The McDonald Quasi Lindley Distribution and Its Statistical Properties with Applications

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1 J. Stat. Appl. Pro. 4, No. 3, Joural of Statstcs Applcatos & Probablty A Iteratoal Joural The McDoald Quas Ldley Dstrbuto ad Its Statstcal Propertes wth Applcatos Rasool Roozegar ad Fatemeh Esfadyar Departmet of Statstcs, Yazd Uversty, P.O. Box , Yazd, Ira Receved: 25 May 25, Revsed: 26 Ju. 25, Accepted: 27 Ju. 25 Publshed ole: Nov. 25 Abstract: A ew fve-parameter dstrbuto so-called the McDoald quas Ldley dstrbuto s proposed. The ew dstrbuto cotas, as specal submodels, several mportat dstrbutos dscussed the lterature, such as the beta quas Ldley, Kumaraswamy quas Ldley, beta Ldley, umaraswamy Ldley ad Ldley dstrbutos, amog others. The propertes of ths ew dstrbuto, cludg hazard fucto, reversed hazard fucto, shapes, momets, etropy ad momet geeratg fucto are derved. We provde the desty fucto of the order statstcs ad ther momets. Method of maxmum lelhood s used to estmate the parameters of the ew ad related dstrbutos. The flexblty ad usefuless of the ew model are llustrated by meas of a applcato to real data set. Keywords: Quas Ldley dstrbuto, McDoald dstrbuto, Maxmum lelhood estmato, Momet geeratg fucto, Kumaraswamy dstrbuto,beta dstrbuto,ldley dstrbuto Itroducto Recetly, several lfetme dstrbutos have bee used to model ad aalyze lfetme data. The Ldley L dstrbuto was orgally proposed by Ldley 5] the cotext of Bayesa statstcs as a couter example of fducal statstcs. Ths dstrbuto s a mxture of expoetal E ad Legth-based expoetal dstrbutos to llustrate the dfferece betwee fducal ad posteror dstrbutos. Ghtay et al. 2] have dscussed the propertes of ths dstrbuto. They have foud that the Ldley dstrbuto performs better tha expoetal model because of ts tme depedet/creasg hazard rate. Zaerzadeh ad Dolat 25] obtaed a geeralzed Ldley GL dstrbuto ad dscussed ts varous propertes ad applcatos. Nadarajah et al. 2] studed the mathematcal ad statstcal propertes of the geeralzed Ldley dstrbuto. Baouch et al. 2] obtaed a exteded Ldley dstrbuto ad dscussed ts varous propertes. Merovc ad Sharma 8] troduced a ew geeralzato of Ldley dstrbuto called beta-ldley BL dstrbuto. Shaer ad Mshra?] troduced ad studed the mathematcal ad statstcal propertes of the quas Ldley QL dstrbuto where t has the L dstrbuto as a partcular case. The cumulatve dstrbuto fucto cdf of the QL dstrbuto s gve by Gx;α,θ= + θx e θx ad the correspodg QL probablty desty fucto pdf s gve by gx;α,θ= θα+ θx e θx, 2 for x >, θ > ad α >. Elbatal ad Elgarly ] studed statstcal propertes of Kumaraswamy quas Ldley KumQL dstrbuto. The QL dstrbuto reduces to L dstrbuto whe α = θ ad at α =, t reduces to the gamma dstrbuto wth parameters2, θ. The desty fucto of QL model s a mxture of expoetal ad gamma dstrbutos, that s Correspodg author e-mal: rroozegar@yazd.ac.r gx;α,θ= p f x;θ+ p f 2 x;θ, Natural Sceces Publshg Cor.

2 376 R. Roozegar, F. Esfadyar: The McDoald Quas Ldley dstrbuto... wth p = α, where f x;θ = θe θx ad f 2 x;θ = θ 2 xe θx. It s also postvely sewed. The hazard ad mea resdual lfe fuctos of the QL dstrbuto are gve by ad hx;α,θ= mx;α,θ= θα+ θx +α+ θx 2+α+ θx θ+α+ θx, respectvely. The hazard fucto hx; α, θ s a creasg fucto whereas the mea resdual lfe fucto mx; α, θ s a decreasg fucto. I recet years, may authors have proposed dstrbutos whch ca arse as specal submodels wth the McDoald Mc geerated or geeralzed beta GB geerated class of dstrbutos. Alexader et al. ] troduce a class of geeralzed beta-geerated dstrbutos that have three shape parameters the geerator. They cosder eleve dfferet parets: ormal, log-ormal, sewed studet-t, Laplace, expoetal, Webull, Gumbel, Brbaum-Sauders, gamma, Pareto ad logstc dstrbutos. Other geeralzatos are McDoald verted beta dstrbuto by Cordero ad Lemote 5], McDoald gamma dstrbuto by Marcao et al. 6], McDoald ormal dstrbuto by Cordero et al. 6] McDoald expoetated expoetal dstrbuto by Cordero et al. 7], McDoald log-logstc dstrbuto by Tahr et al. 24], McDoald arcse dstrbuto by Cordero ad Lemote 8], McDoald Webull dstrbuto by Cordero et al. 9]. ad McDoald Exteded Webull Dstrbuto by Hashmoto et al. 4]. Oe of the ma reasos to cosder the McDoald geerated dstrbuto s ts ablty of fttg sewed data, 9]. The McDoald geerated famly of destes allows for hgher levels of flexblty of ts tals ad has a lot of applcatos varous felds such as ecoomcs, face, relablty, egeerg, bology ad medce. The ma objectve of ths paper s to costruct ad explore the propertes of the fve-parameter model called the McDoald quas Ldley McQL dstrbuto. Ths dstrbuto exhbts the desrable propertes of creasg, decreasg, upsde-dow bathtub ad bathtub shaped hazard fucto. Ths paper s orgazed as follows. The pdf, cdf ad hazard fucto of the McQL dstrbuto are derved Secto 2. Some specal models of the ew dstrbuto are descrbed ths secto. I Secto 3, we preset useful expasos of cdf ad pdf of the McQL dstrbuto. Some propertes of the cdf, pdf, th momet ad momet geeratg fucto of the McQL dstrbuto are dscussed Secto 4. Moreover, the order statstcs, ther momets ad etropy are vestgated ths secto. Maxmum lelhood estmates MLEs of the model parameters are gve Secto 5. A applcato of the McQL dstrbuto by usg a real data set s performed Secto 6. 2 The McQL model The geeralzed beta dstrbuto of the frst d or McDoald dstrbuto deoted wth the prefx Mc for short was troduced by 7]. McDoald 984. The cdf of the McDoald dstrbuto s gve by for a, b, c>, where Iy;a,b= B ya,b Ba,b = Fx=Ix c ;a/c,b, Ba, b beta fucto rato ad the beta fucto, respectvely. The cdf of McQL model s defed by Fx;a,b,c,α,θ=I <x<, y ta t b dt ad Ba,b= ta t b dt are the complete +α+ θx e θx ] c ;a/c,b, x>, 3 where θ > ad α >. The pdf correspodg to 3 s gve by cθα+ θxe θx fx;a,b,c,α,θ = + θx ] a e θx Ba/c,b + θx ] c ] b e θx. 4 For radom varable X wth desty fucto 4, we wrte X McQLa,b,c,α,θ. I fact, the McQL dstrbuto belogs to the ew class of dstrbutos called the McDoald-geerated dstrbutos wth cdf ad pdf as Fx;a,b,c,φ=IG c x;φ;a/c,b= Ba/c, b G c x;φ t a/c t b dt Natural Sceces Publshg Cor.

3 J. Stat. Appl. Pro. 4, No. 3, / ad c fx;a,b,c,φ= Ba/c,b gx;φga x;φ G c x;φ b, respectvely. The cdf s gve 3 ca also be represeted by where Fx;a,b,c,φ= cgx;φa aba/c,b 2 F a c, b, a c + ;Gx;φa, 5 2F a,b;c;x= Γc ΓaΓb Γa+ jγb+ j Γc+ j x j, x <, 6 j! φ =α,θ ad Gx;φ= + θx e θx s the cdf of QL model. Theorem 2.. Let fx; a, b, c, α, θ be the pdf of McQL dstrbuto gve by 4. The lmtg behavor of fx;a,b,c,α,θ for dfferet values of ts parameters s gve below: cθα. If a=, the lm x + fx;a,b,c,α,θ= B/c,b.. If a>, the lm x + fx;a,b,c,α,θ=.. If a<, the lm x + fx;a,b,c,α,θ=. v. lm x + fx;a,b,c,α,θ=. Proof. It s straghtforwared to show the above from the McQL desty equato 4. The hazard rate fucto also ow as the falure rate fucto ht, whch s a mportat quatty characterzg lfe pheomeo, s defed by ht= ft. The hazard rate fucto hrf of the McQL dstrbuto s gve by Ft hx;a,b,c,α,θ = cθα+ θxe θx Ba/c,b B + θx + θx + θx ] a e θx ] e θx ] c a/c,b ] c ] b e θx. 7 The reversed hazard rate fucto rt s defed by rt= ft. The correspodg reversed hazard rate fucto of Ft the McQL dstrbuto s gve as cθα+ θxe θx rx;a,b,c,α,θ = B + θx e θx ] c a/c,b + θx + θx ] a e θx e θx ] c ] b. 8 Fgure llustrates some of the possble shapes of the desty ad hazard fuctos of the McQL dstrbuto for selected values of the parameters. For stace, these plots show the hazard rate fucto of the ew model s much more flexble tha the beta Ldley BL, quas Ldley QL ad Ldley dstrbutos. The hazard rate fucto ca be bathtub shaped, mootocally creasg or decreasg ad upsde-dow bathtub shaped depedg o the parameter values. The McQL dstrbuto cotas as sub-models the beta quas Ldley BQL, the Kumaraswamy quas Ldley KumQL ], ad McDoald Ldley McL dstrbutos for c =, a = c ad α = θ, respectvely. For c = ad α = θ, the McQL dstrbuto reduces to the beta Ldley BL dstrbuto, 8]. The subject dstrbuto also cludes as specal cases the geeralzed quas Ldley dstrbuto GQL, geeralzed Ldley GL dstrbuto proposed by Nadarajah et al. 2] ad McDoald gamma McG dstrbuto. The classes of dstrbutos that are cluded as specal sub-models of the McQL dstrbuto are dsplayed Fgure 2. If the radom varable X has the McQL dstrbuto, the t has the followg propertes:. The radom varable V = + θx e θx ] c satsfes the beta dstrbuto wth parameters a/c ad b. Therefore, +α+ θx the radom varable T = θx l has the BGE or McE dstrbuto, 3]. Furthermore, the radom varable X = G V follows McQL dstrbuto, where G. s gve 3. Ths result helps us smulatg data from McQL dstrbuto. 2. If a= ad b= +, where ad are postve teger values, the the Fx;a,b,c,α,θ s the cdf of the th order statstc of GQL dstrbuto. Natural Sceces Publshg Cor.

4 378 R. Roozegar, F. Esfadyar: The McDoald Quas Ldley dstrbuto... Desty Desty Desty fx a=.5, b=3., α=.5, θ=.5 c=.2 c=.5 c=. c=5. fx a=2.5, c=.5, α=., θ=2 b=.5 b=.2 b=.5 b=. fx a=.5, b=.5, c=.5, α= ~theta=.2 ~theta=.3 ~theta=.5 ~theta= x x x Hazard Hazard Hazard hx b=, c=, α=.5, θ=.3 a=.5 a=. a=. a=3. hx a=.5, b=.5, c=.5, α=. theta=.5 theta=.5 theta=5. theta=. hx a=2, b=.5, c=, α=.5 theta=.5 theta=.3 theta=.55 theta= x Fg. : The pdf ad hrf of McQL model x for some values of parameters. x Fg. 2: Relatoshps of the McQL sub-models Natural Sceces Publshg Cor.

5 J. Stat. Appl. Pro. 4, No. 3, / Expaso of the model I ths secto we derve some represetatos of cdf ad pdf of McQL dstbuto. The bomal seres expaso s defed by z m = j j z j = m j Γm+ z j Γm j+ j!, 9 where z < ad m s a postve real o-teger. The followg proposto reveals that the McQL dstrbuto ca be expressed as a mxture of dstrbuto fucto of GQL dstrbuto, whereas Proposto 2 provdes a useful expaso for the pdf 4. Proposto.The cdf 3 s a mxture of GQL dstrbutos o the form Fx;a,b,c,α,θ= q j G j x, where q j = j Γb Ba/c,bΓb j j!a/c+ j, q j = ad G j x = Gx;α,θ a+ jc s the dstrbuto fucto of a radom varable whch has a GQL dstrbuto wth parameters α, θ ad a + jc. If a s a real o-teger, we ca expad G j x as follows: G j x = Gx;α,θ a+ jc a+ jc = Gx;α,θ] wth so that = = a+ jc Now, equato 3 becomes where If b> s a teger, the Gx;α,θ, Gx;α,θ = r= r G r x;α,θ, r G j x= = r= a+ jc r+ G r x;α,θ. 2 r Fx;a,b,c,α,θ= b j,r = = r= b j,r G r x;α,θ, q j r+ a+ jc b Fx;a,b,c,α,θ=. r q j G j x. Proposto 2.The pdf of McQL model ca be expressed as a fte mxture of GQL destes wth parameters α, θ ad a+ jc gve by where g j x=a+ jcgx;α,θgx;α,θ] a+ jc. fx;a,b,c,α,θ= Smlarly, f b> s a teger, the pdf of McQL model s gve by b fx;a,b,c,α,θ= q j g j x, 3 q j g j x. Natural Sceces Publshg Cor.

6 38 R. Roozegar, F. Esfadyar: The McDoald Quas Ldley dstrbuto... From equatos 3 ad 2, the McQL desty ca be wrtte the form fx;a,b,c,α,θ=gx;α,θ = r= p, j,r G r x;α,θ. where ad, r= p, j,r =. p, j,r = c j+r+ Γb Ba/c,b j!γb j a+ jc r 4 Statstcal propertes I ths secto, we deal wth the basc statstcal propertes of the McQL dstrbuto, partcular, momets ad momet geeratg fucto. 4. Momets ad momet geeratg fucto I ths subsecto we derve the th o-cetral momet ad momet geeratg fucto for the McQL dstrbuto. Momets are ecessary ad mportat ay statstcal aalyss, especally applcatos. Proposto 3.The th momet, EX, of the McQL dstrbuted radom varable X, s gve as ] αγ++ µ X=EX θγ++2 =w, j,r + θr+ ++ θr+ ++2, where w, j,r = j,r= r = c θ b c j+ a Ba/c, b + + j r Proposto 4.If X has the McQL dstrbuto the the momet geeratg fucto mgf of X s gve as follows ] αγ+ M X t=w, j,r θr+ t + + θγ+2 θr+ t r. 4.2 Order Statstcs Order statstcs mae ther appearace may areas of statstcal theory ad practce. Let the radom varable X : be the th order statstc X : X 2: X : a sample of sze from the McQL dstrbuto. The pdf ad cdf of X : for,2,..., are gve by f : x = B, + fxfx] Fx] = B, + fxfx] +, 5 ad F : x= x = f : tdt = B, + = + Fx] +, 6 respectvely, where Fx=Fx;α,θ= b j,rg r x;α,θ. We use throughout a equato by Gradshtey ad Ryzh 27, page 7 for a power seres rased to a postve teger m gve by r= b r u r m = r= c m,r u r, 7 Natural Sceces Publshg Cor.

7 J. Stat. Appl. Pro. 4, No. 3, / 38 where the coeffcets c m,r for r=,2,... are easly determed from the recurrece equato c m,r =r b r = m+ r+]b c m,r, where c m, = b m. Hece, the coeffcets c m,r ca be calculated from c m,,...,c m,r ad therefore, from the quattes b,...,b r. Usg 7, the equatos 5 ad 6 ca be wrtte as r f : x;α,θ= B, + = r= + c +,r gx;α,θgx;α,θ] r, F : x;α,θ= B, + + c +,r Gx;α,θ] r. = r= A explct expresso for the sth momets of X : ca be obtaed as EX: s ] = r + B, + = r= + c +,r t s gt;α,θgt;α,θ] r dt = θγs+ r B, + = r= + c +,r r 2 + θ s α r Γs s = 2 = 3 = Etropy The etropy of a radom varable s defed terms of ts probablty dstrbuto ad ca be show to be a good measure of radomess or ucertaty. The Réy etropy s a exteso of Shao etropy. The Réy etropy s defed to be I R γ= γ R log fx] γ dx, 9 where γ > ad γ. The Réy etropy teds to the Shao etropy as γ. By usg the pdf of McQL model, we have + f γ cθα γ x;a,b,c,α,θdx = Ba/c,b Settg Gx;α,θ] aγ γ = ad = = G c x;α,θ] bγ γ = + + θ α xγ e θγx Gx;α,θ] aγ γ G c x;α,θ] bγ γ dx. ] +θx e θx aγ γ aγ γ α j + θ α x j j +α e θx 2 m l= m= = bγ γ l cl m m l+m α + θ α x +α m e θmx. 2 Natural Sceces Publshg Cor.

8 382 R. Roozegar, F. Esfadyar: The McDoald Quas Ldley dstrbuto... By usg 2 ad 2, we obta + f γ cθα γ m aγ γ x;a,b,c,α,θdx = Ba/c,b = l= m= = j bγ γ cl m +l+m α + j + l m +α +m + θ α xγ+ j+ e θγ++mx dx cθα γ m bγ γ = Ba/c,b,m,l,u= = j l cl m γ+ j+ +l+m α + j u Γu+ l m u θ+α +m γ+ +m u+. 22 Therefore, the Réy etropy for McQL dstrbuto s obtaed by above relato ad 9. The Shao etropy for the McQL dstrbuto s defed as follows: H SH f= E f log fx]= fxlog fxdx. Hece, the Shao etropy for the McQL dstrbuto ca be expressed as Ba/c,b H Sh f = log Elog+ θ cθα α X] + θex+ aeloggx;α,θ]+ belog G c x;α,θ]. We ote that ad Elog G c X;α,θ]= ElogGX;α,θ]= = l= = l EGc x;α,θ] Therefore by usg the results Lemma Nadarajah et al. 2, we have H Sh f = log where Ba/c,b cθα + = θ α EX + θex+ aeloggx;α,θ]+ belog G c x;α,θ], c ElogGx;α,θ] = Ba/c, b ad j = l=, r= l a/c+ b j + j+r θ l+2 +α l+ Ar+c,α,l,α+ Elog G c c x;α,θ] = Ba/c, b ad j =, r= l + j+r θ 2 Ar++c,α,,α, +α Ar,s,t,u= m= m = + p= θ l +α l EX l e θx ]. j r a/c+ b j r m m + p j r m s Γt+p+ +s m sm+u t+p+. Natural Sceces Publshg Cor.

9 J. Stat. Appl. Pro. 4, No. 3, / Estmato Let X,...,X be a radom sample of sze from the McQLa,b,c,α,θ dstrbuto ad Θ = a,b,c,α,θ be the uow parameter vector. The log-lelhood fucto s gve by lθ = logcθ logba/c,b log+ +a +b log log ] logα+ θx θ x ] c ]. 23 The maxmum lelhood estmato MLE of Θ s obtaed by solvg the olear equatos, UΘ=U a Θ,U b Θ,U c Θ,U α Θ,U θ Θ T =, where U a Θ = lθ a U b Θ = lθ b U c Θ = lθ c U α Θ = lθ α b = /cψb+a/c+ψa/c]+ = ψb+a/c ψb+ log log = /c 2 c aψb+a/c+aψa/c]+b = + +a α+ θx c θx 2 ] c ], 24 ] c, 25 θx 2 log + θx ], 26 ] c, 27 U θ Θ = lθ = θ θ x + x α+ θx ] c x c +b x ] c x +a x, 28 where ψ.= Γ. Γ. deotes the dgamma fucto. We eed the observed formato matrx for terval estmato ad hypotheses tests o the model parameters. The 5 5 Fsher formato matrx, J = J Θ, s gve by J aa J ab J ac J aα J aθ J ba J bb J bc J bα J bθ J = J ca J cb J cc J cα J cθ J αa J αb J αc J αα J, αθ J θa J θb J θc J θα J θθ where the expressos for the elemets of J are the appedx. Uder codtos that are fulflled for parameters the teror of the parameter space but ot o the boudary, asymptotcally ˆΘ Θ N 5,IΘ, Natural Sceces Publshg Cor.

10 384 R. Roozegar, F. Esfadyar: The McDoald Quas Ldley dstrbuto... where IΘ s the expected formato matrx. Ths asymptotc behavor s vald f IΘ replaced by J ˆΘ,.e., the observed formato matrx evaluated at ˆΘ Cox ad Hley, Applcato of McQL to a real data set I ths secto, we ft the McQL dstrbuto to a real data set ad compare t wth some models ad submodels such as: the McDoald Dagum McD by Rajasoorya 2], the McDoald Webull McW ad the McDoald log-logstc McLL dstrbutos ad the KumQL, KumG, BQL, QL ad L dstrbutos to show the superorty of the McQL dstrbuto. The data set s gve by Suprawhardaa ad Prayoto 23], that refers to the tme betwee falures thousads of hours of secodary reactor pumps. The data set cossts of 23 observatos. The MLEs of the parameters, -2log-lelhood, AIC Aae Iformato Crtero, BIC Bayesa Iformato Crtero, AICC Cosstet Aae Iformato Crtero, the KS statstc wth ts p-value ad LRT statstc for ths data set are dsplayed Table. From the values of these statstcs, we coclude that the McQL dstrbuto provdes a better ft to ths data tha the McD, McW, McLL, KumQL, BQL, QL ad L dstrbutos. Moreover, the plots of emprcal cdf of the data set ad estmated cdf of seve models are dsplayed Fgure 3. These plots suggest that the McQL dstrbuto s superor to the other dstrbutos terms of model fttg. Table : MLEs of the model parameters for the tme betwee falures data, the correspodg AIC, AICC, BIC, KS ad LRT statstcs. Dst. MLE -2 Log L AIC AICC BIC KS p-value LRT p-value McQLa,b,c,α, θ â =.5594, ˆb =.93,ĉ = 22.84, ˆα = 3.865, ˆθ = 3.56 McDa,b,c,α, θ,δ â = , ˆb = ,ĉ = 6.843, ˆα =.86, ˆθ = 3.326, ˆδ =.362 McWa,b,c,α, θ â = 46.33, ˆb = ,ĉ =.34, ˆα =.288, ˆθ =.95 McLLa,b,c,α, θ â = 2.323, ˆb = 7.38,ĉ = , ˆα =.3, ˆθ =. KumQLa,b,a,α, θ â =.7982, ˆb = 8.296,ĉ =.7982, ˆα = 6.428, ˆθ =.95 BQLa,b,,α, θ â =.7456, ˆb = 8.368,ĉ =, ˆα = 3.228, ˆθ =.642 Desty MQL BQL KumQL KumG QL BL L CDF MQL BQL KumQL KumG QL BL L Tme betwee falures Tme betwee falures Fg. 3: Plots of the estmated pdfs ad cdfs of KumQL, KumG, BQL, QL,BL ad L models usg the tme betwee falures data. Natural Sceces Publshg Cor.

11 J. Stat. Appl. Pro. 4, No. 3, / Acowledgemet The authors are grateful to the Edtor ad aoymous referees for a careful checg of the detals ad for helpful commets that mproved ths paper. Appedx We ca compute the elemets of the observed formato matrx J for the fve parameters a,b,c,α,θ. We obta the followg: J aa = 2 lθ ψ b+a/c ψ a/c a 2 = c 2, J ab = 2 lθ a b J ac = 2 lθ a c J aα = 2 lθ = ψ b+a/c c, = cψb+a/c+cψa/c+a ψ b+a/c+ψ a/c c 3, a α = θx e θx, 2 J aθ = 2 lθ x e θ x a θ = + xe θx, J bb = 2 lθ b 2 J bc = 2 lθ b c J bα = 2 lθ b α = J bθ = 2 lθ b θ = = ψ b+ψ b+a/c, = aψ b+a/c c 2 + log cθx 2, ce θx x e θ x x, ], ] J cc = 2 lθ c 2 + a 2cψb+a/c+ 2cψa/c+ a ψ b+a/c+ ψ a/c c 2 = c 4, J cα = 2 lθ c α =b θx, + 2 θx J cθ = 2 lθ c θ =b e θx x e θ x x ], J αα = 2 lθ α 2 = +α 2 + α+ θx cθ +b 2 x 2 +α 4 + e +a 2θx2 θ 2 x 2 +α 4 2θx 2cθx ], Natural Sceces Publshg Cor.

12 386 R. Roozegar, F. Esfadyar: The McDoald Quas Ldley dstrbuto... J αθ = 2 lθ α θ = x α+ θx 2 +b cθx e θx x e θ x x + +α 2 2 ce θx x e θ x + x e θ x 2 2 x e θ x θx e θ x θx +a 2 2 e θx x e θ x + xe θx, 2 2 J θθ = 2 lθ θ 2 = θ 2 + x 2 α+ θx 2 e θ x x +α +a + e θx x + θx 2 2e θ x x x ce θx x x e θ x +b x 2 ce θx x x e θ x x ce θx 2x e θ x + x 2 ] + +. Refereces ] C. Alexader, G.M. Cordero, E.M.M. Ortega ad J.M. Saraba, Computatoal Statstcs ad Data Aalyss 56, ] H.S. Baouch, B.M. Al-Zahra, A.A. Al-Shomra, V.A. March ad F. Louzada, Joural of the Korea Statstcal Socety 4, ] W. Barreto-Souza, A. H.S. Satos ad G.M. Cordero, Joural of Statstcal Computato ad Smulato 8, ] G.M. Cordero ad M.D. Castro, Joural of Statstcal Computato ad Smulato 8, ] G.M. Cordero ad A.J. Lemote, Joural of the Fral Isttute 349, ] G.M. Cordero, R.J. Ctra, L.C. Rêgo ad E.M. Ortega, Pasta Joural of Statstcs ad Operato Research 8, a. 7] G.M. Cordero, E.M. Hashmoto, E.M. Ortega ad M.A. Pascoa, AStA Advaces Statstcal Aalyss 96, b. 8] G.M. Cordero ad A.J. Lemote, Statstcs 48, ] G.M. Cordero, E.M. Hashmoto ad E.M.M. Ortega, Statstcs 48, ] D. R. Cox ad D. V. Hley, Chapma ad Hall, Lodo, 979. ] I. Elbatal ad M. Elgarhy, Iteratoal Joural of Mathematcs Treds ad Techology, 4, ] M.E. Ghtay, B. Ateh ad S. Nadarajah, Mathematcs ad computers smulato 78, ] I.S. Gradshtey ad I.M. Ryzh, Academc Press, New Yor, 7th edto, 27. 4] E.M. Hashmoto, E.M. Ortega, G.M. Cordero ad M.A. Pascoa, Joural of Statstcal Theory ad Practce 9, ] D.V. Ldley, Joural of the Royal Statstcal Socety, Seres B 2, ] F.W. Marcao, A.D.C. Nascmeto, M. Satos-Neto ad G.M. Cordero, Iteratoal Joural of Statstcs ad Probablty, ] J.B. McDoald, Ecoometrca 52, ] F. Merovc ad V. K. Sharma, Joural of Appled Mathematcs 24. 9] G.S. Mudholar ad H. Wag, Joural of Statstcal Plag ad Iferece 37, ] S. Nadarajah, H.S. Baouch ad R. Tahmasb, Sahya B 73, ] S. Rajasoorya, Electroc Theses ad Dssertatos Paper 45, ] R. Shaer ad A. Mshra, Afrca Joural of Mathematcs ad Computer Scece Research 6, ] M.S. Suprawhardaa ad S. Prayoto, Atom Idoes 25, ] M. Tahr, M. Masoor, M. Zubar ad G. Hameda, Joural of Statstcal Theory ad Applcatos 3, ] H. Zaerzadeh, A. Dolat, Joural of Mathematcal Exteso 3, Natural Sceces Publshg Cor.