Transmuted Weibull Power Function Distribution: its Properties and Applications

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1 Joural of Data Sciece , DOI: /JDS _16(2).0009 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios Muhammad Ahsa ul Haq* 1, M. Elgarhy 2, Sharqa Hashmi 3, Gamze Ozel 4 ad Qurat ul Ai 5 1 Quality Ehacemet Cell (QEC), Natioal College of Arts, Lahore-Pakista. 2 Vice Presidecy for Graduate Studies ad Scietific Research, Uiversity of Jeddah, Jeddah, KSA. 3 Lahore College for Wome Uiversity, Lahore, Pakista. 4 Departmet of Statistics, Hacettepe Uiversity, Turkey. 5 Departmet of Mathematics ad Statistics, Uiversity of Lahore, Lahore. Abstract: I this paper, we itroduce a ew four-parameter distributio called the trasmuted Weibull power fuctio (TWPF) distributio which exteds the trasmuted family proposed by Shaw ad Buckley [1]. The hazard rate fuctio of the TWPF distributio ca be costat, icreasig, decreasig, uimodal, upside dow bathtub shaped or bathtub shape. Some mathematical properties are derived icludig quatile fuctios, expasio of desity fuctio, momets, momet geeratig fuctio, residual life fuctio, reversed residual life fuctio, mea deviatio, iequality measures. The estimatio of the model parameters is carried out usig the maximum likelihood method. The importace ad flexibility of the proposed model are proved empirically usig real data sets. Key words: Weibull distributio, trasmuted family, maximum likelihood, momets, order statistics, etropy. 1. Itroductio There are hudreds of cotiuous distributios i the statistical literature. These distributios have several applicatios i may applied fields such as reliability, life testig, biomedical scieces, ecoomics, fiace, evirometal ad egieerig, amog others. However, these applicatios have prove that the real data followig the well-kow models are more ofte the exceptio rather tha the reality. I order to icrease the flexibility of the well-kow distributios, may authors have proposed differet trasformatios of these models ad used these exteded forms i several areas. The power fuctio (PF) distributio is a flexible model which ca be obtaied from the Pareto distributio by usig a simple trasformatio Y = X 1. The probability desity fuctio (pdf) ad the cumulative distributio fuctio (cdf) of the PF distributio are, respectively, give by

2 398 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios F(x; α, β) = ( x α ) β, (1.1) f(x) = βxβ 1, 0 < x < α, β > 0 (1.2) αβ where β is a shape parameter ad α is a scale parameter. The Power fuctio (PF) distributio has bee used over the past decades for modellig data i egieerig, reliability ad biological studies. Meicoi ad Barry [2] ivestigated the performace of the PF distributio o electrical compoet ad obtaied that the PF distributio is better tha the Weibull, log-ormal ad expoetial models to measure the reliability of electroic compoets. The eed for exteded forms of the PF distributio arises i may applied areas ad hece, some geeralizatios of the PF distributio have bee proposed. Amog these distributios, we refer to the beta power fuctio (BPF) by [3], Weibull power fuctio (WPF) by [4], Kumaraswamy power fuctio (KWPF) distributio by [5], the modified power fuctio distributio by [6], expoetiated power fuctio distributio by [7], the expoetiated Kumaraswamy power fuctio distributio by [8]. The pdf ad cdf of the WPF distributio are, respectively, give by g(x, a, b, α, β) = abβαβ b 1 (α β e a( α ) b+1 β )b, 0 < x < α, α, β > 0 (1.3) ad G(x) = 1 exp ( a { α β }b ) (1.4) where α ad a are the scale parameters, b ad β are the shape parameters. May authors have bee recelty deal with the geeralizatio of some well-kow distributios usig the trasmuted family proposed by [1]. Aryal ad Tsokos [9] defied the trasmuted geeralized extreme value distributio ad the studied some mathematical characteristics of the trasmuted Gumbel distributio. Aryal ad Tsokos [10] also preseted a ew geeralizatio of Weibull distributio called the trasmuted Weibull distributio. Aryal [11] proposed ad studied the various structural properties of the trasmuted log-logistic distributio. The, Kha ad Kig [12] itroduced the trasmuted modified Weibull distributio which exteds recet developmet o trasmuted Weibull distributio by Aryal ad Tsokos [10]. Elbatal [13] preseted trasmuted modified iverse Weibull distributio. Elbatal ad Aryal [14] preseted trasmuted additive Weibull distributio. Recetly, trasmuted geeralized Lidley distributio has bee obtaied by Elgarhy et al. [15], trasmuted power fuctio (TPF) has bee itroduced by Haq et al. [16] ad trasmuted Weibull Frehet by Haq et al. [17], Elbatal ad Elgarhy [18] itroduced Trasmuted quasi Lidley (TQL), ad Elgarhy et al. [19] studied trasmuted geeralized quasi Lidley (TGQL). The aim of this paper is to defie ad study a ew flexible lifetime model called the trasmuted Weibull power fuctio (TWPF) distributio. Usig the trasmuted family proposed by [1], we costruct the four-parameter TWPF model ad give some of its mathematical properties. I fact, the TWPF model ca provide better fits tha other existig models. The rest of the paper is orgaized as follows: I sectio 2 we demostrate trasmuted

3 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 399 probability desity, hazard rate ad reliability fuctios of TWPF distributio. I sectio 3 we studied the statistical properties icludig quatile fuctio, expasio of desity fuctio, momets, momet geeratig fuctio, icomplete momets, coditioal momets, residual life fuctio, reversed residual life fuctio, mea deviatio, iequality measures. The distributio of order statistics is expressed i sectio 4. I sectio 5, we demostrate the maximum likelihood estimates (MLEs) of the ukow parameters. Simulatio study is carried out for TWPF distributio i Sectio 6. Two illustrative applicatios based o real data sets are ivestigated i sectio 7. Fially, cocludig remarks are preseted i sectio The TWPF Distributio Before defiig the TWPF distributio, we explai a trasmuted probability distributio. Let F 1 ad F 2 be the cdfs of two distributios with a commo sample space. The geeral rak trasmutatio is defied by Shaw ad Buckely [1] as G R12 (u) = F 2 (F 1 1 (u)) ad G R21 (u) = F 1 (F 2 1 (u)). Note that the iverse cdf, also kow as quatile fuctio, is defied as F 1 (y) = if x R {F(x) y} for y [0,1] The fuctios G R12 (u) ad G R21 (u) both map the uit iterval I = [0, 1] ito itself. Uder suitable assumptios these fuctios are mutual iverses ad they satisfy G Rij (0) = 0 ad G Rji (0) = 1 A quadratic rak trasmutatio map (QRTM) is defied as G R12 (u) = u + λμ(1 μ), λ 1 (2.1) from which it follows that the cdf s satisfy the relatioship F 2 (x) = (1 + λ)f 1 (x) λf 1 (x) 2 (2.2) After differetiatig (2.2), we have f 2 (x) = f 1 (x)[(1 + λ) 2λF 1 (x)]where f 1 (x) ad f 2 (x) are the correspodig pdfs associated with cdfs F 1 (x)ad F 2 (x) respectively. More iformatio about the quadratic rak trasmutatio map ca be foud i Shaw ad Buckley [1]. Note that we have the baselie distributio for λ = 0. The followig Lemma proves that the fuctio f 2 (x) i give (2.3) satisfies the property of pdfs. Lemma: f 2 (x) give i (2.3) is a well-defied pdf. Proof. Rewritig f 2 (x) as f 2 (x) = f 1 (x)[1 λ{f 1 (x) 1}] we observe that f 2 (x) is o-egative. We eed to show that the itegratio of the support of the radom variable is equal to oe. Cosider the case whe the support of f 1 (x) is 1. I this case, we have f 2 (x) dx = f 1 (x)[(1 + λ) 2λF 1 (x)] dx = (1 + λ) f 1 (x) dx 2λ F 1 (x)f 1 (x) dx = (1 + λ) λ = 1 Similarly, other cases where the support of the radom variable is a part of the real lie follows. Hece, f 2 (x) is a well-defied pdf ad it is called as the trasmuted probability

4 400 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios desity of a radom variable with base desity f 1 (x). Note that whe λ = 0, we have f 2 (x) = f 1 (x). This proves the required result. Now, usig (1.2) ad (1.4), we obtai the cdf of TWPF distributio as F TWPF (x) = (1 exp ( a{ α β }b )){1 + λ λ (1 exp ( a { α β }b ))} (2.4) where α ad a are the scale parameters; b ad β are the shape parameters ad is the trasmuted parameter. The, the pdf of the TWPF distributio is give by { f TWPF(x) = abβαβ b 1 (α β e a( α ) b+1 β )b [(1 + λ) 2λ {1 exp ( a { α β }b )}], } (2.5) 0 < x < α The reliability (survival) fuctio of the TWPF distributio is give by R TWPF (x) = 1 F TWPF (x) = 1 (1 exp ( a { α β }b )){1 + λ λ(1 exp ( a { α β }b ))} by Oe of the characteristics i reliability aalysis is the hazard rate fuctio (hrf) defied h TWPF (x) = f TWPF(x) 1 F TwPF (x) xβ abβα β b 1 e a( α β )b [(1 + λ) 2λ{1 exp ( a { xβ α = β }b )}] (α β ) b+1 (1 (1 exp ( a { xβ α β }b )) {1 + λ λ (1 exp { xβ α β }b ))}) h It is importat to ote that the uits for ( ) TWPF x is the probability of failure per uit of time, distace or cycles. These failure rates are defied with differet choices of parameters. Plots of the pdf ad hrf of the TWPF distributio for some parameter values are displayed i Figure 1 (a) ad (b), respectively. As see from Figure 1(a), the desity fuctio ca take various forms depedig o the parameter values. Icreasig, decreasig, uimodal, upside dow bathtub shaped or bathtub shapes appear to be possible. It is evidet that the TWPF distributio is very flexible. Furthermore, Figure 1(b) shows that the hrf of the TWPF distributio ca have very flexible shapes, such as icreasig, decreasig, upside-dow bathtub, bathtub. This attractive flexibility makes the hrf of the TWPF useful ad suitable for o-mootoe empirical hazard behaviours which are more likely to be ecoutered or observed i real life situatios.

5 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 401

6 402 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios Figure 1 (a): Plots of the pdf of TWPF distributio for selected values of the parameters (left). (b): Plots of the hrf of TWPF distributio for selected values of the parameters (right). 3. Mai Properties This sectio is devoted to mai properties of the TWPF distributio, specifically quatile fuctio, momets ad momet geeratig fuctio Quatile Fuctio The quatile x q of the TWPF distributio is obtaied from (2.4) as x q = α{l [ (λ 1) + (λ 1)2 4λ(μ 1) ] 1 1 a } bβ(1 2λ + {l[ (λ 1) + (λ 1)2 4λ(μ 1) ] 1 a } 1 1 b) β 2λ We simulate the TWPF distributio by solvig the equatio above where u has the uiform distributio U(0, 1) A Useful Expasio Now, a represetatio for the desity fuctio of the TWPF distributio will be preseted. Usig the power series for the expoetial fuctio, we obtai exp( ax) = ( 1)k a k x k k=0 Isertig the expasio (3.1) i (2.5), we have k! (3.1)

7 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 403 f(x) = (1 λ)ab β α ( 1)k (( x α )β ) bk+k 1 (1 ( x α )β ) (bk+b+1) k=0 + 2aλb β α ( 2)k (( x α )β ) bk+b 1 (1 ( x α )β ) (bk+b+1) k=0 Now, usig the geeralized biomial theorem, we obtai k! k! (3.2) (1 z) β β + j 1 = ( ) z j (3.3) j j=0 where β > 0 is real o-iteger. The, by applyig the biomial theorem (3.3) i (3.2) the pdf of the TWPF distributio becomes f(x) = (1 λ)a k+1 b β α 1 ( 1)k Γ(b(k + 1) + j + 1) j! k! Γ(b(k + 1) + 1) ((x α )β ) b(k+1)+j β + 2a k+1 λb β α 1 ( 2)k Γ(b(k + 1) + j + 1) (( x j! k! Γ(b(k + 1) + 1) α )β ) b(k+1)+j 1 β β f(x) = (1 λ)ab β α ( 1)k Γ(b(k + 1) + j + 1) (( x j! k! Γ(b(k + 1) + 1) α )β ) b(k+1)+j 1 + 2aλb β α ( 2)k Γ(b(k + 1) + j + 1) j! k!) k (( x Γ(b(k + 1) + 1) α )β ) b(k+1)+j 1 ad after simplificatio, the TWPF desity ca be expressed as f(x) = ( 1)k a k+1 bβγ(b(k + 1) + j + 1){1 λ + 2 k+1 λ} 1 (( x α )β ) b(k+1)+j 1 β j! k! αβγ(b(k + 1) + 1) or equivaletly, we ca write 1 β f(x) = j,k ( x α )β(b(k+1)+j) 1 (3.4) where j,k = ( 1)k a k+1 bβγ(b(k + 1) + j + 1){1 λ + 2 k+1 λ} 1 j! k! αβγ(b(k + 1) + 1) If β is a iteger, the idex i i the previous sum stops at β 1.

8 404 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios 3.3. Momets This subsectio cocers with the μ r momet ad momet geeratig fuctio for the TWPF distributio. Momets are importat i ay statistical aalysis, especially i applicatios. It ca be used to study the most importat features ad characteristics of a distributio (e.g. tedecy, dispersio, skewess, ad kurtosis). If X has the pdf i (2.5), the its rth momet ca be obtaied through the followig relatio μ r = E(X r ) = x r f(x) dx Substitutig (3.4) i above relatio, we get α μ r = E(X r ) = U j,k x r ( x ) β(b(k+1)+j) 1 dx (3.5) α 0 The, we obtai μ r = j,k r + β(b(k + 1) + j) α r+1 (3.6) Based o the first four momets of the TWPF distributio, the measures of skewess A(φ) ad kurtosis k(φ) of the TWPF distributio ca be obtaied as A(φ) = μ 3 3μ 1 μ μ 1, [μ 2 μ 2 1 ] 3 2 ad k(φ) = μ 4 4μ 1 μ 3 + 6μ 1 2 μ 2 3μ 1 4 [μ 2 μ 1 2 ] 2. Usig the relatio betwee the o-cetral ad cetral momets, we ca obtai the th M cetral momet, deoted by, of TWPF radom variable as follows: The, we ca also write M = E(X μ) = ( r ) ( μ) r E(X r ) r=0 M = ( r ) ( 1) r (μ 1 ) r μ r r=0 ad the cumulats of the radom variable X ca be obtaied as 1 k = μ ( 1 r 1 ) k 1 μ r r=0 where k 1 = μ 1, k 2 = μ 2 (μ 1 ) 2, k 3 = μ 3 3μ 2 μ 1 + (μ 1 ) 3 etc. The skewess ad kurtosis measures ca be calculated from the ordiary momets usig well-kow relatioships.

9 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai Momet Geeratig Fuctio I this subsectio, we derived the momet geeratig fuctio of TWPF distributio. The momet geeratig fuctio is give by the relatio the, we have M x (t) = E(e tx ) = α 0 e tx f(x) dx, M x (t) = U j,k ( x α )β(b(k+1)+j) 1 e tx dx 0 The momet geeratig fuctio of the TWPF distributio is obtaied by t γ(β(k + 1) + j, αt) M x (t) = U j,k (αt) β(b(k+1)+j) 1 (3.7) where γ(s, t) = x s 1 e x dxis the lower icomplete gamma fuctio. 0 Aother formula for momet geeratig fuctio ca be give as M x (t) = tr t r α r+1 r! E(Xr ) = U j,k r! r + β(b(k + 1) + j) r=0 j,k,r= Icomplete ad Coditioal Momets The mai applicatio of the first icomplete momet refers to the Boferroi ad Lorez curves. These curves are very useful i ecoomics, reliability, demography, isurace, ad medicie. The icomplete momets, say φ s (t), is give by Usig (2.5), φ s (t)ca be writte as t φ s (t) = x s f(x) dx. t 0 φ s (t) = U j,k x s ( x α )β(b(k+1)+j) 1 dx. 0 The, t s+β(b(k+1)+j) φ s (t) = U j,k [s + β(b(k + 1) + j)]α β(b(k+1)+j) 1 Further, the coditioal momets, say τ s (t), is give by Hece, by usig pdf (2.5), we ca write α τ s (t) = x s f(x) dx t

10 406 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios α τ s (t) = U j,k x s ( x α )β(b(k+1)+j) 1 dx t The, we have α s+β(b(k+1)+j) t s+β(b(k+1)+j) τ s (t) = U j,k [s + β(b(k + 1) + j)]α β(b(k+1)+j) 1 Additioally, the mea deviatio ca be calculated by the followig relatio q δ 1 (X) = 2μF(μ) 2T(μ) ad δ 2 (X) = μ 2T(M) where, T(q) = xf(x) dx. By usig (3.4) we have the followig equatios: μ μ β(b(k+1)+j)+1 T(μ) = xf(x) dx = U j,k [β(b(k + 1) + j) + 1]α β(b(k+1)+j) 1, 0 M M β(b(k+1)+j)+1 T(M) = xf(x) dx = U j,k [β(b(k + 1) + j) + 1]α β(b(k+1)+j) 1, 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 ( ), see [20-22]. Moreover, the th momet of the residual life, say m (t) = E[(X t) X > t], = 1, 2,.,uiquely determie Fx ( ). The th momet of the residual life of X is give by m (t) = 1 α R(t) (x t) f(x) dx x t Applyig the biomial expasio of t ito the above formula, we get m (t) = 1 R(t) U j,k( t) d ( d ) α +β(b(k+1)+j) t +β(b(k+1)+j) [ + β(b(k + 1) + j)]α β(b(k+1)+j) 1 (3.8) d=0 Aother iterestig fuctio is the mea residual life (MRL) fuctio or the life expectatio at age x defied by m 1 (t) = E[(X t) X > t], which represets the expected additioal life legth for a uit which is alive at age x. The MRL of the TWPF distributio ca be obtaied by settig 1 i (3.8). Furthermore, the th momet of the reversed residual life, say M (t)e[(x t) X t], for t > 0, = 1, 2,..,uiquely determies F(X). Hece, the th momet the reversed residual life of X is give by M (t) = 1 t R(t) (x t) f(x) dx 0

11 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 407 Applyig the biomial expasio of (x t) ito the above formula, we get t +β(b(k+1)+j) M (t) = 1 R(t) U j,k( t) d ( d ) [ + β(b(k + 1) + j]α β(b(k+1)+j) 1 d=0 The mea iactivity time (MIT) or mea waitig time (MWT), also called the mea reversed residual life fuctio, is defied by M 1 (t) = E[(X t) 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). 3.7 Réyi ad q-etropies The etropy of a radom variable X is a measure of variatio of ucertaity ad has bee used i may fields such as physics, egieerig, ad ecoomics. Accordig to Réyi [23], the Réyi etropy is defied by I δ (X) = 1 1 δ log f(x)δ dx, δ > 0 ad δ 1. By applyig the biomial theory (3.3) i the pdf (3.4), the the pdf expressed as follows f(x) δ = W i,k,j x δ(bβ 1)+β(j+k), i,k,j=0 f( x) ca be Where W i,k,j = (abβ)δ ( 1) k (1 λ) δ i (2λ) i [(i + δ)a] k α β[k+bδ+j] k! Therefore, the Réyi etropy of the TWPF distributio is give by ( δ i + 1) + k + j 1 ) (δ(b ). j I δ (X) = 1 1 δ log [ W i,k,j x δ(bβ 1)+β(j+k) dx], i,k,j=0 0 α the, α δ(bβ 1)+β(j+k)+1 I δ (X) = 1 1 δ log [ W i,k,j δ(bβ 1) + β(j + k) + 1 ]. i,k,j=0 The q-etropy is defied by H q (X) = 1 1 δ log (1 f(x)q dx), q > 0 ad q 1. Therefore, the q-etropy of the TWPF distributio is give by α δ(bβ 1)+β(j+k)+1 H q (X) = 1 1 δ log [1 W i,k,j ]. δ(bβ 1) + β(j + k) + 1 i,k,j=0

12 408 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios Whe we replace q with, δ the idex I stop at δ whe δ is a iteger. 4. Order statistics Order statistics have bee extesively applied i may fields of statistics, such as reliability ad life testig. Let X 1, X 2,, X be idepedet ad idetically distributed (i.i.d) radom variables with their correspodig cotiuous distributio fuctio F(x). Let X (1), X (2),, X () the correspodig ordered radom sample from a populatio of size. th David [24] defied the pdf of the k order statistic as k f(x) k f X(k) (x) = B(k, k + 1) ( 1)v ( v ) F(x)v+k 1, (4.1) v=0 where B(.,. ) stads for beta fuctio. The pdf of the k th order statistic for TWPF distributio is derived by substitutig (2.4) ad (3.4) i (4.1), replacig h with v+k-1, where k 1 f X(k) (x) = B(k, k + 1) η ( x α )β(b(k+1)+j) 1 v=0 ((1 exp ( a { α β }b )){1 + λ λ(1 exp ( a { α β }b ))}) v+k 1 η = ( 1) v k ( v ) U j,k This sectio deals with the MLEs of the ukow parameters for the TWPF distributio o the basis of complete samples. Let X 1, X 2,, X be the observed values from the TWPF distributio. The log-likelihood fuctio for parameter vector φ = (a, b, β, λ) T is obtaied as ll(φ) = la + lb + βlα + (βb 1) l(x i ) (b + 1) l(α β i ) a ( α β x β i β x i + l ((1 λ) + 2λexp ( a{ α β } b )) (5.1) x β i The elemets of the score fuctio U(φ) = (U a, U b, U β, U λ ) are give by U a = a ( α β x β )b 2λ i x i β { x i β α β x i β }b exp ( a{ i α β }b ) i 1 λ + 2λexp ( a{ x i β α β x i β }b ) x i β, ) b

13 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 409 U b = b + β l(x i) l(α β i ) a ( α β x β )b l ( i α β x β ) i 2aλ ( x i β α β x i β )b l ( i α β i )exp ( a{ x x i β β i α β }b ) i 1 λ + 2λexp ( a{ x i β α β x i β }b ) U β = β + lα + b l(x i) (b + 1) αβ lα i lx i α β β x i abα β x i bβ (lx i lα) (α β 2abλα β i ) b+1 x i bβ (lx i lα) (α β x i β ) b+1 exp ( a{ x i β x i β α β }b ) i 1 λ + 2λexp ( a{ x i β α β x i β }b ) ad U λ = x i β 2exp ( a { α β }b ) 1 i 1 λ + 2λexp ( a{ x i β α β x i β }b ) Sice x α, the MLE of α is the last-order statistic x (). Settig U a, U b, U β ad U λ equal to zero ad solvig these equatios simultaeously yield the MLE φ = (a, b, β, λ)of φ = (a, b, β, λ) T These equatios caot be solved aalytically ad statistical software ca be used to solve them umerically usig iterative methods. 6. Simulatio study I this sectio, a extesive umerical ivestigatio will be carried out to evaluate the performace of MLE for TWPF model. Performace of estimators is evaluated through their biases, ad mea square errors (MSEs) for differet sample sizes. A umerical study is performed usig Mathematica (9) software. Differet sample sizes are cosidered through the experimets at size = 100, 150 ad 200. I additio, the differet values of parameters ab,,, ad. The experimet will be repeated 1000 times. I each experimet, the estimates of the parameters will be obtaied by maximum likelihood methods of estimatio. The meas, MSEs ad biases for the differet estimators will be reported from these experimets.

14 410 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios Par Parameters MLE Bias MSE Parameters MLE Bias MSE a b α β λ a b α β λ a b α β λ Table 1: The parameter estimatio from TWPF distributio usig MLE 7. Applicatio This sectio provides two applicatios to show how the TWPF distributio ca be applied i practice. For this aim, the TWPF distributio is compared with other competitive distributios. I these applicatios, the model parameters are estimated by the method of maximum likelihood. The Akaike iformatio criterio (AIC), Bayesia iformatio criterio, Aderso-Darlig (A*) ad Cramer vo Mises (W*) statistics are computed to compare the fitted models. I geeral, the smaller the values of these statistics, the better the fit to the data. The plots of the fitted pdfs, cdfs of some distributios give below are displayed for visual compariso. The required computatios are carried out i the R-laguage. The desity fuctios of the compared distributios are give by Beta Expoetial (BE) distributio with the pdf f(x) = λ Beta[a, b] e (λbx) (1 e λx ) a 1 The gamma expoetiated expoetial (GEE) distributio with the pdf f(x) = λaδ Γ(δ) e λx (1 e λx ) α 1 ( log (1 e λx )) δ 1

15 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 411 Weibull Fréchet (WFr) distributio with the pdf f(x) = abβα β x β 1 exp [ b( α x )β ](1 [ b( α x )β ]) b 1 exp [ a(exp [( α x )β ] 1) b ] Weibull power fuctio (WPF) distributio with the pdf f(x) = abβαβ b 1 (α β ) b+1 e a( α β )b, 0 < x < α, a, b, α, β > 0 The first data set represets the stregths of 1.5 cm glass fibers, measured at the Natioal Physical Laboratory, Eglad [25]. The data are as follows: 0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2.0, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.50, 1.54, 1.60, 1.62, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66, 1.70, 1.77, 1.84, 0.84, 1.24, 1.30, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.70, 1.78, The secod data cosistig of 100 observatios o breakig stress of carbo fibres Nichols ad Padgett [26] are give below: 3.7, 3.11, 4.42, 3.28, 3.75, 2.96, 3.39, 3.31, 3.15, 2.81, 1.41, 2.76, 3.19, 1.59, 2.17, 3.51, 1.84, 1.61, 1.57, 1.89, 2.74, 3.27, 2.41, 3.09, 2.43, 2.53, 2.81, 3.31, 2.35, 2.77, 2.68, 4.91, 1.57, 2.00, 1.17, 2.17, 0.39, 2.79, 1.08, 2.88, 2.73, 2.87, 3.19, 1.87, 2.95, 2.67, 4.20, 2.85, 2.55, 2.17, 2.97, 3.68, 0.81, 1.22, 5.08, 1.69, 3.68, 4.70, 2.03, 2.82, 2.50, 1.47, 3.22, 3.15, 2.97, 2.93, 3.33, 2.56, 2.59, 2.83, 1.36, 1.84, 5.56, 1.12, 2.48, 1.25, 2.48, 2.03, 1.61, 2.05, 3.60, 3.11, 1.69, 4.90, 3.39, 3.22, 2.55, 3.56, 2.38, 1.92, 0.98, 1.59, 1.73, 1.71, 1.18, 4.38, 0.85, 1.80, 2.12, The descriptive statistics are preseted i Table 2 for both data sets. Miimum Media Mea Maximum Variace Skewess Kurtosis Table 1: The parameter estimatio from TWPF distributio usig MLE The MLEs of ukow parameters of the distributios for the both data set are give i Tables 3 ad 5. The umerical values of the statistics AIC, BIC, A*, W* are listed i Tables 4 ad 6. Distributio Estimates BE(a, b, λ) GEE(λ, α, δ) WFr(a, b, α, β) WPF(α, β, a, b) TWPF(α, β, a, b, λ) Table 3: The MLEs for the first data set

16 412 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios Model Goodess of fit criteria AIC BIC -L A* W* BE(a, b, λ) GEE(λ, α, δ) WFr(a, b, α, β) WPF(α, β, a, b) TWPF(α, β, a, b, λ) Table 4: Some statistics for models fitted to the first data set Distributio Estimated Parameters BE (a, b, λ) GEE(λ, α, δ) WFr(α, β, a, b) WPF(α, β, a, b) TWPF(α, β, b, a, λ) Table 5: The MLEs for the secod data set Distributio AIC BIC -L A* W* BE (a, b, λ) GEE(λ, α, δ) WFr(α, β, a, b) WPF(α, β, a, b) TWPF(α, β, a, b, λ) Table 6: Some statistics for models fitted to the secod data set Based o the Tables 4 ad 6, we coclude that the ew TWPF model provides adequate fits as compared to other models i both applicatios with small values for AIC, BIC, A*, W*. I the two applicatios, the proposed TWPF model is much better tha the four models. The histograms of the two data sets ad the estimated pdfs ad cdfs of the proposed ad competitive models are displayed i Figure 2. Figure 2 also supports the results i Tables 4 ad 6.

17 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 413 (a) Estimated pdf for the first data set (b) Estimated cdf for the first data set (c) Estimated pdf for the secod data set (d) Estimated cdf for the secod data set Figure 2: Plots of the estimated pdfs ad cdfs of the models for both data sets.

18 Observed Observed Observed Observed Observed Observed Observed Observed 414 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios PP Plot of the GEE Distributio Expected PP Plot of the WPF Distributio PP Plot of the TWPF Distributio Expected Expected Figure 3: PP plots of the TWPF ad other fitted distributios for the first data set PP Plot of the GEE Distributio PP Plot of the WFr Distributio PP Plot of the BE Distributio Expected Expected Expected PP Plot of the WPF Distributio PP Plot of the TWPF Distributio (T-X) Expected Expected Figure 4: PP plots of the TWPF distributio ad other fitted distributios for the secod data set

19 Muhammad Ahsa ul Haq, M. Elgarhy, Sharqa Hashmi, Gamze Ozel ad Qurat ul Ai 415 I Figures 3 ad 4, the probability-probability (P-P) plots of the TWPF distributio ad other fitted distributio are also preseted for both data set. As see i Figures 3 ad 4, both data sets fits well the TWPF distributio tha the other fitted distributios. 8. Cocludig remarks I this paper, we propose ad study the ew distributio called as trasformed Weibull power fuctio (TWPF) distributio. We ivestigate some of its mathematical properties icludig a expasio for the desity fuctio ad explicit expressios for the quatile fuctio, ordiary ad icomplete momets, momet geeratig fuctio, etropies, reliability fuctio ad order statistics. The maximum likelihood method is employed to estimate the model parameters. We fit TWPF model to two real data sets to demostrate the flexibility of it. We hope that the ew distributio will attract wider applicatio i areas such as egieerig, survival ad lifetime data, hydrology, ecoomics, amog others.

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22 418 Trasmuted Weibull Power Fuctio Distributio: its Properties ad Applicatios Muhammad Ahsa ul Haq* 1 ahsashai36@gmail.com M. Elgarhy 2 m_elgarhy85@yahoo.com Sharqa Hashmi 3 sharqa1972@yahoo.com Gamze Ozel 4 gamzeozl@gmail.com Qurat ul Ai 5 aiee.aeem@ymail.come