Exponentiated Generalized. Transformed-Transformer. Family of Distributions

Transcription

1 Journl of Sttistil n Eonometri Methos, vol.6, no.4, 2017, 1-17 ISSN: print), online) Sienpress Lt, 2017 Exponentite Generlize Trnsforme-Trnsformer Fmily of Distributions Sulemn Nsiru 1, Peter N. Mwit 2 n Osr Nges 3 Abstrt Reently, the evelopment of generlize lss of istributions hs beome n issue of interest, to both pplie n theoretil sttistiins, ue to their wier pplition in ifferent fiels of stuies. Thus, the urrent work propose new generlize fmily of istributions lle the exponentite generlize trnsforme-trnsformer fmily. Some members of the new fmily suh s the exponentite generlize hlf logisti fmily ws isusse. Sttistil mesures suh s quntile, moment, moment generting funtion n Shnnon entropy for this new lss of istributions hve been erive. Mthemtis Subjet Clssifition: 62E15; 60E05 Keywors: Exponentite generlize T-X ; quntile; moment; moment generting funtion; Shnnon entropy 1 Pn Afrin University, Institute for Bsi Sienes, Nirobi, Keny. E-mil: sulemnstt@gmil.om/snsiru@us.eu.gh 2 Mhkos University, Deprtment of Mthemtis, Mhkos, Keny. E-mil: petermwit@mksu..ke 3 Tit Tvet University, Mthemtis n Informtis Deprtment. E-mil: osnges@ttu..ke Artile Info: Reeive : June 12, Revise : July 14, Publishe online : Deember 1, 2017

2 2 Exponentite Generlize T-X 1 Introution Myri of problems rise in ifferent fiel of stuies suh s engineering, turil siene, environmentl, biologil stuies, emogrphy, eonomis n finne tht requires moeling using suitble probbility istribution moels. However, the t generting proess is often hrterize with the problems of elongtion n symmetry, whih mkes it iffiult for the lssil istributions to provie equte fit to the rel t. In ition, the t sets my exhibit non-monotoni filure rte suh s the bthtub, unimol n moifie unimol filure rte. Hene, it is often neessry to utilize generl moel tht is likely to inlue moel suitble for the t s speil se. These hve motivte both theoretil n pplie sttistiins to evelop genertors for moifying existing sttistil istributions to mke them more flexible in moeling rel t. For this reson, reserhers in the fiel of istribution theory hve evelope n stuie mny generlize lsses of istributions. Coreiro et l. 4] evelope the exponentite generlize lss of istributions. Given rnom vrible X with umultive istribution funtion CDF) F x), the CDF of the exponentite generlize lss of istributions is efine s Gx) = 1 1 F x)) α ] β. 1) Alztreh et l. 1] reently propose new fmily of istributions lle the trnsforme-trnsformer T-X ) fmily. They use non-negtive ontinuous rnom vrible T s genertor n efine the CDF of their lss of istribution s Gx) = log1 F x)) 0 rt)t = R { log1 F x)), 2) where rt) is the probbility ensity funtion PDF) of the rnom vrible T. Alztreh et l. 1] T-X fmily of istribution extens the bet-generte fmily of 7] by repling the bet rnom vrible with ny non-negtive ontinuous rnom vrible T. The orresponing PDF of the CDF efine in eqution 2) is given by gx) = fx) r { log1 F x)). 3) 1 F x)

3 S. Nsiru, P.N. Mwit n O. Nges 3 Alzghl et l. 2] propose n extension of the T-X fmily by introuing single shpe prmeter to mke the fmily of istributions efine by 1] more flexible. Alzghl et l. 2] lle this new fmily the exponentite T-X fmily. The CDF of the exponentite T-X fmily is efine s Gx) = log1 F x)) The orresponing PDF is given by 0 rt)t = R { log1 F x)). 4) gx) = fx)f 1 x) r { log1 F x)), > 0. 5) 1 F x) It is obvious tht the upper limits use in the T-X fmily n the exponentite T-X fmily re umultive hzr funtions of ertin fmilies of istributions. Thus, new fmilies of the T-X istributions n be efine by employing new umultive hzr funtion s n upper limit. In this stuy, new T-X fmily lle the exponentite generlize EG) T-X fmily is propose by using new upper limit tht generlizes tht of 1] n 2] to provie greter flexibility in moeling rel t. 2 The New Fmily Let rt) n Rt) be the PDF n CDF of non-negtive rnom vrible T with support 0, ) respetively. The CDF of the EG T-X fmily of istributions for rnom vrible X is efine s Gx) = log1 1 F x)) ] 0 rt)t = R { log1 1 F x)) ], 6) where F x) = 1 F x) is the survivl funtion of the rnom vrible X n > 0, > 0 re shpe prmeters. The orresponing PDF of the new fmily is obtine by ifferentiting eqution 6) n is given by gx) = fx)1 F x)) 1 1 F x)) F x)) r { log1 1 F x)) ]. 7) Employing similr nming onvention s T-X istribution, eh member of the new fmily of istribution generte from 7) is nme EG T-X istribution.

4 4 Exponentite Generlize T-X When the prmeter = 1, the PDF in 7) reues to the PDF in eqution 5). In ition, when = = 1, the PDF in 7) reues to the PDF in eqution 3). The CDF n PDF of the EG T-X istribution n be written s Gx) = R { log1 1 F x)) ] = RHx)) n gx) = hx)rhx)), where Hx) n hx) re the umultive hzr n hzr funtions of the rnom vrible X with CDF 1 1 F x)) ] respetively. Thus, the EG T-X istribution n be esribe s fmily of istribution rising from weighte hzr funtion. The hzr funtion of the EG T-X fmily is given by τx) = gx) 1 Gx) = fx)1 F x)) 1 1 F { x)) 1 r log1 1 F x)) ] 1 1 F x)) ) 1 R { log1 1 F x)) ] ). Lemm 1. Let T be rnom vrible with PDF rt), then the rnom vrible X = Q X {1 1 1 e ) ] T 1 1, where Q X ) = F 1 ) is the quntile funtion of the rnom vrible X with CDF F x), follows the EG T-X istribution. Proof. Using the ft tht Gx) = R { log1 1 F x)) ] gives the reltionship between the rnom vrible T n X s T = log1 1 F X)) ]. Thus, solving for X yiels X = Q X {1 1 1 e ) ] T ) Lemm 1 mkes it esy to simulte the rnom vrible X by first generting rnom numbers from the istribution of the rnom vrible T n then omputing X = Q X {1 1 1 e ) ] T 1 1, whih hs the CDF Gx). 3 Some Exponentite Generlize Trnsforme- Trnsformer Fmilies The EG T-X fmily n be tegorize into two bro sub-fmilies. One sub-fmily hs the sme T istribution but ifferent X istributions n the other sub-fmily hs ifferent T istributions but the sme X istribution.

5 S. Nsiru, P.N. Mwit n O. Nges 5 Tble 1 isplys ifferent EG T-X istributions with ifferent T istributions but the sme X istribution. Tble 1: EG T-X Fmilies from Different T Distributions Nme Density rt) EG T-X Fmily ensity gx) Exponentil λe λt λfx)1 F x)) 1 1 F x)) F x)) ] 1 λ 1 Gmm t α 1 e t Γα)β α β fx)1 F x)) 1 1 F x)) F x)) β ] 1 1 Γα)β α { log1 1 F x)) ] 1 α Gompertz θe γt e θ γ eγt 1) θfx)1 F x)) 1 1 F x)) exp 1 θ γ {1 1 1 F x)) ] γ+1 {1 1 1 F x)) ] γ) Hlf logisti Lomx 2λe λt 1+e λt ) 2 λk 1+λt) k+1 2λfx)1 F x)) 1 1 F x)) F x)) ] λ 1 {1+1 1 F x)) ] λ 2 λkfx)1 F x)) 1 1 F x)) 1 {1 λ log1 1 F x)) ] k F x)) ] Burr XII Weibull αkt α 1 1+t α ) k+1 α γ t γ αkfx)1 F x)) 1 1 F x)) 1 { log1 1 F x)) ] α F x)) ]{1+ log1 1 F x)) )] α k+1 { ) α 1 e t γ ) α αfx)1 F x)) 1 1 F x)) 1 exp log1 1 F x)) γ ] α γ1 1 F x)) ]{ log1 1 F X)) ] 1 α 3.1 Exponentite Generlize Hlf Logisti Fmily If the rnom vrible T follows the hlf logisti istribution with prmeter λ, then rt) = 2λe λt, t > 0, λ > 0. Using eqution 7), the PDF of the 1+e λt ) 2 exponentite generlize hlf logisti EGHL) fmily is efine s gx) = 2λfx)1 F x)) 1 1 F x)) F x)) ] λ 1 { F x)) ] λ 2. 9)

6 6 Exponentite Generlize T-X Using the CDF of the hlf logisti istribution, Rt) = 1 e λt 1+e λt n eqution 6), the orresponing CDF of the EGHL fmily is given by Gx) = F x) ) ] λ F x) ) ] λ. The EGHL fmily generlizes ll hlf logisti fmilies of 2] exponentite T-X fmily n 1] T-X fmily. If the rnom vrible X follows Fréthet istribution with CDF F x) = e x) b, x > 0, > 0, b > 0, then the CDF of the EGHL-Fréthet istribution EGHLFD) is given by Gx) = { e ) x) b ] λ { e ) x) b ] λ. 10) The orresponing PDF of the EGHLFD is obtine by ifferentiting 10) n is given by gx) = 2 b bλ 1 e ) x) b e ) x) b ] 1 { e ) x) b ] λ 1 { x b+1 e x) {1 b e ) x) b ] λ 2. Some speil ses of the EGHLFD re: 11) 1. When λ = 1, the EGHLFD reues to EG stnrize hlf logisti Fréthet istribution. 2. When b = 1, the EGHLFD reues to EGHL inverse exponentil istribution. 3. When = = 1, the EGHLFD reues to hlf logisti Fréthet istribution. 4. When = = b = 1, the EGHLFD reues to hlf logisti inverse exponentil istribution. The reltionship between the EGHLFD n the uniform, Fréthet n hlf logisti istributions re given by lemm 2.

7 S. Nsiru, P.N. Mwit n O. Nges 7 Lemm If the rnom vrible Y follows the uniform istribution on the intervl 0, 1), then the rnom vrible X = log Y Y hs EGHLFD with prmeters, b,, n λ. λ 1 2. If the rnom vrible Y follows the Fréthet istribution with prmeters n b, then the rnom vrible X = log e x) b 1 + e x) b hs EGHLFD with prmeters, b,, n λ. 1 λ b 3. If the rnom vrible Y follows the hlf logisti istribution with Proof. The results follow iretly from trnsformtion of rnom vribles. prmeter λ, then the rnom vrible { X = log e ) Y 1 hs EGHLFD with prmeters, b,, n λ. ] 1 b,, 1 b, 4 Sttistil Mesures In this setion, we isuss sttistil mesures suh s quntile, moment, moment generting funtion MGF)n Shnnon entropy of the EG T-X fmily of istributions. Lemm 3. The quntile funtion of the EG T-X fmily for p 0, 1) is given by Qp) = Q X {1 1 1 e ) ] Q T p 1, where Q X ) = F 1 ) is the quntile funtion of the rnom vrible X with CDF F x) n Q T ) = R 1 ) is the quntile funtion of the rnom vrible T with CDF Rt).

8 8 Exponentite Generlize T-X Proof. Using the CDF of the EG T-X fmily efine in eqution 6), the quntile funtion is obtine by solving the eqution R { log 1 1 F Qp)) ) ] = p, for Qp). Thus, the proof is omplete. Corollry 1. Bse on lemm 3, the quntile funtion for the EGHL fmily is given by, Qp) = Q X p 1 + p λ. 1 Proposition 1. The r th non-entrl moment of the EG T-X fmily of istributions is given by i µ r = i, k, l, m=0 j=0 1) j+k+l+m l m δ r,i Γ i + 1) Γ j + 1) Γ k + 1) j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1)E T m ), 12) where δ r,i = ih 0 ) 1 i s=1 s r + 1) i] h s δ r,i s with δ r,0 = h r 0, h i i = 0, 1,...) re suitbly hosen rel numbers tht epen on the prmeters of the F x) istribution, ET m ) is the m th moment of the rnom vrible T, Γ ) is the gmm funtion n r = 1, 2,.... Proof. From lemm 1, X = Q X {1 1 1 e ) ] T 1 1, where Q X ) = F 1 ) is quntile funtion. Thus, Q X ) = F 1 ) n be expresse in terms of power series using the following power series expnsion of the quntile. Q X u) = h i u i, 13) where the oeffiients re suitbly hosen rel numbers tht epen on the prmeters of the F x) istribution. For power series rise to positive i=0 integer r for r 1), we hve ) r Q X u)) r = h i u i = i=0 δ r, i u i, 14) i=0

9 S. Nsiru, P.N. Mwit n O. Nges 9 where the oeffiients δ r, i for i = 1, 2,...) re etermine from the reurrene eqution δ r,i = ih 0 ) 1 i s=1 s r + 1) i] h s δ r,i s n δ r,0 = h r 0 5]. Using equtions 13) n 14), the r th non-entrl moment of the EG T-X fmily of istributions n be expresse s { E X r ) = µ r = E δ r, i 1 Sine 0 < expnsion 1 1 e ) ) T z) η = i=0 1 1 e T ] i. 15) < 1, for T 0, ), pplying the binomil series j=0 for rel non-integer η > 0, thrie, we obtin e ) ) T 1 1 ] i i = 1) j Γ η + 1) j! Γ η j + 1) zη, z < 1, k, l=0 j=0 But the series expnsion of e lt is given by e lt 1) m l m T m =. m! 1 Thus 1 1 e T ] i = m=0 k, l m=0 j=0 1) j+k+l Γ i + 1) Γ j + 1) Γ k + 1) e lt j! k! l! Γ i j + 1) Γ j k + 1) Γ k l + 1). Substituting eqution 16) into 15) n simplifying, we obtin µ r = i i, k, l, m=0 j=0 i 1) j+k+l+m Γ i + 1) Γ j + 1) Γ k + 1) l m T m j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1). 16) 1) j+k+l+m l m δ r,i Γ i + 1) Γ j + 1) Γ k + 1) j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1)E T m ). Corollry 2. Bse on proposition 1, the r th moment of the EGHL fmily is given by µ r = i i, k, l, m=0 j=0 where m = 1, 2,.... 1) j+k+l+m l m δ r,i Γ i + 1) Γ j + 1) Γ k + 1) j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1) { 2 n=0 1) n Γ m + 1) λ m n + 1) m,

10 10 Exponentite Generlize T-X by Proposition 2. The MGF of the EG T-X fmily of istributions is given M X z) = i r, i, k, l, m=0 j=0 1) j+k+l+m z r l m δ r,i Γ i + 1) Γ j + 1) Γ k + 1) r! j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1)E T m ). Proof. By efinition the MGF is given by M X z) = E e zx). Using the series expnsion of e zx, gives us 17) M X z) = r=0 z r µ r. 18) r! Substituting µ r into eqution 18), we hve M X z) = i r, i, k, l, m=0 j=0 whih is the MGF. 1) j+k+l+m z r l m δ r,i Γ i + 1) Γ j + 1) Γ k + 1) r! j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1)E T m ), Corollry 3. Bse on proposition 2, the MGF of the EGHL fmily is M X z) = { r, i, k, l, m=0 j=0 2 n=0 i 1) j+k+l+m z r l m δ r,i Γ i + 1) Γ j + 1) Γ k + 1) r! j! k! l! m! Γ i j + 1) Γ j k + 1) Γ k l + 1) 1) n Γ m + 1) λ m n + 1) m. Entropy is mesure of vrition of unertinty of rnom vrible. Entropy hs been use extensively in severl fiels suh s engineering n informtion theory. Aoring to 3], the entropy of rnom vrible X with PDF gx) is given by η X = E {log gx)).

11 S. Nsiru, P.N. Mwit n O. Nges 11 Proposition 3. The Shnnon s entropy for the EG T-X fmily of istributions is given by η X = log ) µ T + η T E 1 { log f F 1 ) E log 1 1 e ) )] T e ) T 1 1 )] + ) E log 1 e T )],19) where µ T n η T re the men n the Shnnon entropy of the rnom vrible T. Proof. By efinition { η X = 1 )E log 1 1 F X)) ] + 1)E log 1 F X))] { E log fx)] log ) + E log F X)) ) ] E { log r log 1 1 F x)) ))]. 20) From Lemm 1, we know tht T = log1 1 F X)) ] n X = F { e ) ] T 1 1. Hene, we hve E log fx)] = E E log 1 F X))] = E { log f F e ) ] T 1 1 )], 21) log 1 1 e ) ) T 1 1 ], 22) { E log 1 1 F X)) ] = E log 1 e ) ] T 1, 23) { ) ] E log F X) 1 = E T ), 24) n E { log r log 1 1 F x)) ))] = E log r T )]. 25) Substituting 21) through 25) into 20) yiels { η X = log ) µ T + η T E log f F ) E log 1 1 e ) )] T e T + ) )] 1 E log 1 e T. )] +

12 12 Exponentite Generlize T-X Substituting the men n Shnnon entropy of the hlf logisti istribution into 19), gives the Shnnon entropy of the EGHL fmily. is η X Corollry 4. From proposition 3, the Shnnon entropy of the EGHL fmily = 2 log 2λ) 2 log2) λ 1 ) E log E 1 1 e ) T 1 { log f F 1 1 )] 1 1 e T + ) )] 1 E log 1 e T. )] + The men of the hlf logisti istribution is µ T entropy is η T = 2 log2λ). = 2 log2) λ n the Shnnon 5 Prmeter Estimtion of Exponentite Generlize Hlf Logisti Fréthet Distribution Here, the estimtion of the prmeters of the EGHLFD ws one using mximum likelihoo estimtion. Let z i = e ) b n z i = 1 e ) b. Let X 1, X 2,..., X n be rnom smple of size n from EGHLFD, then the loglikelihoo funtion for the vetor of prmeters ϑ = λ,,, b, ) is given by l = n log 2 b bλ ) + 1) λ 1) b 1) log z i ) + 1) log 1 1 z i ) ] 2 log ) log1 z i )+ { log z i ) ] λ ) b. 26) Differentiting eqution 26) with respet to the prmeters λ,,, b n, respetively n equting to zero gives n λ + log 1 1 z i ) ] 1 1 z 2 i ) ] λ log 1 1 z i ) ] z i )] = 0, λ 27)

13 S. Nsiru, P.N. Mwit n O. Nges 13 n + log1 z i ) λ 1) n + log z i ) 1) z i ) log1 z i ) 1 1 z i ) + λ1 z i ) 1 1 z i ) ] λ 1 log1 z i ) z i )] λ = 0, z i log z i ) 1 z i b n2 b λ + 2 b bλ log)) 2bλ ) b ) z i log 1) 1) z i λ 1) 2 + λ 1) 28) z i 1 z i ) 1 log z i ) 1 1 z i ) λ z i 1 z i ) z i ) ] λ 1 log zi ) z i )] λ = 0, 29) log ) z i z 1 i ) b ) z i z 1 i 1 z i ) 1 log 1 1 z i ) ) b ) log + ) b log 1 z i ) λz i z 1 i 1 z i ) z i ) ] ) λ 1 b ) log z i )] = 0, 30) λ ) b 1 ) b 1 bn b bz i + 1) 1) x i x i z i ) b 1 bz i z 1 i 1 z i ) 1 + λ 1) 1 1 z i ) ] 2 ) b 1 bz i z 1 i + 1 z i ) ) b 1 bλz i z 1 i 1 z i ) z i ) ] λ 1 { 1 + = 0. 31) 1 1 z i )] λ The mximum likelihoo estimtes of ϑ = λ,,, b, ) sy ˆϑ = ˆλ, ĉ, ˆ, ˆb, â ), re obtine by solving the non-liner equtions 27), 28), 29), 30) n 31) using numeril methos.

14 14 Exponentite Generlize T-X 6 Applition In this setion, the pplition of the EGHLFD istribution is emonstrte using unensore t on 100 observtion on breking stress of rbon fibers in Gb) obtine from 6]. The t re: 0.39, 0.81, 0.85, 0.98, 1.08, 1.12, 1.17, 1.18, 1.22, 1.25, 1.36, 1.41, 1.47, 1.57, 1.57, 1.59, 1.59, 1.61, 1.61, 1.69, 1.69, 1.71, 1.73, 1.80, 1.84, 1.84, 1.87, 1.89, 1.92, 2.00, 2.03, 2.03, 2.05, 2.12, 2.17, 2.17, 2.17, 2.35, 2.38, 2.41, 2.43, 2.48, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.76, 2.77, 2.79, 2.81, 2.81, 2.82, 2.83, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.51, 3.56, 3.60, 3.65, 3.68, 3.68, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90, 4.91, 5.08, The fit of the EGHLFD ws ompre to tht of trnsmute Mrshll-Olkin Fréthet istribution TMOFD) n Mrshll-Olkin Fréthet istribution MOFD) using ifferent gooness-of-fit tests inluing the Akike informtion riterion AIC), orrete Akike informtion riterion AIC), Byesin informtion riterion BIC), mximize log-likelihoo uner the moel 2ˆl), Anerson-Drling A ) n Crmér-Von Mises W ) sttistis. The PDF of the TMOFD n MOFD re given by gx) = αbb x b+1) e x) b α + 1 α)e x) b ] 2 λ 1, x > 0, 2λe 1 x) b + λ α + 1 α)e, > 0, b > 0, α > 0, x) b n α ) b+1 gx) = b e x) x) b α + 1 α)e ] x) b 2, > 0, b > 0, α > 0, x > 0, respetively. The mximum likelihoo estimtes MLEs) of the prmeters of the fitte istributions with their orresponing stnr errors in brket re given in Tble 2. From Tble 3, it ws ler tht the EGHLFD provies better fit to the rbon fibers t ompre to other fitte istributions sine it hs the smllest vlue for ll the gooness-of-fit sttistis.

15 S. Nsiru, P.N. Mwit n O. Nges 15 Tble 2: MLEs of the prmeters with their stnr errors Moel ˆα â ˆb ĉ ˆ ˆλ EGHLFD ) ) ) TMOFD ) 0.068) 0.206) 0.270) MOFD ) 0.489) 0.160) Tble 3: Gooness-of-fit Sttistis Moel 2ˆl AIC AIC BIC W A EGHLFD TMOFD MOFD The plot of the empiril ensity n the ensity of the fitte moels re shown in Figure 1. Figure 1: Plot of empiril ensity n ensity of fitte moels

16 16 Exponentite Generlize T-X 7 Conlusion This stuy proposes the EG T-X fmily whih is n extension of the T-X fmily of 1] n the exponentite T-X fmily of 2] istributions. The new fmily hs severl sub-fmilies s shown in Figure 2. The two extr shpe prmeters n provies greter flexibility for ontrolling skewness, kurtosis n possibly ing entropy to the enter of the EG T-X ensity funtion. Speifi exmple of member of the EG T-X fmily of istribution, nmely EGHLFD ws given n its reltionship with other bseline istributions estblishe. Some sttistil properties of the new fmily suh s the quntile, moment, moment generting funtion, n Shnnon entropy were erive. Figure 2: Fmilies of EG T-X istributions

17 S. Nsiru, P.N. Mwit n O. Nges 17 ACKNOWLEDGEMENTS. The first uthor wishes to thnk the Afrin Union for supporting his reserh t the Pn Afrin University, Institute for Bsi Sienes, Tehnology n Innovtion. Referenes 1] A. Alztreh, C. Lee n F. Fmoye, A new metho for generting fmilies of ontinuous istributions, Metron, 711), 2013), ] A. Alzghl, F. Fmoye n C. Lee, Exponentite T-X fmily of istributions with some pplitions, Interntionl Journl of Sttistis n Probbility, 23), 2013), ] C.E. Shnnon, A mthemtil theory of ommunition, Bell System Tehnil Journl, 27, 2013), ] G.M. Coreiro, E.M.M. Orteg n C.C.D. Cunh, The exponentite generlize lss of istributions, Journl of Dt Siene, 111), 2013), ] I.S. Grshteyn n I.M. Ryzhik, Tbles of integrls, series n prouts, Aemi Press, New York, ] M.D. Nihols n W. J. Pgett, A bootstrp ontrol hrt for Weibull perentiles, Qulity n Relibility Engineering Interntionl, 22, 2006), ] N. Eugene, C. Lee n F. Fmoye, The bet-norml istribution n its pplitions, Communition in Sttistis-Theory n Methos, 314), 2002),