The Kumaraswamy-Generalized Pareto Distribution

Transcription

1 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 ISSN: Australa Joural of Basc ad Appled Sceces Joural home page: The Kumaraswamy-Geeralzed Pareto Dstrbuto M.E. Habb, T.M. Shams ad 2 E.A. Husse Professor of Statstcs, Departmet of Statstc, Faculty of Commerce, Al-Azhar Uversty 2 Assstat Lecturer, Departmet of Statstc, Faculty of Commerce, Al-Azhar Uversty A R T I C L E I N F O Artcle hstory: Receved 23 Jue 205 Accepted 25 August 205 Avalable ole 2 September 205 Keywords: Hazard fucto, Kumaraswamy dstrbuto, Momet, Maxmum lkelhood estmato, Geeralzed Pareto dstrbuto A B S T R A C T The modelg ad aalyss of lfetmes s a mportat aspect of statstcal work a wdevarety of scetfc ad techologcal felds. For the frst tme, the called Kumaraswamy geeralzedpareto (Kum-GP) dstrbuto, s troduced ad studed.the ew dstrbuto ca have a decreasg ad upsde-dow bathtub falure rate fucto depedg o the value of ts parameters; t's cludg some specal sub-model lke geeralzed Pareto dstrbuto ad ts expoetated. Some structural propertes of the proposed dstrbuto are studed cludg explct expressos for the momets ad the desty fucto of the order statstcs ad obta ther momets wll be obta. The method of maxmum lkelhood s used for estmatg the model parameters ad the observed formato matrx s derved. The formato matrx s easly umercally determed.mote Carlo smulatos ad the applcato oftwo real data setsare performed to llustrate the potetalty of ths dstrbuto. 205 AENSI Publsher All rghts reserved. To Cte Ths Artcle: M.E. Habb, T.M. Shams ad E.A. Husse., The Kumaraswamy-Geeralzed Pareto Dstrbuto. Aust. J. Basc & Appl. Sc., 9(27): 9-30, 205 INTRODUCTION The Pareto dstrbuto s the most popular model for aalyzg skewed data. The Pareto dstrbuto was frst proposed by (Pareto(897) as a model for the dstrbuto of come. It ca be used to represet varous other forms of dstrbutos (other tha come data) that arse huma lfe. Most of authorsgave a extesve hstorcal survey of ts use the cotext of come dstrbuto wth may formulas. There are several forms ad extesos of the Pareto dstrbuto the lterature. (Pckads (975) was the frst to propose a exteso of the Pareto dstrbuto wth the geeralzed Pareto (GP) dstrbuto whe aalyzg the upper tal of a dstrbuto fucto. The GP has bee used for modelg extreme value data because of ts log tal feature (see Choulaka&Stephes, (200). Naturally, the Pareto dstrbuto s a specal case of the GP.The expoetated Pareto (EP) dstrbuto was troduced by (Gupta et al. (998) the same settgs that the geeralzed expoetal(ge) dstrbuto exteds the expoetal dstrbuto (see Gupta &Kudu, (999). For more detals o the GP dstrbuto, ts theory ad further applcatos, we refer the readers to (Leadbetter et al. (987), (Embrechts et al. (997), (Castllo et al. (2005). The four parameters Pareto (geeralzed Pareto) dstrbuto was troduced by (Abdul Fattah et el. The cumulatve dstrbuto fucto cdf of the four parameters Pareto dstrbuto (geeralzed Pareto dstrbuto) s x F x;,,, x () A radom varable x s sad to follow the Pareto dstrbuto wth fourparameters (geeralzed Pareto dstrbuto) f the probablty desty fucto pdf of x s as follows: x x f x;,,, (2) For x,,, 0, 0 ad are the shape parameter where s the locato parameter, s scale parameter ad Correspodg Author: M.E. Habb, Professor of Statstcs, Departmet of Statstc, Faculty of Commerce, Al-Azhar Uversty

2 20 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 I ths cotext, we propose a exteso of the Geeralzed Pareto dstrbuto based o the famly of Kumaraswamy geeralzed (deoted wth the prefx Kw-G for short) dstrbutos troduced by Cordero ad de Castro. Nadarajah et al. studed some mathematcal propertes of ths famly. The Kumaraswamy (Kw) dstrbuto s ot very commo amog statstcas ad has bee lttle explored the lterature. Its cdf (for 0 < x < ) s F x = ( x a ) b, where a > 0 ad b > 0 are shape parameters, ad the desty fucto has a smple formf x = ab x a ( x a ) b, whch ca be umodal, creasg, decreasg or costat, depedg o the parameter values. I ths ote, we combe the works of Kumaraswamy ad Abdul Fattah et al, to derve some mathematcal propertes of a ew model, called the Kumaraswamy Geeralzed Pareto (Kw-GP) dstrbuto, whch stems from the followg geeral costructo: f G deotes the basele cumulatve fucto of a radom varable, the a geeralzed class of dstrbutos ca be defed by F x; a, b = ( G x a ) b (3) where a > 0 ad b > 0 are two addtoal shape parameters whch gover Skewess ad tal weghts. Because of ts tractable dstrbuto fucto (2), the Kw-G dstrbuto ca be used qute effectvely eve f the data are cesored. Correspodgly, ts desty fucto s dstrbutos has a very smple form f x; a, b = ab g(x) G x a ( G x a ) b (4) The desty famly (3) has may of the same propertes of the class of beta-g dstrbutos (see Eugee et al. [4]), but has some advatages terms of tractablty, sce t does ot volve ay specal fucto such as the beta fucto. Ths paper s outled as follows. I secto 2, we defe the Kw-GP dstrbuto ad provde expasos for ts cumulatve ad desty fuctos. A rage of mathematcal propertes of ths dstrbuto s cosdered sectos 3 tll 7. These clude quatle fucto, smulato, skewess ad kurtoss, order statstcs, L-momets ad mea devatos; we also provde expasos for the momets of the order statstcs. The Re y etropy s calculated secto 8. Maxmum lkelhood estmato s performed ad the observed formato matrx s determed secto 9. I secto 0, we provde mote Carlo smulato ad applcato to several real data sets to llustrate the potetalty of ths dstrbuto. Fally, some coclusos are addressed secto. Table : Some Specal Dstrbutos. Dstrbuto λ 3-Parameters Pareto Parameters Burr XII Parameters Lomax Beta-II Compoud Expoetal-Expoetal Compoud Ralegh-Gamma Compoud Webull-Expoetal The Kumaraswamy-Geeralzed Pareto Dstrbuto: If G x; s the Geeralzed Pareto cumulatve dstrbuto wth Parameter =,, λ, the equato (2) yelds the Kw-GP cumulatve dstrbuto F x; ξ = xλ a b x λ (5) whereξ = (a, b,,, λ, ), a, b, ad > 0 are o-egatve shape Parameters, > 0 s the scale parameter s postve,ad λ 0 s the locato parameters real. The correspodg pdf ad Hazard Rate Fucto are xλ ad f x; ξ = ab H x; ξ = a b (6) S x; ξ = F x; ξ = f x; ξ ab xλ S x; ξ = xλ xλ a b xλ a a a

3 H(x) H(x) S(x) S(x) Desty Desty 2 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 Respectvely (a) (b) 0.8 =0.7 ;=.2 ;=0.5; =0.7 ;a=2 ;b=3.5 = ;=.7 ;=0.5 ;=.3 ;a=3 ;b=4.5 =0.7 ;=.2 ;=0.5 ;=0.9 ;a=3 ;b=5 =0.7 ;=.2 ;=0.5 ;=.5 ;a=2.5 ;b=0.5 =0.7 ;=.2;=0.5;=2.5;a=0.5 ;b= = ;= ;=0.5; = ;a=2 ;b=5 =0.9 ;=0.8 ;=0.5 ;=.8 ;a=4 ;b=3 =.9 ;=0.8 ;=0.5 ;=0.7 ;a=4 ;b=0 =.9 ;=2. ;=0.5 ;=0.7 ;a=3 ;b=6 =;=;=0.5;=.5;a=5 ;b= (a) X Fg. : Plots of the Kw-GP desty fucto for some parameter values. (b) X 0.8 =.2;=2.3;=0.5;=.7;a=0.5 ;b=.2 =0.9;=2.;=0.5;=.6;a=.7 ;b=.2 =.2;=2.3;=0.3;=.5;a=3 ;b=.2 =.2;=2.3;=0.5;=.3;a=6 ;b=.2 =.2;=2.3;=0.9;=.5;a=. ;b=.2 =.2;=;=0.5;=;a= ;b=0.5 =0.9;=;=0.5;=;a= ;b= =.2;=;=0.3;=.5;a= ;b=.5 =.2;=;=0.5;=.3;a= ;b=2 =.2;=;=0.9;=.5;a= ;b= Fg. 2: (a) Plots of the Kum-GP survval fucto for some values of fucto for some values of b. 4 (a) =.3;=2;=0.6;=0.;a=2 ;b=3 =.3;=3;=0.3;=;a=0.7 ;b=5 =.3;=;=0.2;=.5;a=2;b=4 =0.7;=4;=0.3;=.7;a=.9;b=4 =0.5;=4;=0.3;=.5;a=.8 ;b=4 4 a. (b) Plots of the Kum-GP survval (b) =2.3;=2;=0.7;=0.;a=0. ;b=.3 =2.3;=2;=0.5;=;a=0.5 ;b=.3 =2.3;=2;=0.3;=.5;a=0.3;b=.5 =.6;=2;=0.5;=.3;a=0.4;b=2 =.2;=2;=0.9;=.5;a=0.6 ;b= Fg. 3: Plots of the Kw-P hazard fucto for some parameter values X 2. Specal Dstrbutos: The followg well-kow ad ew dstrbutos are specal sub-models of the Kum-GP dstrbuto. Expoetated Geeralzed Pareto dstrbuto: If b =, the Kum-GP dstrbuto reduces to f x; ξ = a a

4 22 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 Whch s the expoetated geeralzed Pareto (EGP). Fora = b =, we obta the Geeralzed Pareto dstrbuto, f a = b = ad = the Kum-GP dstrbuto reduces to compoud Webull-Gamma Dstrbuto (Abdul Fattah et al). Kum-Compoud Webull Gamma Dstrbuto: If =, the Kum-GP dstrbuto reduces to f x; ξ = ab a b a whch s the Kum-Compoud Webull Gamma Dstrbuto, For a = we obta the expoetated Webull Gamma Dstrbuto, f a = b = the Kum-GP dstrbuto reduces to compoud Webull-Gamma Dstrbuto (Abdul Fattah et al). Kum-Compoud Raylegh Gamma Dstrbuto: If =, = 2 the Kum-GP dstrbuto reduces to f x; ξ = 2ab a b a Whch s the Kum-Compoud Raylegh Gamma Dstrbuto, for a = we obta the expoetated Raylegh Gamma Dstrbuto, f a = b =,the Kum-GP dstrbuto reduces to compoud Raylegh-Gamma Dstrbuto (Abdul Fattah et al) 2.2Expasos for the cumulatve ad desty fuctos: Here, we gve smple expasos for the Kw-GP cumulatve dstrbuto. By usg the geeralzed bomal theorem (for 0 < a < ) a v v = =0 a where v (v) = (7)! I equato (5), we ca wrte F x; ξ = =0 b a = κ τ x; ζ Where κ = b ad τ x; ζ deotes the EGP cumulatve dstrbuto wth parameters ζ =,, λ,, a, Now, usg the power seres (7) the last term of (6), we obta f x; ξ = ab j j j =0 =0 j b a j =0 j f x; ξ = j =0 w j g(x; ) wherew j = ab (j ) j b a j =0 (8) adg x; deotes the geeralzed Pareto dstrbuto wth parameters = j,, λ,. Thus, the Kw-GP desty fucto ca be expressed as a fte lear combato of Pareto destes. Thus, some of ts mathematcal propertes ca be obtaed drectly from those propertes of the geeralzed Pareto dstrbuto. For example, the ordary, verse ad factoral momets, momet geeratg fucto (mgf) ad characterstc fucto of the Kw-GP dstrbuto follow mmedately from those quattes of the Pareto dstrbuto. 3- Momets: Here, ad heceforth, let x be a Kw-GP radom Varable followg (8), ther t momet of X ca be obtaed from (8) as E X r = j =0 w j x r λ g x; dx

5 23 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 = j =0 w Whereς j l = l=0 I partcular, settg r = ad a = b = (9), the mea of X reduces to j ς l j rl, rl r l λl rl E X = μ = Γ Γ( ) λ Γ() 2 Ths s precsely the mea of the geeralzed Pareto dstrbuto wth Parameters = j,, λ,. 4- Quatle fucto ad smulato: Here, themethod for smulatg from the Kw-GP dstrbuto (6) s preseted. The quatle fucto correspodg to (5) s Q u = F u = u b Smulatg the Kw-GP radom varable s straghtforward. Let U be a uform varate o the ut terval(0, ). Thus, by meas of the verse trasformato method, we cosder the radom varable X gve by X = U b whch follows (6),.e. X~ KW GP (a, b,,, λ, ). 5- Skewess ad Kurtoss: The shortcomgs of the classcal kurtoss measure are well-kow. There are may heavy taled dstrbutos for whch ths measure s fte. So, t becomes uformatve precsely whe t eeds to be. Ideed, our motvato to use quatle-based measures stemmed from the o-exstace of classcal kurtoss for may of the Kw dstrbutos The Bowley sskewess (see Keey ad Keepg [5]) s based o quartles: Q(3 4) 2Q 2 Q( 4) S k = Q(3 4) Q( 4) Ad the Moors kurtoss (see Moors (28)) s based o octles: Q(7 8) Q 5 8 Q 3 8 Q( 8) K u = Q(6 8) Q(2 8) Where Q( ) represets the quatle fucto 6- Order statstcs: Momets of order statstcs play a mportat role qualty cotrol testg ad relablty, where a practtoer eeds to predct the falure of future tems based o the tmes of a few early falures. These predctors are ofte based o momets of order statstcs. We ow derve a explct expresso for the desty fucto of the th order statstcx :, say f : (x), a radom sample of sze from the Kw-GP dstrbuto,wrtte as! f : x =!! f x F x F(x) where (. ) ad F(. ), are the pdf ad cdf of the Kw-GP dstrbuto, respectvely. From the above equato ad usg the seres represetato (7) repeatedly, we obta a useful expaso for f : x, gve by where V (r) : = f : x =! ab!! (r ) (r) V : r =0 l=0 m =0 a a λ g x; (4) lm r () l λ b l m a m m adg x; deotes the geeralzed Pareto dstrbuto wth parameters = j,, λ,. So, the desty fucto of the order statstcs s smply a fte lear combato of geeralzed Pareto destes. The pdf of the t order statstc from a radom sample of the geeralzed Pareto dstrbuto comes by settg a = b = (4). Evdetly, equato (4) plays a mportat role the dervato of the ma propertes of the Kw- GP order statstcs. For example, the S th raw momet of X : ca be expressed as

6 24 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 s (r) E X : = V : r=0, sl s X : λ g x; ϑ dx (r) = r=0 V : j υ l j sl whereυ s l = l=0 l λl sl (5) The L-momets are aalogous to the ordary momets but ca be estmated by lear combatos of order statstcs. They are lear fuctos of expected order statstcs defed by The frst four L-momets are: λ m = m m k=0 k m k E X m k:m, m = 0,, λ = E X :, λ 2 = 2 E X 2:2 X :2 λ 3 = 3 E X 3:3 2X 2:3 X :3, ad λ 4 = 4 E X 4:4 3X 3:4 3X 2:4 X :4 The L-momets have the advatage that they exst wheever the mea of the dstrbuto exsts, eve though some hgher momets may ot exst, ad are relatvely robust to the effects of outlers. From equato (5) wth s =, we ca easly obta explct expressos for the L-momets ofx. 7- Mea Devatos: The mea devatos about the mea ca be used as measure of spread a populato. Let μ = E(X) s the mea of the Kw-GP dstrbuto. The mea devatos about the mea ca be calculated as E X μ r = where j =0 w j λ X μ r g x; dx = w r λ j m rm 2 Γ Γ Γ =0 m =0 w j = ab (j ) =0 j b m a j rm Γ j Γ j Γ rm 8- Re y etropy: The etropy of a radom varable X s a measure of ucertaty varato. The Re y etropy s defed as I R C = log I C WhereI C = C fc (x) dx ; C > 0, C, R we have I C = ac b C C C C Z λ By usg expadg theorem ad Trasformg Varables we obta I C = ab C C C C η j wherez = xλ ad η j = I R C = =0 j =0 C b C Z C Z C a Z a C b dx a C C j j j C, C C C log ab log log log C C C log η j 9 Estmato ad formato matrx: I ths secto, we dscuss maxmum lkelhood estmato ad ferece for the Kw-GP dstrbuto. Let x, x 2,, x be a radom sample from X~KW GP ξ where ξ = (a, b,,, λ, ) be the vector of the model Parameters, the log-lkelhood fucto for ξ reduces to log x λ l ξ = log a log b log log log log x λ b log x λ a The a log x λ (6)

7 25 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 Sce x > λ the the estmate value for λ equal the frst order statstc as λ = x.the score vector U ξ = l a, l b, l, l, l T, where the compoets correspodg to the parameters ξ are gve by dfferetatg (6). By settg z = x λ l a = a l b = b l = log b log z a log z a l = ad l = a b a b log z z a log a a(b ) a z z z log z a z z a z z a a z z The maxmum lkelhood estmates (MLEs) of the parameters are the solutos of the olear equatos l = 0, whch are solved teratvely. The observed formato matrx gve J ξ = J aa J ba J a J a J a J ab J bb J b J b J b J a J b J J J J a J b J J J J a J b J J J whose elemets are, J aa = a 2 b z a log z 2 z a 2 J ab = J a = J a = J a = a log a z log z b z z a z b z z a b a log z z az a log log z z z z z J bb = b 2 J b = a a 2 a a log a 2 z a a log z z a 2 z log z a

8 26 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 J b = a z z a J = 2 a z log z 2 J = where a(b ) z 2 a M = a L = z z J = a b J b = a z z a z log z 2 a a z z 2 a b z z a 2 z 2 z M L a 2 log z a a a z z z A D a 2 z 2 where A = z a z z log z D = z log z z z a z a J = z z a b 2 a 2 z z z z T R z a z 2 Where T = z a z R = z z a z a J = a a b where z 2 log z z z z z G P z a z 2 log z 2 z z z 2 z z z 2 G = a z P = z z a a

9 27 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 J = 2 a 2 a b 2 2 z 2 z z z z log z z z z z z 2 z z Y O a z a z 2 WhereY = z a z O = z z Explct expressos for the remag elemets of J follow by symmetry. 0 - Emprcal Applcatos: I ths secto, we llustrate the usefuless of the KumGP dstrbuto. 0. Smulato study: We coducted Mote Carlo smulato studes to assess o the fte sample behavor of the maxmum lkelhood estmators ofa, b,, ad All results a z a were obtaed from 000 Mote Carlo replcatos ad the smulatos were carred out usg the statstcal software Mathcad, The true parameter values used the data geeratg processes area = 3., b =.2, =.7, = 4ad = 3.2, Table presets the mea maxmum lkelhood estmates of the parameters that dex the KumGPdstrbuto alog wth the respectve root mea squared errors (RMSE) ad bas for sample szes = 30, 50, 80, 00 Table : Mea estmates, root mea squared errorsadbasofa, b,, ad the maxmum lkelhood Estmators of the KW-GP parameters. Parameter Mea RMSE bas 30 a b a 50 b 80 a b 00 a b From the results Table, we otce that the bases ad root mea squared errors of the maxmum lkelhood estmators of a, b,, ad decay toward zero as the sample sze creases, as expected. We also ote that there s small sample bas the estmato of the parameters that dex the KWGP dstrbuto. Future research should obta bas correctos for these estmators. 0.2Real Data Applcatos: I ths secto we use several real data sets to compare the fts of a Kw-GP dstrbuto wth those. AIC = 2l 2q, BIC = 2l qlog()caic = 2l 2q of other sub-models,.e., the Expoetated Geeralzed Pareto (EGP), GP ad Pareto dstrbutos. I each case parameters are estmated va the MLE method descrbed Secto 9 usg the MATHCAD software. Frst we descrbe the data sets. The we report the MLEs (ad the correspodg stadard errors paretheses) of the parameters ad the values of the AIC (Akake Iformato Crtero), CAIC (Cosstet Akake Iformato Crtero) ad BIC (Bayesa Iformato Crtero) statstcs q

10 Carbo Fbers Glass Fbres 28 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 Where l deotes the log-lkelhood fucto evaluated at the maxmum lkelhood estmates, q s the umber of parameters, ad s the sample sze.next, we shall compare the proposed KwGP dstrbuto (ad ther sub-models) wth several other lfetme dstrbutos data set,kumaraswamyfréchet dstrbutokwf (Mead, et al. (204)),the beta Fréchet(BF) (Nadarajah ad Gupta, (2004) ad Souza et al., (20)).Fally, we perform the Kolmogorov-Smrov (K-S) statstc ad 2l tests adplot hstograms of each data set to provde a vsual comparso of the ftted desty fuctos. 0-The Stregths of.5 Cm Glass Fbres: Here,the data set s obtaed from (Fatoet al.(203)).the data are cosstg of 63 of the stregths of.5 cm glass fbres, measured at the Natoal Physcal Laboratory, Eglad. Ufortuately, the uts of measuremet are ot gve the paper. The data are lsted the ext table Table 2: The Stregths of.5 cm Glass Fbres Data Set Ucesored Data Carbo Fbers : Here, the real data set wll use here to compare the fts of the Kum-GP dstrbuto ad those of other sub-models,.e., the Expoetated Geeralzed Pareto (EGP), GP ad Pareto dstrbutos. Cosderg a ucesored data set correspodga ucesored data set from Nchols ad Padgett (2006) cosstg of 00 observatos o breakg stress of carbo fbers ( Gba): Table 3: O breakg stress of carbo fbers set Table 4: MLEs of the model parameters, the correspodg SEs (gve paretheses) ad the statstcs AIC, BIC ad CAIC. Model Estmates Statstc a b AIC BIC CAIC Kum GP (0.4) (60.634) (4.563) (.666) (0.03) EGP (0.85) (8.245) (.44) (0.045) GP (20.244) (3.6) (0.046) KwF (7.982) (53.948) ---- (4.555) (0.07) BF (8.5) (8.238) (.085) (0.8) Kum GP (0.0037) (0.) ( ) (0.0000) ( ) EGP ( ) (0.003) (0.0005) (0.0007) GP (0.0829) (0.0005) (0.0579) KwF (2.393) (6.863) (2.259) )0.028( BF (0.236) (3.552) (2.52) (0.9) Sce the values of the AIC, BIC ad CAIC are smaller for the Kum-GP dstrbuto compared wth those values of the other models, the Kum-GP dstrbuto seems to be a very compettve model to

11 29 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 these data. I summary, the proposed KumGP dstrbuto produces better fts to the data tha ts sub-models. Normal P-P Plot of CarboFbers Observed Cum Prob Fg. 4: The FttedQ-Q Plots ad P-P Plots for the 63 ofthe stregths of.5 cm glass fbres data set& 00 observatos o breakg stress of carbo fbers ad Emprcal CDF. Table 5: K-S ad 2l statstcs for the chose Real data. Data Model KumGP EGP GP Glass Fbres K S l Carbo Fbers K S l Cocludg Remarks: The well-kow geeralzed Pareto dstrbuto s exteded by troducg two extra shape parameters, thus defg the Kumaraswamy geeralzed Pareto (Kum-GP) dstrbuto havg a broader class of hazard rate ad desty fuctos. Ths s acheved by takg (5) as the basele cumulatve dstrbuto of the geeralzed class of

12 30 M.E. Habb et al, 205 Australa Joural of Basc ad Appled Sceces, 9(27) August 205, Pages: 9-30 beta dstrbutos. A detaled study o the mathematcal propertes of the ew dstrbuto s preseted. The ew model cludes as specal submodels as expoetated geeralzed Pareto (EGP) ad tssub models. We obta the ordary momets, ad Re y etropy. The estmato of the model parameters s approached by maxmum lkelhood ad the observed formato matrx s obtaed. A applcato to a real data set dcates that the ft of the ew model s superor to the fts of ts prcpal sub-models. We hope that the proposed model may be terestg for a wder rage of statstcal research. ACKNOWLEDGEMENTS I thak the aoymous referees for useful suggestos ad commets whch have mproved the frst verso of the mauscrpt. REFERENCES Abdul Fattah, A.M., E.A. Elsherpy ad E.A. Husse, A ew geeralzed Pareto dstrbuto" Iterstat Joural, Dec07. Artur J. Lemote, 204. The beta log-logstc dstrbuto, BJPS Joural BJPS Castllo, E., A.S. Had, N. Balakrsha ad J.M. Saraba, ExtremeValue ad Related Models wth Applcatos Egeerg ad Scece.Wley, Hoboke, New Jersey. Cordero, G.M. ad M. de Castro, 20. A ew famly of geeralzed dstrbutos. Joural of Statstcal Computato ad Smulato, 8: Choulaka, V., M.A. Stephes, 200. Goodess-of-ft for the geeralzed Pareto dstrbuto. Techo-metrcs, 43, Cordero, G.M., E.M.M. Ortega ad S. Nadarajah, 200. The Kumaraswamy Webull dstrbuto wth applcato to falure data. Joural of the Frakl Isttute, 347: Eugee, N., C. Lee ad F. Famoye, Betaormal dstrbuto ad ts applcatos.commucatos Statstcs: Theory ad Methods, 3: Embrechts, P., C. Kluppelberg ad T. Mkosch, 997. Modelg ExteralEvets: For Isurace ad Face. Sprger, Berl. Merovc, F., 203. Trasmuted Expoetated Expoetal Dstrbuto, Mathematcal Sceces Ad Applcatos E-Notes, (2): Nadarajah, S. ad A.K. Gupta, The beta Fréchet dstrbuto. Far East Joural of Theoretcal Statstcs, 4: Gupta, R.C., R.D. Gupta, P.L. Gupta, 998. Modelg falure tme data by Lehma alteratves. Commucatos Statstcs: Theory ad Methods, 27, Gupta, R.D., D. Kudu, 999. Geeralzed expoetal dstrbutos. Austral. NZ J. Statst., 4: Gupta, A.K., S. Nadarajah, O the momets of the beta ormal dstrbuto. Commucatos Statstcs - Theory ad Methods, 33, Joes, M.C., Kumaraswamy dstrbuto: A beta-type dstrbuto wth some tractablty advatages. Statstcal Methodology, 6: Keepg, E.S., J.F. Keey, 962. Mathematcs of Statstcs. Part. Kumaraswamy, P., 980. A geeralzed probablty desty fuctos for double-bouded radom processes. Joural of Hydrology, 46: Mead, M.E. ad A.R. Abd-Eltawab, 204. A ote o KumaraswamyFréchet dstrbuto. Australa Joural of Basc ad Appled Sceces, 8(5): Leadbetter, M.R., G. Ldgre ad H. Rootze, 987. Extremes ad Related Propertes of Radom Sequeces ad Processes. Sprger, New York. Moors, J.J., 998. A quatle alteratve for kurtoss. Joural of the Royal Statstcal Socety D, 37: Nadarajah, S., G.M. Cordero ad E.M.M. Ortega, 20. Geeral results for the Kumaraswamy- G dstrbuto. Joural of Statstcal Computato ad Smulato. DOI: 0.080/ Nchols, M.D., W.J. Padgett, A Bootstrap cotrol chart for Webull percetles. Qualty ad Relablty Egeerg Iteratoal, 22: 4-5. Pareto, V., 897. Coursd'ecoomepoltque, Lausaee ad Pars; Rage ad Ce. Pckads, J., 975. Statstcal ferece usg extreme order statstcs. Aals of Statstcs, 3: 9-3. Souza, W.M., G.M. Cordero ad A.B. Smas, 20. Some results for beta Fréchet dstrbuto. Commu.Statst. Theory-Meth., 40: