NA. Modfed Webull model: a Bayes study usng MCMC approach based on progressve censorng data. Relablty Engneerng & System Safety 212;1: [27] Upadhyaya SK Gupta A. A Bayes analyss of modfed Webull dstrbuton va Markov chan Monte Carlo smulaton. Journal of Statstcal Computaton and Smulaton 21;8(3): [28] Webull WA. Statstcal dstrbuton functon of wde applcablty. Journal of Appled Mechancs 1951;18: [29] e M La CD. Relablty analyss usng an addtve Webull model wth bathtub-shaped falure ratefuncton. Relablty Engneerng System Safety 1995;52: [3] e M Tang Y Goh TN. A modfed Webull etenson wth bathtub-shaped falure rate functon. Relablty Engneerng System Safety 22;76(3): [31] Zhang T e M. On the upper truncated Webull dstrbuton and ts relablty mplcatons. Relablty Engneerng System Safety 211;96(1):