Marshall-Olkin Log-Logistic Extended Weibull Distribution : Theory, Properties and Applications

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1 Journal of Data Scence 15(217), Marshall-Olkn Log-Logstc Extended Webull Dstrbuton : Theory, Propertes and Applcatons Lornah Lepetu 1, Broderck O. Oluyede 2, Bokanyo Makubate 3, Susan Foya 4 and Precous Mdlongwa 5 1,3,5 Botswana Internatonal Unversty of Scence and Technology, Palapye, Botswana 2 Georga Southern Unversty, Statesboro, GA, USA 4 Frst Natonal Bank, Goborone, Botswana Marshall and Olkn (1997) ntroduced a general method for obtanng more flexble dstrbutons by addng a new parameter to an exstng one, called the Marshall-Olkn famly of dstrbutons. We ntroduce a new class of dstrbutons called the Marshall - Olkn Log-Logstc Extended Webull (MOLLEW) famly of dstrbutons. Its mathematcal and statstcal propertes ncludng the quantle functon hazard rate functons, moments, condtonal moments, moment generatng functon are presented. Mean devatons, Lorenz and Bonferron curves, Rény entropy and the dstrbuton of the order statstcs are gven. The Maxmum lkelhood estmaton technque s used to estmate the model parameters and a specal dstrbuton called the Marshall-Olkn Log Logstc Webull (MOLLW) dstrbuton s studed, and ts mathematcal and statstcal propertes explored. Applcatons and usefulness of the proposed dstrbuton s llustrated by real datasets. Keywords: Marshall-Olkn, Log-Logstc Webull dstrbuton, Maxmum Lkelhood Estmaton. 1 Introducton Marshall and Olkn (1997) derved an mportant method of ncludng an extra shape parameter to a gven baselne model thus defnng an extended dstrbuton. The Marshall-Olkn transformaton provdes a wde range of behavors wth respect to the baselne dstrbuton (Santos-Neo et al. 214). Addng parameters to a well-establshed dstrbuton s a tme-honored devce for obtanng more flexble new famles of dstrbutons (Cordero and Lemonte 211). Several new models have been proposed that are some way related to the Webull dstrbuton whch s a very popular dstrbuton for modellng data n relablty, engneerng and bologcal studes. Extended forms of the Webull dstrbuton and applcatons n the lterature nclude Xe

2 692 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton et al. (22), Bebbngton et al. (27), Cordero et al. (21) and Slva et al. (21). When modelng monotone hazard rates the Webull dstrbuton may be an ntal choce because of ts negatvely and postvely skewed densty shapes, t however does not provde a reasonable ft for modelng phenomenon wth non-monotone falure rates such as bathtub shaped and unmodal falure rates whch are common n relablty, engneerng and bologcal Scences. Ths paper employs the Marshall-Olkn transformaton to the Log-Logstc Webull dstrbuton to obtan a new more flexble dstrbuton to descrbe relablty data. Marshall and Olkn appled the transformaton and generalzed the exponental and Webull dstrbuton. Subsequently, the Marshall- Olkn transformaton was appled to several well-known dstrbutons: Webull (Ghttany et al. 25, Zhang and Xe 27). More recently, general results have been addressed by Barreto-Souza et al. (213) and Cordero and Lemonte (213). Santos-Nero et al. (214) ntroduces a new class of models called the Marshall-Olkn extended Webull famly of dstrbutons whch defnes at least twenty-one specal models. Chakraborty and Handque (217) presented the generalzed Marshall-Olkn Kumaraswamy-G dstrbuton. Lazhar (216) developed and studed the propertes of the Marshall-Olkn extended generalzed Gompertz dstrbuton. Kumar (216) dscussed the rato and nverse moments of Marshall-Olkn extened Burr Type III dstrbuton based on lower generalzed order statstcs. However these authors do not employ the Marshall-Olkn transformaton n extendng the Log-Logstc Webull dstrbuton (Oluyede et al. 216). The combned dstrbuton of Log- Logstc and Webull s obtaned from the product of the relablty or survval functons of the Log-logstc and Webull dstrbuton. The Marshall-Olkn transformaton s then employed to obtan a new model called Marshall-Olkn Log-Logstc Extended Webull (MOLLEW) dstrbuton. A motvaton for developng ths model s the advantages presented by ths extended dstrbuton wth respect to havng a hazard functon that exhbts ncreasng, decreasng and bathtub shapes, as well as the versatlty and flexblty of the log-logstc and Webull dstrbutons n modelng lfetme data. The results n ths paper are organzed n the followng manner. In secton 2, we present the extended model called Marshall-Olkn Log-Logstc Extended Webull (MOLLEW) dstrbuton, quantle functon and hazard functon. In secton 3, moments, moment generatng functon and condtonal moments are presented. Lorenz and Bonferron curves are also presented n secton 3. Secton 4 contan results on Rény entropy. In Secton 5, maxmum lkelhood estmates of the model parameters are gven. In secton 6, the specal case, that s, the new model called the Marshall-Olkn Log-Logstc Webull (MOLLW) dstrbuton ts sub models, densty expanson, quantle functon, hazard and reverse hazard functon, moments, moment generatng functon, condtonal moments, Lorenz and Bonferron curves and Rény entropy are derved. Max-

3 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 693 mum lkelhood estmates of the model parameters are gven n secton 7. A Monte Carlo smulaton study to examne the bas and mean square error of the maxmum lkelhood estmates are presented n secton 8. Secton 9 contans applcatons of the new model to real data sets. A short concluson remark s gven n secton 1. 2 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton In ths secton, the model, hazard functon and quantle functon of the Marshall- Olkn Log-Logstc Extended Webull (MOLLEW) dstrbuton are presented. Frst, we present the class of extended Webull dstrbutons, Log-Logstc Webull dstrbuton and the Marshall-Olkn Log-Logstc Extended Webull dstrbuton. Gurvch et al. (1997) poneered the class of extended Webull (EW) dstrbutons. The cumulatve dstrbuton functon (cdf) s gven by G(x; α, ξ) = 1 exp[ αh(x; ξ)], x D, α >, (2.1) where D s a subset of R, and H(x; ξ) s a non-negatve functon that depends on the vector of parameters ξ. The correspondng probablty densty functon (pdf) s gven by g(x; α, ξ) = α exp[ αh(x; ξ)]h(x; ξ), (2.2) where h(x; ξ) s the dervatve of H(x; ξ). The choce of the functon H(x; ξ) leads to dfferent models ncludng for example, exponental dstrbuton wth H(x; ξ) = x, Raylegh dstrbuton s obtaned from H(x; ξ) = x 2 and Pareto dstrbuton from settng H(x; ξ) = log(x/k). Now consder a general case called log-logstc extended Webull famly of dstrbutons. The dstrbuton s obtaned va the use of competng rsk model and s gven by combnng both the log-logstc and extended Webull famly of dstrbutons as gven below (Oluyede et al. 216). The correspondng pdf s gven by f(x; c, α, ξ) = e αh(x;ξ) (1 + x c ) 1 { αh(x; ξ) + cx c 1 (1 + x c ) 1}, (2.3) for c, α, ξ > and x. The cumulatve dstrbuton functon (cdf) of the dstrbuton s gven by F (x; c, α, β) = 1 (1 + x c ) 1 e αh(x;ξ), (2.4) for c, α, ξ >. Santos-Neto et al. (214) proposed a new class of models called the Marshall-Olkn extended Webull (MOEW) famly of dstrbutons based

4 694 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton on the work by Marshall and Olkn (1997). The survval functon s gven as G(x; α, δ, ξ) = δ exp[ αh(x; ξ)] 1 δ exp[ αh(x; ξ)], (2.5) where δ = 1 δ and for α, δ >. The assocated hazard rate reduces to h(x; α, δ, ξ) = δh(x; ξ), x D, α >, δ >. (2.6) 1 δ exp[ αh(x; ξ)] The correspondng pdf s gven by g(x; α, δ, ξ) = δαh(x; ξ) exp[ αh(x; ξ)], x D, α >, δ >, (2.7) [1 δ exp[ αh(x; ξ)]] 2 where δ = 1 δ. The general case of Marshall-Olkn Log-Logstc Extended Webull (MOLLEW) famly of dstrbutons has survval functon that s gven by G MOLLEW (x; c, α, δ, ξ) = δ(1 + xc ) 1 e αh(x;ξ). (2.8) 1 δ(1 + x c ) 1 e αh(x;ξ) The pdf s gven by g MOLLEW (x; c, α, δ, ξ) = δ(1 + xc ) 1 e αh(x;ξ) {αh(x; ξ) + cx c 1 (1 + x c ) 1 } [1 δ(1 + x c ) 1 e αh(x;ξ) ] 2 (2.9) for c, α, ξ, δ > and x. The parameter α control the scale of the dstrbuton and c and ξ controls the shape of the dstrbuton and δ s the tlt parameter. The hazard and reverse functons of the MOLLEW dstrbuton are gven by and h MOLLEW (x; c, α, ξ, δ) = αh(x; ξ) + cxc 1 (1 + x c ) 1 1 δ(1 + x c ) 1 e αh(x;ξ), (2.1) τ MOLLEW (x; c, α, ξ, δ) = αh(x; ξ) + cxc 1 (1 + x c ) 1 1 (1 + x c ) 1 e αh(x;ξ), (2.11) for c, α, ξ, δ > and x, respectvely. 2.1 Quantle Functon The MOLLEW quantle functon can be obtaned by nvertng G(x) = 1 u, where G(x) = u, u 1, and G MOLLEW (x; c, α, β, δ, ξ) = δ(1 + xc ) 1 e αh(x;ξ). (2.12) 1 δ(1 + x c ) 1 e αh(x;ξ)

5 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 695 The quantle functon of the MOLLEW dstrbuton s obtaned by solvng the non-lnear equaton [ ] αh(x; ξ) + log (1 + x c 1 u ) + log =, (2.13) δ + (1 u)δ usng numercal methods. based on equaton (2.13). Consequently, random number can be generated 2.2 Expanson for the Densty Functon Usng the generalzed bnomal expresson (1 z) k = j= Γ (k + j) z j, for z < 1, (2.14) Γ (k)j! the MOLLEW pdf can be rewrtten as follows: g MOLLEW (x) = (j + 1)δδ j [(1 + x c ) 1 e αh(x;ξ) ] j (1 + x c ) 1 e αh(x;ξ) j= { αh(x; ξ) + cx c 1 (1 + x c ) 1} = δδ j (1 + x c ) (j+1) 1 e α(j+1)h(x;ξ) j= { α(j + 1)h(x; ξ)(1 + x c ) + c(j + 1)x c 1 ) } = w(j, δ)f BW (x, c, j + 1, α(j + 1), ξ), (2.15) j= where w(j, δ) = δδ j and f BW (x, c, j + 1, α(j + 1), ξ) s the Burr XII Extended Webull pdf wth parameters for c, α(j + 1), ξ, δ > and x, respectvely. Thus MOLLEW pdf can be wrtten as a lnear combnaton of Burr XII Extended Webull densty functons. The mathematcal and statstcal propertes of the MOLLEW densty functon follows drectly from those of the Burr XII Extended Webull densty functon. 3 Moments, Moment Generatng Functon and Condtonal Moments In ths secton, moments, moment generatng functon and condtonal moments are gven for the MOLLEW dstrbuton. The r th moment of the MOLLEW

6 696 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton dstrbuton s gven by E(X r ) = = x r g MOLLEW (x)dx x r δ(1 + x c ) 1 e αh(x;ξ) { αh(x; ξ) + cx c 1 (1 + x c ) 1} [1 δ(1 + x c ) 1 e αh(x;ξ)] 2 dx [ = x r (j + 1)δδ j [(1 + x c ) 1 e αh(x;ξ) ] j (1 + x c ) 1 e αh(x;ξ) j= { αh(x; ξ) + cx c 1 (1 + x c ) 1} ] dx = [ δδ j (α(j + 1)) j= + (c(j + 1) Applyng the power seres expanson x r (1 + x c ) (j+1) e α(j+1)h(x;ξ) h(x; ξ)dx x r+c 1 (1 + x c ) (j+1) 1 e α(j+1)h(x;ξ) ]dx. e α(j+1)h(x;ξ) = ( 1) [α(j + 1)H(x; ξ)],! we have E(X r ) = [ δδ j (α(j + 1)) j= + (c(j + 1)) = j= δδ j ( 1)! x r (1 + x c ) (j+1) x r+c 1 (1 + x c ) (j+1) 1 ( 1) [α(j + 1)H(x; ξ)] h(x; ξ)dx! ] ( 1) [α(j + 1)H(x; ξ)] dx! [ (α (j + 1) ) x r (1 + x c ) (j+1) h(x; ξ)h(x; ξ) dx + ( cα ) x r+c 1 (1 + x c ) (j+1) 1 H(x; ξ) ]dx. (3.1)

7 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa Condtonal Moments The r th condtonal moment for MOLLEW dstrbuton s gven by E(X r X > t) = = 1 x r g MOLLEW (x)dx G MOLLEW (t) t 1 G MOLLEW (t) [ δδ j ( 1) (j + 1) +1! j= [ (α ) +1 x r (1 + x c ) (j+1) h(x; ξ)h(x; ξ) dx t + ( cα ) x r+c 1 (1 + x c ) (j+1) 1 H(x; ξ) ]dx. t (3.2) Note that once H(x; ξ) s specfed, the moment and condtonal moments can be readly obtaned. 3.2 Bonferron and Lorenz Curves In ths subsecton, we present Bonferron and Lorenz curves. Bonferron and Lorenz curves have applcatons not only n economcs for the study ncome and poverty, but also n other felds such as relablty, demography, nsurance and medcne. Bonferron and Lorenz curves for the MOLLEW dstrbuton are gven by B(p) = 1 q xg pµ MOLLEW (x)dx = 1 [µ T (q)], pµ and L(p) = 1 µ q xg MOLLEW (x)dx = 1 [µ T (q)], µ respectvely. The specal cases for specfed H(x; ξ) can be readly computed. 4 Rény Entropy 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 for the MOLLEW dstrbuton. Rény entropy s an extenson of Shannon entropy. Rény entropy s defned to be I R (v) = 1 1 v log ( ) [g(x; c, α, ξ, δ)] v dx, v 1, v >. (4.1)

8 698 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton Rény entropy tends to Shannon entropy as v 1. Note that [g(x; c, α, ξ, θ)] v = g v (x) can be wrtten as [ g v (x) = δ(1 + x c ) 1 e { αh(x;ξ) αh(x; ξ) + cx c 1 (1 + x c ) 1} Thus, [1 δ(1 + x c ) 1 e αh(x;ξ) ] 2 ] v = g v (x)dx = v j= p= w= ( v w )( ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p Γ (2v)j!p! α p+v w (1 + x c ) j v w x cw w h(x; ξ) v w H(x; ξ) p. v j= p= w= ( v w )( ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p Γ (2v)j!p! α p+v w (1 + x c ) j v w x cw w h(x; ξ) v w H(x; ξ) p dx. Consequently, Rény entropy s gven by ( ) ( 1 v ( )( v ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p I R (v) = log 1 v w Γ (2v)j!p! j= p= w= ) α p+v w (1 + x c ) j v w x cw w h(x; ξ) v w H(x; ξ) p dx for v 1, and v >. 5 Maxmum Lkelhood Estmaton Let X MOLLEW (c, α, ξ, δ) and = (c, α, ξ, δ) T be the parameter vector. The log-lkelhood functon l = l( ) based on a random sample of sze n s gven by l( ) = n log δ log (1 + x c ) α H(x ; ξ) + ( ) log αh(x ; ξ) + cx c 1 (1 + x c ) 1 2 The elements of the score vector U = ( ) l, l, l, l δ α ξ k c l δ = n δ 2 (1 + x c) 1 e αh(x ;ξ) 1 δ(1 + x c) 1 e, αh(x ;ξ) ) log (1 δ(1 + x c ) 1 e αh(x ;ξ). are gven by

9 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 699 l ξ k = and l c l α = H(x ; ξ) + = 2δ α H(x ; ξ) + α ξ k H(x 2δα ; ξ) ξ k + xc 1 αh(x ; ξ) αh(x ; ξ) + cx c 1 (1 + x c ) 1 H(x ; ξ)(1 + x c ) 1 e αh(x ;ξ) 1 δ(1 + x c ) 1 e αh(x ;ξ), x c log x (1 + x c ) 2α h(x ; ξ) ξ k 1 αh(x ; ξ) + cx c 1 (1 + x c ) 1 (1 + x c ) 1 e αh(x;ξ) 1 δ(1 + x c ) 1 e αh(x ;ξ), δ(1 + x c ) 1 e αh(x ;ξ) 1 δ(1 + x c ) 1 e αh(x ;ξ) log x (1 + x c) 1 cx 2c 1 (1 + x c) 2 log x + cx c 1 (1 + x c) 1 log x. αh(x ; ξ) + cx c 1 (1 + x c) 1 The equatons are obtaned by settng the above partal dervatves to zero are not n closed form and the values of the parameters c, α, ξ and δ must be found va teratve methods. The maxmum lkelhood estmates of the parameters, denoted by ˆ s obtaned by solvng the nonlnear equaton ( l, l, l, l c α ξ k δ )T =, usng a numercal method such as Newton-Raphson procedure. The Fsher nformaton matrx s gven by I( ) = [I θ,θ j ] 4X4 = E( 2 l θ θ j ),, j = 1, 2, 3, 4, can be numercally obtaned by MATLAB or NLMIXED n SAS or R software. The total Fsher nformaton matrx ni( ) can be approxmated by [ ] J n ( ˆ ) 2 l θ θ j,, j = 1, 2, 3, 4. (5.1) = ˆ 4X4 For a gven set of observatons, the matrx gven n equaton (5.1) s obtaned after the convergence of the Newton-Raphson procedure va NLMIXED n SAS or R software. 6 Marshal-Olkn Log-Logstc Webull (MOLLW) Dstrbuton Now consder the Log-Logstc-Webull dstrbuton. Ths dstrbuton s obtaned va the use of competng rsk model and s gven by combnng both the

10 7 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton Log-Logstc and Webull dstrbuton. In ths case, we take H(x; ξ) = x β n the MOLLEW dstrbuton. The MOLLW cdf s gven by F (x; c, α, β) = 1 (1 + x c ) 1 e αxβ, (6.1) for c, α, β >. The correspondng pdf s gven by f(x; c, α, β) = e αxβ (1 + x c ) 1 { αβx β 1 + cx c 1 (1 + x c ) 1} (6.2) for c, α, β > and x. Recall the extended case, Marshall-Olkn Log Logstc Extended Webull (MOLLEW) dstrbuton has survval functon gven by G MOLLEW (x; c, α, β, δ, ξ) = δ(1 + xc ) 1 e αh(x;ξ) (6.3) 1 δ(1 + x c ) 1 e αh(x;ξ) wth correspondng pdf gven by g MOLLEW (x; c, α, β, δ, ξ) = δ(1 + xc ) 1 e { αh(x;ξ) αβx β 1 + cx c 1 (1 + x c ) 1}. [1 δ(1 + x c ) 1 e αxβ ] 2 (6.4) In ths secton, we study a specal case of the famly, namely Marshall- Olkn Log-Logstc Webull (MOLLW) dstrbuton, by takng H(x; ξ) = x β. The MOLLW survval functon s gven by G MOLLW (x; c, α, β, δ) = δ(1 + xc ) 1 e αxβ. (6.5) 1 δ(1 + x c ) 1 αxβ e The correspondng pdf s gven by g MOLLW (x; c, α, β, δ) = δ(1 + xc ) 1 e αxβ { αβx β 1 + cx c 1 (1 + x c ) 1} [1 δ(1 + x c ) 1 e αxβ ] 2 (6.6) for c, α, β, δ > and x. Note that the parameter α control the scale of the dstrbuton, c and β controls the shape of the dstrbuton and δ s the tlt parameter. The plots of the MOLLW pdf s gven n Fgure 6.1. The plots shows that the pdf can be L-shaped, decreasng, unmodal dependng on selected parameter values. 6.1 Quantle Functon The quantle functon of the MOLLW dstrbuton s obtaned by solvng the non-lnear equaton [ ] αx β + log (1 + x c 1 u ) + log =. (6.7) δ + (1 u)δ

11 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 71 Fgure 6.1: Plots of MOLLW PDF The quantle functon of the MOLLW dstrbuton s obtaned by solvng the non-lnear equaton usng numercal methods. Consequently, random number can be generated based on equaton (6.7). Table 6.1 lsts the quantle for selected values of the parameters of the MOLLW dstrbuton. Table 6.1: MOLLW Quantle for Selected Values (c, α, β, δ) u (2.,2.,.5,.5,.1) (2,.5,2.,.5) (.5,.5,1.5,1.5) (.4,1.,.4,.7) (1.,1.5,2.,2.)

12 72 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton 6.2 Hazard and Reverse Hazard Functons The hazard and reverse functons of the MOLLW dstrbuton are gven by and h MOLLW (x; c, α, β, δ) = αβxβ 1 + cx c 1 (1 + x c ) 1 1 δ(1 + x c ) 1 e αxβ, (6.8) τ MOLLW (x; c, α, β, δ) = (x; c, α, β, δ) = αβxβ 1 + cx c 1 (1 + x c ) 1 1 (1 + x c ) 1 e αxβ, (6.9) for c, α, β, δ > and x, respectvely. Plots of the MOLLW hazard functon are gven n Fgure 6.2. The plots shows that the hazard functon of the MOLLW can ether be decreasng, bathtub followed by upsde down bathtub or ncreasng-decreasng dependng on selected parameter values. Fgure 6.2: Plots of MOLLW Hazard 6.3 Some Sub-models There are several new as well as well known dstrbutons that can be obtaned from the MOLLW dstrbuton. The sub-models nclude the followng dstrbutons: When δ = 1, we obtan the Log-Logstc Webull (LLW) dstrbuton. When β = 2, we obtan the Marshall-Olkn Log-Logstc Raylegh (MOLLR) dstrbuton.

13 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 73 If α +, we obtan the Marshall-Olkn Log-Logstc (MOLL) dstrbuton. If c = 1, then the Marshall-Olkn Log-Logstc Webull dstrbuton reduces to the 3 parameter dstrbuton wth survval functon G MOLLW (x; α, δ, β) = δ(1 + x) 1 e αxβ. (6.1) 1 δ(1 + x) 1 αxβ e If c = 1 and β = 1, then the Marshall-Olkn Log-Logstc Webull dstrbuton reduces to the 2 parameter dstrbuton wth survval functon G MOLLW (x; α, δ) = δ(1 + x) 1 e αx. (6.11) 1 δ(1 + x) 1 e αx If c = 1 and β = 2, then the Marshall-Olkn Log-Logstc Webull dstrbuton reduces to the 2 parameter dstrbuton wth survval functon gven by G MOLLW (x; α, δ) = δ(1 + x) 1 e αx2. (6.12) 1 δ(1 + x) 1 αx2 e 6.4 Expanson for the Densty Functon Usng the generalzed bnomal expresson (1 z) k = j= Γ (k + j) z j, for z < 1, Γ (k)j! the MOLLW pdf can be rewrtten as follows: g MOLLW (x) = w(j, δ)f BW (x, c, j + 1, α(j + 1), β), (6.13) j= where w(j, δ) = δδ j and f BW (x, c, j + 1, α(j + 1), β) s the Burr XII Webull pdf wth parameters for c >, j + 1 >, α(j + 1) >, and β > respectvely. Thus MOLLW pdf can be wrtten as a lnear combnaton of Burr XII-Webull densty functons. The mathematcal and statstcal propertes of the MOLLW dstrbuton follows drectly from those of the Burr XII Webull densty functon.

14 74 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton 6.5 Moments, Moment Generatng Functon and Condtonal Moments In ths secton, moments, moment generatng functon and condtonal moments are gven for the MOLLW dstrbuton. The r th moment of the MOLLW dstrbuton s gven by E(X r ) = [ δδ j (α(j + 1)β) j= + (c(j + 1) Applyng the power seres expanson x r+β 1 (1 + x c ) (j+1) e α(j+1)xβ dx x r+c 1 (1 + x c ) (j+1) 1 e α(j+1)xβ ]dx. e α(j+1)xβ = ( 1) [α(j + 1)x β ],! we have [ E(X r ) = δδ j (α(j + 1)β) x r+β 1 (1 + x c ) (j+1) ( 1) [α(j + 1)x β ] dx j=! ] + (c(j + 1)) x r+c 1 (1 + x c ) (j+1) 1 ( 1) [α(j + 1)x β ] dx! [ = δδ j ( 1) (α (j + 1) β ) x r+β+β 1 (1 + x c ) (j+1) dx! j= + ( cα ) x r+c+β 1 (1 + x c ) ]dx. (j+1) 1 Let y = (1 + x c ) 1, then x = ( 1 y ) 1 c, and dx = c 1 y 2 (1 y) 1 c 1 y 1 1 c dy, y (6.14)

15 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 75 so that E(X r ) = j= 1 j= [ δδ j ( 1) (α (j + 1) β )! ) 1 c ) r+β+β 1 y j+1 y 2 c 1 (1 y) 1 c 1 y 1 1 c dy (( 1 y y + ( cα ) 1 (( 1 y y [ = δδ j ( 1) (c (j + 1) +1 1 α +1 β ) 1! + α 1 ) 1 ) r+c+β 1 ] c y j+2 y 2 c 1 (1 y) 1 c 1 y 1 1 c dy y r c β c +j (1 y) r c + β c dy ] y r c β c β c +j (1 y) r c + β c + β c 1 dy [ = δδ j ( 1) (c (j + 1) +1 1 α +1 β ) B (j 1c! (r + β + β c), 1c ) (r + β + β j= + α B (j 1c (r + β c), 1c ) ] (r + β c, (6.15) where B(a, b) = 1 ta 1 (1 t) b 1 dt s the complete beta functon. The moment generatng functon (MGF) of the MOLLW dstrbuton s gven by M X (t) = E(e tx ) = n= t n n! E(Xn ), where E(X n ) s gven above. Table 6.2 lsts the frst fve moments together wth the standard devaton (SD or σ), coeffcent of varaton (CV), coeffcent of skewness (CS) and coeffcent of kurtoss (CK) of the MOLLW dstrbuton for selected values of the parameters, by fxng β = 1.5 and δ = 1.5. Table 6.3 lsts the frst fve moments, SD, CV, CS and CK of the MOLLW dstrbuton for selected values of the parameters, by fxng c = 1. and α = 1.5. These values can be determned numercally usng R and MATLAB. The SD, CV, CS and CK are gven by σ = µ 2 µ 2, and CV = σ µ = µ 2 µ 2 µ = µ 2 µ 2 1, CS = E [(X µ)3 ] [E(X µ) 2 ] 3/2 = µ 3 3µµ 2 + 2µ 3 (µ 2 µ 2 ) 3/2, CK = E [(X µ)4 ] [E(X µ) 2 ] 2 = µ 4 4µµ 3 + 6µ 2 µ 2 3µ 4 (µ 2 µ 2 ) 2, respectvely. Plots of the skewness and kurtoss for selected choces of the parameter β as a functon of c as well as for some selected choces of c as a functon of β are

16 76 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton dsplayed n fgure 6.3. The plots show that the skewness and kurtoss depend on the shape parameters c and β. Fgure 6.3: Plots of skewness and kurtoss for selected parameter values

17 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 77 Table 6.2: Moments of the MOLLW dstrbuton for some parameter values; β = 1.5 and δ = 1.5. µ s c =.5, α =.5 c =.5, α =.5 c = 2., α = 2. c = 2., α = 2. µ µ µ µ µ SD CV CS CK Table 6.3: Moments of the MOLLW dstrbuton for some parameter values; c = 1. and α = 1.5. µ s β =.7, δ = 1.2 β = 1., δ = 1. β = 2.5, δ = 2.5 β = 1.5, δ =.4 µ µ µ µ µ SD CV CS CK Condtonal Moments The r th condtonal moment for MOLLW s gven by E(X r X > t) = 1 G MOLLW (t) = 1 G MOLLW (t) t t x r g MOLLW (x)dx (1 δ(1 + x c ) 1 e αxβ ) 2 dx. x r δ(1 + x c ) 1 e αxβ { αβx β 1 + cx c 1 (1 + x c ) 1}

18 78 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton Let y = (1 + x c ) 1, then x = ( 1 y ) 1 c, and dx = c 1 y 2 (1 y) 1 c 1 y 1 1 c dy,. Now y [ E(X r 1 X > t) = δδ j ( 1) (j + 1) +1 G MOLLW (t) j= [ (c 1 α +1 β ) B (1+tc) 1! (j 1c (r + β + β c), 1c ) (r + β + β + α B (1+t c ) 1 ( j 1 c (r + β c)), 1 c (r + β c ) ], (6.16) where B y (a, b) = y xa 1 (1 x) b 1 dx s the ncomplete beta functon. The mean resdual lfetme functon of the MOLLW dstrbuton s gven by E(X X > t) t. 6.7 Bonferron and Lorenz Curves In ths subsecton, we present Bonferron and Lorenz Curves. Bonferron and Lorenz curves have applcatons not only n economcs for the study ncome and poverty, but also n other felds such as relablty, demography, nsurance and medcne. Bonferron and Lorenz curves for the MOLLW dstrbuton are gven by B(p) = 1 q xg pµ MOLLW (x)dx = 1 [µ T (q)], pµ and L(p) = 1 µ q xg MOLLW (x)dx = 1 [µ T (q)], µ respectvely, where T (q) = xg(x)dx, s gven by q τ(q) = q xg MOLLW (x)dx = and q = G 1 (p), p 1. δδ j ( 1) (j + 1) ( +1 c 1 α +1 β )! j= [B (j 1c (r + β + β c), 1c ) (r + β + β α (1+qc) 1 + B (1+q c ) 1 ( j 1 c (r + β c), 1 c (r + β c ) ] (6.17) 6.8 Rény Entropy 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 subsecton,

19 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 79 Rény entropy of the MOLLW dstrbuton s derved. An entropy s a measure of uncertanty or varaton of a random varable. Rény entropy s an extenson of Shannon entropy. Note that [g(x; c, α, β, θ)] v = g v (x) can be wrtten as { g v (x) = [δ(1 + x c ) 1 e αxβ αβx β 1 + cx c 1 (1 + x c ) 1} ] v [1 δ(1 + x c ) 1 e αxβ ] 2 Now, = v j= p= w= ( v w )( ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p Γ (2v)j!p! α p+v w β v w c w (1 + x c ) j v w x βp βw+βv+cw v. g v (x)dx = v j= p= w= ( v w )( ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p Γ (2v)j!p! α p+v w β v w c w (1 + x c ) j v w x βp βw+βw+cw v dx v ( )( v ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p α p+v w β v w c w 1 = w Γ (2v)j!p! = = j= p= w= 1 x βv βw+βp+cw v (1 + x c ) j v w dx v j= p= w= 1 B ( v w )( ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p α p+v w β v w c w 1 Γ (2v)j!p! y 1 c (βw βv βp+v+jc+vc c 1) (1 y) 1 c (βv βw+βp+cw v+1 c) dy v j= p= w= ( v w )( ( 1) p Γ (j + 2v)δ v δ j ) (v + j) p α p+v w β v w c w 1 Γ (2v)j!p! ( 1 c (βw βv βp + vc + jc + v + 1), 1 (βv βw + βp + cw v + 1) c Consequently, Rény entropy s gven by ). (6.18) ( ) [ 1 v ( )( v ( 1) p Γ (j + 2v)δ v δ j )] (v + j) p α p+v w β v w c w 1 I R (v) = log 1 v w Γ (2v)j!p! j= p= w= ( 1 B c (βw βv βp + vc + jc + v + 1) ; 1 ) (βv βw + βp + cw v + 1) c (6.19) for v 1, and v >.

20 71 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton 7 Maxmum Lkelhood Estmaton Let X MOLLW (c, α, β, δ) and = (c, α, β, δ) T be the parameter vector. The log-lkelhood functon l = l( ) based on a random sample of sze n s gven by l( ) = n log δ + log (1 + x c ) α log { αβx β 1 x β + cx c 1 (1 + x c ) 1} 2 The elements of the score functon are gven by log(1 δ(1 + x c ) 1 e αxβ ). l δ = n δ 2 (1 + x c ) 1 e αxβ 1 δ(1 + x c ) 1 e αxβ, l α = x β βx β 1 + αβx β 1 + cx c 1 (1 + x c) 1 x β (1 + x c 2δ ) 1 e αxβ, 1 δ(1 + x c) 1 e αxβ l β = α x β log x + 2δα αx β 1 x β(1 + xc) 1 e αxβ log x, 1 δ(1 + x c) 1 e αxβ + αβx β 1 log x αβx β 1 + cx c 1 (1 + x c ) 1 and l c = + xc 1 x c log x (1 + x c) 2α δ(1 + x c ) 2 e αxβ x c log x 1 δ(1 + x c ) 1 e αxβ log x (1 + x c) 1 cx 2c 1 (1 + x c) 2 log x + cx c 1 (1 + x c) 1 log x. αβx β 1 + cx c 1 (1 + x c) 1 Note that the expectatons n the Fsher Informaton Matrx (FIM) can be obtaned numercally. Let ˆ = (ĉ, ˆα, ˆβ, ˆδ) be the maxmum lkelhood estmate of = (c, α, β, δ). Under the usual regularty condtons and that the parameters are n the nteror of the parameter space, but not on the boundary, we have: n( ˆ ) N 4 (, I 1 ( )), where I( ) s the d expected

21 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 711 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 4 (, 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 c, k, α, β and δ are gven by: ĉ ± Z η 2 I 1 cc ( ), α ± Z η 2 I 1 αα( ), and δ ± Z η 2 I 1 δδ ( ), β ± Z η I 1 2 ββ ( ), respectvely, where I 1 cc ( ), I 1 αα( ), I 1 ββ ( ), and I 1 δδ ( ), are the dagonal elements of I 1 n ( ˆ ) = (ni( ˆ )) 1, and Z η s the upper η th percentle of a standard 2 2 normal dstrbuton. 8 Smulaton Study In ths secton, we study the performance and accuracy of maxmum lkelhood estmates of the MOLLW model parameters by conductng varous smulatons for dfferent sample szes and dfferent parameter values. Equaton 6.7 s used to generate random data from the MOELLW dstrbuton. The smulaton study s repeated for N = tmes each wth sample sze n = 25, 5, 75, 1, 2, 4, 8 and parameter values I : c =.3, α = 1.3, β = 1.8, δ = 1.4 and II : c = 8.5, α =.3, β =.3, δ =.4. Four quanttes are computed n ths smulaton study. (a) Average bas of the MLE ˆϑ of the parameter ϑ = c, α, β, δ : 1 N N ( ˆϑ ϑ). =1 (b) Root mean squared error (RMSE) of the MLE ˆϑ of the parameter ϑ = c, α, β, δ : 1 N N ( ˆϑ ϑ) 2. =1 (c) Coverage probablty (CP) of 95% confdence ntervals of the parameter ϑ = c, α, β, δ,.e., the percentage of ntervals that contan the true value of parameter ϑ.

22 712 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton (d) Average wdth (AW) of 95% confdence ntervals of the parameter ϑ = c, α, β, δ. Table 8.1 presents the Average Bas, RMSE, CP and AW values of the parameters c, α, δ and β for dfferent sample szes. From the results, we can verfy that as the sample sze n ncreases, the RMSEs decay toward zero. We also observe that for all the parametrc values, the bases decrease as the sample sze n ncreases. Also, the table shows that the coverage probabltes of the confdence ntervals are qute close to the nomnal level of 95% and that the average confdence wdths decrease as the sample sze ncreases. Consequently, the MLE s and ther asymptotc results can be used for estmatng and constructng confdence ntervals even for reasonably small sample szes. Table 8.1: Monte Carlo Smulaton Results: Average Bas, RMSE, CP and AW I II Parameter n Average Bas RMSE CP AW Average Bas RMSE CP AW c α β δ Applcatons In ths secton, we present examples to llustrate the flexblty of the MOLLW dstrbuton and ts sub-models for data modelng. We ft the densty functon

23 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 713 of the MOLLW, Marshall-Olkn log logstc exponental (MOLLE), Marshall- Olkn log logstc Raylegh (MOLLR), Marshall-Olkn log logstc and log logstc Webull (LLW) dstrbutons. We also compare the MOLLW dstrbuton to other models ncludng the gamma-dagum (GD) (Oluyede et al. 216) and beta Webull (Lee et al. 27) dstrbutons. The GD and BW pdfs are gven by and g GD (x) = λβδx δ 1 (1 + λx δ ) β 1 ( log[1 (1 + λx δ ) β ]) α 1, x >, Γ (α) g BW (x) = kλk B(a, b) xk 1 e b(λx)k (1 e (λx)k ) a 1, x >, respectvely. The maxmum lkelhood estmates (MLEs) of the MOLLW parameters c, α, β, δ are computed by maxmzng the objectve functon va the subroutne NLMIXED n SAS as well (bbmle) package n R. The estmated values of the parameters (standard error n parenthess), -2log-lkelhood statstc, Akake Informaton Crteron, AIC = 2p 2 ln(l), Consstent Akake Informaton Crteron, (AICC = AIC + 2p(p+1) n p 1 ) and Bayesan Informaton Crteron, BIC = p ln(n) 2 ln(l), 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 n Tables 9.2 and 9.4 for the MOLLW dstrbuton and ts sub-models MOLLE, MOLLR, MOLL, LLW, LLE and LW dstrbutons and alternatves (non-nested) GD and BW dstrbutons. The Cramer von Mses and Anderson-Darlng goodness-of-ft statstcs W and A, are also presented n these tables. These statstcs can be used to verfy whch dstrbuton fts better to the data. In general, the smaller the values of W and A, the better the ft. The AdequacyModel package was used to evaluate the statstcs stated above. We can use the lkelhood rato (LR) test to compare the ft of the MOLLW dstrbuton wth ts sub-models for a gven data set. For example, to test β = 1, the LR statstc s ω = 2[ln(L(ĉ, ˆα, ˆβ, ˆδ)) ln(l( c, α, 1, δ))], where ĉ, ˆα, ˆβ and ˆδ are the unrestrcted estmates, and c, α and δ are the restrcted estmates. The LR test rejects the null hypothess f ω > χ 2, where ɛ χ2 denote ɛ the upper 1ɛ% pont of the χ 2 dstrbuton wth 1 degrees of freedom. 9.1 Glass Fbers Data Specfcally we consder the followng data set whch conssts of 63 observatons of the strengths of 1.5 cm glass fbers, orgnally obtaned by workers at the UK Natonal Physcal Laboratory. The data was also studed by (Smth and Naylor 1987). The data observatons are gven below:

24 714 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton Table 9.1: Glass fbers data Estmates of the parameters of MOLLW dstrbuton and ts related submodels (standard error n parentheses), 2 ln(l), AIC, AICC, BIC, W and A are gven n Table 9.2. Intal values for the MOLLW model n R code are (c =.6, α =.15, β =.15, δ = 1.4). Table 9.2: Estmates of Models for Glass fbers Data Estmates Statstcs Model ĉ ˆα ˆβ ˆδ 2 log L AIC AICC BIC W A MOLLW (1.8197) (.6353) (1.9556) ( ) MOLLE (1.5263) (.935) - ( ) MOLLR (2.1648) (.4351) - (39.869) MOLL (.8725) (12.727) LLW ).885) (.3961) (.53) (.8923) ˆλ ˆβ ˆδ ˆα GD ( ) (.1939) (3.12) (3.1763) ˆλ ˆk â ˆb BW (.119).8567 (.3594) The asymptotc covarance matrx of the MLE s for the MOLLW model parameters whch s the observed Fsher nformaton matrx I 1 n ( ) s gven by: ,

25 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 715 and the 95% confdence ntervals for the model parameters are gven by c (2.3671± ), α (.3779± ), β (3.7953± ) and δ ( ± ), respectvely. The LR test statstc of the hypothess H : MOLLE aganst H a : MOLLW, H : MOLLR aganst H a : MOLLW, and H : MOLL aganst H a : MOLLW and LLW aganst H a : MOLLW, are (p-value =.45), (p-value =.251), (p-value= ) and (p-value=.1). We can conclude that there are sgnfcant dfferences between MOLLW and ts sub-models: MOLLE, MOLLR, MOLL, and LLW dstrbutons at the 5% level of sgnfcance. The values of the statstcs: AIC, AICC and BIC also shows that the MOLLW dstrbuton s a better ft than the non-nested GD and BW dstrbutons for the glass fbers data. There s also clear evdence based on the goodness-of-ft statstcs W and A that the MOLLW dstrbuton s by far the better ft for the glass fbers data. Plot of the ftted denstes, and the hstogram of the data are gven n Fgure 9.1. Fgure 9.1: Ftted denstes for glass fber data 9.2 Breakng Stress of Carbon Fbres (n Gba) Data Ths data set conssts of 66 uncensored data on breakng stress of carbon fbres (n Gba) (Ncholas and Padgett 26). The data values are gven Table 9.3. Estmates of the parameters of MOLLW dstrbuton and ts related submodels (standard error n parentheses), 2 ln(l), AIC, AICC, and BIC are gven n Table 9.2. Intal values for the MOLLW model n R code are (c = 5.326, α =.15, β = 1.56, δ = 1.4).

26 716 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton Table 9.3: Carbon fbers data Table 9.4: Estmates of Models for Carbon fbers Data Estmates Statstcs Model ĉ ˆα ˆβ ˆδ 2 log L AIC AICC BIC W A MOLLW (1.551) (.6144) (1.1526) ( ) MOLL (.5137) ( ) LLW (.173) (.442) (.55) LLE (.1473) (.4461) ˆλ ˆβ ˆδ ˆα GD (2.5145) (1.4628) (.2773) (.473) ˆλ ˆk â ˆb BW (.1883) (1.2754) (.3697) (2.3168) The asymptotc covarance matrx of the MLE s for the MOLLW model parameters whch s the observed Fsher nformaton matrx I 1 n ( ) s gven by: , and the 95% confdence ntervals for the model parameters are gven by c (1.6752± ), α (.2786± ), β (1.9393± ) and δ (5.171 ± ), respectvely. The LR test statstc of the hypothess H : LLW aganst H a : MOLLW, H : LLE aganst H a : MOLLW, and H : MOLL aganst H a : MOLLW and are 83.3 (p-value =.1), (p-value =.1), 13.9 (p-value=.959).

27 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 717 We can conclude that there are sgnfcant dfferences between MOLLW and the sub-models: LLW, LLE, and MOLL dstrbutons at the 5% level of sgnfcance. The values of the statstcs: AIC, AICC and BIC also shows that the MOLLW dstrbuton s a better ft than the non-nested GD and BW dstrbutons for the carbon fber data. Infact, there s also clear evdence based on the Cramer von Mses and Anderson-Darlng goodness-of-ft statstcs W and A that the MOLLW dstrbuton s by far the better ft for the carbon fber data. Plot of the ftted denstes, and the hstogram of the data are gven n Fgure 9.2. Fgure 9.2: Ftted denstes for carbon fber data 1 Concludng Remarks A new class of dstrbutons called the MOLLEW dstrbuton and ts specal case MOLLW are presented. Ths general and specfc class of dstrbutons and some of ts structural propertes ncludng hazard and reverse hazard functons, quantle functon, moments, condtonal moments, Bonferron and Lorenz curves, Rény entropy, maxmum lkelhood estmates, asymptotc confdence ntervals are presented. Applcatons of the specfc MOLLW model to real data sets are gven n order to llustrate the applcablty and usefulness of the proposed dstrbuton. MOLLW dstrbuton has a better ft than some of ts sub-models and the non-nested GD and BW dstrbutons for the glass fbers data and carbon fbers data.

28 718 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton References [1] Barreto-Souza, W., and Bakouch., H. S. (213). A new lfetme model wth decreasng falure rate, Statstcs: A Journal of Theoretcal and Appled Statstcs, 47, [2] Bebbngton M., La C. D., and Ztks R. (27). A flexble Webull extenson, Relablty Engneerng System Safety, 92, [3] Cordero, G. M., Ortega E. M., and Nadarajah S. (21). The Kumaraswamy Webull Dstrbuton wth Applcatons to Falure Data, Journal of Frankln Insttute, 347, [4] Cordero, G. M., and Lemonte A. J. A. (211). On the Marshall-Olkn extended Webull dstrbuton, Stat. Paper, 54, [5] Chakraborty, S., and Handque, L. (217). The Generalzed Marshall- Olkn-Kumaraswamy-G Famly of Dstrbutons, Journal of Data Scence, 15(3), [6] Lee C., Famoye F., and Olumolade O. (27). Beta-Webull Dstrbuton: Some Propertes and Applcatons, Journal of Modern Appled Statstcal Methods,6, [7] Ghtany, M. E, AL-Hussan, E. K., and AL-Jarallah. (25). Marshall- Olkn Extended Webull dstrbuton and ts applcaton to Censored data, Journal of Appled Statstcs, 32(1), [8] Gurvch, M. R., DBenedetto, A. T., and Ranade, S. V. (1997). A new statstcal dstrbuton for characterzng the random strength of brttle materals, Journal of Materals Scence 32, [9] Hjorth, U. (198). A Relablty Dstrbuton Wth Increasng, Decreasng, Constant and Bathtub Falure Rates, Technometrcs, 22, [1] Kumar, D. (216). Rato and Inverse Moments of Marshall-Olkn Extended Burr Type III Dstrbuton Based on Lower Generalzed Order Statstcs, Journal of Data Scence, 14(1), [11] Lazhar, B. (217). Marshall-Olkn Extended Generalzed Gompertz Dstrbuton, Journal of Data Scences, 15(2), [12] Marshall A. W., and Olkn I. (1997). A New Method for Addng a Parameter to a Famly of Dstrbutons wth Applcaton to the Exponental and Webull Famles, Bometrka 84:

29 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 719 [13] Nchols, M. D., and Padgett, W. J. (26). A Bootstrap control chart for Webull percentles, Qualty and Relablty Engneerng Internatonal, 22, [14] Oluyede, B. O., Huang, S., and Parara, M. (214). A New Class of Generalzed Dagum Dstrbuton wth Applcatons to Income and Lfetme Data, Journal of Statstcal and Econometrc Methods, 3(2), [15] Oluyede, B. O., Foya, S., Warahena-Lyanage, G., and Huang, S. (216). The Log-logstc Webull Dstrbuton wth Applcatons to Lfetme Data, Autran Journal of Statstcs, 45, [16] Rény, A. (196). On Measures of Entropy and Informaton, Proceedngs of the Fourth Berkeley Symposum on Mathematcal Statstcs and Probablty, 1, [17] Santos-Neo, M., Bourgugnon, M., Zea, L. M., and Nascmento, A. D. C. (214). The Marshall-Olkn Extended Webull Famly of Dstrbutons, Journal of Statstcal Dstrbutons and Applcatons, 1-9. [18] Slva, G. O., Ortega, E. M., and Cordero, G. M. (21). The Beta Modfed Webull Dstrbuton, Lfetme Data Analyss, 16, [19] Smth, R. L., and Naylor, J. C. (1987). A comparson of maxmum lkelhood and Bayesan estmators for the three-parameter Webull dstrbuton. Appled Statstcs, 36, [2] Xe, M., Tang Y., Goh T. N. (29). A modfed Webull extenson wth bathtub-shaped falure rate functon, Relablty and Engneerng System Safety, 76, [21] Zhang, T., and Xe M. (27). Falure data analyss wth extended Webull dstrbuton, Commun. Stat. Smul.,

30 72 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton A R Code: Defne Functons # MOLLW pdf MOLLW pdf=f u n c t o n ( c, alpha, beta, delta, x ){ ( d e l t a exp( alpha ( xˆ beta )) (1+ xˆc )ˆ( 1) ( alpha beta x ˆ( beta 1)+(c x ˆ( c 1) (1+xˆc )ˆ( 1)))/ (1 (1 d e l t a ) (1+xˆ c )ˆ( 1) exp( alpha ( xˆ beta ) ) ) ˆ 2 ) } # MOLLW cdf MOLLW cdf=f u n c t o n ( c, alpha, beta, delta, x ){ (1 (1+xˆ c )ˆ( 1) exp( alpha ( xˆ beta ) ) ) / (1 (1 d e l t a ) (1+xˆ c )ˆ( 1) exp( alpha ( xˆ beta ) ) ) } # MOLLW hazard MOLLW hazard=f u n c t o n ( c, alpha, beta, delta, x ){ MOLLW pdf( c, alpha, beta, delta, x )/ (1 MOLLW cdf( c, alpha, beta, delta, x ))} #MOLLW q u a n t l e MOLLW quantle=f u n c t o n ( c, alpha, beta, delta, u){ f=f u n c t o n ( x ){ alpha ( xˆ beta)+ l o g (1+xˆc)+ l o g (1 u) l o g ( d e l t a +(1 u) (1 d e l t a ) ) } y=u n r o o t ( f, c (, 1 ) ) $root return ( y ) $ } #MOLLW moments MOLLW moments=f u n c t o n ( c, alpha, beta, delta, r ){ f=f u n c t o n ( x, c, alpha, beta, delta, r ){ ( xˆ r ) (MOLLW pdf( x, c, alpha, beta, d e l t a ) ) } y= n t e g r a t e ( f, lower =, upper=inf, s u b d v s o n s =1, c=c, alpha=alpha, beta=beta, d e l t a=delta, r=r ) return ( y$value ) $ }

31 Lornah Lepetu, Broderck O. Oluyede, Bokanyo Makubate, Susan Foya and Precous Mdlongwa 721 Lornah Lepetu Department of Mathematcs and Statstcal Scences Botswana Internatonal Unversty of Scence and Technology, Palapye, BW Broderck O. Oluyede Department of Mathematcal Scences Georga Southern Unversty, GA 346, USA Bokanyo Makubate Department of Mathematcs and Statstcal Scences Botswana Internatonal Unversty of Scence and Technology, Palapye, BW Susan Foya Frst Natonal Bank Gaborone, BW Precous Mdlongwa Department of Mathematcs and Statstcal Scences Botswana Internatonal Unversty of Scence and Technology, Palapye, BW

32 722 Marshall-Olkn Log-Logstc Extended Webull Dstrbuton