Exponentiated Transmuted Modified Weibull Distribution

Transcription

1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 1, 2015, 1-14 ISSN Exponentated Transmuted Modfed Webull Dstrbuton Mansha Pal 1,, Montp Tensuwan 2 1 Department of Statstcs, Unversty of Calcutta, Inda 2 Department of Mathematcs, Faculty of Scence, Mahdol Unversty, Thaland Abstract. The paper ntroduces the exponentated transmuted modfed Webull dstrbuton, whch contans a number of dstrbutons as specal cases. The propertes of the dstrbuton are dscussed and explct expressons for the quantles, mean devatons and the relablty are derved. The dstrbuton and moments of order statstcs are also studed. Estmaton of the model parameters by the methods of least squares and maxmum lkelhood are dscussed. Fnally, the usefulness of the dstrbuton for modelng data s llustrated usng real data Mathematcs Subject Classfcatons: 90B25; 62N05 Key Words and Phrases: Relablty functon, moment generatng functon, quantles, mean devaton, order statstcs, least square estmaton, maxmum lkelhood estmaton. 1. Introducton Modellng and analyss of lfetme data have become very crucal n dfferent areas of research, lke engneerng, medcne, relablty, etc. In ths regard, t s observed that the Webull dstrbuton s extensvely used as t s found to provde reasonable ft n many practcal stuatons. Attempts at generalzaton of the dstrbuton have led to the exponentated Webull dstrbuton, the modfed Webull dstrbuton and the exponentated modfed Webull dstrbuton. An nterestng dea of generalzng a dstrbuton, whch s known n the lterature as transmutaton has been used to develop further dstrbutons. A random varable T s sad to have a transmuted dstrbuton f ts dstrbuton functon s gven by Z(t)=(1λ)G(t) λ[g(t)] 2, (1) where G(t) denotes the base dstrbuton, and λ [-1, 1] denotes the transmutng parameter. Aryall and Tsokos[2] ntroduced the transmuted Webull dstrbuton, Ebrahem[4] studed the exponentated transmuted Webull dstrbuton, Pal and Tensuwan[7] developed the beta transmuted Webull dstrbuton, Khan and Kng[6] nvestgated the transmuted modfed Correspondng author. Emal addresses: manshapal2@gmal.com (M. Pal),montp.te@mahdol.ac.th (M. Tensuwan) 1 c 2015 EJPAM All rghts reserved.

2 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Webull dstrbuton, and Ashour and Eltehwy[3] proposed the transmuted exponentated modfed Webull dstrbuton. In ths paper, we ntroduce and study several mathematcal propertes of a new relablty model referred to as the exponentated transmuted modfed Webull dstrbuton. The modfed Webull dstrbuton, ntroduced by Sarhan and Zandn[8], has the cumulatve dstrbuton functon (c.d.f.) F MW (t)=1 ex p( αt γt β ), t 0,α,β,γ>0. (2) The c.d.f. of the transmuted modfed Webull dstrbuton[6] s gven by F T MW (t)=[1 ex p( αt γt β )][1λex p( αt γt β )], t 0,α,β,γ>0. (3) The above dstrbuton has three shape parametersα,β andγ, andλdenotes the transmutng parameter. The exponentated transmuted modfed Webull dstrbuton generalzes ths dstrbuton by ntroducng another shape parameter. The paper s organzed as follows. In Secton 2 we ntroduce the dstrbuton. In Sectons 3, we obtan the quantle functon of the dstrbuton. The moment generatng functon and the moments are derved n Sectons 4. Mean devaton s dscussed n Secton 5. Order statstcs and ther moments are studed n Sectons 6. In Secton 7, the stress-strength relablty s obtaned. Estmaton of parameters by the least square method and the maxmum lkelhood method are dscussed n Sectons 8 and 9. In Secton 10, a smulaton study s carred out to compare the two methods of estmaton. The usefulness of the dstrbuton for modelng real lfe data s llustrated n Secton 11. Fnally, n Secton 12, we make some concludng remarks on our study. 2. Exponentated Transmuted Modfed Webull Dstrbuton The fve parameter exponentated transmuted modfed Webull (ETMW) dstrbuton s gven by the c.d.f. F(t)=[1 ex p( αt γt β )1λex p( αt γt β )] δ, t 0,α,β,γ>0,λ [ 1,1], (4) whereα,β,γ,δare all shape parameters, andλs the transmutng parameter. The densty functon of the dstrbuton s obtaned as f(t)=δ[{1 ex p( αt γt β )}{1λex p( αt γt β )}] δ 1 (αβγt β 1 )ex p( αt γt β ) [1 λ2λex p( αt γt β )], t 0. The hazard rate and the hazard functon of the dstrbuton are as follows: r(t)=δ[{1 ex p( αt γt β )}1λex p( αt γt β )] δ 1 (αtβγt β 1 )ex p( αt γt β ) [1 λ2λex p( αt γt β )][1 {ex p( αt γt β )} ( δ){1λex p( αt γt β )} δ ] ( 1), t 0, H(t)= ln[1 λ2λex p( αt γt β )][1 { ex p( αt γt β )} δ {1λex p( αt γt β )} δ ], t 0. (5)

3 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Plots of the p.d.f. and the hazard rate are gven n Fgure 1. Fgure 1a exhbts the dverse shapes of the exponentated transmuted modfed Webull densty for dfferent choces of the parameters. Fgure 1b shows that for almost all the parameter combnatons consdered the dstrbuton exhbts monotonc hazard rate. (a) Probablty densty functon. (b) Hazard rate functon. Fgure 1: Exponentated Transmuted Modfed Webull Dstrbuton whenα=1. By proper selecton of the model parameters we can get a number of dstrbutons as specal cases as shown below: Parameters Dstrbuton δ=1 Transmuted modfed Webull δ=1,α=0 Transmuted Webull δ = 1, β = 1 Transmuted exponental dstrbuton α=0 Exponentated transmuted Webull β= 1 Exponentated transmuted exponental λ=0 Exponentated modfed Webull λ = 0, α = 0 Exponentated Webull λ = 0, β = 1 Exponentated exponental δ=1,λ=0 Modfed Webull δ=1,λ=0,β= 2 Lnear falure rate dstrbuton δ=1,λ=0,α=0 Webull δ=1,λ=0,α=0,β= 2 Raylegh δ=1,λ=0,β= 1 Exponental 3. Quantle Functon and Smulaton For any random varable X wth cumulatve dstrbuton functon F( ), for a gven q, 0 q 1, the quantle functon returns the threshold value x such that F(x)=q. Ths functon s useful n statstcal applcatons and Monte Carlo smulaton. For statstcal applcatons, one needs to know the key percentage ponts of a gven dstrbuton, lke the medan and the 25% and 75% quartles. The quantle functon s also helpful n examnng the ft of a gven data set to a theoretcal dstrbuton. Ths s done by usng the quantle-quantle (Q-Q) plot.

4 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), The quantle functon correspondng to the ETMW dstrbuton (4) s gven by t q = F 1 (q)= G 1 (q 1/δ ), (6) where G(x)=[1 ex p( αt γt β )][1λex p( αt γt β )] s the cumulatve dstrbuton functon of the transmuted modfed Webull dstrbuton nvestgated by Khan and Kng[6]. Thus, t q s the(q 1/δ )-th quantle of a transmuted modfed Webull dstrbuton, and, from Khan and Kng[6], t q s the real soluton to the equaton where Forβ= 2, t q s gven by γt β q αt q ln(z )=0, (7) z = 1 (1λ) (1λ) 2 4λq 1/δ. (8) 2λ t q = α α 2 4γ ln(z ), (9) 2γ where z s gven by (8). Thus, forβ= 2, the medan of the dstrbuton has the form t 0.5 = α α 2 4γ ln[{ (1λ) /δ λ (1 λ)}/2λ]. 2γ In order to smulate from the ETMW dstrbuton, we have to solve for t q from (7) for a random proporton q. However, forβ= 2, smulaton s straght forward from (9). 4. Moment Generatng Functon We can express the moment generatng functon M(t ) of the ETMW dstrbuton n terms of the moment generatng functon of the modfed Webull dstrbuton as follows: M(t )= 0 ex p(t t)f(t)dt 1 λ =δ A(, j;δ,λ)[ ( j 1) M MW(t ;( j 1)α,( j 1)γ,β) 2λ ( j 2) M MW(t ;( j 2)α,( j 2)γ,β)], where M MW (t ;α,γ,β) denotes the moment generatng functon of a modfed Webull dstrbuton wth c.d.f. (2), and A(, j;δ,λ)=( 1) {Γ(δ 1)} 2 Γ()Γ(j)Γ(δ 1)Γ(δ j 1) λj.

5 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), From Sarhan and Zandn[8], we have M MW (t ( γ) k αγ(kβ 1) γβγ(k 1)β ;α,γ,β)= k! (α t ) kβ1, forα,γ>0,α> t (α t ) (k1)β = t k Γ( k β1 ) γ k β forα=0,γ>0 = α, forγ=0,α> t α t Thus, M(t 1 λ ( ( j 1)γ) k ( j 1)αΓ(kβ 1) )=δ A(, j;δ,λ)[ { ( j 1) k! (( j 1)α t ) kβ1 = ( j 1)γβΓ(k 1)β (( j 1)α t ) (k1)β 2λ ( j 2) ( ( j 2)γ) k k! ( j 2)αΓ(kβ 1) ( j 2)γβΓ(k 1)β { (( j 2)α t ) kβ1 (( j 2)α t ) (k1)β}], forα,γ>0,α> t, 1 λ t k Γ( k β1 =δ A(, j;δ,λ)[ ) ( j 1) {( j 1)γ} k β 2λ t k Γ( k β1 ), for alpha=0,γ>0, ( j 2) {( j 2)γ} k β 1 λ ( j 1)α =δ A(, j;δ,λ)[ ( j 1) {( j 1)α t } 2λ ( j 2)α ( j 2) {( j 2)α t } ], forγ=0,α> t (10) The moments can be ndependently derved as follows: µ r =E{X r } =δ A(, j;δ,λ)[ 1 λ j 1 µmw r (( j 1)α,( j 1)γ,β) 2λ j 2 µmw k (( j 2)α,( j 2)γ,β), whereµ MW r (α,γ,β) denotes the r-th moment of the modfed Webull dstrbuton (2). From Sarhan and Zandn[8] we have µ MW r (α,γ,β)= ( β) k k! Γ( r γ 1) β r γ Γ(kγr1) α kγr βγ Γ(rkγγ), forα,β> 0 α kγγr, forα=0,β> 0 Γ(r1) α r, forα>0,β= 0. (11)

6 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Hence, we get µ r =δ A(, j;δ,λ)[ 1 λ j 1 ( β) k k! Γ(k( j 1)γ r 1) { {( j 1)α} k(j1)γr Γ(r{k( j 1)1}γ) 2λ ( β) k ( j 1)βγ {( j 1)α} {k(j1)1}γr} j 2 k! Γ(k( j 2)γ r 2) Γ(r{k( j 2)1}γ) {( j 2)α} k(j2)γr( j 2)βγ {( j 2)α} {k(j2)1}γr},forα,β> 0 =δ A(, j;δ,λ)[ 1 λ j 1 ( r ( j 1)γ 1)/β r (j1)γ 2λ j 2 ( r ( j 2)γ 1)/β r (j2)γ, forα=0,β> 0 =δ A(, j;δ,λ)[ 1 λ Γ(r 1) 2λ Γ(r 1) j 1{( j 1)α} r j 2{( j 2)α} r], forα>0,β= 0 5. Mean Devaton The amount of scatter n a populaton s evdently measured to some extent by the totalty of devatons from the mean and the medan. If X has a ETMW dstrbuton, then we can derve the mean devatons about the meanµ= E(X) and about the medan M as η 1 = 0 x µ f(x)d x,η 2 = 0 x M f(x)d x. The mean of the dstrbuton s obtaned from (12) by puttng r= 1, and the medan s obtaned by solvng the equaton γm β αm= ln(ν 0 ), whereν 0 s gven by ν 0 = (1 λ) (1 λ) 2 4λ(1 (0.5) 1/δ ), fλ 0 2λ = 1 (0.5) 1/δ, fλ=0. (12)

7 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Order Statstcs Let T (1) < T (2) <...< T (n) be the ordered observatons n a random sample of sze n drawn from the exponentated transmuted modfed Webull dstrbuton wth cdf F(t), gven by (4) and densty f(t), gven by (5). The pdf of T(r), 1 r n, s gven by n! f (r) (t)= (r 1)!(n r)! [F(t)]r 1 [1 F(t)] n r f(t) n! = (r 1)!(n r)! δ[{1 ex p( αt γtβ )}{1λex p( αt γt β }] δr 1 [1 {1 ex p( αt γt β )} δ {1λex p( αt γt β )} δ ] n r (αtβγt β 1 )ex p( αt γt β ) [1 λ2λex p( αt γt β )], t 0,α,β,δ>0,λ [ 1,1]. Hence the pdf of the smallest and the largest order statstcs are as follows: f (1) (t)=nδ[{1 ex p( αt γt β )}{1λex p( αt γt β )}] δ 1 [1 {1 ex p( αt γt β )} δ {1λex p( αt γt β )} δ ] n 1 (αtβγt β 1 )ex p( αt γt β ) [1 λ2λex p( αt γt β )], f (n) (t)=nδ[{1 ex p( αt γt β )}{1λex p( αt γt β )}] δn 1 (αtβγt β 1 ) [1 λ2λex p( αt γt β ], t 0,α,β,δ>0,λ [ 1,1]. The densty of the(r 1)-th order statstc can be expressed as a functon of the densty of the r-th order statstc from the followng relaton: f (r1) (t)= n r r {1 (1 ex p( αt γt β )) δ (1λex p( αt γt β )) δ }] 1 1 f (r) (t). The moments of the order statstcs can be easly wrtten n terms of the moments of the modefed Webull dstrbuton by proceedng as follows: We can wrte f (r) (t) as n! f (r) (t)= (r 1)!(n r)! δ =0 ( 1) n r [{1 ex p( αt γt β )}{1λex p( αt γt β )}] δ(r) 1 (αγβ t β 1 )ex p( αt γt β )[1 λ2λex p( αt γt β )] n! n r = ( 1) g(t;α,γ,β,λ,δ(r)), t 0,α,β,γ,δ>0,λ [ 1,1], (r 1)!(n r)! =0 where g(t;α,γ,β,λ,δ(r)) denote the densty functon of an exponentated transmuted modfed Webull dstrbuton wth shape paameters(α, β, γ), transmutng parameter λ and exponentatnf parameterδ(r).

8 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Hence, usng (12) we have E T r (s) n! = (us)δ (s 1)!(n s)! u=0 [ A(, j;(us)δ,λ)[ 1 λ j 1 ( β) k k! Γ(k( j 1)γ r 1) { {( j 1)α} k(j1)γr Γ(r{k( j 1)1}γ) 2λ ( β) k ( j 1)βγ {( j 1)α} {k(j1)1}γr} j 2 k! Γ(k( j 2)γ r 2) Γ(r{k( j 2)1}γ) { {( j 2)α} k(j2)γr( j 2)βγ {( j 2)α} {k(j2)1}γr}], forα,β> 0 n! = (us)δ[ A(, j;(us)δ,λ) (s 1)!(n s)! u=0 { 1 λ j 1 ( r ( j 1)γ 1)/β r 2λ (j1)γ j 2 ( r ( j 2)γ 1)/β r (j2)γ}], forα=0,β> 0 n! = (us)δ[ A(, j;(us)δ,λ){ 1 λ Γ(r 1) (s 1)!(n s)! j 1{( j 1)α} r u=0 2λ Γ(r 1) j 2{( j 2)α} r}]tex t, f orα>0,β= 0. In addton, we can calculate the L-moments[5], whch are summary statstcs for probablty dstrbutons and data samples but have several advantages over ordnary moments. For example, they apply for any dstrbuton havng a fnte mean and no hgher-order moments need be fnte. The rth L-moment s computed from the lnear combnatons of the ordered data values as gven below: ρ r = r 1 r u 1 ( 1) r u 1 θ u, u u u=0 whereθ u = E[T F(T) u ]. Thus,ρ 1 =θ 0,ρ 2 = 2θ 1 θ 0,ρ 3 = 6θ 2 6θ 1 θ 0, andρ 4 = 20θ 3 30θ 2 12θ 1 θ 0. In general,θ k =(k 1) 1 E(T k1:k1 ), whch can be computed from (6) by substtutng n=s=k 1 and r= 1.

9 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Relablty A stress-strength model descrbes the lfe of a component havng a random strength X 1 and subjected to a random stress X 2. The component functons satsfactorly for X 1 > X 2 and fals when X 1 < X 2. The probablty R= Pr(X 1 > X 2 ) defnes the component relablty. Stressstrength models have many applcatons especally n engneerng concepts such as structures, deteroraton of rocket motors, statc fatgue of ceramc components, fatgue falure of arcraft structures and the agng of concrete pressure vessels. Consder X 1 and X 2 to be ndependently dstrbuted, wth X 1 ET MW(α 1,γ 1,β,λ 1,δ 1 ) and X 2 ET MW(α 2,γ 2,β,λ 2,δ 2 ). The c.d.f. F 1 of X 1 and the pdf f 2 of X 2 are obtaned from (4) and (5), respectvely. Then, where and A(k, l)= = 0,j=0 R=Pr(X 1 > X 2 )= =1 w (1) k,l =( 1)k δ1 k k,l=0 δ1 l 0 w (1) A(k, l), k,l f 2 (y)ex p( (kl)α 1 y (kl)γ 2 y β )d y f 2 (y)[1 F 1 (y)]d y λ l 1, w(2) k,l=( 1) k δ2 1 δ2 1 λ l 2 k l, w (2) (α,j 2 γ 2 β y β 1 )ex p( {(kl)α 1 ( j 1)α 2 }y 0 {(kl)γ 1 ( j 1)γ 2 }y β )(1 λ 2 2λ 2 ex p( α 2 y γ 2 y β ))d y = w (2),j [α 2{(1 λ 2 )µ MW 1 ((kl)α 1 ( j 1)α 2,(kl)γ 1 ( j 1)γ 2,β),j=0 2λ 2 µ MW 1 ((kl)α 1 ( j 2)α 2,(kl)γ 1 ( j 2)γ 2,β)} γ 2 {(1 λ 2 )µ MW β ((kl)α 1 ( j 1)α 2,(kl)γ 1 ( j 1)γ 2,β) 2λ 2 µ MW β ((kl)α 1 ( j 2)α 2,(kl)γ 1 ( j 2)γ 2,β)}], wthµ MW r (α, γ, β) gven by (11). In partcular, fα 1 =α 2 andγ 1 =γ 2, we have A(k, l)=,j=0 w (2),j [ (1 λ 2 ) ( jkl 1) 2λ 2 ( jkl 2) ].

10 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Least Squares Estmaton Let T (1) < T (2) <...< T (n) denote the ordered observatons n a random sample of sze n drawn from the ETMW(α, γ, β, λ, δ) dstrbuton wth dstrbuton functon F( ), gven by (4). Then, E[F(T () )]=, = 1,2,..., n. n1 The least square estmators are obtaned by mnmzng D(θ)= = F(T () ) 2 n1 (1 ex p( αt () γt β () ))δ (1λex p( αt () γt β () ))δ 2. n1 Wrtng V = ex p( αt () γt β ), the normal equatons to be satsfed by the estmators are as () follows: (1 V ) δ (1λV ) δ n1 (1 V ) δ (1λV ) δ n1 (1 V ) δ (1λV ) δ n1 (1 V ) δ (1λV ) δ n1 (1 V ) δ (1λV ) δ n1 (1 V ) δ 1 (1λV ) δ 1 T () V (1 λ2λv )=0 (1 V ) δ 1 (1λV ) δ 1 T β () V (1 λ2λv )=0 (1 V ) δ 1 (1λV ) δ 1 T β () ln(t ())V (1 λ2λv )=0 (1 V ) δ (1λV ) δ 1 V = 0 (1 V ) δ (1λV ) δ [ln(1 V )ln(1λv )]=0. 9. Maxmum Lkelhood Method of Estmaton Consder a random sample(t 1, T 2,..., T n ) of sze n taken from the dstrbuton ETMW(α,γ,β,λ,δ) wth densty functon (5). For gven T = t, = 1,2,..., n, the log-lkelhood functon forθ=(α,γ,β,λ,δ) s l(θ)=n lnδ(δ 1){ ln(1 ex p( αt γt β )) ln(1λex p( αt γt β ))} ln(αγβ t β 1 ) ln(1 λ2λex p( αt γt β )) (αt γt β ).

11 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), Wrtngν = ex p( αt γt β )), = 1,2,..., n, the log-lkelhood equatons are obtaned as follows: t = 2λ t β = 2λ t β ln t = 2λ t ν (1 λ2λν ) (1 δ) t β ν (1 λ2λν ) (1 δ) (t β ln t )ν (1 λ2λν ) (1 δ) t ν (1 λ2λν ) (1 ν )(1λν ) t β ν (1 λ2λν ) (1 ν )(1λν ) β 1 (αγβ t β 1 ) (13) (t β ln t )ν (1 λ2λν ) (1 ν )(1λν ) t β 1 (αγβ t β 1 ) (14) (15) t β 1 (1β ln t ) (αγβ t β 1 ) (16) ν 1 2ν (δ 1) = 1λν 1 λ2λν (17) ln(1 ν ) ln(1λex pν )= n δ. (18) Solvng the non-lnear system of equatons (13)-(18) we obtan the maxmum lkelhood estmate ˆθ=(ˆα, ˆγ, ˆβ, ˆλ, ˆδ) ofθ. Under certan regularty condtons, n(ˆθ θ) d Normal(0, I 1 (θ)) (here d stands for convergence n dstrbuton), where I(θ) denotes the nformaton matrx gven by 2 l(θ) I(θ)= E. (19) θ θ Ths nformaton matrx I(θ) may be approxmated by the observed nformaton matrx 2 l(θ) I(ˆθ)= E. (20) θ θ Hence, usng the approxmaton n(ˆθ θ) Normal(0, I 1 (ˆθ)), one can carry out tests and fnd confdence regons for functons of some or all of the parameters nθ. θ=ˆθ 10. Smulaton Study A smulaton study s carred out to nvestgate the performance of the least square (LS) estmators and the ML estmators. We take sample szes to be n=15,25,50,100,250,500 and generate observatons from a ETMW dstrbuton wth parametersα=1,γ=1,β= 2.5, λ=0.5,δ=0.75. For each sample we compute the estmates of the parameters usng the

12 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), two methods. The process s repeated N= 1000 tmes, and the average mean squared error s computed as follows: AMSE(T)= 1 N N (ˆθ (T) θ) (ˆθ (T) θ), (21) where T denotes the method used to estmate θ =(α, γ, β, λ, δ), denotes the sample repetton number, = 1,2,..., N, and ˆθ (T) s the correspondng estmate. Table 1 gves the AMSEs for the two methods of estmaton. Table 1: Average Mean Squared Errors for the Method of Least Squares and Maxmum Lkelhood Method for Estmatngθ Sample sze AMSE(LS) AMSE(ML) The above table shows that as the sample sze ncreases, the average mean squared errors decrease. Ths verfes the consstency propertes of the estmates. Further, for each sample sze AMSE(LS)>AMSE(M L). Thus, we may conclude that the maxmum lkelhood method provdes better estmators than the least square method. However, the dfference between AMSE(LS) and AMSE(M L) decrease as the sample sze ncreases. 11. Applcaton of the Exponentated Transmuted Modfed Webull Dstrbuton In ths secton we llustrate the usefulness of the exponentated transmuted modfed Webull dstrbuton for modellng real lfe data. The data set relates to the tme-to-falure of 50 devces, and s taken from Aarset[1]: Table 2: The tme-to-falure of 50 devces The Webull (W), modfed Webull (M), transmuted modfed Webull(T) and exponentated transmuted modfed Webull (E) dstrbutons are ftted to the data and the MLEs of the

13 M. Pal and M. Tensuwan/Eur. J. Pure Appl. Math, 8 (2015), parameters are gven n Table 3. The values of the log-lkelhood, Kolmogorov-Smrnov statstc (K-S), Akake nformaton crteron (AIC) and Bayesan nformaton crteron (BIC) for the dfferent ftted dstrbutons are also gven, and show that the ETMW dstrbuton gves a better ft than the others. The same s also evdent from Fgure 2, whch compares the cumulatve dstrbuton curves of the ftted dstrbutons wth that of the emprcal dstrbuton. Table 3: The estmated parameters and the log-lkelhood, K-S, AIC, BIC values for the dfferent ftted dstrbutons α γ β λ δ -l(θ) K-S AIC BIC W M T E Fgure 2: Comparson of the CDFs of the Ftted Dstrbutons wth the Emprcal CDF 12. Dscusson In ths paper, we ntroduce a new generalzaton of the Webull dstrbuton called exponentated transmuted modfed Webull dstrbuton and dscuss ts ntrnsc propertes. The dstrbuton s very flexble n the sense that t exhbts both ncreasng and decreasng falure rates dependng on ts parameters. ACKNOWLEDGEMENTS The authors thank the anonymous referees for ther frutful suggestons, whch mmensely helped to mprove the presentaton of the paper.

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