The Modified Burr III G family of Distributions

Transcription

1 Joural of Data Scece 5(07), 4-60 The Modfed Burr III G famly of Dstrbutos Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al Natoal College of Busess Admstrato ad Ecoomcs, Lahore, Pasta. Abstract: We troduce a ew famly of dstrbutos based o a geeralzed Burr III geerator called Modfed Burr III G famly ad study some of ts mathematcal propertes. Its desty fucto ca be bell-shaped, left-sewed, rght-sewed, bathtub, J or reversed-j. Its hazard rate ca be creasg or decreasg, bathtub, upsde-dow bathtub, J ad reversed-j. Some of ts specal models are preseted. We llustrate the mportace of the famly wth two applcatos to real data sets. Key words: Burr famly of dstrbutos, geeralzed famly, hazard rate, maxmum lelhood estmato.. Itroducto Burr (94) developed a system of dstrbutos for fttg cumulatve frequecy dstrbutos. The system cludes twelve types of cumulatve dstrbuto fuctos whch accommodate a varety of desty shapes. The attractveess of ths famly of dstrbutos for model fttg s that t combes a smple mathematcal expresso for the cumulatve frequecy fucto wth wde coverage the sewess-urtoss plae. May stadard dstrbutos, cludg the Webull, expoetal, logstc, geeralzed logstc, Gompertz, ormal, extreme value ad uform dstrbutos are specal cases or lmtg cases of the Burr system of dstrbutos. Amog the twelve dstrbuto fuctos Burr type XII (BXII) ad ts verse, type III (BIII) have gathered specal atteto physcs, actuaral studes, relablty ad appled statstcs. Nadarajah ad Kotz (007) appled t to fracture toughess ad fracture stress data. Gove et al. (008) ftted Burr III to data related to forestry. Mele (973) proposed t as a model for precptato amouts the meteorologcal data. Shao et al. (008) proposed ad ftted a exteded Burr III dstrbuto low-flow frequecy aalyss where ts lower tal was of ma terest. Durg the last couple of decades ew famles of probablty dstrbutos have bee defed to exted well-ow famles of dstrbutos. The purpose behd s to provde greater flexblty modelg practcal data or the physcal or statstcal eed to expla the mechasm of the geerated data. Oe such approach was by Marshall ad Ol (997) whch oe parameter was added to the survval fucto of a basele dstrbuto. Later o Gupta et al. (998), Gupta ad Kudu (999; 00a; 00b; 00; 003), Nadarajah ad Kotz (006) ad Nadarajah (0) to ame a few who used ths approach to geeralze ther dstrbutos. I aother geeralzato, poeered by Eugee et al. (00) ad Joes (004) the beta-geerated class from the logt of the beta dstrbuto was defed. Cordero ad de Castro (0), Cordero

2 4 The Modfed Burr III G famly of Dstrbutos et al. (0) used the beta-geerated class approach. Kumar (06) derved ew expressos for rato ad verse momets of lower geeralzed order-statstcs for the Marshall-Ol exteded Burr type x dstrbuto. Yousof et al.(06) troduced a ew Burr X-G famly of dstrbutos by addg a shape parameter. Gurvch et al. (997) poeered a class of exteded Webull dstrbutos whch has acheved a promet posto lfetme models. Its cdf s defed by F( x;, ) exp H x,, x, 0, (.) where H(x;ξ) s a o-egatve mootocally creasg fucto whch depeds o the parameter vector ξ. The correspodg pdf s gve by f ( x;, ) exp H x, h x,, (.) where h(x; ξ ) s the dervatve of H(x; ξ ). I ths class of dstrbutos H(x; ξ) s tae to be the odds rato of the basele dstrbuto. Aother mportat class of uvarate dstrbutos were developed by Zografos ad Balarsha (009). They exteded Stacy s geeralzed gamma dstrbuto by replacg x wth log[-g(x)] ad defed ther famly of dstrbutos havg cdf F( x; ) {, log[ G( x)]}, x, 0 (.3) wth pdf gve by z f ( x; ) log G x g x, z t t e dt / where 0 deotes the complete gamma fucto ad G(.) s the gamma fucto. Rstc ad Balarsha (0) defed a dfferet type of Gamma-G famly of dstrbuto wth pdf gve by g( x; ) log F x f x x, 0 (.5) where G(x) s ay basele cdf ad g(x)=dg(x). Alzaatreh et al. (03a) proposed a trasformed-trasformer (T-X) famly of dstrbutos. der ths famly Gamma-Pareto, Webull-Pareto ad Gamma-Normal dstrbutos were developed by Alzaatreh et al. (03b), (0) ad (04) respectvely. I a recet wor. Al et al. (04) defed Modfed Burr III as f ( x ) x x, x 0 (.4) (.6)

3 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 43 wth dstrbuto fucto gve by F( x) x where,, 0 are shape parameters of the Modfed Burr III dstrbuto. Bourgugo et al. (04) defed Webull-G, Nascmeto et al. (04) proposed Gamma exteded Webull famly, Satos-Neto et al. (04) troduced Marshall-Ol exteded Webull famly, Alzadeh et al. (05) defed Kumaraswamy odd log-logstc famly, Tahr et al. (05a) proposed a ew Webull-G famly ad Tahr et al. (05b) troduced the odd geeralzed expoetal famly of dstrbutos. Kormaz ad Geç (05) troduced a geeralzato of Webull dstrbuto based o trasformato of the two-sded Power dstrbuted radom varate. Kormaz ad Geç (06)studed detal Two-sded geeralzed Normal dstrbuto. Mat et al (05) defed a Odds Expoetal- Expoetal dstrbuto for modelg lfetme models. Affy et al. (06) troduced as Kumaraswamy Trasmuted-G Famly of dstrbutos, by replacg x of the cdf of Kumaraswamy dstrbuto wth the cdf of the Trasmuted Class of dstrbutos. I ths paper, we troduce ad study geeralty a famly of Modfed Burr III G dstrbuto, usg the Modfed Burr III geerator appled to the odds rato of a basele dstrbuto. By the term geerator we mea that for each basele dstrbuto G(x) we have a dfferet dstrbuto. The objectve of wrtg ths paper s to study a ew famly of dstrbutos whch s flexble because of ts creasg, decreasg, bathtub, upsde-dow bathtub, J, reversed- J shaped hazard rate fucto. Wth so may practcal stuatos arsg dfferet felds of applcatos oe of these models may provde better ft a specfc practcal stuato. We also derve some of ts mathematcal propertes ad llustrate ts usefuless as a better model. Ths paper s outled as follows. I Secto, we troduce the Modfed Burr III G (MBIII G) famly of dstrbutos ad Secto 3 some specal models geerated by MBIII G famly are gve. Secto 4 relates to some geeral mathematcal propertes. I Secto 5 estmato of parameters s performed by the method of maxmum lelhood. Secto 6 llustrates two applcatos based o real data sets. Fally cocludg remars are preseted Secto7.. Modfed Burr III G (MBIII G) Famly of Dstrbutos Cosder a cotuous dstrbuto G(x;ξ) wth desty g(x;ξ) ad the cdf of Modfed Burr III, gve (.7). Based o ths desty we replace x wth the odds rato H ( x; ) G x; / G x; of a basele dstrbuto, where ξ s the parameter vector of ths basele dstrbuto. The the cdf of the Modfed Burr III G Famly s defed by H( x; ) F( x;,,, ) t dt 0 F ( x ;,,, ) ;,,, 0 H x x D (.7) (.)

4 44 The Modfed Burr III G famly of Dstrbutos The pdf of the Modfed Burr III G famly reduces to g x; f ( x;,,, ) H x; H x;, x 0 G x; (.) Here F(x;α,β,γ,ξ) ad f(x;α,β,γ,ξ) deotes the cumulatve dstrbuto fucto ad probablty desty fucto respectvely of the Modfed Burr III G famly whereas G(x;ξ), g(x;ξ) ad H(x;ξ) are the cdf, pdf ad odds rato of the basele dstrbuto respectvely. The survval fucto of Modfed Burr III G famly s defed as ad hazard rate fucto s S( x;,,, ) H x; H x; H x; hx ( ;,,, ) H x; Lmtg cases of the famly g x; G x; (.3). (.4) For γ 0 MBIII G famly teds to Geeralzed Iverse Webull G famly. For β= ad γ 0 MBIII G famly teds to Geeralzed Iverse Expoetal G famly. Whe β=, α=, γ 0 MBIII G famly approaches Iverse expoetal G famly of dstrbutos. For α=, γ 0 Iverse Webull G famly of dstrbutos s a lmtg case of MBIII G famly. All of the above metoed famles of dstrbutos are lmtg case of Modfed Burr III G famly. Therefore the proposed famly s a geeralzed famly of dstrbutos ad all the lmtg cases are ew famles of dstrbutos 3. Specal Models of MBIII G dstrbuto I ths secto we gve some examples of Modfed Burr III G famly of dstrbutos. These specal models geeralze several classcal models, for example, log-logstc, Lomax, expoetal, form amog other dstrbutos. 3. Modfed Burr III Normal dstrbuto Frst example refers to the Normal dstrbuto. The Modfed Burr III Normal (MBIII N) desty s obtaed from (.) by tag G(x) ad g(x) to be the cdf ad pdf of the ormal N(µ,s) dstrbuto. The MBIII N dstrbuto has cdf x F ( ) MBIII N x, x, x

5 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 45 where, 0 ad f(.) ad F(.) are the pdf ad cdf of the stadard ormal dstrbuto respectvely. Its correspodg pdf s f MBIII N x x x ( x), x x x x. 3. Modfed Burr III Webull Dstrbuto G x The Webull dstrbuto has pdf ( ) exp x H x wth odds rato x The MB III Webull dstrbuto has cdf x x x g x x exp x, x 0 ad cdf ( ) exp. Here λ s rate ad ν s scale parameter. ( ) x FMBIII W x e f ( ) MBIII W x e e x e, x Modfed Burr III Kumaraswamy Dstrbuto Cosder the Kumaraswamy dstrbuto wth pdf a x b a a b wth ts pdf as g x abx x, 0 x ad cdf as a G x H x x. wth odds rato Where a ad b are ts shape parameters. The MB III Kumaraswamy dstrbuto has cdf ad pdf respectvely as a b FMBIII Kum x x a a a a b b b b fmbiii Kum x x x abx x, x Modfed Burr III Burr XII dstrbuto s The pdf of Burr XII s ad ts odds rato be c c g x cx x c x G x, where c ad are shape parameters. Its cdf

6 46 The Modfed Burr III G famly of Dstrbutos The MBIII BXII s defed as c x H x. c c c c fmbiii BXII x cx x x x, x 0. Wth correspodg cdf c FMBIII BXII x x. The MBIII BXII dstrbuto cludes log-logstc whe α=β=γ= ad = wth pdf c c f x cx x. For α=β=γ= ad c= we obta Lomax dstrbuto wth pdf f x x. Ad whe all the Modfed Burr III parameters are oe we get Burr XII dstrbuto wth pdf c c f x cx x. To study the shape of the Modfed Burr III G famly of dstrbutos we tae some selected values of the parameters of the above metoed specal models to plot ther destes ad hazard rate fucto. (a) (b) f x f x.5,.5,4.5,0,,.5,.5,0,.5,.5,3,0.5,.5,3,, 0.5, ,.5,0.5,0.5,,3,3,0.5,.5,3,.5,,0.5,3,3.5,0.5, x x (c) (d)

7 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 47 f x f x ,0.5,0.,0.5,0.5,,.5,4.5,5,,,5,,,,0.5, ,.5,.5,,,,,3,3 0.5,0.5,.5,3,0.5,3,3.5,, x x Fgure : Plots of the destes of (a) MBIII Normal (b) MBIII Webull (c) MBIII Kumaraswamy ad (d) MBIII Burr XII dstrbutos for varous values of the parameters. Fgure dcates that the proposed Modfed Burr III G famly geerates dstrbutos wth varous shapes such as bell-shaped, left-sewed, rght-sewed, bathtub, J ad reversed-j. It s a clear dcato that the MBIII G famly s very flexble ad ca be used approprately to ft dfferet datasets havg varous shapes. (a) (b) h x h x 4.5,.5,4.5,0,,.5,.5,0,.5,.5,3,0.5,.5,3,, 0.5, ,.5,0.5,0.5,,3,3,0.5,.5,3,.5,,0.5,3,3.5,0.5,0.5 3 x x (c) (d)

8 48 The Modfed Burr III G famly of Dstrbutos h x h x ,0.5,0.,0.5,0.5,,.5,4.5,5,,,5,,,,0.5, ,.5,.5,,,,,4,4 0.5, 0.5,.5, 3,0.5,3,3.5,.5, x x Fgure : Plots of the hazard rate of (a) MBIII Normal (b) MBIII Webull (c) MBIII Kumaraswamy ad (d) MBIII Burr XII dstrbutos for varous values of the parameters. Fgure shows that the hazard rate of Modfed Burr III G famly of dstrbutos ca be mootoc or o-mootoc. The shapes of hazard rate ca be creasg, decreasg, bathtub, upsde-dow bathtub, J ad reversed-j. Ths s dcatve of ts usefuless survval aalyss, bomedcal, egeerg ad socal sceces. 4. Mathematcal Propertes We ow derve some of the geeral mathematcal propertes of Modfed Burr III G dstrbuto. 4. Expaso for the Modfed Burr III G famly of dstrbutos Expaso for the pdf of MBIII G famly ca be derved by usg the bomal seres expaso (.). Let f(x;α,β,γ;ξ)=f(x) f ( x) H x H x g x G x sg bomal expaso for the secod term the above equato we have H x H x 0 As H x G x / G x equato (4..) becomes (4..) (4..)

9 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 49 f ( x) 0 sg bomal seres aga (4..3) G x G x G x g x j G x j0 j ad substtutg (4..3) we get j ( ) j f x G x g x j, 0 j where ad The above equato ca be rewrtte as w j, f ( x) wjh x j 0, j j! j! j j a (4..3) (4..4) (4..5) (4..6) ha x ag x G x. I other words we ca wrte the MBIII G famly desty as a lear combato of w j, expoetated-g desty (exp-g) fucto wth weghts. Therefore some mathematcal propertes of the proposed famly ca be derved from (4..5) smlar to those of exp-g propertes such as the ordary ad complete momets ad momet geeratg fucto. For mathematcal propertes of the exp-g dstrbutos the reader s referred to Gupta ad Kudu (00), Gupta et al. (998) ad Nadarajah ad Kotz(006) amog others. 4. Momets The sth momet of Modfed Burr III G famly of dstrbutos ca be obtaed from (4..5) as t ca be expressed as a fte lear combato of the expoetated-g desty fuctos as f ( x) wj, h x j 0 s s j, j, E X w E Z j, 0

10 50 The Modfed Burr III G famly of Dstrbutos Z, where j j. The er quattes (4..5) are absolutely tegrable. The complete momets ad momet geeratg fucto ca be evaluated as where MGF s deotes the expoetated-g dstrbuto wth power parameter y X j, j, j, 0 s I y x f x dx w I y y I s y x h x dx j, ; j tz,, j j M X t w E e j, Quatle fucto ad Radom Number Geerato The Modfed Burr III G famly of dstrbutos ca easly be smulated from (.6) as follows. X G S / ( S ) If q has a uform dstrbuto (0,), the soluto has the MB III G / S / ( q ) dstrbuto where. For q=0.5 we get meda of MB III G dstrbuto ad radom umber geerator f q s tae as uform. 5. Maxmum Lelhood Estmato Now we fd the maxmum lelhood estmates (MLEs) of the uow parameters of the Modfed Burr III G famly of dstrbutos. Let x, x,,x be a sample of sze from MBIII G famly gve by (.) wth parameters α, β, γ ad ξ. Let Q=(α, β, γ, ξ)t be the p x parameter vector. The log-lelhood fucto ca be expressed as l log log log[ G x ; ] log[ G x ; ] log ( ; ) H x log( ; ) g x Where H(x; ξ) = G(x; ξ)/(-g(x; ξ)). The compoets of the score fucto (Q)=(α, β, γ, ξ) are gve by log H ( x ; ) G x G x log[ ; ] log[ ; ]. H x H x log H x ; ; ( ; )

11 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 5 H x ; log ( ; ) H x H( x ; ) ad G x ; / ; / G x ; G x G x ; ( ; ) ( ; ) / ; / H x H x g x ( ; ) ; H x g x Settg these score equatos equal to zero ad solvg them smultaeously yelds the MLE ˆ?,?,, T,,, T of.these equatos caot be solved aalytcally ad recourse to teratve procedure such as Newto-Raphso algorthm s suggested. For terval estmato of the model parameters, we eed the observed formato matrx Whose elemets are J H x ; log H x ; H x ; H x ; log ; H x H x ;

12 5 The Modfed Burr III G famly of Dstrbutos H x ; H ; x H x ; log H x ; H x ; H x ; H x H x log H x ; ; ; ; ; ; G x; H x ; H x ; log H x ; H x ; G x G x G x H x ; H x ; H x ; log ; 3 H x ; H x H x ; ad H x ; H x ; H x ; H x ;

13 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 53 l G x G x ; G ; ; ; ; x Gl x g x g x gl x ; l ; ; G x g x g x; G ; ; ; l x G x Gl x Gl x ; ; G x ; H x ; Hl x ; H x ; ; H x Hl x ; ; H x ; H x H x ; ; H x Hl x ;. H x ; G x; G l Where ad 6. Applcatos x ; deotes frst ad secod dervatve of x wth respect to ξ. I ths secto, we provde two applcatos to real data to llustrate the mportace of the Modfed Burr III G famly by meas of Modfed Burr III Webull (MBIII We) ad Modfed Burr III Burr XII (MBIII BXII) models preseted Secto 3. Applcato The frst data cossts of 63 observatos of the stregths of.5 cm glass fbers. It was orgally obtaed be worers at the K Natoal Physcal Laboratory. These data have also bee aalyzed by Smth ad Naylor (987). The uts of measuremet are ot gve the paper. The data are: 0.55, 0.74, 0.77, 0.8, 0.84, 0.93,.04,.,.3,.4,.5,.7,.8,.9,.30,.36,.39,.4,.48,.48,.49,.49,.50,.50,.5,.5,.53,.54,.55,.55,.58,.59,.60,.6,.6,.6,.6,.6,.6,.63,.64,.66,.66,.66,.67,.68,.68,.69,.70,.70,.73,.76,.76,.77,.78,.8,.8,.84,.84,.89,.00,.0,.4. For ths data we compare Modfed Burr III Webull dstrbuto wth Modfed Burr III ad Webull model. Table : Summary statstcs of stregths of.5 cm glass fbers data Data Mea Meda S.D Varace Sewess Kurtoss Stregth of glass fbers

14 54 The Modfed Burr III G famly of Dstrbutos The requred umercal calculatos of MLEs, egatve log-lelhood (W), Aae Iformato Crtero (AIC), Cumulatve Aae Iformato Crtero (CAIC) ad Bayesa Iformato Crtero (BIC) values are carred out usg SAS through PROC NLMIXED commad. sg several dfferet tal values of the parameters we fd the best ft for each model. The tal guess that gves the mmum value of the lelhood crtero s cosdered to be the best ft of parameter estmates. Followg results were obtaed. Table : MLEs of the parameters, ther stadard errors ad varous formato crtera for the stregths of.5 cm glass fbers dataset Model estmates S.E. Lower boud pper boud W AIC CAIC BIC ˆ = ˆ = MBIII Webull ˆ = ˆ = ˆ = ˆ = MBIII ˆ = ˆ = Webull ˆ = ˆ = Negatve log-lelhood crtero s smaller for MBIII We tha MBIII ad Webull therefore, t provdes better ft tha the rest. The CDF plot of these models s gve below.

15 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 55 CDF of MBIII Webull, MBIII, Webull ad Emprcal data MBIII Webull MBIII Webull Emprcal F(x) x Fgure 3: Cdf of MBIII W, MBIII, Webull ad emprcal data ftted o glass fber data The above plot dcates that Modfed Burr III Webull model gves a close ft of the data tha the other models. Applcato I the secod applcato we compare Modfed Burr III Burr XII (MBIII BXII) wth Modfed Burr III (MBIII) ad Burr XII (BXII) dstrbuto. We use fracture toughess data ( the uts of MPa m/) of Aluma (AlO3) that Nadarajah ad Kotz (007) ft to Burr III (B III) ad Burr XII (BXII) dstrbutos. Al et al. (04) also appled ths data. The data are also avalable ole at Table 3: Summary statstcs of Fracture toughess data of Aluma (AlO3) Data Mea Meda S.D Varace Sewess Kurtoss Fracture Toughess ( the ut MPa m / ) The dfferet formato crtera computed are gve as. Table 4: MLEs of the parameters, ther stadard errors ad varous formato crtera for Fracture toughess data of Aluma (AlO3)

16 56 The Modfed Burr III G famly of Dstrbutos Model estmates S.E. Lower boud pper boud W AIC CAIC BIC ˆ = ˆ = MBIII BXII ˆ = ĉ = ˆ = ˆ = MBIII ˆ = ˆ = BXII ĉ = ˆ = Clearly MB III BXII dstrbuto provdes best ft amog MB III ad B XII dstrbutos, because statstc value all crtera are smaller for MB III BXII dstrbuto. To llustrate the ft of these models we plot the cdfs of these dstrbutos ad the emprcal data.

17 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 57 CDF of MBIII BXII, MBIII ad Emprcal data F(x) MBIII BXII MBIII Emprcal x Fgure 4: Cdf of MBIII BXII, MBIII, Burr XII dstrbutos ad emprcal data ftted o fracture toughess data of Aluma (AlO3) The ft of the plot shows that MBIII BXII dstrbuto outperforms other models to the emprcal data. Hece we ca say that the proposed dstrbuto provdes us best ft ad s more flexble ad has wder applcato lfe tme data sets. 7. Cocludg Remars I ths paper, we propose a ew Modfed Burr III G famly of dstrbutos. Modfed Burr III s the geerator where the radom varable s replaced by the odd rato of the basele dstrbuto. We study some of ts specal dstrbutos. It s observed that the desty fucto ca be left-sewed, rght-sewed, bell shaped, bathtub, J ad reversed-j shaped. We derve some of the geeral propertes of the famly ad maxmum lelhood method s used to estmate ts parameters. We ft two models of the ew famly to demostrate usefuless ad flexblty of the proposed famly. We hope that the ew famly ad ts models wll be useful for practtoers varous felds of appled sceces. Referece [] Affy, A. Z., Cordero, G. M., Yousof, H. M., Nofal, Z. M. ad Alzaatreh, A. (06). The Kumaraswamy Trasmuted-G Famly of Dstrbutos: Propertes ad applcatos. Joural of Data Sceces. 4(),

18 58 The Modfed Burr III G famly of Dstrbutos [] Al, A., Hasa, S. A. ad Ahmed, M. (05). Modfed Burr III dstrbuto: Propertes ad applcatos. Pasta Joural of Statstcs, Vol. 3(6), [3] Alzadeh, M., Emad, M., Doostparast, M., Cordero, G.M., Ortega, E.M.M. ad Pescm, R.R.(05). A ew famly of dstrbutos: the Kumaraswamy odd log-logstc, propertes ad applcatos. Hacettepa Joural of Mathematcs ad Statstcs (to appear). [4] Alzaatreh, A., Famoye, F. ad Lee, C. (0). Gamma-Pareto dstrbuto ad ts applcatos. Joural of Moder Appled Statstcal Methods, (), [5] Alzaatreh, A., Lee, C. ad Famoye, F. (03a). A ew method for geeratg famles of cotuous dstrbutos. Metro. 7(), dx.do.org/0.007/s y [6] Alzaatreh, A. Famoye, F. ad Lee, C. (03b). Webull-Pareto dstrbuto ad ts applcatos. Commucato Statstcs-Theory ad Methods, 4(9), dx.do.org/0.080/ [7] Alzaatreh, A. Famoye, F. ad Lee, C. (04). The gamma-ormal dstrbuto: Propertes ad applcatos. Computatoal Statstcs ad Data Aalyss, 69(), dx.do.org/0.06/j.csda [8] Bourgugo, M., Slva, R.B. ad Cordero, G.M.(04). The Webull-G Famly of Probablty Dstrbutos. Joural of Data Scece, (), [9] Burr, I. W. (94) Cumulatve frequecy fuctos. Aals of Mathematcal Statstcs, 3, 5-3. [0] Cordero, G.M., de Castro, M. (0).A ew famly of geeralzed dstrbutos. Joural of Statstcal Computato ad Smulato. 8, [] Cordero, G.M., Hashmoto, E.M. ad Ortega, E.M.M. (0). The McDoald Webull model. Statstcs, Frst, -3. [] Eugee, N., Lee, C. ad Famoya, F. (00). Beta-Normal dstrbuto ad ts applcato. Commucato Statstcs-Theory ad Methods,3,Issue [3] Gove, J. H., Ducey, M. J., Lea, W. B. ad Zhag, L. (008). Rotated sgmod structure maaged ueve-aged orther hardwood stads: a loo at the Burr Type III dstrbuto. Iteratoal Joural of Forestry Resources., 8(), [4] Gupta, R.C.,Gupta, P.I. ad Gupta, R.D. (998). Modelg falure tme data by Lehma alteratves, Commucatos Statstcs-Theory ad Methods, 7, [5] Gupta, R.D. ad Kudu, D. (00). Expoetated expoetal famly:a alteratve to gamma ad Webull dstrbutos, Bometrcal Joural, 43, [6] Gurvch, M.R., DBeedetto, A.T. ad Raade, S.V. (997). A ew statstcal dstrbuto for characterzg the radom stregth of brttle materals. Joural of Materal Scece, Vol.3, Issue 0, [7] Joes, M.C. (004). Famles of dstrbutos arsg from dstrbutos of order statstcs. Test, 3, -43. [8] Kormaz, M. Ç. ad Geç, A. I. (05). A Lfetme Dstrbuto based o a trasformato of a Two-Sded Power Varate. Joural of Statstcal Theory ad Applcatos, Vol. 4, No.3, [9] Kormaz, M. Ç. ad Geç, A. I. (06). A New Geeralzed Two-sded Class of Dstrbutos wth Emphass o Two-sded Geeralzed Normal Dstrbuto.

19 Shahzad Arfa *, Mohammad Zafar Yab, Azeem Al 59 Commucatos Statstcs-Smulato ad Computato, DOI: 0.080/ [0] Kumar, D. (06). Rato ad Iverse Momets of Marshall-Ol exteded Burr type XII Dstrbuto based o Lower Geeralzed Order-Statstcs. Joural of Data Sceces. 4(), [] Mat, S. S. ad Prama, S. (05). Odds Geeralzed Expoetal-Expoetal Dstrbuto. Joural of Data Sceces. 3() [] Marshall, A.W. ad Ol, I. (997). A ew method for addg a parameter to a famly of dstrbutos wth applcato to the expoetal ad Webull famles. Bometra, 84, 3, [3] Mele, P.W. (973). Aother famly of dstrbutos for descrbg ad aalyzg precptato data. Joural of Appled Meteorology,, [4] Nadarajah, S. ad Kotz, S. (006). The expoetated type dstrbutos, Acta Applcadae Mathematca, 9, 97-. [5] Nadarajah, S. ad Kotz, S. (007). O the alteratve to the Webull fucto. Egeerg Fracture Mechacs, Vol. 74, ssue 3, [6] Nascmeto, A.D.C., Bourgugo, M., Zea, L.M., Satos-Neto, M.S., Slva, R.B. ad Cordero, G.M. (04). The Gamma Exteded Webull famly of dstrbutos. Joural of Statstcal Theory ad Applcatos, Vol. 3, No., -6. [7] Rstc, M. M. ad Balarsha, N. (0). The gamma-expoetated expoetal dstrbuto. Joural of Statstcal Computato ad Smulato, 8, [8] Satos-Neto, M.S., Bourgugo, M., Zea, L.M., Nascmeto, A.D.C., ad Cordero, G.M. (04). The Marshall-Ol exteded Webull famly of dstrbutos.joural of Statstcal Dstrbutos ad Applcatos, :9, DOI:0.86/ [9] Shao, Q., Che, Y. D. ad Zhag, L. (008). A exteso of three-parameter Burr III dstrbuto for low-flow frequecy aalyss. Computatoal Statstcs ad Data Aalyss, 5, [30] Smth, R.L. ad Naylor, J.C. (987). A comparso of Maxmum Lelhood ad Baysa Estmators for the Three-Parameter Webull Dstrbuto. Joural of the Royal Statstcal Socety. Seres C (Appled Statstcs), Vol.36, No.3, [3] Tahr, M.H., Zubar, M., Masoor, M., Cordero, G.M., Alzadeh, M. ad Hameda, G.G. (05a). A ew Webull-G Famly of Dstrbutos. Hacettepe Joural of Mathematcs ad Statstcs (to appear). [3] Tahr, M.H., Cordero, G.M., Alzadeh, M., Masoor, M., Zubar, M. ad Hameda, G.G. (05b). The odd geeralzed expoetal famly of dstrbutos wth applcatos. Joural of Statstcal Dstrbutos ad Applcatos, DOI 0.86/s [33] Yousof, H.M., Affy, A.Z., Hamda, G.G. ad Aryal, G.(06). The Burr X Geerator of Dstrbutos for Lfetme Data. Joural of Statstcal Theory ad Applcatos, Vol [34] Zografos, K. ad Balarsha, N. (009). O famles of beta- ad geeralzed gammageerated dstrbutos ad assocated ferece. Statstcal Methodology,6,

20 60 The Modfed Burr III G famly of Dstrbutos Shahzad Arfa Departmet of Socal Sceces Natoal College of Busess Admstrato ad Ecoomcs Lahore, Pasta shahzadarfa@gmal.com Mohammad Zafar Yab Departmet of Socal Sceces Natoal College of Busess Admstrato ad Ecoomcs Lahore, Pasta drzafaryab@yahoo.com Azeem Al Departmet of Socal Sceces Natoal College of Busess Admstrato ad Ecoomcs Lahore, Pasta syedazeemal@gmal.com