Georga Southern Unversty Dgtal Commons@Georga Southern Mathematcal Scences Faculty Publcatons Mathematcal Scences, Department of 1-215 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Broderck O. Oluyede Georga Southern Unversty, boluyede@georgasouthern.edu Shujao Huang Georga Southern Unversty Tantan Yang Georga Southern Unversty Follow ths and addtonal works at: https://dgtalcommons.georgasouthern.edu/math-sc-facpubs Part of the Mathematcs Commons Recommended Ctaton Oluyede, Broderck O., Shujao Huang, Tantan Yang. 215. "A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons." Austran Journal of Statstcs, 44 (3): 43-68. do: 1.17713/ajs.v443.36 https://dgtalcommons.georgasouthern.edu/math-sc-facpubs/38 Ths artcle s brought to you for free and open access by the Mathematcal Scences, Department of at Dgtal Commons@Georga Southern. It has been accepted for ncluson n Mathematcal Scences Faculty Publcatons by an authorzed admnstrator of Dgtal Commons@Georga Southern. For more nformaton, please contact dgtalcommons@georgasouthern.edu.
Austran Journal of Statstcs October 215, Volume 44, 45 68. AJS http://www.ajs.or.at/ do:1.17713/ajs.v443.36 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Broderck O. Oluyede Georga Southern Unversty Shujao Huang Georga Southern Unversty Tantan Yang Georga Southern Unversty Abstract A new fve parameter gamma-generalzed modfed Webull (GGMW) dstrbuton whch ncludes exponental, Raylegh, Webull, modfed Webull, gamma-modfed Webull, gamma-modfed Raylegh, gamma-modfed exponental, gamma-webull, gamma-raylegh, gamma-lnear falure rate and gamma-exponental dstrbutons as specal cases s proposed and studed. Some mathematcal propertes of the new class of dstrbutons ncludng hazard functon, quantle functon, moments, dstrbuton of the order statstcs and Rény entropy are presented. Maxmum lkelhood estmaton technque s used to estmate the model parameters and applcatons to real datasets n order to llustrate the usefulness of the proposed class of models are presented. Keywords: Gamma dstrbuton, Modfed Webull dstrbuton, Maxmum lkelhood estmaton. 1. Introducton Webull dstrbuton has been wdely used for modelng data n a wde varety of areas ncludng relablty, engneerng, stochastc processes, survval analyss and renewal theory. In ths paper, we present and study the mathematcal propertes of the gamma-generalzed modfed Webull dstrbuton. Ths class of dstrbutons s flexble n accommodatng all forms of hazard rate functons and contans several well known and new sub-models such as Webull, Raylegh, exponental, modfed Webull, gamma-modfed Webull, gamma-modfed exponental, gamma-webull, gamma-raylegh, gamma-lnear falure rate, gamma-extreme value, gamma-addtve exponental and gamma-exponental dstrbutons. There are several extensons of the Webull dstrbuton and ts sub-models ncludng the exponentated Webull (Mudholkar, Srvastava, and Kolla 1996), whch s a specal case of the beta Webull dstrbuton proposed by (Lee, Famoye, and Olumolade 27), generalzed Raylegh (Kundu and Rakab 25), exponentated exponental (Gupta and Kundu 1999), (Gupta and Kundu 21), modfed Webull (Mudholkar, Srvastava, and Fremer 1995), exponentated modfed Webull (Sarhan and Zandn 29), and a host of other dstrbutons, some of whch are presented n secton 2 of ths paper. Addtonal generalzatons of Webull dstrbuton nclude (Famoye, Lee, and Olumolade 25) where the authors dscussed and presented results on the beta-webull dstrbuton. (Nadarajah 25) presented results on the modfed Webull
46 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons dstrbuton. A host of researchers have also developed several parameter Webull, modfed Webull and flexble Webull dstrbutons over the years. The two parameter Webull extensons nclude (Bebbngton, La, and Ztks 27), (Zhang and Xe 211). The three parameter Webull extensons nclude (Marshall and Olkn 1997), (Xe, Tang, and Goh 22), (Nadarajah and Kotz 25). Some of these extensons enable the accommodaton of bathtub shape hazard rate functon. (Carrasco, Ortega, and Cordero 28) generalzed the modfed Webull dstrbuton of (La, Moore, and Xe 1998) to obtan the exponentated modfed Webull dstrbuton. The four parameter generalzatons nclude the addtve Webull dstrbuton of (Xe and La 1995), modfed Webull (Sarhan and Zandn 29), beta-webull proposed by (Famoye et al. 25) and Kumaraswamy Webull by (Cordero, Ortega, and Nadarajah 21). The fve parameter modfed Webull dstrbuton nclude those ntroduced by (Phan 1987), beta modfed Webull by (Slva, Ortega, and Cordero 21) and (Nadarajah, Cordero, and Ortega 211). Addtonal results on the generalzaton of the Webull dstrbuton nclude work by (Sngha, Jan, and Kumar 212), as well as (Almalk and Yuan 213) where results on a new modfed Webull dstrbuton was presented. (Barlow and Campo 1975) dscussed total tme on test processes wth applcaton to falure data analyss. (Choudhury 25) presented moments of the exponentated Webull dstrbuton. The exponentated Webull dstrbuton was also studed by (Nassar and Essa 23). (Haupt and Schabe 1992) presented a model for bathtub shaped falure rate functon. (Hjorth 198) studed a relablty functon wth ncreasng, decreasng and bathtub shaped falure rate functons, and (Rajarsh and Rajarsh 1988) gave a comprehensve revew of bathtub shaped dstrbutons. For any contnuous baselne cdf F (x), and x R, (Zografos and Balakrshnan 29) defned the dstrbuton (when ψ 1 n equaton (1)) wth pdf g(x) and cdf G(x) (for δ > ) as follows: 1 g(x) ψ δ [ log(f (x))]δ 1 (1 F (x)) 1/ψ 1 f(x), (1) and G(x) 1 ψ δ log(f (x)) t δ 1 e t/ψ dt γ(δ, ψ 1 log(f (x))), (2) respectvely, where g(x) dg(x)/dx, t δ 1 e t dt s the gamma functon, and γ(z, δ) z tδ 1 e t dt s the ncomplete gamma functon. The correspondng hazard rate functon (hrf) s h G (x) [ log(1 F (x))]δ 1 f(x)(1 F (x)) 1/ψ 1 ψ δ ( γ( ψ 1. (3) log(1 F (x)), δ)) When ψ 1, ths dstrbuton s referred to as the ZB-G famly of dstrbutons. Also, (when ψ 1), (Rstć and Balakrshnan 211) proposed an alternatve gamma-generator defned by the cdf and pdf and G 2 (x) 1 g 2 (x) 1 ψ δ log F (x) t δ 1 e t/ψ dt, x R, δ >, (4) 1 ψ δ [ log(f (x))]δ 1 (F (x)) 1/ψ 1 f(x), (5) respectvely. Note that f ψ 1 and δ n + 1, n equatons (1) and (2), we obtan the cdf and pdf of the upper record values U gven by G U (u) 1 n! log(1 F (u)) y n e y dy, (6) and g U (u) f(u)[ log(1 F (u))] n /n!. (7)
Austran Journal of Statstcs 47 Smlarly, from equatons (4) and (5), the pdf of the lower record values T s gven by g L (t) f(t)[ log(f (t))] n /n!. (8) In ths paper, we wll consder and present a generalzaton of the generalzed modfed Webull dstrbuton va the famly of dstrbutons gven n equaton (5). (Zografos and Balakrshnan 29) motvated the ZB-G model as follows. Let X (1), X (2),..., X (n) be upper record values from a sequence of ndependent and dentcally dstrbuted (..d.) random varables from a populaton wth pdf f(x). Then, the pdf of the n th upper record value s gven by equaton (1) when ψ 1. A logarthmc transformaton of the parent dstrbuton F transforms the random varable X wth densty (1) to a gamma dstrbuton. That s, f X has the densty (1), then the random varable Y log[1 F (X)] has a gamma dstrbuton GAM(δ; 1) wth densty k(y; δ) 1 yδ 1 e y, y >. The opposte s also true, f Y has a gamma GAM(δ; 1) dstrbuton, then the random varable X G 1 (1 e Y ) has a ZB-G dstrbuton. In addton to the motvatons provded by (Zografos and Balakrshnan 29), we are nterested n the generalzaton of the generalzed modfed Webull dstrbuton va the gamma-generator and establshng the relatonshp between weghted dstrbutons and equatons (1) and (5), respectvely. Weghted dstrbutons apples to a varety of areas and provdes an approach to dealng wth model specfcaton and data nterpretaton problems. It adjusts the probabltes of actual occurrence of events to arrve at a specfcaton of the probabltes when those events are recorded. (Fsher 1934) ntroduced the concept of weghted dstrbuton, n order to study the effect of ascertanment upon estmaton of frequences. (Patl and Rao 1978) used weghted dstrbuton as stochastc models n the study of harvestng and predaton. (Rao 1965) unfed concept of weghted dstrbuton and use t to dentfy varous samplng stuatons. The usefulness and applcatons of weghted dstrbuton to based samples n varous areas ncludng medcne, ecology, relablty, and branchng processes can also be seen n (Nanda and Jan 1999), (Gupta and Keatng 1985), (Oluyede 1999) and n references theren. Let Y be a non-negatve random varable wth ts natural pdf f(y; θ), where θ s a vector of parameters, then the pdf of the weghted random varable Y w s gven by: f w w(y, β)f(y; θ) (y; θ, β), (9) ω where the weght functon w(y, β) s a non-negatve functon, that may depend on the vector of parameters β, and < ω E(w(Y, β)) < s a normalzng constant. In general, consder the weght functon w(y) defned as follows: w(y) y k e ly F (y)f j (y). (1) Settng k ; k j ; l j ; k l ; 1; j n ; k l and k l j n ths weght functon, one at a tme, mples probablty weghted moments, moment-generatng functons, moments, order statstcs, proportonal hazards and proportonal reversed hazards, respectvely, where F (y) P (Y y) and F (y) 1 F (y). If w(y) y, then Y Y w s called the sze-based verson of Y. (Rstć and Balakrshnan 211) provded motvatons for the famly of dstrbutons gven n equaton (4) when ψ 1, that s for n N, equaton (4) s the pdf of the n th lower record value of a sequence of..d. varables from a populaton wth densty f(x). (Rstć and Balakrshnan 211) used the exponentated exponental (EE) dstrbuton wth cdf F (x) (1 e βx ) α, where α > and β >, to obtaned and study the gamma-exponentated exponental (GEE) model. See references theren for addtonal results on the GEE model. In ths note, we obtan a natural extenson of the generalzed modfed Webull dstrbuton, whch we refer to as gamma-generalzed modfed Webull (GGMW) dstrbuton. In secton 2, some basc results, the gamma-generalzed modfed Webull (GGMW) dstrbuton, seres expanson and ts sub-models, quantle functon, hazard and reverse hazard
48 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons functons are presented. Moments and moment generatng functon are gven n secton 3. Secton 4 contans some addtonal useful results on the dstrbuton of order statstcs and Rény entropy. In secton 5, results on the estmaton of the parameters of the GGMW dstrbuton va the method of maxmum lkelhood are presented. Applcatons are gven n secton 6, and concludng remarks n secton 7. 2. GGMW dstrbuton, seres expanson and sub-models In ths secton, the GGMW dstrbuton and some of ts sub-models are presented. Frst consder the generalzed modfed Webull (GMW) dstrbuton (Sarhan and Zandn 29) gven by F GMW (x, α, β, θ, λ) 1 exp( αx βx θ e λx ), x, α, β, θ, λ. (11) We note that n (Sarhan and Zandn 29) paper, the parameter λ was taken to be zero. The parameters α and β control the scale of the dstrbuton, θ controls the shape, whereas λ can be consdered to be an acceleratng factor n the mperfecton tme and a factor of fraglty n the survval of the ndvdual as tme ncreases. By nsertng the GMW dstrbuton n equaton (4), the survval functon G GGMW (x) 1 G GGMW (x) of the GGMW dstrbuton s obtaned as follows: G GGMW (x) 1 ψ δ log(1 e αx βx θ e λx ) t δ 1 e t/ψ dt γ( ψ 1 log(1 e αx βxθ e λx ), δ), (12) where x >, α, β, θ, λ, δ >, ψ >, and γ(x, δ) x tδ 1 e t dt s the lower ncomplete gamma functon. The correspondng pdf s gven by g GGMW (x) 1 ψ δ [ log(1 e αx βxθ e λx )] δ 1 (α + βx θ 1 e λx [θ + λx])e αx βxθ e λx [1 e αx βxθ e λx ] (1/ψ) 1. (13) If F (x) [1 e αxη βx θ e λx ] φ, then the correspondng generalzed gamma-generalzed modfed Webull pdf s gven by g GGMW (x) φ ψ δ [ log(1 e αxη βx θ e λx ) φ ] δ 1 (αηx η 1 + βx θ 1 e λx [θ + λx])e αxη βx θ e λx [1 e αxη βx θ e λx ] φ+(1/ψ) 2. (14) In ths note, we take φ η ψ 1. The pdf n equaton (14) s now gven by g GGMW (x) 1 [ log(1 e αx βxθ e λx )] δ 1 (α + βx θ 1 e λx [θ + λx])e αx βxθ e λx. (15) If a random varable X has the GGMW densty gven n equaton (15), we wrte X GGMW (α, β, θ, λ, δ). The parameter δ s an extra shape parameter n the GGMW dstrbuton. Let y e αx βxθ e λx, < y < 1, α, β, θ, δ >, and λ, then usng the seres representaton log(1 y) y +1 +1, we have [ ] δ 1 [ log(1 y) y δ 1 m ( δ 1 m ) ( y m y s ) m ]. s + 2 s
Austran Journal of Statstcs 49 Applyng the result on power seres rased to a postve nteger, wth a s (s + 2) 1, that s, ( ) m a s y s s b s,m y s, (16) where b s,m (sa ) 1 s l1 [m(l + 1) s]a lb s l,m, and b,m a m, (Gradshteyn and Ryzhk 2), the GGMW pdf can be wrtten as g GGMW (x) 1 1 m s m s ( δ 1 m ( δ 1 m s ) b s,m y m+s+δ (α + βx θ 1 e λx [θ + λx]) ) b s,m e α(m+s+δ)x β(m+s+δ)xθ e λx m + s + δ m + s + δ (α + βxθ 1 e λx [θ + λx]) ( ) δ 1 b s,m m (m + s + δ) g (x; α(m + s + δ), β(m + s + δ), θ, λ), m s where g (x; α(m + s + δ), β(m + s + δ), θ, λ) s the generalzed modfed Webull pdf wth parameters α(m + s + δ) >, β(m + s + δ) >, θ >, and λ. Let C {(m, s) Z 2 +}, then the weghts n the GGMW pdf above are ( ) δ 1 b s,m w ν m (m + s + δ), and g GGMW (x) ν C w ν g (x; α(m + s + δ), β(m + s + δ), θ, λ), (17) for x >, δ >, α(m + s + δ), β(m + s + δ), θ >, and λ. It follows therefore that the GGMW densty s lnear combnaton of the generalzed modfed Webull (GMW) denstes. The statstcal and mathematcal propertes of the GGMW dstrbuton can be readly obtaned from those of the generalzed modfed Webull dstrbuton. For the convergence of equatons (16) and (17), as well as elsewhere n ths paper, note that for δ >, so that [ 1 + y [ log(1 y)] δ 1 k s convergent f and only f < y k ] δ 1 k + 2 [ ( y 1 + y ( δ 1 k k s y s )] δ 1 s + 2 ) y k ( s y s ) k s + 2 ( y k y k k k+2) < 1 y (, 1), snce < y e αx βxθ e λx < 1, for x >, α, β, θ >, and λ. Now, y y k k k+2 log(1 y) y 1, so we must have < log(1 y) y 1 < 1. Ths leads to 1 y > exp( 2y), and on the other hand exp( y) ( 1) k y k k k! > 1 y. Thus, we have the system of nequaltes 1 y > exp( 2y) and exp( y) > 1 y, whch s satsfed y (,.7968). The mplcaton here s that the nequalty ( < y k y k k k+2) < 1 s not vald for all values of < y e αx βxθ e λx < 1, and equatons (16) and (17), and elsewhere n ths paper are convergent only y (,.7968). The seres n equatons (16) and (17), and elsewhere n ths paper are not vald for all values of <
5 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Fgure 1: Graphs of GGMW pdf y e αx βxθ e λx < 1, but are convergent y (,.7968), and not vald (convergent) for y >.7986. Note that n general, g GGMW (x) s a weghted pdf wth the weght functon that s, w(x) [ log(1 F (x))] δ 1 [1 F (x)] 1 ψ 1, (18) g GGMW (x) [ log(1 F (x))]δ 1 [1 F (x)] 1 ψ 1 ψ δ f(x) w(x)f(x) E F (w(x)), (19) where < E F {[ log(1 F (x))] δ 1 [1 F (x)] 1 ψ 1 } ψ δ <, s the normalzng constant. Graphs of the pdf of GGMW dstrbuton are gven n the Fgure 1 for selected values of the parameters. The plots show that the GGMW pdf can be decreasng or rght skewed among several other possble shapes as seen n Fgure 1. The dstrbuton has postve asymmetry. 2.1. Quantle functon The quantle functon of the GGMW dstrbuton s gven by the soluton of the nonlnear equaton γ( log[1 e αx βxθ e λx ], δ) 1 u. (2) That s, log[1 e αx βxθ e λx ] γ 1 ((1 u), δ) and αx + βx θ e λx log(1 exp( γ 1 ((1 u), δ))). (21) We can smulate from the GGMW by solvng the nonlnear equaton αx + βx θ e λx + log(1 exp( γ 1 ((1 u), δ))), (22) where u s a unformly dstrbuted random varable on the nterval [, 1]. The nverse ncomplete gamma functon can be mplemented by usng numercal methods. Consequently, random numbers can be generated based the equaton above. Table 1 lsts the quantle for selected parameter values of the GGMW dstrbuton.
Austran Journal of Statstcs 51 Table 1: GGMW quantle for selected values (α, β, θ, λ, δ) u (1,1,1,1,1) (2,1,2,1,1) (6,4,3,6,1) (5,3,3,5,6) (.1,.3,.4,.2,.3).1.5132855.512954.175568.1875 1.26742.2.156817.153998.3714788.7372 2.594722.3.1637671.1627524.5924798.18145 3.826921.4.226598.224198.844759.3758 4.95229.5.295724.29269.11359275.6965 6.1953.6.3735554.3632644.14752769.123419 7.7161.7.464656.4463389.18721167.21965 8.161678.8.578369.5466338.23367423.47477 9.3737.9.742499.685397.2944828.87428 1.926297 2.2. Some sub-models of the GGMW dstrbuton The proposed model has several new and well known sub-models. Some of the sub-models of the GGMW dstrbuton are lsted n Table 2. They nclude the gamma-generalzed modfed Raylegh (GGMR), gamma-generalzed modfed exponental (GGME), gamma-modfed Webull (GMW), gamma-modfed exponental (GME), gamma-addtve exponental (GAE), gamma-extreme value (GEV), gamma-webull (GW), modfed Webull (MW), Sardn and Zandn modfed Webull (S-ZMW), modfed Raylegh (MR), modfed exponental (ME), gamma-lnear falure rate (GLFR), lnear falure rate (LFR), extreme value (EV), Webull (W) and exponental (E) dstrbutons. 2.3. Hazard and reverse hazard functons In ths secton, we present the hazard and reverse hazard functons, as well as graphs of the hazard functon for selected values of the model parameters. Let X be a contnuous random varable wth dstrbuton functon G, and probablty densty functon (pdf) g, then the hazard functon, reverse hazard functon and mean resdual lfe functons are gven by h G (x) g(x)/g(x), τ G (x) g(x)/g(x), and δ G (x) x G(u)du/G(x), respectvely. The functons λ G (x), δ G (x), and G(x) are equvalent. (Shaked and Shanthkumar 1994). The hazard and reverse hazard functons are of the GGMW dstrbuton are gven by and h G (x) { log(1 e αx βxθ e λx )} δ 1 e αx βxθ e λx (α + βx θ 1 e λx [θ + λx]), (23) γ( log(1 e αx βxθ e λx ), δ) τ G (x) { log(1 e αx βxθ e λx )} δ 1 e αx βxθ e λx (α + βx θ 1 e λx [θ + λx]), (24) γ( log(1 e αx βxθ e λx ), δ) respectvely. Plots of the hazard rate functon for dfferent combnatons of the parameter values are gven n Fgure 2. The plot shows varous shapes ncludng monotoncally ncreasng, monotoncally ncreasng and bathtub shapes for fve combnatons of the values of the parameters. Ths flexblty makes the GGMW hazard rate functon sutable for both monotonc and non-monotonc emprcal hazard behavors that are lkely to be encountered n real lfe stuatons. 3. Moments and moment generatng functon In ths secton, we obtan moments and moment generatng functon of the GGMW dstrbuton. Let X GGMW (α, β, θ, λ, δ), and Y GMW (α, β, θ, λ). Note that the r th moment of the random varable Y s obtaned as follows. By Taylor seres expanson of the functons
52 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Table 2: Sub-models of the gamma generalzed modfed Webull dstrbuton Model α β θ λ δ G(x) Reference GGMR - - 2 - - GGME - - 1 - - GMW - - - - GME - 1 - - GAE - - 1 - GEV 1 - - γ( log[1 e αx βx2 e λx ],δ) γ( log[1 e αx βxeλx ],δ) γ( log[1 e βxθ e λx ],δ) γ( log[1 e βxeλx ],δ) γ( log[1 e αx βx ],δ) γ( log[1 e eλx ],δ) GW - - - γ( log[1 e βxθ ],δ) Pnho, Cordero, and Nobre (212) MW - - - 1 1 e βxθ e λx La, Xe, and Murthy (23) S-ZMW - - - - 1 1 e αx βxθ e λx Sarhan and Zandn (29) LFR - - 2 1 1 e αx βx2 Ban (1974) EV 1-1 1 e eλx Ban (1974) Webull - - 1 1 e βxθ Webull (1951) Exponental - 1 1 e αx Ban (1974) New New New New New New Fgure 2: Graphs of GGMW hazard functon
Austran Journal of Statstcs 53 e βxθ e λx and e kλx, we have: E(Y r ) k,n k,n y r d(1 e αy βxθ e λy ) ry r 1 e αy βyθ e λy dy r( β) n (nλ) k k!n! ry r+nθ+k 1 e αy dy r( β) n (nλ) k α (r+θn+k) Γ (r + θn + k). (25) k!n! Consequently, that the r th raw moment of GGMW dstrbuton s gven by: µ r E(X r ) ν C w ν E(Y r ), where Y GMW (α(m + s + δ), β(m + s + δ), θ, λ). Note that, snce r tr r! xr g GGMW (x) converges and each term s ntegrable for all t close to zero, say (for t < 1), the moment generatng functon (MGF) of the GGMW dstrbuton s gven by: M X (t) ν C ν C j k,n,j w ν t j j! E(Y j ) w ν t j j( β(k + s + δ)) n (nλ) k Γ (j + θn + k), (26) k!n!j!(α(k + s + δ)) (j+θn+k) where Γ (a) b a t a 1 e t dt s the gamma functon, and r 1, 2,... Table 3 lsts the frst sx moments for selected parameter values of GGMW dstrbuton, where V arance E(Y 2 ) E(Y ) 2, Skewness E(Y 3 ) 3E(Y )σ 2 E(Y ) 3, and Kurtoss E(Y 4 ) 3. σ 3 σ 4 Theorem 3.1. Proof: E{[ log(1 F (X))] r [(1 F (X)) s ]} ψr+δ Γ (r + δ) (sψ + 1) δ ψ δ. (27) E{[ log(1 F (X))] r [(1 F (X)) s ]} If s n equaton (28), then we have f(x) ψ δ [ log(1 F (x))]r+δ 1 [1 F (x)] s+(1/ψ) 1 dx ψ r+δ Γ (r + δ) (sψ + 1) δ ψ δ. (28) E[ log(1 F (X)) r ] ψr+δ Γ (r + δ) ψ δ. (29) Let ψ s + 1 ψ, then wth r n equaton (28), we obtan E[(1 F (X)) s ] ( ) 1 δ (ψ ) δ f(x) ψψ [ log(1 F (x))] δ 1 [1 F (x)] ψ 1 dx [sψ + 1] δ. (3)
54 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Table 3: GGMW moments for selected values (α, β, θ, λ, δ) Moments (1,2,.5,.5,1) (1,2,.5,1.5,2) (1,4,2,1,6) (1,1.5,2,1,2.5) (2,.9,1,1,3) E(Y ).179884.36182.13699.144246.52597 E(Y 2 ).883142.5822.7539.394158.7137 E(Y 3 ).649863.11492.767.142427.16577 E(Y 4 ).68936.3385.17.6942.521 E(Y 5 ).674815.1191.18.29347.221 E(Y 6 ).848496.477.4.15457.917 Varance.559832.37849.5831.18689.4611 Skewness 2.1873821 2.978169 3.663231 1.256166 2.668344 Kurtoss 16.4292658 2.6287 28.43215 14.5986436 21.5646 4. Order statstcs and Rény entropy Order statstcs play an mportant role n probablty and statstcs. The concept of entropy plays a vtal role n nformaton theory. The entropy of a random varable s defned n terms of ts probablty dstrbuton and can be shown to be a good measure of randomness or uncertanty. In ths secton, we present Rény entropy and the dstrbuton of the order statstcs for the GGMW dstrbuton. 4.1. Rény entropy Rény entropy s an extenson of Shannon entropy. Rény entropy s defned to be I R (v) 1 ( ) 1 v log [g GGMW (x; α, β, θ, λ, δ)] v dx, v 1, v >. (31) Rény entropy tends to Shannon entropy as v 1. Note that ( ) 1 v g v (x)dx ((α + βx θ 1 e λx [θ + λx])e αx βxθ e λx ) v GGMW [ log(1 e αx βxθ e λx )] v(δ 1) dx. (32) Let < y e αx βxθ e λx <.7968. Note that ((α + βx θ 1 e λx [θ + λx])) v v j v ( ) v α v j β j x jθ j j j j r n ( v j )( j r n ( ) j (jλx) n r n! j θ j r (λx) r r )α v j β j θ j r λ r (jλ)n x n+r+jθ j. n! Now, for < e vβxθ e λx < 1, v >, and applyng Taylor seres expanson, we have e vβxθ e λx ( 1) l (vβ) l (lλ) w x lθ+w, (33) l!w! l w
Austran Journal of Statstcs 55 so that, g v (x) [] v v j j r n,l,w,m,s ( 1) l ( v j )( )( ) j δ(v 1) r m α v j β j θ j r λ r (jλ)n (vβ) l (lλ) w b s,m n! l! w! x n+r+jθ j+lθ+w e (m+s+vδ v)αx (m+s+vδ v)βxθ e λx e vαx v j ( )( v j [] v ( 1) l+k j r m j r n,l,w,m,s,k, α v j β j+l θ j r λ r+n+w (j)n (v) l (l) w n!l!w! )( δ(v 1) ) b s,m (m + s + vδ v)k β k (kλ) x n+r+jθ j+lθ+w+kθ+ e (m+s+vδ)αx. k!! Usng the fact that t a 1 e t dt Γ (a) b, we have a g v GGMW (x)dx [] v v j j r n,l,w,m,s,k, ( 1) l+k ( v j )( j r )( δ(v 1) α v j β j+l+ θ j r λ r+n+w+ (j)n (v) l (l) w k (m + s + vδ v) k n!l!w!k!! Γ (n + r + w + + θ(j + l + k) j + 1) (m + s + vδ) n+r+w++θ(j+l+k) j+1, for v >, v 1. Consequently, Rény entropy for the GGMW dstrbuton s gven by I R (v) [ 1 v 1 v log [] v j j r n,l,w,m,s,k, ( 1) l+k ( v j )( j r m )( δ(v 1) α v j β j+l+ θ j r λ r+n+w+ (j)n (v) l (l) w k (m + s + vδ v) k n!l!w!k!! ] Γ (n + r + w + + θ(j + l + k) j + 1) (m + s + vδ) n+r+w++θ(j+l+k) j+1, for v >, v 1. m ) b s,m ) b s,m 4.2. Order statstcs In ths secton, the pdf of the th order statstc and the correspondng moments are presented. Let X 1, X 2,..., X n be ndependent and dentcally dstrbuted GGMW random varables. The pdf of of the th order statstc for a random sample of sze n for any gamma G famly wth densty (5) can be expressed as an nfnte weghted sum of gamma G denstes. The pdf of the th order statstc from the GGMW pdf g GGMW (x) s gven by g :n (x) n!g(x) ( 1)!(n )! [G(x)] 1 [1 G(x)] n n!g(x) 1 ( 1 ( 1) j ( 1)!(n )! j j j ) [G(x)] n +j n!g(x) 1 ( )[ ] 1 γ( log(1 e ( 1) j αx βx θ e λx )) n +j. ( 1)!(n )! j where < y e αx βxθ e λx <.7968, x >, α, β, θ, δ >, and λ. Usng the fact that γ(x, δ) ( 1) m x m+δ m (m+δ)m!, and settng c m ( 1) m /((m + δ)m!), we can wrte the pdf of the
56 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons th order statstc from the GGMW dstrbuton as follows: g :n (x) n!g(x) 1 ( ) 1 ( 1) j ( 1)!(n )! j [] n +j [ log(1 e λx e αx βxθ )] δ(n +j) j [ ( 1) m (log(1 e αx βxθ e λx )) m ] n +j (m + δ)m! m n!g(x) 1 ( ) 1 ( 1) j ( 1)!(n )! j [] n +j [ log(1 e λx e αx βxθ )] δ(n +j) j d m,n +j ( log(1 e αx βxθ e λx )) m, m where d c (n +j), d m,n +j (mc ) 1 m l1 [(n + j)l m + l]c ld m l,n +j. We note that g :n (x) where n!g(x) 1 ( 1)!(n )! j m ( n n![ log(1 e αx βxθ e λx )] δ 1 f(x) 1 ( 1)!(n )! j ) ( 1) j d m,+j 1 [] n +j [ log(1 e αx βxθ e λx )] δ(n +j)+m j m ( n [ log(1 e αx βxθ e λx )] δ(n +j)+m n! 1 ( ) 1 ( 1) j d m,n +j ( 1)!(n )! j [] n +j j m j ) ( 1) j d m,n +j [] n +j Γ (δ(n + j) + m + δ) [ log(1 e αx βxθ e λx )] δ(n +j)+m+δ 1 Γ (δ(n + j) + m + δ) (α + βx θ 1 e λx [θ + λx])e αx βxθ e λx n! 1 ( 1)!(n )! j m ( ) 1 ( 1)j d m,n +j Γ (δ(n + j) + m + δ) [] n +j+1 f GGMW (x), f GGMW (x) j 1 Γ (δ(n + j) + m + δ) [ log(1 e αx βxθ e λx )] δ(n +j)+m+δ 1 (α + βx θ 1 e λx [θ + λx])e αx βxθ e λx (34) s the GGMW pdf wth parameters α, β, θ >, λ, and shape parameter δ δ(n + j) + m + δ >. It follows therefore that the r th moment s gven by E(X j :n ) ν C 1 j m,k,n r( β) n (k + s + δ )(nλ) k w ν l,j,m k!n![α(k + s + δ ) r+nθ+k Γ (r + nθ + k), ] where l,j,m n! ( 1)!(n )! ( 1) j d m,n +j Γ (δ(n +j)+m+δ) [] n +j+1, and δ δ(n + j) + m + δ >. We note that these moments are often used n several areas ncludng relablty, survval analyss, bometry, engneerng, nsurance and qualty control for the predcton of future falures tmes from a set of past or prevous falures.
Austran Journal of Statstcs 57 5. Maxmum lkelhood estmaton be the parameter vector. The log- Let X GGMW (α, β, θ, λ, δ) and (α, β, θ, λ, δ) T lkelhood for a sngle observaton x of X s gven by l l( ) (δ 1) log( log(1 e αx βxθ e λx )) + log(α + βx θ 1 e λx [θ + λx]) αx βx θ e λx log(). (35) The frst dervatve of the log-lkelhood functon wth respect to the parameters (α, β, θ, λ, δ) T are gven by l β l α x(δ 1)e αx βxθeλx (1 e αx βxθ e λx ) log(1 e αx βxθ e λx ) + 1 α + βx θ 1 e λx x, (36) [θ + λx] x θ e λx (δ 1)e αx βxθeλx (1 e αx βxθ e λx ) log(1 e αx βxθ e λx ) + xθ 1 e λx (θ + λx) α + βx θ 1 e λx [θ + λx] xθ e λx, (37) l θ (δ 1)x θ log(x)βe λx e αx βxθ e λx (1 e αx βxθ e λx ) log(1 e αx βxθ e λx ) + βxθ 1 e λx [(θ + λx) log(x) + 1] α + βx θ 1 e λx [θ + λx] βx θ e λx log(x), (38) l λ (δ 1)x θ+1 βe λx e αx βxθeλx (1 e αx βxθ e λx ) log(1 e αx βxθ e λx ) + βxθ e λx (θ + λx + 1) α + βx θ 1 e λx [θ + λx] βxθ+1 e λx, (39) and l δ log( log(1 e αx βxθ e λx )) Γ (δ). (4) The total log-lkelhood functon based on a random sample of n observatons: x 1, x 2,..., x n drawn from the GGMW dstrbuton s gven by l n l( ) n l ( ), where l ( ), 1, 2,..., n s gven by equaton (35). The equatons obtaned by settng the above partal dervatves to zero are not n closed form and the values of the parameters α, β, θ, λ, δ must be found by usng teratve methods. The maxmum lkelhood estmates of the parameters, denoted by ˆ s obtaned by solvng the nonlnear equatons ( l α, l β, l θ, l λ, l δ )T. It s convenent to apply or use nonlnear optmzaton algorthm such as quas-newton algorthm to numercally maxmze the log-lkelhood functon. We maxmze the lkelhood functon usng NLmxed n SAS as well as the functon nlm n R (The R Development Core Team (211)). These functons were appled and executed for wde range of ntal values. Ths process often results or lead to more than one maxmum, however, n these cases, we take the MLEs correspondng to the largest value of the maxma. In a few cases, no maxmum was dentfed for the selected ntal values. In these cases, a new ntal value was tred n order to obtan a maxmum. The ssues of exstence and unqueness of the MLEs are theoretcal nterest and has been studed by several authors for dfferent dstrbutons ncludng Seregn (21), Santos Slva and Tenreyro (21), Zhou (29), and Xa, M, and Zhou (29). At ths pont we are not able to address the theoretcal aspects (exstence, unqueness) of the MLE of the parameters of the GGMW dstrbuton. Note that for the fve parameters of the GGMW dstrbuton, all second order partal dervatves of the log-lkelhood functon ext, and are gven n appendx A. The Fsher nformaton matrx s gven by I( ) [I θ,θ j ] 5X5 E( 2 l θ θ j ),, j 1, 2, 3, 4, 5, can be numercally
58 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons obtaned by MATHLAB, R or MAPLE software. The total Fsher nformaton matrx ni( ) can be approxmated by J n ( ˆ ) [ 2 l θ θ j ˆ ],, j 1, 2, 3, 4, 5. (41) 5X5 For a gven set of observatons, the matrx gven n equaton (41) s obtaned after the convergence of the Newton-Raphson procedure n MATHLAB or R software. Elements of the observed nformaton matrx are gven n the appendx. 5.1. Asymptotc confdence ntervals In ths secton, we present the asymptotc confdence ntervals for the parameters of the GGMW dstrbuton. The expectatons n the Fsher Informaton Matrx (FIM) can be obtaned numercally. Let ˆ (ˆα, ˆβ, ˆθ, ˆλ, ˆδ) be the maxmum lkelhood estmate of (α, β, θ, λ, δ). Under the usual regularty condtons and that the parameters are n the nteror of the parameter space, but not on the boundary, (Ferguson 1996) we have: n( ˆ d ) N 5 (, I 1 ( )), where I( ) s the expected Fsher nformaton matrx. The asymptotc behavor s stll vald f I( ) s replaced by the observed nformaton matrx evaluated at ˆ, that s J( ˆ ). The multvarate normal dstrbuton N 5 (, J( ˆ ) 1 ), where the mean vector (,,,, ) T, can be used to construct confdence ntervals and confdence regons for the ndvdual model parameters and for the survval and hazard rate functons. That s, the approxmate 1(1 η)% two-sded confdence ntervals for α, β, θ λ, and δ are gven by: and δ ± Z η 2 α ± Z η Iαα( 1 ˆ ), 2 β ± Z η I 1 2 ββ ( ˆ ), θ ± Z η I 1 2 θθ ( ˆ ), λ ± Z η I 1 2 λλ ( ˆ ), I 1 δδ ( ˆ ), respectvely, where I 1 αα( ˆ ), I 1 ββ ( ˆ ), I 1 θθ ( ˆ ), I 1 λλ ( ˆ ) and I 1 the dagonal elements of In 1 dstrbuton. ( ˆ ), and Z η 2 s the upper η 2 δδ ( ˆ ) are th percentle of a standard normal The maxmum lkelhood estmates (MLEs) of the GGMW parameters α, β, θ, λ, and δ are computed by maxmzng the objectve functon va the subroutne NLmxed n SAS and the functon nlm n R. The estmated values of the parameters (standard error n parenthess), -2log-lkelhood statstc, Akake Informaton Crteron, AIC 2p 2 ln(l), Bayesan Informaton Crteron, BIC p ln(n) 2 ln(l), and Consstent Akake Informaton Crteron, AICC AIC + 2 p(p+1) n p 1, where L L( ˆ ) s the value of the lkelhood functon evaluated at the parameter estmates, n s the number of observatons, and p s the number of estmated parameters are presented. In order to compare the models, we use the crtera stated above. Note that for the value of the log-lkelhood functon at ts maxmum (l n ), larger value s good and preferred, and for AIC, AICC and BIC, smaller values are preferred. GGMW dstrbuton s ftted to the data sets and these fts are compared to the fts of the GGME, GGMR, GMW, GW, beta exponentated Webull (BEW) and beta Webull (BW) dstrbutons. We can use the lkelhood rato (LR) test to compare the ft of the GGMW dstrbuton wth ts sub-models for a gven data set. For example, to test λ, δ 1, the LR statstc s ω 2[ln(L(ˆα, ˆβ, ˆθ, ˆλ, ˆδ)) ln(l( α, β, θ,, 1))], where ˆα, ˆβ, ˆλ, ˆθ and ˆδ, are the unrestrcted estmates, and α, β, and θ are the restrcted estmates. The LR test rejects the null hypothess f ω > χ 2, where ɛ χ2 denote the upper 1ɛ% pont of the ɛ χ2 dstrbuton wth 2 degrees of freedom. 6. Applcatons In ths secton, we present examples to llustrate the flexblty and applcablty of the GGMW dstrbuton and ts sub-models for data modelng. The GGMW dstrbuton s also compared
Austran Journal of Statstcs 59 Table 4: Estmaton of GGMW model for watng tmes data Estmates Statstcs Dstrbuton α β θ λ δ -2LogLkelhood AIC AICC BIC SS GGMW.3529.5611 1.6133.2399.1687 637.7 647.7 648.3 66.7.574 (.1679) (.2513) (.1642) (.178) (.1386) GGME.223.4152 1.2139.1875 64.5 648.5 649 659.929 (.292) (.2142) (.6622) (.2897) GMW.2887 1.3239.17.1629 634.8 642.8 643.2 653.2.271 (.7322) (.162) (.5437) (.1887) GME.3892 1.1897.258 64.9 646.9 647.2 654.7.124 (.9978) (.6636) (.3738) GAE.218.188 1.2776 647.9 653.9 654.1 661.7.3585 (.9663) (.9657) (.141) GEV 1.9372.3111 727.1 731.1 731.2 736.3 1.1924 (.96) (.2987) GW.2443 1.3435.1829 635.3 641.3 641.6 649.2.329 (.7656) (.8978) (.236) BW k λ a b 1.2455 2.1348 1.7298.159 634.2 642.2 642.7 652.7.239 (.18) (.4812) (.5264) (.1871) BEW k λ α a b 1.135 2.6752 1.6422 1.3894.2681 633.9 643.9 644.6 657.159 (.2224) (1.4164) (1.3948) (.7837) (.2345) wth the non-nested beta exponentated Webull (BEW), and beta Webull (BW) dstrbutons. The pdf of the BEW dstrbuton (Cordero, Gomes, da Slva, and Ortega 213) s gven by g(x) αkλk B(a, b) xk 1 e (λx)k (1 e (λx)k ) aα 1 [1 (1 e (λx)k ) α ] b 1, x >. (42) When α 1, we have the BW dstrbuton. The frst data set s watng tmes (n mnutes) of 1 bank customers before servce. See (Ghtany, Ateh, and Nadarajah 28) for addtonal detals. The second data set s falure tmes of a sample of n 3 devces, see (Meeker and Escobar 1998). The thrd data set represent the survval tmes of 121 patents wth breast cancer obtaned from a large hosptal n a perod from 1929 to 1938, (Lee 1992). Estmates of the parameters of GGMW dstrbuton (standard error n parentheses), Akake Informaton Crteron (AIC), Consstent Akake Informaton Crteron (AICC) and Bayesan Informaton Crteron (BIC) are gven n Table 4 for the frst data set, n Table 5 for the second data set and n Table 6 for the thrd data set. The estmated covarance matrx for the GGMW dstrbuton (Watng Tmes Data) s gven by.28.11.3.2.54.11.63.294.4.1.3.294.2697.141.48.2.4.141.12.3.54.1.48.3.19 The 95% asymptotc confdence ntervals for the GGMW model (Watng Tmes Data) parameters are: α (.32,.3529), β (.69,.154), θ (1.2915, 1.9351), λ (.187,.235), and δ (.1415,.1959), respectvely. The estmated covarance matrx for the GGMW dstrbuton (Meeker Data) s gven by.15.1.399..1.1.8.4.5.2.399.4.7974.14.11..5.14.7.1.1.2.11.1.145
6 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Fgure 3: Graphs for watng tmes data Table 5: Estmaton of GGMW model for meeker data Estmates Statstcs Dstrbuton α β θ λ δ -2LogLkelhood AIC AICC BIC SS GGMW.5354.411.4549.2772.6625 345.2 355.2 357.7 362.2.1885 (.392) (.2771) (.893) (.2629) (.123) GGME.856.2468 1.5281.9626 418.7 426.7 428.3 432.3 4.7185 (.8775) (.2176) (.289) (.1758) GMW.5256.3819.6687.636 355.8 363.8 365.4 369.4.2396 (.6536) (.5393) (.585) (.195) GAE.6438.2438 1.677 37.2 376.2 377.1 38.4.343 (.3377) (.3435) (.4661) GEV 1.124.17 366.3 37.3 37.7 373.1.2661 (.1629) (.484) GW.221 1.453 1.766 368.3 374.3 375.2 378.5.3412 (.1258) (.672) (2.1364) BW k λ a b.6935 1.819 1.1725.486 378.7 386.7 388.3 392.3.6369 (.3331) (.4279) (1.1345) (.938) BEW k λ α a b.9895 8.76.7834.9181.426 371.2 381.2 383.7 388.2.294 (.3881) (1.1182) (.2485) (.5873) (.799) Fgure 4: Graphs for meeker data
Austran Journal of Statstcs 61 Table 6: Estmaton of GGMW model for breast cancer data Estmates Statstcs Dstrbuton α β θ λ δ -2LogLkelhood AIC AICC BIC SS GGMW.1249.247.1573.5334.1657 1155.3 1165.3 1165.9 1179.3.1153 (.6267) (.1812) (.8788) (.2297) (.6779) GGME.1293.2883 1.2348.1476 1155.8 1163.8 1164.2 1175..927 (.1342) (.1374) (.2869) (.1646) GMW.7926.9986.3633.2248 1157.1 1165.1 1165.4 1176.3.65 (.1335) (.1543) (.2969) (.3898) GME.7876 1.361.2253 1157.1 1163.1 1163.3 1171.5.64 (.1223) (.1698) (.3784) GAE.3852.3352 1.3249 1163.3 1169.3 1169.5 1177.7.1943 (.257) (.257) (.189) GEV 1.2336.2439 1243.4 1247.4 1247.5 1252.9 1.1443 (.181) (.2174) GW.2764 1.3964 1.3417 1158. 1164. 1164.2 1172.4.527 (.142) (.6238) (2.2681) BW k λ a b.7573 1.2899.241.6145 1251.7 1259.7 126.1 127.9 3.5347 (.495) (.362) (.3752) (.6166) BEW k λ α a b.7958 1.6244 1.2491.3575.6836 1223.6 1233.6 1234.1 1247.6 2.5138 (.741) (.2646) (.2518) (.694) (.6678) Fgure 5: Graphs for breast cancer data
62 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons The estmated covarance matrx for the GGMW dstrbuton (Breast Cancer Data) s gven by.3927.59.493.1194.414.59 3.28E 6.9.16.6.493.9.7724.1968.552.1194.16.1968.528.122.414.6.552.122.4595 Plots of the ftted denstes, the hstogram of the data are gven n Fgure 3, Fgure 4 and Fgure 5. For the probablty plot, we plotted G GGMW (x (j) ; ˆα, ˆβ, ˆθ, ˆλ, ˆδ) aganst j.375 n +.25, j 1, 2,, n, where x (j) are the ordered values of the observed data. We also computed a measure of closeness of each plot to the dagonal lne. Ths measure of closeness s gven by the sum of squares SS j1 [ G GGMW (x (j) ; ˆα, ˆβ, ˆθ, ˆλ, ˆδ) ( )] j.375 2. n +.25 For watng tmes data set, the LR test statstc of the hypothess H : GGME aganst H a : GGMW s ω 2.8. The p-value.94. Therefore, there s no sgnfcant dfference between GGMW and GGME dstrbutons at the 5% level. However, there s a sgnfcant dfference between GGME and GGMW dstrbutons at the 1% level. The LR statstc of the hypothess H : GEV aganst H a : GGMW for watng tmes data s ω 89.4 The p-value <.1, we can conclude that there s a sgnfcance dfference between GGMW and GEV dstrbutons. There s no sgnfcant dfference between the GGMW and GMW dstrbutons. Also, there s no sgnfcant dfference between the GW and GMW dstrbutons. The values of the statstcs AIC, AICC and BIC shows that the sub-model GW s a good ft for ths data. Based on these statstcs, the GW dstrbuton could be chosen as the best model among these dstrbutons. The values of the statstcs are comparable to those of the non-nested BW dstrbuton and those correspondng to the BEW dstrbuton. For Meeker data set, the LR test statstcs of the hypothess H : GGME aganst H a : GGMW s ω 73.5. The p-value <.1. Therefore, there s sgnfcant dfference between GGMW and GGME dstrbutons. The LR statstc of the hypothess H : GMW aganst H a : GGMW s ω 1.6. The p-value.11, we can conclude that there s a sgnfcance dfference between GGMW and GMW dstrbutons. The values of the statstcs AIC, BIC, and AICC are smaller for the GGMW dstrbuton. The values of these statstcs ponts to the GGMW dstrbuton as the better ft for Meeker data. Also, the values of AIC, BIC and AICC are better for the GMW and GGMW dstrbutons when compared to the non-nested BW and BEW dstrbutons. For breast cancer data set, there s no sgnfcant dfference between GGMW, GGME, GMW, GW and GME dstrbutons based on the correspondng LR tests. The sub-models GME and GW seem to be the best fts for ths data. The values of the statstcs AIC, BIC and AICC are smaller for the GME dstrbuton. The values of SS from the probablty plots are.64 and.527 for the GME and GW dstrbutons, respectvely. The values of these statstcs ponts to and supports the GW as well as the GME dstrbutons as the better fts among the nested dstrbutons. Also, the values of the statstcs: AIC, BIC and AICC are far better for the GMW and GGMW dstrbutons when compared to those of the non-nested BW and BEW dstrbutons. The conclusons based on the LR tests, ftted pdfs, the hstograms of the data, and probablty plots are n agreement wth the statstcs AIC, AICC and BIC for the selected models. The GW dstrbuton provdes a better fts for the watng tmes data, whle the GGMW dstrbuton and GME as well as the GW dstrbutons provdes better fts for the Meeker and Escobar, and breast cancer data, respectvely.
Austran Journal of Statstcs 63 7. Concludng remarks A new class of generalzed modfed Webull dstrbuton called the gamma-generalzed modfed Webull (GGMW) dstrbuton s proposed and studed. The GGMW dstrbuton has several sub-models such as the GGMR, GGME, GAE, GLFR, LFR, GMW, GME, MW, MR, ME, Webull, Ralegh and exponental dstrbutons as specal cases. The densty of ths new class of dstrbutons can be expressed as a lnear combnaton of GMW densty functons. The GGMW dstrbuton possesses hazard functon wth flexble behavor. We also obtan closed form expressons for the moments, dstrbuton of order statstcs and Reny entropy. Maxmum lkelhood estmaton technque was used to estmate the model parameters. Fnally, the GGMW dstrbuton and ts sub-models was ftted to real data sets to llustrate the applcablty and usefulness of ths class of dstrbutons. Acknowledgements The authors would lke to thank the edtor and the referee for carefully readng the paper and for ther valuable comments, whch greatly mproved the presentaton n ths paper. References Almalk SJ, Yuan J (213). A New Modfed Webull Dstrbuton. Relablty Engneerng and System Safety, 111, 164 17. Ban L (1974). Analyss for the Lnear Falure Rate Lfe Testng Dstrbuton. Technometrcs, 16(4), 551 559. Barlow R, Campo R (1975). Total Tme on Test Processes and Applcatons to Falure Data Analyss. Socety for Industral and Appled Mathematcs. Bebbngton M, La C, Ztks R (27). A Flexble Webull Extenson. Relablty Engneerng and System Safety, 92(6), 719 726. Carrasco M, Ortega EM, Cordero G (28). A Generalzed Webull Dstrbuton for Lfetme Modelng. Computatonal Statstcs and Data Analyss, 53(2), 45 462. Choudhury A (25). A Smple Dervaton of Moments of the Exponentated Webull Dstrbuton. Metrka, 62(1), 17 22. Cordero G, Gomes A, da Slva C, Ortega M (213). The Beta Exponentated Webull Dstrbuton. Journal of Statstcal Computaton and Smulatons, 38(1), 114 138. Cordero G, Ortega E, Nadarajah S (21). The Kumaraswamy Webull Dstrbuton wth Applcatons to Falure Data. Journal of Frankln Insttute, 347(8), 1399 1429. Famoye F, Lee C, Olumolade O (25). The Beta-Webull Dstrbuton. Journal of Statstcal Theory and Applcatons, pp. 121 138. Ferguson T (1996). A Course n Large Sample Theory. Chapman and Hall. Fsher R (1934). The Effects of Methods of Ascertanment Upon the Estmaton of Frequences. Annals of Human Genetcs, 6(1), 439 444. Ghtany M, Ateh B, Nadarajah S (28). Lndley Dstrbuton and Its Applcatons. Mathematcs and Computers n Smulatons, 78(4), 493 56. Gradshteyn I, Ryzhk I (2). Tables of Integrals, Seres and Products. Academc Press. Gupta R, Keatng J (1985). Relaton for Relablty Measures under Length Based Samplng. Scandnavan Journal of Statstcs, 13, 49 56.
64 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons Gupta R, Kundu D (1999). Generalzed Exponental Dstrbutons. Australan and New Zealand Journal of Statstcs, 43, 117 13. Gupta R, Kundu D (21). Exponentated Exponental Dstrbuton: An Alternatve to Gamma and Webull Dstrbutons. Bometrcal Journal, 43, 117 13. Haupt E, Schabe H (1992). A New Model for A Lfetme Dstrbuton wth Bathtub Shaped Falure Rate. Mcroelectroncs and Relablty, 32, 633 639. Hjorth U (198). A Relablty Dstrbuton wth Increasng, Decreasng, Constant and Bathtub Falure Rates. Technometrcs, 22, 99 17. Kundu D, Rakab M (25). Generalzed Raylegh Dstrbuton: Dfferent Methods of Estmaton. Computatonal Statstcs and Data Analyss, 49, 187 2. La C, Moore T, Xe M (1998). The Beta Integrated Model. Proceedngs Internatonal Workshop on Relablty Modelng and Analyss-From Theory to Practce, pp. 153 159. La C, Xe M, Murthy D (23). A Modfed Webull Dstrbuton. IEEE Transactons on Relablty, 52, 33 37. Lee C, Famoye F, Olumolade O (27). Beta Webull Dstrbuton, Propertes and Applcatons to Censored Data. Journal of Mod. Appl. Statst. Meth, 6, 173 186. Lee E (1992). Statstcal Methods for Survval Data Analyss. John Wley. Marshall AW, Olkn I (1997). A New Method for Addng a Parameter to a Famly of Dstrbutons wth Applcatons to the Exponental and Webull Famles. Bometrka, 84(3), 641 652. Meeker W, Escobar L (1998). Statstcal Methods for Relablty Data. John Wley. Mudholkar G, Srvastava D, Fremer M (1995). The Exponentated Webull Famly: A Reanalyss of the Bus-motor-falure Data. Technometrcs, 37, 436 445. Mudholkar G, Srvastava D, Kolla G (1996). A Generalzaton of the Webull Dstrbuton wth Applcaton to the Analyss of Survval Data. Journal of the Amercan Statstcal Assocaton, 91, 1575 1583. Nadarajah S (25). On the Moments of the Modfed Webull Dstrbuton. Relablty Engneerng and System Safety, 9, 114 117. Nadarajah S, Cordero GM, Ortega EMM (211). General Results for the beta-modfed Webull Dstrbuton. Journal of Statstcal Computaton and Smulaton, 81(1), 1211 1232. Nadarajah S, Kotz S (25). On Some Recent Modfcatons of Webull Dstrbuton. IEEE Transactons Relablty, 54, 561 562. Nanda K, Jan K (1999). Some Weghted Dstrbuton Results on Unvarate and Bvarate Cases. Journal of Statstcal Plannng and Inference, 77(2), 169 18. Nassar M, Essa F (23). On the Exponentated Webull Dstrbuton. Communcatons n Statstcs - Theory and Methods, 32(7), 1317 1336. Oluyede B (1999). On Inequaltes and Selecton of Experments for Length-Based Dstrbutons. Probablty n the Engneerng and Informatonal Scences, 13(2), 129 145. Patl G, Rao C (1978). Weghted Dstrbutons and Sze-Based Samplng wth Applcatons to Wldlfe and Human Famles. Bometrcs, 34(6), 179 189.
Austran Journal of Statstcs 65 Phan KK (1987). A New Modfed Webull Dstrbuton Functon. Communcatons of the Amercan Ceramc Socety, 7(8), 182 184. Pnho L, Cordero G, Nobre J (212). The Gamma-Exponentated Webull Dstrbuton. Journal of Statstcal Theory and Applcatons, 11(4), 379 395. Rajarsh S, Rajarsh M (1988). Bathtub Dstrbutons: A Revew. Communcatons n Statstcs-Theory and Methods, 17, 2521 2597. Rao C (1965). On Dscrete Dstrbutons Arsng out of Methods of Ascertanment. The Indan Journal of Statstcs, 27(2), 32 332. Rstć M, Balakrshnan N (211). The Gamma-Exponentated Exponental Dstrbuton. J. Statst. Comp. and Smulaton, 82(8), 1191 126. Santos Slva JMC, Tenreyro S (21). On the Exstence of Maxmum Lkelhood Estmates n Posson Regresson. Economcs Letters, 17, 31 312. Sarhan AM, Zandn M (29). Modfed Webull Dstrbuton. Appled Scences, 11, 123 136. Seregn A (21). Unqueness of the Maxmum Lkelhood Estmator for K-monotone Denstes. Proceedngs of the Amercan Mathematcal Socety, 138(12), 4511 4515. Shaked M, Shanthkumar J (1994). Stochastc Orders and Ther Applcatons. Academc Press. Slva G, Ortega E, Cordero G (21). The Beta Modfed Webull Dstrbuton. Lfetme Data Analyss, 16, 49 43. Sngha N, Jan K, Kumar SS (212). The Beta Generalzed Webull Dstrbuton: Propertes and Applcatons. Relablty Engneerng and System Safety, 12, 5 15. The R Development Core Team (211). A Language and Envronment for Statstcal Computng. R Foundaton for Statstcal Computng. Webull WA (1951). Statstcal Dstrbuton Functon of Wde Applcablty. Journal of Appled Mechancs, 18, 293 296. Xa J, M J, Zhou YY (29). On the Exstence and Unqueness of the Maxmum Lkelhood Estmators of Normal and Log-normal Populaton Parameters wth Grouped Data. Journal of Probablty and Statstcs. Xe M, La C (1995). Relablty Analyss Usng an Addtve Webull Model wth Bathtubshaped Falure Rate Functon. Relablty Engneerng and System Safety, 52, 87 93. Xe M, Tang Y, Goh T (22). A Modfed Webull Extenson wth Bathtub Falure Rate Functon. Relablty Engneerng and System Safety, 76, 279 285. Zhang T, Xe M (211). On the Upper Truncated Webull Dstrbuton and Its Relablty Implcatons. Relablty Engneerng and System Safety, 96(1), 194 2. Zhou C (29). Exstence and Consstency of the Maxmum Lkelhood Estmator for the Extreme Index. J. Multvarate Analyss, 1, 794 815. Zografos K, Balakrshnan N (29). On Famles of Beta- and Generalzed Gamma-Generated Dstrbuton and Assocated Inference. Stat. Method, 6, 344 362.
66 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons APPENDIX Let A(x ; α, β, θ, λ) (1 e αx βx θ eλx ) log(1 e αx βx θ eλx ), B(x ; α, β, θ, λ) e αx βx θ eλx + log(1 e αx βx θ eλx ), and C(x ; α, β, θ, λ) (1 e αx βx θ eλx βx θ eλx ) log(1 e αx βx θ eλx ) βx θ e (α λ)x βx θ eλx. Elements of the observed nformaton matrx of the GGMW dstrbuton are gven by α 2 (1 δ)x 2 e αx βx θ eλx B(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) 1 [α + βx θ 1 e λx (θ + λx )]. (43) 2 α β (1 δ)x θ+1 e (α λ)x βx θ eλx B(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) x θ 1 e λx (θ + λx ) [α + βx θ 1 e λx (θ + λx )] 2. (44) α θ (1 δ)βx θ+1 e (α λ)x βx θ eλx log(x )B(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) βx θ 1 e λx [(θ + λx ) log(x ) + 1] [α + βx θ 1 e λx (θ + λx )] 2. (45) α λ (1 δ)βx θ+2 e (α λ)x βx θ eλx B(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) βx θ eλx (θ + λx + 1) [α + βx θ 1 e λx (θ + λx )] 2. (46) α δ e αx βx θ eλx x A(x ; α, β, θ, λ). (47) β 2 (1 δ)x 2θ e (α 2λ)x βx θ eλx B(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) x 2θ 2 e 2λx (θ + λx ) 2 [α + βx θ 1 e λx (θ + λx )] 2. (48) β θ + (δ 1)x θ e (α λ)x βx θ eλx log(x )C(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) x θ 1 e λx [(θ + λx ) log(x ) + 1]α [α + βx θ 1 x θ e λx (θ + λx )] 2 e λx log(x ). (49)
Austran Journal of Statstcs 67 β λ + (δ 1)x θ+1 e (α λ)x βx θ eλx C(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) x θ eλx (θ + λx + 1)α [α + βx θ 1 e λx (θ + λx )] 2 x θ+1 e λx. (5) β δ e (α λ)x βx θ eλx x θ A(x ; α, β, θ, λ). (51) θ 2 + (δ 1)βx θ e (α λ)x βx θ eλx (log(x )) 2 C(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) { [(θ βx θ 1 e λx + λx )(log(x )) 2 + 2 log(x ) ] } α βx θ 1 e λx β [α + βx θ 1 e λx (θ + λx )] 2 x θ e λx (log(x )) 2. (52) θ λ + β (δ 1)βx θ+1 e (α λ)x βx θ eλx log(x )C(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) βx θ eλx { [(θ + λx + 1) log(x ) + 1] α βx θ 1 [α + βx θ 1 e λx (θ + λx )] 2 e λx x θ+1 e λx log(x ). (53) } θ δ βe (α λ)x βx θ eλx x θ log(x ). (54) A(x ; α, β, θ, λ) λ 2 + (δ 1)βx θ+2 e (α λ)x βx θ eλx C(x ; α, β, θ, λ) A 2 (x ; α, β, θ, λ) βx θ+1 e λx [(θ + λx + 2)α βx θ 1 e λx ] [α + βx θ 1 β e λx (θ + λx )] 2 x θ+2 e λx. (55) λ δ βe (α λ)x βx θ eλx x θ+1. (56) A(x ; α, β, θ, λ) δ 2 nψ d log() (δ), where Ψ(δ) Γ (δ) dδ. (57)
68 A New Class of Generalzed Modfed Webull Dstrbuton wth Applcatons ## defne GGMW pdf GGMW_pdf <- functon(alpha, beta, theta, lambda, delta, x){ (1/gamma(delta)) * ((-log(1-exp(-alpha * x - beta * (x^theta) * (exp(lambda * x)))))^(delta-1)) * (alpha + beta * (x^(theta - 1)) * (exp(lambda * x)) * (theta + lambda * x)) * (exp(-alpha * x - beta * (x^theta) * (exp(lambda * x)))) } ## defne GGMW cdf GGMW_cdf <- functon(alpha, beta, theta, lambda, delta, x){ 1 - pgamma(-log(1 - exp(-alpha * x - beta * (x^theta) * exp(lambda * x))), delta) } ## defne GGMW Hazard GGMW_hazard <- functon(alpha, beta, theta, lambda, delta, x){ GGMW_pdf(alpha, beta, theta, lambda, delta, x) / (1 - GGMW_cdf(alpha, beta, theta, lambda, delta, x)) } ## defne GGMW moments GGMW_moments <- functon(alpha, beta, theta, lambda, delta, k){ f <- functon(alpha, beta, theta, lambda, delta, k, x){ (x^k) * (GGMW_pdf(alpha, beta, theta, lambda, delta, x)) } y <- ntegrate(f, lower, upper Inf, subdvsons 1, alpha alpha, beta beta, theta theta, lambda lambda, delta delta, k k) return(y) } ## defne GGMW quantle GGMW_quantle <- functon(alpha, beta, theta, lambda, delta, u){ f <- functon(x){alpha * x + beta * (x^theta) * (exp(lambda * x)) + log(1 - exp(-qgamma(1 - u, delta))) } rc <- unroot(f, lower, upper1, tol 1e-9) result <- rc$root # check error <- GGMW_cdf(alpha, beta, theta, lambda, delta, result) - u return(lst("result" result, "error" error)) } Afflaton: Broderck O. Oluyede Department of Mathematcal Scences Georga Southern Unversty Statesboro, GA 346 E-mal: boluyede@georgasouthern.edu Austran Journal of Statstcs publshed by the Austran Socety of Statstcs http://www.ajs.or.at/ http://www.osg.or.at/ Volume 44 Submtted: 214-5-12 October 215 Accepted: 215-1-27