Transmuted Lindley-Geometric Distribution and its Applications

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1 J. Stat. Appl. Pro. 3, No. 1, ) 77 Joural of Statstcs Applcatos & Probablty A Iteratoal Joural Trasmuted Ldley-Geometrc Dstrbuto ad ts Applcatos Fato Merovc 1, ad Ibrahm Elbatal 2 1 Departmet of Mathematcs, Uversty of Prshta Hasa Prshta, Republc of Kosovo 2 Isttute of Statstcal Studes ad Research, Departmet of Mathematcal Statstcs, Caro Uversty, Egypt Receved: 2 Nov. 213, Revsed: 1 Ja. 214, Accepted: 21 Ja. 214 Publshed ole:... Abstract: A fuctoal composto of the cumulatve dstrbuto fucto of oe probablty dstrbuto wth the verse cumulatve dstrbuto fucto of aother s called the trasmutato map. I ths artcle, we wll use the quadratc rak trasmutato map QRTM) order to geerate a flexble famly of probablty dstrbutos takg Ldley-geometrc dstrbuto as the base value dstrbuto by troducg a ew parameter that would offer more dstrbutoal flexblty. It wll be show that the aalytcal results are applcable to model real world data. Keywords: Ldley geometrc dstrbuto, momets,order Statstcs,Trasmutato map, Maxmum Lkelhood Estmato, Relablty Fucto. 1 Itroducto ad Motvato The Ldley dstrbuto was orgally proposed by Ldley 23 the cotext of Bayesa statstcs, as a couter example of fudcal statstcs. More detals o the Ldley dstrbuto ca be foud Ghtay et al. 1. A radom varable X s sad to have the Ldley dstrbuto wth parameter θ f ts probablty desty s defed as f L x,θ)= θ 2 The correspodg cumulatve dstrbuto fucto c.d.f.) s: x ; x>,θ >. 1) F L x,θ)=,x>,θ >. 2) May authors gves geeralzed Lldey dstrbuto lke Sakara 27 troduced the dscrete Posso-Ldley, Mahmoud ad Zakerzadeh 14 troduced geeralzed Ldley dstrbuto, Bakouch et al. 5 troduced exteded Ldley EL) dstrbuto, Adamds ad Loukas 4 troduced expoetal geometrc EG) dstrbuto. Recetly, Hoatollah ad Mahmoud 29 troduced Ldley-geometrc dstrbuto where the cdf ad pdf of ths dstrbuto are gve by F LG x,θ, p)= >,< p<1, 3),x>,θ ad f LG x,θ, p)= θ 2 1 p)x 2, 4) respectvely. I ths paper, we troduce a ew lfetme dstrbuto by trasmuted ad compoudg Ldley ad geometrc dstrbutos amed trasmuted Ldley-geometrc dstrbuto. The cocept of trasmuted explaed the followg subsecto. Correspodg author e-mal: fmerovc@yahoo.com

2 78 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto Trasmutato Map I ths subsecto we demostrate trasmuted probablty dstrbuto. Let F 1 ad F 2 be the cumulatve dstrbuto fuctos, of two dstrbutos wth a commo sample space. The geeral rak trasmutato s defed as G R12 u)=f 2 F 1 1 u)) ad G R21 u)=f 1 F 1 2 u)). Note that the verse cumulatve dstrbuto fucto also kow as quatle fucto s defed as F 1 y)= f x R {Fx) y} for y,1. The fuctos G R12 u) ad G R21 u) both map the ut terval I =,1 to tself, ad uder sutable assumptos are mutual verses ad they satsfy G R )= ad G R )=1. A quadratc Rak Trasmutato Map QRTM) s defed as G R12 u)=u+λu1 u), λ 1, 5) from whch t follows that the cdf s satsfy the relatoshp whch o dfferetato yelds, F 2 x)=λ)f 1 x) λf 1 x) 2 6) f 2 x)= f 1 x)λ) 2λF 1 x) 7) where f 1 x) ad f 2 x) are the correspodg pdfs assocated wth cdf F 1 x) ad F 2 x) respectvely. A extesve formato about the quadratc rak trasmutato map s gve Shaw et al. 31. Observe that at λ = we have the dstrbuto of the base radom varable. The followg lemma proved that the fucto f 2 x) gve 7) satsfes the property of probablty desty fucto. Lemma: f 2 x) gve 7) s a well defed probablty desty fucto. May authors dealg wth the geeralzato of some well- kow dstrbutos. Aryal ad Tsokos 1 defed the trasmuted geeralzed extreme value dstrbuto ad they studed some basc mathematcal characterstcs of trasmuted Gumbel probablty dstrbuto ad t has bee observed that the trasmuted Gumbel ca be used to model clmate data. Also Aryal ad Tsokos 2 preseted a ew geeralzato of Webull dstrbuto called the trasmuted Webull dstrbuto. Recetly, Aryal 213) proposed ad studed the varous structural propertes of the trasmuted Log-Logstc dstrbuto, ad Kha ad Kg 13 troduced the trasmuted modfed Webull dstrbuto whch exteds recet developmet o trasmuted Webull dstrbuto by Aryal et al. 2, Merovc 19,2,21troduced the trasmuted Raylegh dstrbuto, trasmuted geeralzed Raylegh dstrbuto, trasmuted Ldley dstrbuto ad they studed the mathematcal propertes ad maxmum lkelhood estmato of the ukow parameters. 1.2 Trasmuted Ldley Geometrc Dstrbuto I ths secto we studed the trasmuted Ldley geometrc TLG) dstrbuto. Now usg 5)ad 6) we have the cdf of trasmuted Ldley-geometrc TLG) dstrbuto ) F T LG x,θ, p,λ)= λ λ 8) where λ s the trasmuted parameter. The correspodg probablty desty fucto pdf) of the trasmuted Ldleygeometrc s gve by f T LG x,θ, p,λ)= f LG x)λ) 2λF LG x) = θ 2 1 p)x 2 { )} λ) 2λ, 9)

3 J. Stat. Appl. Pro. 3, No. 1, ) / 79 respectvely. Fgure 1 ad fgure 2 llustrates some of the possble shapes of the pdf ad cdf of TLG dstrbuto for selected values of the parameters θ, p ad λ, respectvely. fx) θ = 4.1 p =.1 λ =.3 θ = 2.1 p =.2 λ =.5 θ = 3.1 p =.3 λ =.8 θ = 4.1 p =.4 λ =.8 θ = 5.1 p =.5 λ = x Fg. 1: The pdf s of varous TLG dstrbutos. Fx) θ = 4.1 p =.1 λ =.3 θ = 2.1 p =.2 λ =.5 θ = 3.1 p =.3 λ =.8 θ = 4.1 p =.4 λ =.8 θ = 5.1 p =.5 λ = x Fg. 2: The cdf s of varous TLG dstrbutos.

4 8 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto... The relablty fucto RF) of the trasmuted Ldley-geometrc dstrbuto s deoted by R T LG x) also kow as the survvor fucto ad s defed as R T LG x)=1 F T LG x) = 1 ) λ λ. 1) Fgure 3 llustrates some of the possble shapes of the survval fucto of trasmuted Ldley geometrc dstrbuto for selected values of the parameters θ, p ad λ, respectvely. Sx) θ = 4.1 p =.1 λ =.3 θ = 2.1 p =.2 λ =.5 θ = 3.1 p =.3 λ =.8 θ = 4.1 p =.4 λ =.8 θ = 5.1 p =.5 λ = x Fg. 3: The survval fucto of varous trasmuted Ldely geometrc dstrbutos. It s mportat to ote that R T LG x)+f T LG x)=1. Oe of the characterstc relablty aalyss s the hazard rate fucto HF) defed by h T LG x)= f T LGx) 1 F T LG x) 11) Fgure 4 llustrates some of the possble shapes of the hazard fucto of trasmuted Ldley-geometrc dstrbuto for selected values of the parameters θ, p ad λ, respectvely.

5 J. Stat. Appl. Pro. 3, No. 1, ) / 81 hx) θ = 4.1 p =.1 λ =.3 θ = 2.1 p =.2 λ =.5 θ = 3.1 p =.3 λ =.1 θ = 4.1 p =.4 λ =.8 θ = 5.1 p =.5 λ = x Fg. 4: The survval fucto of varous trasmuted Ldely geometrc dstrbutos. It s mportat to ote that the uts for h T LG x) s the probablty of falure per ut of tme, dstace or cycles. These falure rates are defed wth dfferet choces of parameters. The cumulatve hazard fucto of the trasmuted Ldley-geometrc dstrbuto s deoted by H T LG x) ad s defed as ) H T LG x)= l λ λ It s mportat to ote that the uts for H T LG x) s the cumulatve probablty of falure per ut of tme, dstace or cycles. We ca show that. For all choce of parameters the dstrbuto has the decreasg patters of cumulatve stataeous falure rates. 12) 2 Statstcal Propertes Ths secto s devoted to studyg statstcal propertes of thet LG) dstrbuto. 2.1 Momets I ths subsecto we dscuss the r th momet for T LG) dstrbuto. Momets are ecessary ad mportat ay statstcal aalyss, especally applcatos. It ca be used to study the most mportat features ad characterstcs of a dstrbuto e.g., tedecy, dsperso, skewess ad kurtoss). Theorem 3.1). If X has T LG Φ,x),Φ =θ, p,λ) the the r th momet of X s gve by the followg µ rx)=a Γr+ +1) r+ +1 g θ + 1)) r++1 θ + 1)) { Γr+ +1) Γr+ +1) B θ θ + 1)) r++1 Γr+ +2) θ + 2)) r++2 r+ +1 θ + 1)) ) r+ +2 θ + 2)) r+ +1 θ + 2)) r++1 θ + 2)) }, 13)

6 82 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto... where ad A g = θ 2 λ) 1 p) B = λθ2 1 p) = = = = ) ) θ + 1)p, ) ) θ + 1) + 2)p. Proof: Let X be a radom varable wth desty fucto 9). The r th ordary momet of the T LG) dstrbuto s gve by µ rx)=ex r = x r fx,φ)dx = θ 2 λ) 2λθ2 usg the seres expaso 1 p) 1 p) where z <1 ad k>. Equato 14) ca be demostrated by µ rx)= θ 2 λ) 1 p) { x r + x r+1 )e x r + x r+1 )e 1 z) k = = λθ 2 1 p) + 1) + 2)p = 2 dx ) 3 dx. 14) Γk+ ) = Γk)! z, 15) + 1)p x r + x r+1 ) ) e θ +1)x dx x r + x r+1 ) ) ) e θ +1)x dx also applyg the bomal expresso for ) where ) = =, 16) ) ) θ x, 17) substtutg from 17) to 16) we get { µ rx)= θ 2 ) ) λ) θ 1 p) + 1)p = = x r+ + x r++1 )e θ +1)x dx { λθ 2 ) ) θ 1 p) + 1) + 2)p = = x r+ + x r++1 ) ) e θ +1)x dx = A g I 1 B I 2

7 J. Stat. Appl. Pro. 3, No. 1, ) / 83 where A g = θ 2 λ) 1 p) = = ) ) θ + 1)p, B = λθ2 1 p) = = ) ) θ + 1) + 2)p, I 1 = x r+ + x r++1 )e θ +1)x dx Γr+ +1) Γr+ +2) = + r++1 θ + 1)) θ + 1)) r++2 =, Γr+ +1) θ + 1)) r++1 r+ +1 θ + 1)) ad I 2 = x r+ + x r++1 ) Γr+ +1) = θ + 1)) r++1 θ Γr+ +2) θ + 2)) r++2 r+ +1 θ + 1)) ) r+ +2 θ + 2)) ) e θ +1)x dx Γr+ +1) θ + 2)) r++1, r+ +1 θ + 2)) thus the r th momet s gve by Whch completes the proof. µ r x)=θα 2 = m= 1) ) m α m Γr+ m+2) α + 1)) r+m+2 λ) ) θ 1 2λ 2θ 1). We otce that f we put λ =, we get the r th momet of Ldley geometrc see Hoatollah ad Mahmoud 212)). Based o the frst four momets of the T LG) dstrbuto, the measures of skewess AΦ) ad kurtoss kφ) of the T LG) dstrbuto ca obtaed as AΦ)= µ 3θ) 3µ 1 θ)µ 2 θ)+2µ 1 3θ), µ2 θ) µ 1 2θ) 2 3 ad kφ)= µ 4θ) 4µ 1 θ)µ 3 θ)+6µ 2 1 θ)µ 2θ) 3µ 4 1 θ) µ2 θ) µ 2 1 θ) Momet Geeratg fucto I ths subsecto we derved the momet geeratg fucto oft LG) dstrbuto.

8 84 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto... Proof. Theorem 3.2): If X has T LG) dstrbuto, the the momet geeratg fucto M X t) has the followg form M X t)= A gγ+1) +1 θ + 1) t) +1 θ + 1) t) { Γ+1) +1 B θ + 1) t) +1 θ + 1) t) Γ+1) +1 θ + 2) t) +1 θ + 2) t) ) θ } Γ+2) +2 θ + 2) t) +2 18) θ + 2) t) We start wth the well kow defto of the momet geeratg fucto gve by M X t)=ee tx )= e tx f T LG x,φ)dx = θ 2 λ) 2λθ2 1 p) 1 p) substtutg from 15) ad 17) to 19) we get Whch completes the proof. x)e xθ t) x)e xθ t) M X t)=a g x + x +1 )e xθ +1) t dx B 2 dx x + x +1 )e xθ +1) t = A gγ+1) θ + 1) t) +1 { Γ+1) B θ + 1) t) +1 Γ+1) θ + 2) t) +1 θ Γ+2) θ + 2) t) θ + 1) t) +1 θ + 1) t) +1 θ + 2) t) ) ) 3 dx. 19) +2 θ + 2) t) ) } 2) 3 Dstrbuto of the order statstcs I ths secto, we derve closed form expressos for the pdfs of the r th order statstc of the T LG dstrbuto, also, the measures of skewess ad kurtoss of the dstrbuto of the r th order statstc a sample of sze for dfferet choces of ;r are preseted ths secto. Let X 1,X 2,...,X be a smple radom sample from T LG) dstrbuto wth pdf ad cdf gve by 8) ad 9), respectvely.

9 J. Stat. Appl. Pro. 3, No. 1, ) / 85 Let X 1,X 2,...,X deote the order statstcs obtaed from ths sample. We ow gve the probablty desty fucto of X r:, say f r: x,φ) ad the momets of X r:,r = 1,2,...,. Therefore, the measures of skewess ad kurtoss of the dstrbuto of the X r: are preseted. The probablty desty fucto of X r: s gve by f r: x,φ)= 1 Br, r+ 1) Fx,Φ)r 1 1 Fx,Φ) r fx,φ) 21) where Fx,Φ) ad fx,φ) are the cdf ad pdf of the T LG) dstrbuto gve by 8), 9), respectvely, ad B.,.) s the beta fucto, sce <Fx,Φ)<1, for x>, by usg the bomal seres expaso of 1 Fx,Φ) r, gve by 1 Fx,Φ) r r = = ) r 1) Fx,Φ), 22) we have r f r: x,φ)= = ) r 1) Fx,Φ) r+ 1 fx,φ), 23) substtutg from 8) ad 9) to 23), we ca express the k th ordary momet of the r th order statstcs X r: say EX k r:) as a ler combato of the k th momets of thet LG) dstrbuto wth dfferet shape parameters. Therefore, the measures of skewess ad kurtoss of the dstrbuto of X r: ca be calculated. 4 Estmato ad Iferece 4.1 Least Squares ad Weghted Least Squares Estmators I ths subsecto we provde the regresso based method estmators of the ukow parameters of the trasmuted Ldley-geometrc dstrbuto, whch was orgally suggested by Swa, Vekatrama ad Wlso 1988) to estmate the parameters of beta dstrbutos. It ca be used some other cases also. Suppose Y 1,...,Y s a radom sample of sze from a dstrbuto fucto G.) ad suppose Y ) ; =1,2,..., deotes the ordered sample. The proposed method uses the dstrbuto of GY ) ). For a sample of sze, we have E GY ) ) ) = +1,V GY ) ) ) + 1) = +1) 2 +2) ad Cov GY ) ),GY k) ) ) k+ 1) = +1) 2 ;for < k, +2) see Johso, Kotz ad Balakrsha 1995). Usg the expectatos ad the varaces, two varats of the least squares methods ca be used. Method 1 Least Squares Estmators). Obta the estmators by mmzg =1 GY ) ) 2, 24) +1 wth respect to the ukow parameters. Therefore case of T LG dstrbuto the least squares estmators of θ, p ad λ, say, θ LSE, p LSE ad λ LSE respectvely, ca be obtaed by mmzg =1 wth respect to θ, p ad λ. ) 2 λ λ +1

10 86 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto... Method 2 Weghted Least Squares Estmators). The weghted least squares estmators ca be obtaed by mmzg w GY ) ) 2, 25) +1 wth respect to the ukow parameters, where w = =1 1 V GY ) ) ) = +1)2 +2) + 1). Therefore, case of T LG dstrbuto the weghted least squares estmators of θ, p ad λ, say, θ WLSE, p WLSE ad λ WLSE respectvely, ca be obtaed by mmzg =1 wth respect to the ukow parameters oly. ) w 2 λ λ MAxmum lkelhood estmato I ths subsecto we determe the maxmum lkelhood estmates MLEs) of the parameters of the T LG) dstrbuto from complete samples oly. Let X 1,X 2,...,X be a radom sample of sze from T LGθ, p,λ,x).the lkelhood fucto for the vector of parameters Φ =θ, p,λ) ca be wrtte as L fx ),Φ)=Π =1 fx ),Φ) ) θ 2 = 1 p) Π=1 x ) e θ Π =1x =1 2 { )} Π=1 λ) 2λ. 26) Takg the log-lkelhood fucto for the vector of parameters Φ =θ, p,λ) we get l=logl=2logθ logθ)+log1 p)+ 2 + =1 =1 =1 log { log λ) 2λ logx ) θ )} x ) =1. 27) The log-lkelhood ca be maxmzed ether drectly or by solvg the olear lkelhood equatos obtaed by dfferetatg 27). The compoets of the score vector are gve by l p = 2λ 1 p + 2 =1 =1 { λ) 2λ 1 p ) 2 1 p )} =, 28)

11 J. Stat. Appl. Pro. 3, No. 1, ) / 87 l θ = 2 θ θ x 2p =1 2λ =1 { λ) 2λ 1 p) =1 1 p x e ) 1 x )} e θ) 2 ) 1 θ) 2 2 = 29) ad l λ = = p { λ) 2λ ) 1 p )} =. 3) We ca fd the estmates of the ukow parameters by maxmum lkelhood method by settg these above o-lear equatos 29)- 3) to zero ad solve them smultaeously. Therefore, we have to use mathematcal package to get the MLE of the ukow parameters. Applyg the usual large sample approxmato, the MLE ˆΦ ca be treated as beg approxmately trvarate ormal ad varace-covarace matrx equal to the verse of the expected formato matrx,.e. ˆΦ Φ) N,I 1 Φ) ), where I 1 Φ) s the lmtg varace-covarace matrx of ˆΦ. The elemets of the 3 3 matrx IΦ) ca be estmated by I ˆΦ)= l Φ Φ Φ= ˆΦ,, {1,2,3}. Approxmate two sded 11 α)% cofdece tervals for θ, p ad for λ are, respectvely, gve by ˆθ ± z α/2 I11 1 ˆθ), ˆp±z α/2 I22 1 ˆp) ad ˆλ ± z α/2 I 1 33 ˆλ), where z α s the upper αth quatle of the stadard ormal dstrbuto. Usg R we ca easly compute the Hessa matrx ad ts verse ad hece the stadard errors ad asymptotc cofdece tervals. 5 Applcato I ths secto, we use a real data set to show that the trasmuted Ldley dstrbuto ca be a better model tha oe based o the Ldley geometrc dstrbuto ad Ldley dstrbuto. The data set gve Table 1 represets the watg tmes mutes) before servce of 1 bak customers. Table 1: The watg tmes mutes) before servce of 1 bak customers

12 88 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto... Table 2: Estmated parameters of the Ldley, Ldley-geometrc ad trasmuted Ldley geometrc dstrbuto for the watg tmes mutes) before servce of 1 bak customers. Model Parameter Estmate Stadard Error l ;x) Ldley ˆθ = Ldley ˆθ = Geometrc ˆp = Trasmuted ˆθ = Ldley ˆp = Geometrc ˆλ = The varace covarace matrx of the MLEs uder the trasmuted Ldley geometrc dstrbuto s computed as I ˆθ) 1 = Thus, the varaces of the MLE of θ, p ad λ s var ˆθ) =.12,var ˆp) =.326,varâ) =.368. Therefore, 95% cofdece tervals for θ, p ad λ are.12,.24,.32, 1, ad.577, 1 respectvely. Table 3: Crtera for comparso. Model K-S 2l AIC AICC Ldley Ldley Geometrc TLG I order to compare the two dstrbuto models, we cosder crtera lke K-S, 2l, AIC Akake formato crtero)ad AICC corrected Akake formato crtero) for the data set. The better dstrbuto correspods to smaller K-S, 2l, AIC ad AICC values: AIC=2k 2l, ad AICC=AIC+ 2kk+ 1) k 1, where k s the umber of parameters the statstcal model, the sample sze adls the maxmzed value of the loglkelhood fucto uder the cosdered model. Also, here for calculatg the values of KS we use the sample estmates of θ,α,a,b ad c. Table 2 shows the MLEs uder both dstrbutos, Table 3 shows the values of K-S, 2l, AIC ad AICC values. The values table 3 dcate that the trasmuted Ldley geometrc dstrbuto leads to a better ft tha the Ldley geometrc dstrbuto ad Ldely dstrbuto. A desty plot compares the ftted destes of the models wth the emprcal hstogram of the observed data Fg. 4). The ftted desty for the trasmuted Lldey geometrc model s closer to the emprcal hstogram tha the fts of the Ldley geometrc ad Ldley sub-models.

13 J. Stat. Appl. Pro. 3, No. 1, ) / 89 Desty TLG LG L The watg tmes mutes) before servce of 1 bak customers Fg. 5: Estmated destes of the models for the watg tmes mutes) before servce of 1 bak customers. Fx) Emprcal Lldey LG TLG x Fg. 6: Emprcal, ftted Ldley, Ldley geometrc ad trasmuted Ldley geometrc cdf of the the watg tmes mutes) before servce of 1 bak customers. 6 Cocluso Here we propose a ew model, the so-called the trasmuted Ldley geometrc dstrbuto whch exteds the Ldley geometrc dstrbuto the aalyss of data wth real support. A obvous reaso for geeralzg a stadard dstrbuto s because the geeralzed form provdes larger flexblty modelg real data. We derve expasos for

14 9 F. Merovc, I. Elbatal: Trasmuted Ldley-Geometrc Dstrbuto... momets ad for the momet geeratg fucto. The estmato of parameters s approached by the method of maxmum lkelhood, also the formato matrx s derved. A applcato of the trasmuted Ldley geometrc dstrbuto to real data show that the ew dstrbuto ca be used qute effectvely to provde better fts tha Ldley geometrc ad Ldley dstrbuto. Ackowledgemet The authors are grateful to the aoymous referee for a careful checkg of the detals ad for helpful commets that mproved ths paper. Refereces 1 Aryal, G. R., & Tsokos, C. P., O the trasmuted extreme value dstrbuto wth applcato. Nolear Aalyss: Theory, Methods & Applcatos, 71, e141-e147 29). 2 Aryal, G. R., & Tsokos, C. P., Trasmuted Webull Dstrbuto: A Geeralzato of thewebull Probablty Dstrbuto. Europea Joural of Pure ad Appled Mathematcs, 4, ). 3 Adamds K., Dmtrakopoulou T., ad Loukas S., O a geeralzato of the expoetal-geometrc dstrbuto, Statst. Probab. Lett., 73, ). 4 Adamds, K., ad Loukas, S., A lfetme dstrbuto wth decreasg falure rate. Statstcs ad Probablty Letters, 39, ). 5 Bakouch, H. S., Al-Zahra, B. M., Al-Shomra, A. A., March, V. A., ad Louzada, F., A exteded Ldley dstrbuto. Joural of the Korea Statstcal Socety, 41, ). 6 Barreto-Souza, W., ad Crbar-Neto, F., A geeralzato of the expoetal-posso dstrbuto. Statstcs ad Probablty Letters, 79, ). 7 Barreto-Souza, W., de Moras, A. L., ad Cordero, G. M., The Webull-geometrc dstrbuto. Joural of Statstcal Computato ad Smulato, 81, ). 8 Cacho, V. G., Louzada-Neto, F., ad Barrga, G. D., The Posso-expoetal lfetme dstrbuto. Computatoal Statstcs ad Data Aalyss, 55, ). 9 Chahkad, M., ad Gaal, M., O some lfetme dstrbutos wth decreasg falure rate. Computatoal Statstcs ad Data Aalyss, 53, ). 1 Ghtay, M. E., Ateh, B., ad Nadaraah, S.,. Ldley dstrbuto ad ts applcato. Mathematcs ad Computers Smulato, 78, ). 11 Johso, N. L., Kotz, S., & Balakrsha, N., Cotuous uvarate dstrbutos, vol. 2Wley. New York, 1995). 12 Kus, C., A ew lfetme dstrbuto. Computatoal Statstcs ad Data Aalyss, 51, ). 13 Kha, M. S., & Kg, R., Trasmuted Modfed Webull Dstrbuto: A Geeralzato of the Modfed Webull Probablty Dstrbuto. Europea Joural of Pure ad Appled Mathematcs, 6, ). 14 Mahmoud, E., ad Zakerzadeh, H., Geeralzed PossoLdley dstrbuto. Commucatos StatstcsTheory ad Methods, 39, ). 15 Mahmoud, E., ad Tork, M., Geeralzed verse Webull-Posso dstrbuto ad ts applcatos. Submted to Joural of Statstcal Computato ad Smulato, 211). 16 Mahmoud, E., ad Sepahdar, A., Expoetated Webull-Posso dstrbuto ad ts applcatos. Submted to Mathematcs ad Computer Smulato, 211). 17 Mahmoud, E., ad Jafar, A. A., Geeralzed expoetalpower seres dstrbutos. Computatoal Statstcs ad Data Aalyss, 56, ). 18 Marshall, A. W., ad Olk, I., A ew method for addg a parameter to a famly of dstrbutos wth applcato to the expoetal ad Webull famles. Bometrka, 84, ). 19 Merovc, F., Trasmuted Raylegh dstrbuto. Austra Joural of Statstcs, 42, ). 2 Merovc, F., Trasmuted geeralzed Raylegh dstrbuto. Joural of Statstcs Applcatos ad Probablty, 2, ). 21 Merovc, F., Trasmuted Ldley dstrbuto. Iteratoal Joural of Ope Problems Computer Scece ad Mathematcs, 6, ). 22 Moras, A. L., ad Barreto-Souza, W. A compoud class of Webull ad power seres dstrbutos. Computatoal Statstcs ad Data Aalyss, 55, ). 23 Ldley, D. V., Fducal dstrbutos ad Bayes theorem. Joural of the Royal Statstcal Socety. Seres B Methodologcal), ). 24 Ldley, D. V., Itroducto to probablty ad statstcs from bayesa vewpot. part 2 ferece. CUP Archve, 1965). 25 Louzada, F., Roma, M., ad Cacho, V. G., The complemetary expoetal geometrc dstrbuto: Model, propertes, ad a comparso wth ts couterpart. Computatoal Statstcs ad Data Aalyss, 55, ).

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