The Transmuted Weibull-Pareto Distribution

Transcription

1 Marquette Uiversity Mathematics, Statistics ad Computer Sciece Faculty Research ad Pulicatios Mathematics, Statistics ad Computer Sciece, Departmet of The Trasmuted Weiull-Pareto Distriutio Ahmed Z. Afify Beha Uiversity Haitham M. Yousof Beha Uiversity Nadeem Shafique Butt Kig Adul Aziz Uiversity Gholamhossei G. Hamedai Marquette Uiversity, Pulished versio. Pakista Joural of Statistics, Vol. 3, No. 3 (016): Permalik. 016 Pakista Joural of Statistics. Used with permissio.

2 Pak. J. Statist. 016 Vol. 3(3), THE TRANSMUTED WEIBULL-PARETO DISTRIBUTION Ahmed Z. Afify 1, Haitham M. Yousof 1, Nadeem Shafique Butt ad G.G. Hamedai 3 1 Departmet of Statistics, Mathematics ad Isurace Beha Uiversity, Egypt. ahmed.afify@fcom.u.edu.eg haitham.yousof@fcom.u.edu.eg Departmet of Family ad Commuity Medicie Faculty of Medicie i Raigh, Kig Adulaziz Uiversity Jeddah, Saudi Araia. shafique@kau.edu.sa 3 Departmet of Mathematics, Statistics ad Computer Sciece Marquette Uiversity, Milwaukee, USA. gholamhoss.hamedai@marquette.edu ABSTRACT A ew geeralizatio of the Weiull-Pareto distriutio called the trasmuted Weiull-Pareto distriutio is proposed ad studied. Various mathematical properties of this distriutio icludig ordiary ad icomplete momets, quatile ad geeratig fuctios, Boferroi ad Lorez curves ad order statistics are derived. The method of maximum likelihood is used for estimatig the model parameters. The flexiility of the ew lifetime model is illustrated y meas of a applicatio to a real data set. KEY WORDS Trasmuted family, Weiull-Pareto Distriutio, Boferroi ad Lorez curves, Order Statistics, Likelihood Estimatio. 1. INTRODUCTION Recetly, several families of distriutios have ee proposed via extedig commo families, y addig oe or more parameters to the aselie model, of cotiuous distriutios. These ew families provide more flexiility i modelig ad aalyzig real life data i may applied areas. For that reaso the statistical literature cotais a good umer of ew families. For example, the geeralized trasmuted-g family proposed y Nofal et al. (015), the trasmuted expoetiated geeralized-g class defied y Yousof et al. (015), the trasmuted geometric-g family itroduced y Afify et al. (016a) ad the Kumaraswamy trasmuted-g family itroduced y Afify et al. (016). Aother example is the Weiull-G (W-G for short) class defied y Bourguigo et al. (014). Usig the W-G family, Tahir et al. (015) defied ad studied the Weiull-Pareto (WPa) distriutio extedig the Pareto (Pa) distriutio. The cumulative distriutio fuctio (cdf) of the WPa distriutio is give (for x > > 0 ) y 016 Pakista Joural of Statistics 183

3 184 The Trasmuted Weiull-Pareto Distriutio x Gx ( ) = 1 exp 1, (1.1) where is a positive scale parameter ad ad are positive shape parameters. The correspodig proaility desity fuctio (pdf) is give y 1 1 x x g( x) = x 1 exp 1. (1.) I this paper, we defie ad study a ew distriutio y addig oe extra shape parameter i equatio (1.) to provide more flexiility to the geerated model. I fact, ased o the trasmuted-g (TG) family of distriutios proposed y Shaw ad Buckley (007), we costruct a ew model called the trasmuted Weiull-Pareto (TWPa) distriutio ad give a comprehesive descriptios of some of its mathematical properties. We hope that the ew model will attract wider applicatios i reliaility, egieerig ad other areas of applicatios. Recetly, may authors used the TG family to costruct ew distriutios. For example, Afify et al. (015a) itroduced the trasmuted Marshall-Olki Fréchet, Afify et al. (015) proposed the trasmuted Weiull Lomax, Afify et al. (014) defied the trasmuted complemetary Weiull geometric, Kha ad Kig (013) itroduced the trasmuted modified Weiull ad Aryal ad Tsokos (011) proposed the trasmuted Weiull distriutios. For a aritrary aselie cdf with cdf ad pdf give y ad = 1 Gx, Shaw ad Buckley (007) defied the TG family F x G x G x (1.3) f x = g x 1 G x, (1.4) respectively, where 1. The TG desity is a mixture of the aselie desity ad the expoetiated-g (Exp-G) desity with power parameter two. For = 0, (1.3) gives the aselie distriutio. The rest of the paper is outlied as follows. I Sectio, we defie the TWPa distriutio ad provide the graphical presetatio of its pdf ad hazard rate fuctio (hrf). A useful mixture represetatio for its pdf ad cdf is provided i Sectio 3. Sectio 4 provides the mathematical properties icludig ordiary ad icomplete momets, quatile ad geeratig fuctios, Boferroi, Lorez ad Zega curves, momets of residual life ad momets of the reversed residual life are derived. I Sectio 5, the order statistics ad their momets are discussed. The proaility weighted

4 Afify, Yousof, Butt ad Hamedai 185 momets (PWMs) are discussed i Sectio 6. Certai characterizatios are preseted i Sectio 7. The maximum likelihood estimates (MLEs) for the model parameters are demostrated i Sectio 8. I Sectio 9, simulatio results to assess the performace of the proposed maximum likelihood estimatio method are discussed. The TWPa distriutio is applied to a real data set to illustrate its usefuless i Sectio 10. Fially, some cocludig remarks are give i Sectio 11.. THE TWPa DISTRIBUTION By isertig Equatio (1.1) ito Equatio (1.3), the cdf of the TWPa model is give (for x > ) y x x Fx ( ) = 1 exp 1 1 exp 1. The correspodig pdf is otaied y (.1) 1 1 x x x f x= x 1 exp 1 1 exp 1, (.) where is a scale parameter ad, ad are positive shape parameters. The radom variale X is said to have a TWPa distriutio, deoted y X TWPa (,,, ), if its cdf is give y Equatio (.1). It is clear that Equatio (.) reduced to the WPa model for =0. The TWPa distriutio due to its flexiility i accommodatig all forms of the hrf as show from Figure seems to e a importat model that ca e used. A physical iterpretatio of the cdf of TWPa model is possile if we take a system cosistig of two idepedet compoets fuctioig idepedetly at a give time. So, if the two compoets are coected i parallel, the overall system will have the TWPa cdf with = 1. Aother motivatio for the TWPa distriutio follows y takig two iid radom variales, say Z 1 ad Z, with cdf x = 1 exp 1. G x Let Z1: = mi( Z1, Z ) ad Z: = max( Z1, Z ). Now, cosider the radom variale X defied y

5 186 The Trasmuted Weiull-Pareto Distriutio Z X = Z 1: : 1, with proaility ; 1, with proaility. The, the cdf of X is give y (.1). Figure 1 provide some plots of the TWPa desity curves for differet values of the parameters,, ad. Plots of the hrf of TWPa for selected parameter values are give i Figure. Some possile shapes for the TWPa cdf are displayed i Figure 3. Figure 1: Plots of the TWPa Desity Fuctio for some Parameter Values

6 Afify, Yousof, Butt ad Hamedai 187 Figure : Plots of the hrf of the TWPa for some Parameter Values

7 188 The Trasmuted Weiull-Pareto Distriutio Figure 3: Plots of the cdf of the TWPa for some Parameter Values

8 Afify, Yousof, Butt ad Hamedai MIXTURE REPRESENTATION The TWPa desity fuctio give i Equatio (.) ca e expressed as ( ) ( ) ( ) ( ) 1 G x G x f ( x) = 1 g( x) exp G x 1 G x 1 1 G( x) G( x) G( x) G( x) g( x) exp g( x) exp. 1 1 G( x) G( x) G( x) G( x) Usig Equatios (1.1) ad (1.) ad after a power series, the we have k 1 k 1 j 1 x = 1 1!! 1 1 x f x x k j k k x 1 k 1 j 1 x 1 x k! j! k 1 1 k x k 1 j 1 x 1 x k! j! k 1 1 k j1 k j1 k j1. (3.1) Usig the geeralized iomial series expasio ad after some algera, the TWPa desity ca e expressed as f x = or equivaletly where ad k k 1 1 k 1 j 1 1 k! j! k 1 1 k 1 j k 1 j1 x x k 1 j 1, x f ( x ) = k, j h,, k1 j ( x ), (3.) k, j k k1 1 k 1 j =. k! j! k 1 j k 1 1 x x h,, 1 ( x) = k 1 j 1 k j x k1 j1

9 190 The Trasmuted Weiull-Pareto Distriutio is the expoetiated Pareto (EPa) desity with parameters, ad k 1 j. This meas that the TWPa desity ca e expressed as a mixture of EPa desities. So, several of its properties ca e derived form those of the EPa model. Itegratig (3.), the cdf of TWPa ca e expressed as F ( x ) = k, j H,, k1 j ( x ), where H,, k 1 j ( x) is the cdf of EPa with parameters, ad 1 4. PROPERTIES k j. Estalished algeraic expasios to determie some structural properties of the TWPa distriutio ca e more efficiet tha computig those directly y umerical itegratio of its desity fuctio. The mathematical properties of the TWPa distriutio icludig ordiary ad icomplete momets, factorial momets, quatile ad geeratig fuctios, Boferroi, Lorez ad Zega curves, residual life fuctio ad reversed residual life fuctio are provided i this sectio. 4.1 Momets Usig (.), the r th momet of X, deoted y r r = k, j EY,, k1 j, r, is give y where E Y r r,, k1 j = 0 x h,, k1 j ( x ) dx. The, we otai (for r ) r r r = k 1 j k, jb1, k 1 j, (4.1) 1 m1 where, = B m t t dt is the eta fuctio. The skewess ad kurtosis measures ca e calculated from the ordiary momets usig well-kow relatioships. Corollary 1 Usig the relatio etwee the cetral momets ad o-ceteral momets, we ca otai the th cetral momet of a TWPa radom variale, deoted y M, as follows the, r r 1 M = E X = E X, r=0r

10 Afify, Yousof, Butt ad Hamedai 191 M r r = 1 1 r r=0r ad cumulats ( ) of X are otaied from (4.1) as 1 1 = r r, r=0r 1 3 where 1 = 1 hece = 1, 3 = etc. The skewess ad kurtosis measures ca e calculated from the ordiary momets usig well-kow relatioships. 4. Quatile ad Geeratig Fuctios The quatile fuctio (qf) of X, where X TWPa (,,, ), is otaied y ivertig (.1) to otai x u 1 x = F u u as 1/ 1/ 1 1 4u = 1 log. Simulatig the TWPa radom variale is straightforward. If U is a uiform variate o the uit iterval (0,1), the the radom variale X x follows (.). = U Here, we will provide two formulas for the momet geeratig fuctio (mgf) of the r t r TWPa distriutio. The mgf is defied y M X t = r=0 E X t r M X t = k 1 j k, jb 1, k 1 j. k, j, r=0 r! r r!. The The secod ca e computed usig Maple. For t 0, p >0 ad q >0 Let p tx q J q, p, t = x e dx. Usig this software, we ca otai ad,, = 1 J q p t csc tq p p p p e p p, tq q p 1 tq p qt tq tq M X t = d J, m 1 1, t, k, j m=0 where z1 y s ( z, s) = y e dy is the complemetary icomplete gamma fuctio ad 1 m m k 1 j 1 dk, j = 1 k, j k 1 j. m

11 19 The Trasmuted Weiull-Pareto Distriutio 4.3 Icomplete Momets The s th icomplete momets, say t, is give y t s 0 s t = x f x dx. s Usig equatio (3.) ad the lower icomplete eta fuctio, we otai (for s ) s s s t = k, j k 1 j Bt, k 1 j, (4.) z a1 where, = 1 1 Bz a w w dw is the icomplete eta fuctio. The first 0 icomplete momet of the TWPa distriutio ca e otaied y settig s =1 i (4.). The mai applicatio of the first icomplete momet refers to Boferroi ad Lorez curves. These curves are very useful i ecoomics, reliaility, demography, isurace ad medicie. The aswers to may importat questios i ecoomics require more tha just kowig the mea of the distriutio, ut its shape as well. This is ovious ot oly i the study of ecoometrics ut i other areas as well. The importat applicatio of the first icomplete momet is related to the Lorez ad Boferroi curves. These curves are very useful i ecoomics, reliaility, demography, isurace ad medicie. The Lorez curve, say LF x, ad Boferroi curve, say B F x, are defied, respectively, y The, ad x 1 x 0 1 x LF LF x = tf t dt ad B F x = tf 0 tdt =. E X E X F x F x L F x = 1 Bt 1, k 1 j r B1, k 1 j k, j k, j 1 k, j Bt 1, k 1 j B F x =. r F x k, jb1, k 1 j Aother applicatio of the first icomplete momet is related to mea residual life ad mea waitig time give y m1 t = 1 1t / R( t) t ad M1t = t 1t / F t, respectively.

12 Afify, Yousof, Butt ad Hamedai Residual Life Fuctio Several fuctios are defied related to the residual life. The failure rate fuctio, mea residual life fuctio ad the left cesored mea fuctio, also called vitality fuctio. It is well kow that these three fuctios uiquely determie Fx ( ). Moreover, the momets of the residual life, m ( t) = E[( X t) X > t], = 1,,..., uiquely determie Fx ( ). The th momet of the residual life of X is give y 1 m ( t) = ( x t) df( x). t 1 Ft ( ) The, we ca write (for r < ) 1 r r m ( t) = t x df( x), t 1 Ft ( ) r=0r 1 m ( t) = k 1 j t Rt () r=0 r r k, j r r B, k 1 j Bt, k 1 j. r Aother iterestig fuctio is the mea residual life (MRL) fuctio or the life expectatio at age x defied y m 1 ( x) = E ( X x) X > x, which represets the expected additioal life legth for a uit which is alive at age x. The MRL of the TWPa distriutio ca e otaied y settig =1 i the last equatio. 4.5 Reversed Residual Life Fuctio The momets of the reversed residual life, M( t) = E ( t X ) X t =1,,... uiquely determie Fx ( ). We otai for t >0, t 0 M( t) = ( t x) df( x). Therefore, the th momet of the reversed residual life of X give that r < ecomes 1 r r t r M ( t) = 1 t x df( x), 0 Ft () r=0r r 1 r M ( t) = k 1 jk, jbt 1, k 1 j. r Ft () r=0 r t The mea iactivity time (MIT) or mea waitig time (MWT) also called the mea reversed residual life fuctio is defied y M 1 ( t) = E[( t X ) X t], ad it represets the waitig time elapsed sice the failure of a item o coditio that this failure had occurred i (0, x ). The MIT of X ca e otaied y settig =1 i the last equatio.

13 194 The Trasmuted Weiull-Pareto Distriutio 5.ORDER STATISTICS If X1, X,..., X is a radom sample of size from the TWPa distriutio ad X 1, X,..., X are the correspodig order statistics, the the pdf of i th order statistic deoted y where f i : x is give y f( x) 1 fi : x = 1 F ( x). (5.1) B i, 1 1 j ij1 i j=0 j Usig Equatios (.1) (.) ad (5.1), we ca write ij1,,, 1 f x F( x) = t h ( x). t kl, = mw, =0 The q th momet of X i : is kl k l kl, =0 k mw m i jm1 k 1 1 k 1 l 1 k! l! k 1 l w 1 k 1 1 i j 1i j m 1 k k 1 w 1 w. m w j 1 1 q i i j 1 q E Xi: = tk, l E Y.,, k1 l k, l=0 j=0 B, 1 (5.) Usig (4.1), we ca write 1 j 1 j q q B i, i 1 1 q E Xi : = k 1 l B 1, k 1 l. k, l=0 j=0 Based upo the momets i Equatio (5.), we ca derive explicit expressios for the L-momets of X as ifiite weighted liear comiatios of the meas of suitale TWPa distriutio. They are liear fuctios of expected order statistics defied y r1 1 k r 1 r = 1 E X rk: r, r 1. r k=0 k The first four L-momets are give y: = E X, = E X X 1 1:1 1, : 1: 1 3 = 3:3 :3 1:3 3 E X X X ad 1 4 = 4:4 3 3:4 3 :4 1:4 4 E X X X X. Oe simply ca otai the 's for X from equatio (5.) with q =1.

14 Afify, Yousof, Butt ad Hamedai PROBABILITY WEIGHTED MOMENTS The PWMs are expectatios of certai fuctios of a radom variale. They ca e derived for ay radom variale whose ordiary momets exist. The PWM approach ca e used for estimatig parameters of ay distriutio whose iverse form caot e expressed explicitly. The ( sr, ) th PWM of X, say sr,, is defied y s r s r sr, = E X F( X ) = x F( x) f x dx. Usig Equatios (.1) ad (.), we ca write f k mw k!! w 1 k 1 j 1 r m rm r r m x F( x) = 1 m, w, k j k k j m w k 1 j1 k k x x 1 w 1 w k 1 j 1. x The, we have where r,,, 1 f x F ( x ) = d h ( x ), d k, j = mw, =0 k j k j k mw m rm k 1 1 k 1 j 1 k! j! k 1 j w 1 k 1 1 r r m k k 1 w 1 w. m w The, the ( sr, ) th PWM of X ca e expressed as s = d E Y. s, r k, j,, k 1 j Usig (4.1), we ca write s s s, r = dk, j k 1 j B1, k 1 j. 7. CHARACTERIZATIONS The prolem of characterizig a distriutio is a importat prolem i various fields which has recetly attracted the attetio of may researchers. These characterizatios have ee estalished i may differet directios. This sectio deals with two characterizatios of TWPa distriutio. These characterizatios are ased o (i) a simple relatioship etwee two trucated momets ad (ii) o coditioal expectatio of a

15 196 The Trasmuted Weiull-Pareto Distriutio fuctio of the radom variale. It should e metioed that for our characterizatio (i), the cdf eed ot have a closed form. We elieve, due to the ature of the cdf of TWPa, there may ot e other possile characterizatios of this distriutio tha the oes preseted here. 7.1 Characterizatios Based o Two Trucated Momets I this susectio we preset characterizatios of TWPa distriutio i terms of a simple relatioship etwee two trucated momets. Our first characterizatio result orrows from a theorem due to Gläzel (1987), see Theorem A elow. We refer the iterested reader to Gläzel (1990) for a proof of Theorem A. Note that the result holds also whe the iterval H is ot closed. Moreover, as metioed aove, it could e also applied whe the cdf F does ot have a closed form. As show i Gläzel (1990), this characterizatio is stale i the sese of weak covergece. Theorem 7.1. Let,,P e a give proaility space ad let H = d, e e a iterval for some d < e ( d =, e = might as well e allowed). Let X : H e a cotiuous radom variale with the distriutio fuctio F ad let g ad h e two real fuctios defied o H such that E g X X x = E h X X x x, x H, 1 is defied with some real fuctio. Assume that g, h C H, C H ad F is twice cotiuously differetiale ad strictly mootoe fuctio o the set H. Fially, assume that the equatio h = g has o real solutio i the iterior of H. The F is uiquely determied y the fuctios g, h ad, particularly x u uhu F x = C exp s u du, a g u ' ' where the fuctio s is a solutio of the differetial equatio s = h / h g C is the ormalizatio costat, such that =1 H df. Propositio 7.1. X :, e a cotiuous radom variale ad let Let ad ad x hx = 1 exp 1 1

16 Afify, Yousof, Butt ad Hamedai 197 x g x = hxexp 1 for x >. The radom variale X elogs to TWPa family (.) if ad oly if the fuctio defied i Theorem 7.1 has the form 1 x x = exp 1, x >. Proof. Let X e a radom variale with desity (.), the ad ad fially x 1 F x E h x X x = exp 1, x >, 1 x 1 F x E g x X x = exp 1, x >, 1 x xhx g x = hxexp 1 < 0, for x >. Coversely, if is give as aove, the (7.1) xhx xhx g x 1 1 x sx = = x 1, x >, ad hece x = 1, x >. s x Now, i view of Theorem 7.1, X has desity (.). Corollary 7.1. X :, e a cotiuous radom variale ad let Let hx e as i Propositio 7.1. The pdf of X is (6) if ad oly if there exist fuctios g ad defied i Theorem 7.1 satisfyig the differetial equatio

17 198 The Trasmuted Weiull-Pareto Distriutio xhx xhx g x 1 1 x = x 1, x >. The geeral solutio of the differetial equatio i Corollary 7.1 is (7.) x 1 1 x x 1 x = exp 1, x 1 exp 1 h x g x dx D where D is a costat. Note that a set of fuctios satisfyig the differetial equatio (7.) is give i Propositio 7.1 with D = 0. However, it should e also oted that there are other triplets hg,, satisfyig the coditios of Theorem Characterizatios Based o Coditioal Expectatio of a Fuctio of the Radom Variale I this susectio we preset a characterizatio result i terms of a fuctio of the radom variale X. The followig Propositio has appeared i our previous work which will e used to characterize TWPa distriutio Propositio 7.. Let X : a, e a cotiuous radom variale with cdf F ad correspodig pdf f. Let x e a differetiale fuctio greater tha 1 o, that limxa x =1 ad =1 if ad oly if Remark 7.1. limx x c. The, for 0 < c <1, a such E X X x = c 1 c x, (7.3) x 1c c 1 Fx =. c Takig, e.g., a, =, ad x x x = 1 c 1 exp 1 1 exp 1. Propositio 7.3 gives a characterizatio of TWPa distriutio. c 1c (7.4)

18 Afify, Yousof, Butt ad Hamedai ESTIMATION The maximum likelihood estimators (MLEs) for the parameters of the TWPa distriutio are discussed i this sectio. Let x1,..., x e a radom sample of of size from the TWPa( x; ) distriutio, where is the ukow parameter vector =,,,. T The, the log-likelihood fuctio for the parameters vector, say expressed as i=1 i i=1 i s p = l l l 1 l x s xi s where si = 1 ad p = i i e. 1 l l 1, i=1 i i=1 i, ca e Assumig kow, therefore the score vector compoets, U = =,, ad T are give y 1 zi 1 zi pi si i=1 i i=1 i=1 i i i=1 si 1 pi = l l x 1 z s, pi si l si i=1 si i=1 si si i=1 1 i = l l p pi 1 = i=1, 1 p xi xi where zi = l. i We ca fid the estimates of the ukow parameters y settig the score vector to zero, U ˆ = 0, ad solvig them simultaeously to otai the ML estimators ˆ, ˆ ad ˆ. These equatios caot e solved aalytically ad statistical software ca e used to solve them umerically y meas of iterative techiques such as the Newto-Raphso algorithm. For the four-parameter TWPa distriutio all the secod order derivatives exist. For iterval estimatio of the model parameters, we require the 3 3 oserved J = J for r, s =,,, whose elemets are give y iformatio matrix, rs

19 00 The Trasmuted Weiull-Pareto Distriutio zi xi 1 xi J = 1 =1 l 1 =1 1 l i i zisi si s i x 1 i 1 pi 1 zisi si l zi pi si i=1 1 pi, 1 zi 1 zi pi si l si = i=1 i=1 i i 1 l i i=1 si 1 pi J z s s J 1 zi pi si 1l si si l s i i=1 1 pi, 1 1 zi pi si 1 pi zi pisi i=1 1 pi =, l 1 l 1 p isi si si pi si si s i i=1 i i i=1 i=1 1 pi 1 pi J = s l s, J pi 1 pi 1 i=1 J i=1 1 pi 1 pi = ad =. Uder stadard regularity coditios, the multivariate ormal N 1 3 0, J ˆ distriutio ca e used to costruct approximate cofidece itervals for the model J ˆ is the total oserved iformatio matrix evaluated at ˆ. parameters. Here, Therefore, approximate 100(1 )% cofidece itervals for, ad ca e determied as: ˆ Z Jˆ, ˆ Z Jˆ, ad ˆ Z J ˆ, where th percetile of the stadard ormal distriutio. Z is the upper 9. SIMULATION STUDY Here, we assess the performace of the maximum likelihood estimatio procedure for estimatig the TWPa parameters usig Mote Carlo simulatio. A ideal techique for simulatig from (5) is the iversio method. For differet comiatios of,, ad samples of sizes = 100, 00, 500 ad 1000 are geerated from the TWPa model. We repeated the simulatio k = 100 times ad calculated the MLEs ad the stadard deviatios of the parameter estimates. We use three comiatios for the parameter values (I: = = = 1.5 ad = 1, II: =1.5, = = ad =0 ad III: = =1, = ad =3). The empirical results are give i Tale 1. It is evidet

20 Afify, Yousof, Butt ad Hamedai 01 that the estimates are quite stale ad are close to the true value of the parameters for these sample sizes. Additioally, as the sample size icreases, the iases ad the stadard deviatios of the MLEs decrease as expected. Tale 1 MLEs ad Stadard Deviatios for various Parameter Values Sample Estimated Values (Stadard Deviatios) size ( ) ˆ ˆ ˆ ˆ I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) II ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) III ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The old values are comied results of ( = 1800).

21 0 The Trasmuted Weiull-Pareto Distriutio 10. APPLICATION I this sectio, we provide a applicatio of the TWPa distriutio to show its flexiility ad importace. We shall compare the TWPa model with other related models, amely Weiull Pareto (WPa), McDoald Lomax (McL) (Lemote ad Cordeiro 013), trasmuted Weiull Lomax (TWL) (Afify et al. 015) ad trasmuted complemetary Weiull geometric (Afify et al. 014) distriutios. The desity fuctios (for x > 0) associated to these models are give y The McL pdf give y a1 1 x x x f( x) = B a, The TWL pdf give y 1 a x x f ( x) = 1 exp a 1 1 The TCWG pdf give y 1 x x exp a x x x f x= x e 1 e 1 1 e. The parameters of the aove desities are all positive real umers except for the TWL ad TCWG distriutios for which 1. We make use of the data set of gauge legths of 10 mm from Kudu ad Raqa (009). This data set cosists of 63 oservatios: 1.901,.13,.03,.8,.57,.350,.361,.396,.397,.445,.454,.474,.518,.5,.55,.53,.575,.614,.616,.618,.64,.659,.675,.738,.740,.856,.917,.98,.937,.937,.977,.996, 3.030, 3.15, 3.139, 3.145, 3.0, 3.3, 3.35, 3.43, 3.64, 3.7, 3.94, 3.33, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.56, 3.68, 3.85, 3.871, 3.886, 3.971, 4.04, 4.07, 4.5, 4.395, These data have ee used y Afify et al. (015) to fit the expoetiated trasmuted geeralized Rayleigh distriutio. We cosider some criteria like (where is the maximized log-likelihood), AIC (Akaike iformatio criterio) ad CAIC (the cosistet Akaike iformatio criterio), HQIC (Haa-Qui iformatio criterio) ad BIC (Bayesia iformatio criterio) i order to compare the distriutios. I geeral, the smaller the values of these statistics, the etter the fit to the data. 3 1

22 Afify, Yousof, Butt ad Hamedai 03 Tale provides the MLEs ad their correspodig stadard errors (SEs) of the model parameters ad the umerical values of the, AIC, CAIC, HQIC ad BIC. These umerical results are otaied usig the Mathcad program. It is show from Tale that the TWPa model has the lowest values for the, AIC, CAIC, HQIC ad BIC statistics amog all fitted models (except for HQIC ad BIC of the WPa distriutio). So, the TWPa model could e chose as the est model. Figure 4 displays the fitted pdf ad cdf for the TWPa model ad Figure 5 displays the QQ-plot ad estimated survival fuctio of the TWPa distriutio. It is clear from these plots that the TWPa provides good fit to this data set. Tale MLEs, their Stadard Errors (SEs) ad Goodess-of-Fit Statistics Model Estimates SEs AIC CAIC HQIC BIC TWPa ˆ = ˆ = ˆ = ˆ = WPa ˆ = ˆ = ˆ = TWL ˆ = ˆ = ˆ = ˆ = ˆ = McL ˆ = ˆ = ˆ = ˆ = ˆ = TCWG ˆ = ˆ = ˆ = ˆ =

23 04 The Trasmuted Weiull-Pareto Distriutio 11. CONCLUSIONS I this paper, We propose a ew four-parameter model, called the trasmuted Weiull-Pareto (TWPa) distriutio, which exteds the Weiull-Pareto (WPa) distriutio itroduced y Tahir et al. (015). We provide some of its mathematical properties. The TWPa desity fuctio ca e expressed as a mixture of expoetiated Pareto desities. We derive explicit expressios for the ordiary ad icomplete momets, quatile ad geeratig fuctios, Lorez, Boferroi ad Zega curves, momets of residual life ad momets of the reversed residual life. We also otai the desity fuctio of order statistics ad their momets. Further, the Proaility weighted momets are ivestigated ad certai characterizatio results are provided. We discuss the maximum likelihood estimatio of the model parameters. The proposed distriutio is applied to a real data set ad provides a etter fit tha several ested ad o-ested models. Figure 4: Estimated pdf ad cdf of the TWPa Distriutio Figure 5: QQ-Plot ad Estimated Survival Fuctio of the TWPa Distriutio

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