On the Transmuted Additive Weibull Distribution

Transcription

1 AUSTRIAN JOURNAL OF STATISTICS Volume , Number 2, On the Transmuted Addtve Webull Dstrbuton Ibrahm Elbatal 1 and Gokarna Aryal 2 1 Mathematcal Statstcs, Caro Unversty, Egypt 2 Mathematcs, CS, & Statstcs, Purdue Unversty Calumet, USA Abstract: In ths artcle a contnuous dstrbuton, the so-called transmuted addtve Webull dstrbuton, that extends the addtve Webull dstrbuton and some other dstrbutons s proposed and studed. We wll use the quadratc rank transmutaton map proposed by Shaw and Buckley 29 n order to generate the transmuted addtve Webull dstrbuton. Varous structural propertes of the new dstrbuton ncludng explct expressons for the moments, random number generaton and order statstcs are derved. Maxmum lkelhood estmaton of the unknown parameters of the new model for complete sample s also dscussed. It wll be shown that the analytcal results are applcable to model real world data. Zusammenfassung: In desem Artkel wrd ene stetge Vertelung vorgeschlagen und untersucht, de sogenannte addtve umgewandelte Webull- Vertelung, welche de addtve Webull-Vertelung und enge andere Vertelungen erwetert. Wr verwenden de quadratsche Rang Transmutatons- Abbldung, vorgeschlagen n Shaw and Buckley 29, um de addtve umgewandelte Webull-Vertelung zu erzeugen. Verschedene strukturelle Egenschaften der neuen Vertelung enschleßlch explzte Ausdrücke für de Momente, Erzeugung von Zufallszahlen, und Ordnungsstatstken werden hergeletet. Maxmum Lkelhood Schätzung der unbekannten Parameter deses neuen Modells für de vollständge Stchprobe wrd ebenfalls dskutert. Es wrd gezegt, dass dese analytschen Ergebnsse verwendbar snd um reale Daten zu modelleren. Keywords: Addtve Webull Dstrbuton, Order Statstcs, Transmutaton Map, Maxmum Lkelhood Estmaton, Relablty Functon. 1 Introducton Snce the qualty of the procedures used n statstcal analyss depends heavly on the assumed probablty model or dstrbutons, a sgnfcant contrbuton has been made on generalzaton of some well-known dstrbutons and ther successful applcatons. For many mechancal and electronc components, the hazard falure rate functon has a bathtub shape. It s well-known that, because of desgn and manufacturng problems, the falure rate s hgh at the begnnng of a product lfe cycle and decreases toward a constant level. After reachng a certan age, the product enters the wear-out phase and the falure rate starts to ncrease. Despte the fact that ths phenomenon has been presented n many relablty engneerng texts, few practcal models possessng ths property have appeared n the lterature. Because of ths, only a part of the bathtub curve s consdered at any one tme. Another common fact s that most engneers may be nterested only n a part of

2 118 Austran Journal of Statstcs, Vol , No. 2, the lfetme, because at component level, they only see one part of the falure rate functon. However, t wll be helpful to have a model that s reasonably smple and good for the whole product lfe cycle for makng overall decsons. Furthermore, for complex systems, both the decreasng and ncreasng parts of the falure rate fall nto the ordnary product lfetme. Lfetme dstrbutons for many components usually have a bathtub-shaped falure rate n practce. However, there are very few standard dstrbutons to model ths type of falure rate functon. The Webull dstrbuton has been used n many dfferent felds wth many applcatons, see for example La, Xe, and Murthy 23. The hazard functon of the Webull dstrbuton can only be ncreasng, decreasng or constant.thus t can not 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 extensons and modfed forms of the Webull dstrbuton, wth dfferent number of parameters. A state-of-the-art survey on the class of such dstrbutons can be found n La, Xe, and Murthy 21 and Nadarajah 29. Xe and La 1995 proposed a fourparameter addtve Webull AW dstrbuton as a compettve model. A random varable X s sad to have a AW dstrbuton f ts cumulatve dstrbuton functon cdf s F x 1 e αx γx, x, 1 where α,, γ, and are non-negatve, wth < 1 < or < 1 <. Note that and are the shape parameters and α and γ are scale parameters. The probablty densty functon pdf of the AW dstrbuton s and the hazard rate functon s gven by fx αx 1 + γx 1 e αx γx, 2 hx fx 1 F x αx 1 + γx 1 e αx γx e αx γx αx 1 + γx 1. The change pont x where the hazard rate functon hx acheves ts mnmum s at 1 1 α x, 1γ when < 1 <. It s mportant to note that the change pont remans the same when < 1 <. In ths artcle the four parameter AW dstrbuton s embedded n a larger famly obtaned by ntroducng an addtonal parameter. We wll call the generalzed dstrbuton as the transmuted addtve Webull TAW dstrbuton. The rest of the artcle s organzed as follows. In Secton 2 we present the expresson of the pdf and cdf of the subject dstrbuton and some specal sub-models. In Secton 3 we study the statstcal propertes ncludng quantle functons, moments, moment generatng functon etc. The relablty functons of the subject model are gven n Secton 4. The mnmum, maxmum and medan order statstcs models are dscussed n Secton 5. In Secton 6 we demonstrate the maxmum lkelhood estmates and the asymptotc confdence ntervals of the unknown parameters. Fnally, n Secton 7 we present a real world data analyss to llustrate the usefulness of the proposed dstrbuton.

3 I. Elbatal, G. Aryal Transmuted Addtve Webull Dstrbuton A random varable X s sad to have a transmuted probablty dstrbuton wth cdf F x, f F x 1 + λgx λgx 2, λ 1, where Gx s the cdf of the base dstrbuton. Observe that at λ we have the dstrbuton of the base random varable. Aryal and Tsokos 211 studed the TW as a generalzaton of Webull dstrbuton. Khan and Kng 213 extended the MW to a TMW dstrbuton. In ths secton we present the TAW dstrbuton and the sub-models of ths dstrbuton. Now, usng 1 and 2 we have the cdf of the TAW dstrbuton F T AW x 1 e αx γx 1 + λe αx γx, 3 where and are the shape parameters representng the dfferent patterns of the TAW dstrbuton and are postve, α and γ are scale parameters representng the characterstc lfe and are also postve, and λ s the transmuted parameter. The probablty densty functon pdf of a TAW dstrbuton s gven by f T AW x αx 1 + γx 1 e αx γx 1 λ + 2λe αx γx. 4 The TAW dstrbuton s a very flexble model that approaches to dfferent dstrbutons when ts parameters vary. The flexblty of the TAW dstrbuton s explaned n Table 1. Table 1: Transmuted addtve Webull dstrbuton and some of ts sub-models Model α γ λ Cumulatve dstrbuton functon TAW 1 exp αx γx 1 + λ exp αx γx TMW 1 1 exp αx γx 1 + λ exp αx γx TLF exp αx γx λ exp αx γx 2 TME exp αx γx1 + λ exp αx γx AW 1 exp αx γx MW 1 1 exp αx γx MR exp αx γx 2 ME exp αx γx TW 1 exp γx 1 + λ exp γx TR 2 1 exp γx λ exp γx 2 TE 1 1 exp γx1 + λ exp γx W 1 exp γx R 2 1 exp γx 2 E 1 1 exp γx Abbrvatons: T Transmuted, A Addtve, M Modfed, W Webull, E Exponental, LF Lnear Falure, R Raylegh. The subject dstrbuton ncludes as specal cases the transmuted modfed Webull TMW, the transmuted modfed exponental TME, transmuted lnear falure rate TLFR,

4 12 Austran Journal of Statstcs, Vol , No. 2, transmuted Webull TW, transmuted Raylegh TR and transmuted exponental, Webull, Raylegh and exponental dstrbutons. Fgure 1 shows some of the the sub-models of the TAW dstrbuton that approach several dfferent lfetme dstrbutons for partcular choces of ts parameters. Fgure 2 llustrates the graphcal behavor of the pdf of TAW dstrbuton for selected values of the parameters. Note that the fgure on the left exhbts the shape of the dstrbuton for dfferent choce of all the parameters whereas the fgure on the rght exhbts the behavor shape as λ vares from 1 to 1 whle keepng all other four parameters fxed. Fgure 1: Sub-models of transmuted addtve Webull dstrbutons fx λ α.5 γ λ 1 α 1 γ λ.5 α.5 γ λ.5 α 1 γ λ 1 α.2 γ λ 1 α.2 γ fx λ λ 1 λ.5 λ.5 λ γ 1 α x x Fgure 2: Densty functons of varous transmuted addtve Webull dstrbutons

5 I. Elbatal, G. Aryal Statstcal Propertes In ths secton we dscuss few statstcal propertes of the TAW dstrbuton. 3.1 Quantles The quantle x q of the T AW D α,,, γ, λ, x s the real soluton of the equaton αx q + γx q + log 1 λ + 1 λ λq 2λ. 5 Snce the above equaton has no closed form soluton n x q, we have to use numercal methods to get the quantles. We can use 5 to derve quantles for some specal cases of the TAW dstrbuton. Let hλ, q log 1 λ + 1 λ λq 2λ then the followng results can be derved. Settng 1, the x q quantle of the T MW α,, γ, λ, x dstrbuton s the soluton of αx q + γx q + hλ, q. Settng 1 and 2, the x q quantle of the T LF RD α, γ, λ, x dstrbuton s x q α + α 2 4γhλ, q 2γ Settng 1 and α, the x q quantle of the T W D, γ, λ, x dstrbuton s x q 1γ hλ, q 1.., v Settng 1, α, and 2, the x q quantle of the T RD α, γ, λ, x dstrbuton s x q 1 hλ, q. γ v Settng 1, α, and 1, the x q quantle of the T ED α, γ, λ, x dstrbuton s x q 1 hλ, q. γ In all the above cases the medan can be obtaned by settng q.5.

6 122 Austran Journal of Statstcs, Vol , No. 2, Random Number Generaton In order to generate random numbers from a T AW D α,,, γ, λ, x dstrbuton we need to solve [ 1 exp αx γx ] [ 1 + λ exp αx γx ] u, where u U, 1. Ths yelds, where hλ, u log αx + γx + hλ, u, 6 1 λ + 1 λ λu 2λ. Equaton 6 does not have a closed form soluton so we generate u from U, 1 and solve for x n order to generate random numbers from a TAW dstrbuton. 3.3 Moments The rth order moment µ r EX r of a TAW dstrbuton s gven by Theorem 3.1 below. Theorem 3.1 If X s from a T AW D α,,, γ, λ, x dstrbuton wth λ 1, then the rth moment of X s gven by 1 k µ r k 1 k + k γ k α r+k α k γ r+k [1 λ + λ 2k where denotes the gamma functon,.e. a 2 r+k [1 λ + λ 2k 2 r+k ] r + k + ] t a 1 exp tdt. r + k +, Proof: µ r 1 λ +2λ x r f T AW α,,, γ, λ, xdx x r αx 1 + γx 1 e αx γx 1 λ + 2λe αx γx dx αx r+ 1 + γx r+ 1 e αx γx dx αx r+ 1 + γx r+ 1 e 2αx 2γx dx 1 λi 1 + 2λI 2. 7

7 I. Elbatal, G. Aryal 123 Now, usng e γx 1 k γ k x k k, e αx 1 k α k x k, k we have I 1 αx r+ 1 + γx r+ 1 e αx γx dx αx r+ 1 e αx e γx dx + 1 k k [ 1 k k Smlarly, for I 2 we get I [ 1 k k γ k α r+k γ k α r+k 2γ k 2α r+k Substtutng 8 and 9 n 7 we get 1 k µ r k 1 k + k Ths completes the proof. γx r+ 1 e αx e γx dx r + k + 1 k α k r + k + + k γ r+ ] r + k + + αk r + k +. 8 γ k α r+k γ r+k r + k + + 2αk α k γ r+k [1 λ + λ 2k 2 r+k [1 λ + λ 2k 2 r+k 2γ r+k ] r + k +. 9 ] r + k + ] r + k +. Note that n partcular, f α, we have Smlarly, f γ, we get µ r µ r γ r α r ] [1 λ + λ2 r r + 1. αx r+ 1 e αx 1 λ + 2λe αx dx [ ] 1 λ + λ2 r r + 1.

8 124 Austran Journal of Statstcs, Vol , No. 2, Moment Generatng Functon The moment generatng functon mgf of the TAW dstrbuton s gven by Theorem 3.2. Theorem 3.2 If X s from a T AW α,, γ,, λ, x dstrbuton wth λ 1, then ts mgf s 1 k t j M X t 1 λ γ k j+k+ j+k+ + α k j! j k α j+k γ j+k 2 k t j + j! λ γ k j+k+ j+k+ + α k. j k 2α j+k 2γ j+k Proof: We have M X t I 1 1 λ +2λ e tx f T AW α,,, γ, λ, xdx e tx αx 1 + γx 1 e αx γx 1 λ + 2λe αx γx dx e tx αx 1 + γx 1 e αx γx dx e tx αx 1 + γx 1 e 2αx 2γx dx 1 λi 1 + 2λI 2. 1 j k j k Smlarly, for I 2 we get I e tx αx 1 + γx 1 e αx γx dx αx 1 e tx e αx e γx dx + 1 k t j γ k j! 1 k j k t j j! 2 k j+k+ + α j+k γ k j+k+ t j j! α j+k γx 1 e tx e αx e γx dx 1 k t j α k j k + α k γ k j+k+ 2α j+k j! j+k+ γ j+k + α k j+k+ γ j+k. 11 j+k+ 2γ j+k. 12

9 I. Elbatal, G. Aryal 125 Now, substtutng 11 and 12 n 1 we get M X t j k + 1 k j k Ths completes the proof. t j j! 2 k In partcular, for α, we have M X t 1 λ t j j! λ t j [ j j! γ j γ k j+k+ α j+k γ k j+k+ 2α j+k 1 λ + λ2 j + α k + α k j+k+ γ j+k j+k+ 2γ j+k ] j + 1. and for γ, we get M X t t j [ j j! α j 1 λ + λ2 j ] j Relablty Analyss Because of the analytcal structure of the TAW dstrbuton, t can be a useful model to characterze falure tme of a system. The relablty functon also known as survval functon of the TAW dstrbuton s denoted by R T AW t and s gven as R T AW t 1 F T AW t 1 1 e αt γt [ 1 + λe αt γt e αt γt 1 λ + λe αt γt. One of the most mportant quanttes characterzng lfe phenomenon n lfe testng analyss s the hazard rate functon defned by ht ft 1 F t. The hazard rate functon for a TAW dstrbuton s gven by h T AW t αt 1 + γt 1 1 λ + 2λe αt γt 1 λ + λe αt γt. 13 It s mportant to note that the unt for h T AW t s the probablty of falure per unt of tme, dstance or cycles. The falure rates for several dfferent dstrbuton can be

10 126 Austran Journal of Statstcs, Vol , No. 2, ht λ α 1 γ λ 1 α 5 γ λ.5 α.5 γ λ.5 α 1 γ λ 1 α 2 γ λ 1 α 8 γ ht λ λ 1 λ.5 λ.5 λ γ.2 α t t Fgure 3: Hazard rate functons of varous transmuted addtve Webull dstrbutons obtaned by smply changng the parameters. Fgure 3 llustrates the graphcal behavor of the hazard rate functon of the TAW dstrbuton for selected values of the parameters. Note that the fgure on the left dsplays the hazard rate functon for dfferent choce of all parameters whereas the fgure on the rght exhbts the behavor of the hazard rate functon as λ vares from 1 to 1 whle keepng all other four parameters fxed. In Table 2 we summarze few specal cases of the hazard rate functon of the TAW dstrbuton 13. In addton to the cases presented n Table 2 we can also easly obtan hazard rate functons of the Webull, Raylegh and exponental dstrbuton from the TAW hazard rate functon. Also t should be noted that the relablty behavor remans the same when we consder the case < 1 <. The cumulatve hazard functon, whch descrbes how the rsk of a partcular outcome changes wth tme, s gven by t H T AW t h T AW xdx log [e αt γt 1 λ + λe αt γt ] αt + γt log 1 λ + λe αt γt. Notce that the unt for H T AW t s the cumulatve probablty of falure per unt of tme, dstance or cycles. It descrbes how the rsk of a partcular outcome changes wth tme for a TAW dstrbuton.

11 I. Elbatal, G. Aryal 127 Table 2: Hazard rates of some sub-models of the transmuted addtve Webull dstrbuton Model Parameters Hazard rate functon TMWD α, γ,, λ α + γt 1 1 λ + 2λe αt γt 1 λ + λe αt γt AWD α,, γ, αt 1 + γt 1 TLFRD TMED TWD TED TRD α, γ, λ α, γ, λ γ,, λ γ, λ γ, λ α + 2γt1 λ + 2λe αt γt2 1 λ + λe αt γt2 α + γ1 λ + 2λe αt γt 1 λ + λe αt γt γt 1 1 λ + 2λe γt 1 λ + λe γt γ1 λ + 2λe γt 1 λ + λe γt 2γt1 λ + 2λe γt2 1 λ + λe γt2 MWD α, γ, α + γt 1 MRD α, γ α + 2γt MED α, γ α + γ 5 Order Statstcs Order statstcs are among the most fundamental tools n non-parametrc statstcs and statstcal nference. These statstcs also arse n the study of relablty and lfe testng. Some dstrbutonal propertes of the maxmum and the mnmum of random varables have been extensvely studed n the lterature. Let X 1,..., X n be a smple random sample from a T AW D α,,, γ, λ, x dstrbuton wth cdf and pdf as n 3 and 4, respectvely. Let X 1:n X n:n denote the order statstcs obtaned from ths sample. In relablty lterature, X :n denotes the lfetme of an n + 1 out of n system whch conssts of n ndependent and dentcally dstrbuted components. If 1 or n, such systems are better known as seres or parallel systems, respectvely. Consderable attenton has been gven to establsh several relablty propertes of such systems. It s well known that the cdf and pdf of X :n for 1,..., n are gven by F :n x k F x n F x k 1 F x n k k n! 1!n! t 1 1 t n dt

12 128 Austran Journal of Statstcs, Vol , No. 2, and f :n x n! 1!n! F x 1 1 F x n fx. Defne the mnmum X 1 mnx 1,..., X n, the maxmum X n maxx 1,..., X n, and the medan as X m+1 wth m n/ Dstrbuton of Mnmum, Maxmum and Medan Let X 1,..., X n denote a random sample from a TAW dstrbuton, then the pdf of the the th order statstc s gven by n! f :n x αx 1 + γx 1 e n +1αx +γx 1 e αx γx 1 1!n! 1 + λe αx γx 1 1 λ + λe αx γx n 1 λ + 2λe αx γx. Therefore, the pdfs of the mnmum, the maxmum and the medan are f 1:n x n αx 1 + γx 1 e nαx +γx 1 λ + λe αx γx n 1 1 λ + 2λe αx γx, f n:n x n αx 1 + γx 1 e αx γx [1 e αx γx 1 + λe αx γx ] n 1 1 λ + 2λe αx γx, 2m + 1! [ f m+1:n x 1 e αx γx 1 + λe αx γx ] m e αx γx [ m!m! 1 1 e αx γx 1 + λe αx γx ] m αx 1 + γx 1 1 λ + 2λe αx γx. Also note that the mnmum, maxmum and medan order statstcs of the fve parameters TAW dstrbuton converges to the order statstcs of several lfe tme dstrbutons when ts parameters are changed. 6 Parameter Estmaton We now consder the maxmum lkelhood estmators MLEs of the parameters α,,, γ, λ. Let x 1,..., x n be an observed random sample of sze n from a T AW D α,,, γ, λ dstrbuton then the lkelhood functon can be wrtten as Lα,,, γ, λ x n αx 1 + γx 1 e αx γx 1 λ + 2λe αx γx.

13 I. Elbatal, G. Aryal 129 Hence, the log-lkelhood functon l log L becomes lα,,, γ, λ x + log αx 1 + γx 1 α log 1 λ + 2λe αx γx x γ x. 14 Dfferentatng 14 wth respect to the parameters, the components of the score vector are l α l l γ l l λ x 1 x 2λe αx γx x + γx 1, 1 λ + 2λe αx γx logx + 1 α x 2αλe αx γx x αx 1 + γx 1 logx logx, 1 λ + 2λe αx γx x 1 x + γx 1 2λe αx γx x, 1 λ + 2λe αx γx logx + 1 γ x αx 1 + γx 1 logx αx 1 x 1 αx 1 x 1 2e αx γx 1 1 λ + 2λe αx γx. 2λγe αx γx x logx, 1 λ + 2λe αx γx The maxmum lkelhood estmators ˆα, ˆ, ˆγ, ˆ, ˆλ of α,, γ,, λ are obtaned by settng the score vector to zero and solvng the system of nonlnear equatons. It s usually more convenent to use nonlnear optmzaton algorthms such as the quas-newton algorthm to numercally maxmze the log-lkelhood functon gven n 14. For the fve parameters TAW dstrbuton all second order dervatves exst. Thus, we have wth ˆα α ˆ ˆ ˆγ Normal γ, Σ ˆλ λ V αα V α V α V αγ V αλ V α V V V γ V λ Σ E V α V V V γ V λ V γα V γ V γ V γγ V γλ V λα V λ V λ V λγ V λλ Here V.. denotes the second dervatve of the log-lkelhood functon wth respect to the two parameters n the ndex. 1.

14 13 Austran Journal of Statstcs, Vol , No. 2, By calculatng ths nverse matrx ths wll yeld asymptotc varance and covarances of the MLEs. Approxmate 11 ϕ% confdence ntervals for the parameters can be determned as ˆα ± z ϕ 2 ˆV αα, ˆ ± z ϕ 2 ˆV, ˆ ± z ϕ 2 ˆV, ˆγ ± z ϕ 2 ˆV γγ, ˆλ ± z ϕ ˆV λλ, 2 where z ϕ s the upper ϕth percentle of the standard normal dstrbuton. We can compute the maxmum values of the unrestrcted and restrcted log-lkelhood functons to obtan lkelhood rato test statstcs for testng the sub-model of the new dstrbuton. For example, we can then use the test statstc to check whether the TAW dstrbuton s statstcally superor to an AW dstrbuton for a gven data set. In ths case we can compare the frst model aganst the second model by testng H : λ versus H 1 : λ. 7 Applcaton We now provde a lttle data analyss n order to assess the goodness-of-ft of the model. The data refers to the lfetmes of n 5 devces provded by Aarset 1987 and s often cted as an example wth bathtub shaped falure rate. Table 3 gves a descrptve summary of the data. Table 3: Descrptve statstcs of the lfetme data n Mean Medan Varance Mnmum Maxmum We have used the Webull, the AW and the TAW dstrbuton to analyze the data. In addton to the MLEs, also the Akake Informaton Crteron AIC by Akake 1974, whch s used to select the best model among several models, s provded n Table 4. The AIC s gven as AIC 2l + 2p, where p s the number of parameters n the model. The result ndcates that a TAW dstrbuton has the lowest AIC so t fts best to ths data. The statstcal software R Ihaka and Gentleman, 1996 has been used to perform all necessary calculatons and to produce all graphcs. 8 Concludng Remarks In the present study, we have ntroduced a new generalzaton of the AW dstrbuton called the TAW dstrbuton. The 4-parameter AW dstrbuton s embedded n the proposed dstrbuton. Some mathematcal propertes along wth estmaton ssues are addressed. The hazard rate functon and relablty behavor of the TAW dstrbuton shows that the subject dstrbuton can be used to model relablty data. We have presented an example where a TAW dstrbuton fts better than the AW dstrbuton. We beleve that the subject dstrbuton can be used n several dfferent areas. We expect that ths study wll serve as a reference and help to advance future research n the subject area.

15 I. Elbatal, G. Aryal 131 Table 4: MLEs under the consdered models and correspondng AIC values Model MLEs l x AIC W γ, ˆγ.27, ˆ AW α,, γ, ˆα.4, ˆ ˆγ.1118, ˆ.2889 T AW α,, γ,, λ ˆα.1, ˆ ˆγ.834, ˆ.4152 ˆλ.77 Fx Emprcal Webull Addtve Webull Transmuted Addtve Webull x Fgure 4: Emprcal cdf compared wth Webull, addtve Webull, and transmuted addtve Webull fts of the lfe tme data Acknowledgements The authors would lke to thank the anonymous revewer for carefully readng the manuscrpt and makng valuable suggestons. References Aarset, M. V How to dentfy bathtub hazard rate. IEEE Transactons on Relablty, R-36,

16 132 Austran Journal of Statstcs, Vol , No. 2, Akake, H A new look at statstcal model dentfcaton. IEEE Transactons on Relablty, 19, Aryal, G. R., and Tsokos, C. P Transmuted Webull dstrbuton: A generalzaton of the Webull probablty dstrbuton. European Journal of Pure and Appled Mathematcs, 4, Ihaka, R., and Gentleman, R R: A language for data analyss and graphcs. Journal of Computatonal and Graphcal Statstcs, 5, Khan, M. S., and Kng, R Transmuted modfed Webull dstrbuton: A generalzaton of the modfed Webull probablty dstrbuton. European Journal of Pure and Appled Mathematcs, 6, La, C. D., Xe, M., and Murthy, D. N. 21. Bathtub shaped falure rate dstrbutons. In N. Balakrshnan and C. R. Rao Eds., Handbook n Relablty Vol. 2, p La, C. D., Xe, M., and Murthy, D. N. 23. A modfeded Webull dstrbuton. IEEE Transactons on Relablty, 52, Nadarajah, S. 29. Bathtub-shaped falure rate functons. Qualty and Quantty, 43, Shaw, W. T., and Buckley, I. R. C. 29. The alchemy of probablty dstrbutons: beyond Gram-Charler expansons, and a skew-kurtotc-normal dstrbuton from a rank transmutaton map. arxv preprnt arxv: Xe, M., and La, C. D Relablty analyss usng an addtve Webull model wth bathtub-shaped falure rate functon. Relablty Engneerng and System Safety, 52, Authors addresses: Ibrahm Elbatal Insttute of Statstcal Studes and Research Department of Mathematcal Statstcs Caro Unversty Egypt E-mal: elbatal@staff.cu.edu.eg Gokarna Aryal Department of Mathematcs, Computer Scence, and Statstcs Purdue Unversty Calumet th Street CLO 38 Hammond, IN USA E-mal: aryalg@purduecal.edu