A Generalization of the Weibull Distribution with Applications

Transcription

1 Journal of Modern Appled Statstcal Methods Volume 5 Issue 2 Artcle A Generalzaton of the Webull Dstrbuton wth Applcatons Maalee Almhedat Unversty of Petra, Amman, Jordan, malmhedat@uop.edu.jo Carl Lee Central Mchgan Unversty, carl.lee@cmch.edu Felx Famoye Central Mchgan Unversty, felx.famoye@cmch.edu Follow ths and addtonal wors at: Part of the Appled Statstcs Commons, Socal and Behavoral Scences Commons, and the Statstcal Theory Commons Recommended Ctaton Almhedat, Maalee; Lee, Carl; and Famoye, Felx (26) "A Generalzaton of the Webull Dstrbuton wth Applcatons," Journal of Modern Appled Statstcal Methods: Vol. 5 : Iss. 2, Artcle 47. DOI:.22237/jmasm/47843 Avalable at: Ths Emergng Scholar s brought to you for free and open access by the Open Access Journals at DgtalCommons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCommons@WayneState.

2 Journal of Modern Appled Statstcal Methods November 26, Vol. 5, No. 2, do:.22237/jmasm/47843 Copyrght 26 JMASM, Inc. ISSN A Generalzaton of the Webull Dstrbuton wth Applcatons Maalee Almhedat Unversty of Petra Amman, Jordan Carl Lee Central Mchgan Unversty Mount Pleasant, MI Felx Famoye Central Mchgan Unversty Mount Pleasant, MI The Lomax-Webull dstrbuton, a generalzaton of the Webull dstrbuton, s characterzed by four parameters that descrbe the shape and scale propertes. The dstrbuton s found to be unmodal or bmodal and t can be sewed to the rght or left. Results for the non-central moments, lmtng behavor, mean devatons, quantle functon, and the mode(s) are obtaned. The relatonshps between the parameters and the mean, varance, sewness, and urtoss are provded. The method of maxmum lelhood s proposed for estmatng the dstrbuton parameters. The applcablty of ths dstrbuton to modelng real lfe data s llustrated by three examples and the results of comparsons to other dstrbutons n modelng the data are also presented. Keywords: Estmaton, moments, quantle functon, Shannon s entropy, T- Webull{Y} famly Introducton The Webull dstrbuton s a popular dstrbuton for modelng phenomena wth monotonc falure rates (Webull, 939; 95). It s used to model lfetme data. However, t cannot capture the behavor of lfetme data sets that exhbt bathtub or upsde-down bathtub (unmodal) falure rate, often encountered n relablty and engneerng studes. A number of new dstrbutons were developed as generalzatons or modfcatons of the Webull dstrbuton. e and La (995) ntroduced the addtve Webull model, whch was obtaned by addng two Webull survval functons. Mudholar and Srvastava (993) proposed the exponentated Webull dstrbuton. e, Tang, and Goh (22) studed the modfed Webull extenson. Bebbngton, La, and Zts (27) proposed a flexble Webull Dr. Almhedat s an Assstant Professor n the Department of Basc Scences. Emal her at: malmhedat@uop.edu.jo. Dr. Lee s a Professor n the Department of Mathematcs. Emal hm at: carl.lee@cmch.edu. Dr. Famoye s a Professor n the Department of Mathematcs. Emal hm at: felx.famoye@cmch.edu. 788

3 ALMHEIDAT ET AL. dstrbuton and dscussed ts propertes. For a revew of some generalzed Webull dstrbutons, one may refer to La (24). Dfferent methods to generate probablty dstrbutons contnue to appear. Eugene, Lee, and Famoye (22) ntroduced the beta-generated famly and some propertes of the famly were studed by Jones (24). Many beta-generated dstrbutons were studed (e.g., Eugene et al., 22; Nadarajah & Kotz, 24; Famoye, Lee, & Eugene, 24; Famoye, Lee, & Olumolade, 25; Nadarajah & Kotz, 26; Ansete, Famoye, & Lee, 28; Barreto-Souza, Santos, & Cordero, 2; Mahmoud, 2; Alshawarbeh, Lee, & Famoye, 22). For a revew of betagenerated dstrbutons and other generalzatons, see Lee, Famoye, and Alzaatreh (23). Alzaatreh, Lee, and Famoye (23) extended the dea of beta-generated dstrbutons to usng any contnuous random varable T wth probablty densty functon (PDF) r(t) as a generator and developed a new class of dstrbutons called the T- famly. Gven a random varable wth cumulatve dstrbuton functon (CDF) F(x), the CDF of the T- famly of dstrbutons s defned by Alzaatreh, Lee, and Famoye (23) as G x x WF r t dt () a where W(F(x)) s a monotonc and absolutely contnuous functon of the CDF F(x). Alzaatreh, Lee, and Famoye (23) studed n detals the case when W(F(x)) = -log( F(x)). Some members of the famly have been nvestgated, ncludng gamma-pareto dstrbuton (Alzaatreh, Famoye, & Lee, 22), Webull- Pareto dstrbuton (Alzaatreh, Famoye, & Lee, 23), and gamma-normal dstrbuton (Alzaatreh, Famoye, & Lee, 24a). Aljarrah, Lee, and Famoye (24) used the quantle functon QY of a random varable Y to defne the transformaton W(.) n the T- famly n () and called t the T-R{Y} famly. Followng the notaton proposed by Alzaatreh, Famoye, and Lee (24b), the CDF of the T-R{Y} famly, as defned by Aljarrah et al. (24), s gven by QYFR F x x ft tdt FT QY FR x (2) a where FT(x), FR(x), and FY(x) are, respectvely, the CDFs of the random varables T, R, and Y. The PDF correspondng to (2) s 789

4 A GENERALIZATION OF THE WEIBULL DISTRIBUTION x x fr f x ft QY FR x (3) f Q F Y Y R Almhedat, Famoye, and Lee (25) used the T-R{Y} framewor to defne and study dfferent approaches to the generalzaton of the Webull dstrbuton, the T-Webull{Y} famly. The authors defned the T-Webull{Y} famly by tang R n (2) to be a Webull random varable wth CDF F x x R e and usng the quantle functon of the random varable Y, where Y has unform, exponental, loglogstc, Fréchet, logstc, or extreme value dstrbuton. When Y follows loglogstc dstrbuton wth parameters θ and β, the CDF and PDF of the T-Webull{log-logstc} (T-Webull{LL}) famly are, respectvely, gven by f x F x FR x FT FR x fr x FR x ft F F FR x R x R x (4) (5) Settng β = = θ and tang T n (4) to be a Lomax random varable wth CDF FT(x) = ( + (x/θ)) -α, Almhedat et al. (25) defned the Lomax-Webull{LL} dstrbuton (LWD) as an example of T-Webull{LL} famly. The purpose of ths study s to nvestgate the LWD as a generalzaton of the Webull dstrbuton and a member of T-Webull{Y} famly. Defnton and Some Propertes of the LWD The CDF of the LWD defned n Almhedat et al. (25) s gven by and the PDF correspondng to (6) s x F x e (6) 79

5 ALMHEIDAT ET AL. x x x f x e e, x,,,, (7) Specal cases of the LWD are as follows: Lemma : when θ = α =, the LWD reduces to the Webull dstrbuton wth parameters and λ. when θ = =, the LWD reduces to the exponental dstrbuton wth mean λ/α. when α = /2, θ =, and = 2, the LWD reduces to the Raylegh dstrbuton wth parameter λ. (Transformatons). If a random varable T follows a Lomax dstrbuton wth parameters α and θ, then the random varable = λ{ln(t + )} / follows the LWD. 2. If a random varable T follows an exponental dstrbuton wth mean /α, then the random varable = λ{ln(θe T θ + )} / follows the LWD. 3. If a random varable T follows a standard unform dstrbuton, then the random varable = λ{ln[θ( T) -/α θ + )} / follows the LWD. Proof: Usng the transformaton technque, t s easy to show that the random varable follows the LWD as gven n (7). Hazard Functon The hazard functon assocated wth the LWD n (7) s x f x x x x h x e e F (8) The followng Lemma addresses the lmtng behavors of the hazard functon n (8). 79

6 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Lemma 2: The lmts of the LWD hazard functon as x and as x are, respectvely, gven by,, lm h x,, lm h x, x x,, (9) Proof: Ths result s obtaned by tang the lmt of the hazard functon n (8). The followng theorem s on the lmtng behavors of the PDF n (7). Theorem : gven by The lmt of the LWD as x s and the lmt as x s, lm f x, x, () Fgure. The PDFs of LWD for varous values of α, θ,, and λ 792

7 ALMHEIDAT ET AL. Proof: The x lm f. If, the result follows from Lemma 2 and the x fact that f(x) = h(x)( F(x)). If >, usng L Hôptal s rule, we have lm f x x lm x x e x e lm x x x x x e x e x lm x x e lm x x x x e e Ths completes the proof of the lmt as x. The result n () follows drectly by tang the lmt of the LWD. In Fgures and 2, varous graphs of f(x) are provded for dfferent values of the parameters. The graphs n Fgure ndcate that the LWD s unmodal wth dfferent shapes such as left-sewed, rght-sewed wth long rght tal, or monotoncally decreasng (reversed J- shape). The graphs n Fgure 2 show that the LWD can be bmodal wth two postve modal ponts (when > ) or one postve mode and the other mode at zero (when < ). The parameters α and are shape parameters whch characterze the sewness, urtoss, and bmodalty of the dstrbuton. However, the parameter λ s a scale parameter and the parameter θ s a shape and scale parameter. 793

8 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Fgure 2. The PDFs of LWD for varous values of α, θ, and when λ = Fgure 3. Hazard functon of LWD for varous values of α, θ,, and λ Dsplayed n Fgure 3 are dfferent graphs of the hazard functon related to the LWD for varous values of α, θ, and λ. When =, the LWD falure rate s ether constant (when θ = ) or frst ncreases (when θ > ) or decreases (when θ < ) and then becomes a constant. When <, the falure rate of the LWD s ether monotoncally decreasng or decreasng followed by unmodal (reflected N- shape). When >, the falure rate of the LWD s ether ncreasng or unmodal followed by ncreasng (N-shape). These dfferent falure rate shapes provde more flexblty to the LWD over the Webull dstrbuton, whch has only ncreasng, decreasng, or constant falure rate. 794

9 ALMHEIDAT ET AL. Quantle Functon The quantle functon s commonly used n general statstcs (Stenbrecher & Shaw, 28). Many dstrbutons do not have a closed form quantle functon. For the LWD, the quantle functon has a closed form as gven n the followng lemma. Lemma 3: The quantle functon of the LWD s gven by Q p ln p, p () Proof: The result follows drectly by usng part () of Lemma 2 n Almhedat et al. (25) when the random varable T follows a Lomax dstrbuton. Usng the formula n (), the quantle functon of the LWD s an ncreasng functon of λ when α, θ, and are held fxed. a decreasng functon of α when θ, λ, and are held fxed. an ncreasng functon of θ when α,, and λ are held fxed. a decreasng (ncreasng, or constant) functon of, f θ < B (θ > B, or θ = B), when α, θ, and λ are held fxed, where B = (e )/[( p) - (/α) ]. The closed form quantle functon n () maes smulatng the LWD random varates straghtforward. If U s a unform random varate on the unt nterval (, ), then the random varable = Q(U) follows the LWD. Note that the medan (M) can be calculated by settng p =.5 n the quantle functon n (). The medan of the LWD s gven by M = Q(.5) = λ{ln[θ(.5) -/α θ + ]} /. Mode(s) From Almhedat et al. (25), the mode(s) of T-Webull{LL} famly satsfy the mplct equaton 795

10 A GENERALIZATION OF THE WEIBULL DISTRIBUTION ft FR x FR x, FR xft FR x FR x x f T FR x FR x log 2 F 2 R x, FR x ft FR x FR x (2) where FR(x) and F R(x) are, respectvely, the CDF and the survval functon of the Webull dstrbuton. When T s a Lomax random varable, (2) can be smplfed to FR x FR x, FR x FR x x 2 FR x 2FR x log, FR x FR x FR x (3) Thus, the mode(s) of the LWD satsfy (3). Consder the varatonal behavor wth respect to changes n the parameter values. When, (3) can be smplfed to x e x x e (4) Rewrtng (4), x e e x x (5) Settng u = (x/λ) n (5), 796

11 ALMHEIDAT ET AL. u e e u u (6) Both x and u have the same varatonal behavors wth respect to changes n the parameters α and θ. The frst dervatves of u wth respect to α and θ are, respectvely, gven by u u e u e, u e e u 2 u 2 (7) From (7), the mode s a decreasng functon of α when > and an ncreasng functon of α when <. On the other hand, the mode s an ncreasng functon of θ when > and a decreasng functon of θ when <. When =, (3) can be smplfed as x 2 e x log e x e x (8) or, equvalently, e x x 2 e x x e e On smplfyng (8), x log (9) Therefore, when =, the mode s an ncreasng functon of θ and a decreasng functon of α. The mode s an ncreasng functon of the scale parameter λ. However, t s not easy to determne ncreasng/decreasng behavor of the mode wth respect to changes n parameter. From Fgures and 2, the LWD can be unmodal or bmodal dependng on the parameter values. Ths property gves more flexblty to the LWD over the 797

12 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Webull dstrbuton, whch s only unmodal. The followng theorem shows some cases when the LWD s only unmodal. Theorem 2: θ. The LWD s unmodal whenever () = or () < and ) If =, then the mode s at the pont x = whenever θ α and the mode s at the pont x = λln[(θ )/α] whenever θ > α. ) If < and θ, the mode s at the pont x =. Proof: The dervatve wth respect to x of the PDF n (7) s gven by 2 2 x x e x f x e m 2 x (2) where x x m x e x e (2) By usng (2) when, the crtcal ponts of f(x) are x = and x = x where m(x) =. Hence, f there s a mode of the LWD, then t wll be ether at x = or at x = x where m(x) =. Note that the sgnal of x s the same as that of m(x). If =, then m(x) = (θ ) αe (x/λ). Equatng m(x) to zero and solvng for x we get x = λlog[(θ )/α], the same result we obtaned n (9). If θ > α, then the modal pont s at x = λlog[(θ )/α], otherwse the mode s at x =. If <, t s easy to see that m(x) < whenever θ, therefore f x, so f(x) s strctly decreasng. From Theorem, lm f x x f and x lm f. Thus f(x) has a unque mode at x =. Graphcal dsplays of the LWD for many combnatons of the parameters when < and θ >, and when > ndcate that the LWD s unmodal or bmodal dependng on the parameter values. However, no analytcal method has been used to show when the dstrbuton s unmodal or bmodal. Numercal methods are appled to study the regons of unmodalty and bmodalty. To study the modes of the LWD, the number of turnng ponts of f(x) x 798

13 ALMHEIDAT ET AL. n (7) s examned, whch s equvalent to examnng the sgn of f x. Ths s equvalent to studyng the sgn of the equaton m(x) n (2). Consder the stuaton when <. Select a fxed value of < ( =.5,.7,.9) and allow the values of α and θ to change from. to 5 at an ncrement of. and the values of x to change from -6 to 3 at an ncrement of.. A matrx M s constructed wth two entres {, 2} whch ndcates the number of turnng ponts of f(x). For each combnaton of α and θ, f the sgn of m(x) s negatve for all values of x between -6 and 3, then t s ndcated by n the matrx M. If the sgn of m(x) starts as beng negatve, turns postve, then turns negatve, t s ndcated by 2 n the matrx M. Ths leads to the followng two regons: In the frst regon (the values correspondng to n the matrx M ), f(x) contans no turnng ponts. Ths regon ndcates that the dstrbuton has only one mode, whch s at zero (reversed J-shape). In the second regon (correspondng to 2 n the matrx M ), f(x) contans two turnng ponts. Ths regon ndcates that the dstrbuton has two modes (one of them at zero). By usng the boundary between the two regons, we draw a regresson lne whch s a lnear functon relatng α to θ for each value of n the set {.5,.7,.9}. The regresson lnes all have R 2 = %. Shown n Fgure 4 s the regon when LWD s unmodal or bmodal for dfferent values of and three PDFs for the bmodal case when s.5,.7, and.9. Values of <, =. to.9 are also consdered at an ncrement of., and the relatonshp between α and θ on the boundary ponts of the bmodalty regon remans lnear. For the case >, a matrx M 2 s constructed wth entres {, 3}. If the sgn of m(x) starts as beng postve then turns negatve for x values between -6 and 3, then t s ndcated by n the matrx M 2. If the sgn of m(x) starts as beng postve, turns negatve, then turns postve agan and fnally becomes negatve, t s ndcated by 3 n the matrx M 2. Ths leads to the followng regons: In the frst regon (where the values n the matrx M 2 are ), f(x) contans one turnng pont. Ths regon ndcates that the dstrbuton has only one postve mode. In the second regon (where the value n the matrx M 2 are 3), f(x) contans three turnng ponts. Ths regon ndcates that the dstrbuton has two postve modes. By usng the boundary between the two regons, we draw two regresson lnes whch are non-lnear functons relatng α to θ for each value of n the set {2, 4, 6}. Each regresson lne has R 2 = %. 799

14 A GENERALIZATION OF THE WEIBULL DISTRIBUTION (a) (b) (c) (d) Fgure 4. Regons of modalty of LWD when λ = and =.5 (a); =.7 (b); =.9 (c); Some PDFs of LWD when λ = and = {.5,.7,.9} (d) Shown n Fgure 5 are the regons when LWD s unmodal or bmodal and three PDFs for the bmodal case when s 2, 4 and 6. Note that, from Fgures 4 and 5, the bmodal regon ncreases as ncreases when < and the bmodal regon decreases as ncreases when >. Notce when s large ( > 2), the regon of bmodalty does not change wth respect to changes n the value of parameter. 8

15 ALMHEIDAT ET AL. (a) (b) (c) (d) Fgure 5. Regons of modalty of LWD when λ = and = 2 (a); = 4 (b); = 6 (c); Some PDFs of LWD when λ = and = {2, 4, 6} (d) Moments, Mean Devatons, and Shannon s Entropy Moments The n th non-central moment E( n ) of the LWD can be computed by usng an nfnte sum as shown n the followng theorem: Theorem 3: expresson The n th non-central moment of the LWD s gven by the 8

16 A GENERALIZATION OF THE WEIBULL DISTRIBUTION n n j E w, j! j j, j j2 2 n j2 j2! j2 (22) where, log j n j, (a)r = a(a + ) (a + r ) s the, 2 2 ascendng factoral, Γ(a, x) s the ncomplete gamma functon gven n Abramowtz and Stegun (972) by and a t a, x t e dt x w, j m m j n m! m n log m Proof: By defnton, E n n f x x dx n x n x x e e dx (23) Usng the substtuton u = (x/λ), the ntegral n (23) can be smplfed as 82

17 ALMHEIDAT ET AL. n u n n u e E u e du u log n n u e u e n n u u u e e du log u I I 2 du e (24) Usng the generalzed bnomal expanson u u e e! the ntegral I n (24) reduces to I u du! log n u u e e (25) where (α + ) s the ascendng factoral. Usng the bnomal expanson equaton (25) can be smplfed as j u e e j ju j log j n j u e (26) I u du! j j On usng the seres representaton for the exponental functon 83

18 A GENERALIZATION OF THE WEIBULL DISTRIBUTION 84 e! m j u m j u m equaton (26) becomes log!! log!! m n m j j m m n j m j m j I u du j m j n j m m,! j j j w j (27) where, j w s as defned after equaton (22) n Theorem 3. By usng the generalzed bnomal expanson e e! u u the ntegral I2 n (24) reduces to 2 log e e! u n u I u du (28) Usng the generalzed bnomal expanson e! j ju u j e j equaton (28) reduces to

19 ALMHEIDAT ET AL. j 2 n 2 e j u I2 u du log! j2 j2!, j j2 2 n! j2! j2 j2 (29) where s as defned after equaton (22) n Theorem 3. Substtutng I gven by, j2 (27) and I2 gven by (29) nto (24) completes the proof of the result n (22). Table. Mean and varance of LWD for some values of α, θ, and =.5 =. = 7. =. θ α Mean Var Mean Var Mean Var Mean Var Gven n Table are the mean and the varance of LWD for varous combnatons of α, θ, and when λ =.5. Many parameter combnatons were used but, to save space, only a few of them are reported n Table. For fxed θ and, the mean s a decreasng functon of α. The mean s an ncreasng functon of θ when 85

20 A GENERALIZATION OF THE WEIBULL DISTRIBUTION α and are fxed. For fxed α and θ, the mean decreases frst and then ncreases as ncreases. However, there s no clear pattern for the varance wth respect to changes n the parameter values. The sewness (S) and urtoss (Ku) of LWD are gven n Table 2 for some values of α, θ, and. For fxed α and θ the sewness of LWD decreases as ncreases. For fxed values of α and, the sewness of LWD decreases as θ ncreases. Note that when θ =, at whch the LWD reduces to the Webull dstrbuton wth shape parameter and scale parameter λα -/, the sewness and the urtoss do not depend on α. However, there s no clear pattern for the urtoss wth respect to changes n the parameter values. Table 2. Sewness and urtoss of LWD for some values of α, θ, and =.5 =. = 7. =. θ α S Ku S Ku S Ku S Ku

21 ALMHEIDAT ET AL. Fgure 6. Galton s sewness and Moors urtoss for LWD when α =.5 A measure of sewness and urtoss, based on the quantle functon, s obtaned by usng Galton s sewness (Galton, 883) and Moors urtoss (Moors, 988). By usng the quantle functon defned n (), Galton s sewness and Moors urtoss for LWD, respectvely, are gven by Q 2Q Q Q Q Q Q S, K Q Q Q Q (3) Presented n Fgure 6 are three dmensonal graphs of Galton s sewness and Moors urtoss for the same parameter values as n Table 2. To save space, these values are not reported but are compared wth the values n Table 2. The results show smlar patterns to those n Table 2. Mean Devatons Let be a random varable wth mean μ and medan M. The mean devaton from the mean s defned as f D E x x dx f f x x dx x x dx x 2F 2 f x dx (3) 87

22 A GENERALIZATION OF THE WEIBULL DISTRIBUTION where F f x dx can be calculated usng (6). Smlarly, the mean devaton from the medan can be defned as f D E M x M x dx M M f f M x x dx x M x dx M 2 x f x dx M (32) The ntegrals M xf x dx and xf can be obtaned as follows: Let u = (x/λ). Then x dx from (3) and (32), respectvely, u f e u e x x dx u du (33) If e, usng a smlar approach as n Theorem 3, (33) reduces to x f x dx m j j j j! m m! m u du! j j m m! m m m j j (34) If e, then 88

23 ALMHEIDAT ET AL. u f e * * 2 log log u e x x dx u du I I u u u e e u du u u e e du (35) Agan, usng the approach n Theorem 3, the ntegrals as where j * I and m * j I log! j j m j 2 2 e! m! m * I 2 can be smplfed m * j u I2 u du log! j2 j2! j! j2 j j *, j, j 2 2 log, j2 *, j2 The ntegral M xf x dx can be obtaned n a smlar fashon. Shannon s Entropy The entropy of a random varable s a measure of varaton of uncertanty. Shannon (948) defned the entropy of a random varable wth PDF g(x) to be 89

24 A GENERALIZATION OF THE WEIBULL DISTRIBUTION. Entropy has varous applcatons n many felds ncludng scence, engneerng, and economcs. Usng Theorem 2 n Almhedat et al. (25), the Shannon s entropy of LWD s gven by E ln g log Elog log T T (36) where s the th non-central moment of the LWD and T ln s the Shannon s entropy of the Lomax random varable. Thus, from (36), the Shannon s entropy of LWD can be smplfed as ln I, (37) where Parameter Estmaton I, ln ln t t Let, 2,, n be a random sample from LWD wth parameters α, θ,, and λ.,,, for the PDF n (7) s gven by The log-lelhood functon dt n log log log log n x x j j x j e log log j (38) On tang the frst partal dervatves of the log-lelhood functon n (38) wth respect to the parameters α, θ,, and λ, n e e j n log (39) j 8

25 ALMHEIDAT ET AL. n x j n e j (4) (4) n n xj xj xj xj e e log j n n xj x x j j e e j (42) By settng (39) to (42) equal to zero and solvng them smultaneously, obtan ˆ, ˆ, ˆ, and ˆ, the maxmum lelhood estmates (MLEs) for the parameters α, θ,, and λ, are respectvely obtaned. The computatons are done usng the NLMIED procedure n SAS. In ths procedure the ntal estmates of α, θ,, and λ can be obtaned as follows: Frst, assume that the sample data (x, x2,, xn) s from a Webull dstrbuton. The parameter estmates gven n Johnson, Kotz, and Balarshnan (994, pp ) are used for and λ as the ntal estmates, whch are, expw w 6 s where w = log(x), w and sw are respectvely the mean and the standard devaton of w random sample, and γ = -Γ() s the Euler s constant. By usng Lemma, the sample data (x, x2,, xn) can be transformed to a data set from Lomax dstrbuton by usng y x exp The ntal estmates for α and θ are the moment estmates of α and θ from the Lomax dstrbuton and they are gven by 8

26 A GENERALIZATION OF THE WEIBULL DISTRIBUTION 2v, y y 2 vy y where y and vy are, respectvely, the mean and the varance of (y, y2,, yn). Applcatons Three applcatons of the LWD usng real lfe data sets are consdered. Each of the three data sets exhbts rght sewed, left sewed, or bmodal dstrbuton shape. In these applcatons, the maxmum lelhood estmates of the parameters of the ftted dstrbutons are obtaned. The LWD s compared wth other dstrbutons based on the maxmzed log-lelhood, the Kolmogorov-Smrnov (K-S) test along wth the correspondng p-value, and Aae Informaton Crteron (AIC). In addton, the hstogram of the data and the PDFs of the ftted models are presented for graphcal llustraton of the goodness of ft. Wheaton Rver Data The data set n Table 3, from Choulaan and Stephens (2), s the exceedances of flood peas (n m 3 /s) of Wheaton Rver, Yuon Terrtory, Canada. The data conssts of 72 exceedances for the years , rounded to one decmal place. It s a rght-sewed data (sewness =.5 and urtoss = 3.9) wth a long rght tal. The data set was analyzed usng several dstrbutons. Ansete et al. (28) used ths data set as an applcaton of beta-pareto dstrbuton (BPD). Alshawarbeh et al. (22) ftted the data set to beta-cauchy dstrbuton (BCD). It was also used by Al-Aqtash, Famoye, and Lee (24) to llustrate the flexblty of Gumbel- Webull dstrbuton (GWD) to ft dfferent data sets. We ft the LWD to the data set. The MLEs and the goodness of ft statstcs are presented n Table 4. The results for BPD, BCD and GWD are taen from Al-Aqtash et al. (24). The goodness of ft statstcs ndcate that the BCD, GWD, and LWD provde good ft based on the p-value of K-S statstc. But the LWD seems to provde the best ft among these dstrbutons n Table 4, snce t has the smallest AIC and K-S statstcs and the largest log-lelhood value. The LWD seems to be very compettve to other dstrbutons n fttng the data. Ths suggests that LWD fts hghly rght-sewed data wth a long tal very well. Fgure 7 contans the hstogram of the data wth the ftted dstrbuton and supports the results n Table 4. 82

27 ALMHEIDAT ET AL. Table 3. Exceedances of the Wheaton Rver data Table 4. MLEs for Wheaton Rver data (standard errors n parentheses) Dstrbuton BPD BCD GWD LWD Parameter estmates ˆα = ˆα = ˆμ = ˆα =.449 ˆb = ( ) (.24) (.472) ˆθ =. ˆb =.4584 ˆσ = ˆθ =.324 ˆ =.28 (.4899) (.7295) (.383) ˆθ = ˆα =.4848 ˆ =.6396 (.23) (.3665) (.2842) ˆλ =.967 ˆλ = ˆλ = (.688) (2.826) (2.724) Log Lelhood AIC K-S (p-value) (.446) (.235) (.9) (.9652) Fgure 7. The hstogram and the PDFs of the Wheaton Rver data 83

28 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Strengths of.5cm Glass Fbers Data The second applcaton represents fttng the LWD to the strength of.5 cm glass data set gven n Table 5. The data set s sample of Smth and Naylor (987) and deals wth the breang strength of 63 glass fbers of length.5 cm, orgnally obtaned by worers at the UK Natonal Physcal Laboratory. Barreto-Souza et al. (2) appled the beta generalzed exponental dstrbuton (BGED) to ft the data and Barreto-Souza, Cordero, and Smas (2) ftted beta Fréchet dstrbuton (BFD) to the data. Recently, Alzaghal, Famoye, and Lee (23) used the data n an applcaton of the exponentated Webull-exponental dstrbuton (EWED). Table 5. Strength of.5 cm glass fbers data Table 6. MLEs for the strength of.5 cm glass fbers data (standard errors n parentheses) Dstrbuton BFD BGE EWED LWD Parameter estmates ˆα =.396 ˆα =.425 ˆα = ˆα =.97 (.74) (.32) (3.954) (.7232) ˆb = ˆb = ˆγ = ˆθ = (64.476) (2.85) (.994) (9.467) ˆλ =.32 ˆα = ĉ =.33 ˆ = (.27) (2.925) (.3) (.2329) ˆσ = ˆλ =.9227 ˆλ =.889 (.992) (.5) (.35) Log Lelhood AIC K-S (p-value) (.6) (.588) (.95) (.5373) 84

29 ALMHEIDAT ET AL. Fgure 8. The hstogram and the PDFs for the glass fbers data The LWD s ftted to the data and the estmaton results and goodness of ft statstcs are presented n Table 6. From Table 6, the BGE, EWED, and LWD provde an adequate ft to the data wth the LWD provdng the best ft among all dstrbutons n Table 6 based on every crteron. The dstrbuton of the data s sewed to the left (sewness = -.95 and urtoss =.). Ths suggests that the LWD performs well n modelng left sewed data. Contaned n Fgure 8 are the hstogram of the data and the PDFs of the ftted dstrbutons. Australan Athletes Data In ths example, a data set reported by Coo and Wesberg (994) about Australan Athletes s consdered. It contans 3 varables on 2 male and female athletes collected at the Australan Insttute of Sport. Jamalzadeh, Arabpour, and Balarshnan (2) used the heghts for the female athletes and the hemoglobn concentraton levels for the 22 athletes to llustrate the applcaton of a generalzed sew two-pece sew-normal dstrbuton. Choudhury and Abdul Matn (2) also used percentage of the hemoglobn blood cell for the male athletes to llustrate the applcaton of an extended sew generalzed normal dstrbuton. In ths example we consder the percentage of body fat (%Bfat) varable for the 22 athletes. 85

30 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Table 7. MLEs for the %Bfat data (standard error n parentheses) Dstrbuton WD BND LND LWD Parameter estmates ˆ = ˆα =.896 ˆλ =.3 ˆα =.265 (.25) (.549) (.235) (.448) ˆλ = 5.33 ˆβ =.253 ˆμ = ˆθ =.65 (.4852) (.24) (.369) (.62) ˆμ = ˆσ = ˆ = (.286) (.682) (.635) ˆσ = ˆλ = (.65) (.36) Log Lelhood AIC K-S (p-value) (.63) ( ) ( ) (.7676) Fgure 9. The hstogram and the PDFs for %Bfat data The LWD, the beta-normal dstrbuton (BND) defned by Eugene et al. (22), the logstc-normal{logstc} dstrbuton (LND) defned by Alzaatreh et al. (24b), and the Webull dstrbuton (WD) are appled to ft the data set. Table 7 contans the estmates, standard errors of the estmates, log-lelhood values, AIC, K-S test statstc, and the correspondng p-values. 86

31 ALMHEIDAT ET AL. The hstogram and the denstes of the ftted dstrbutons are provded n Fgure 9. From Fgure 9, the dstrbuton of ths data appeared to be bmodal and sewed to the rght (sewness =.759, urtoss = 2.827). From Table 7, LWD has the smallest AIC and K-S statstcs and the largest log-lelhood value, whch ndcates that LWD seems to be superor to the other dstrbutons n fttng the data. Even though the BND has the ablty to ft bmodal data, t could not capture the bmodalty property n fttng the data. On the other hand, the LND capture the bmodalty property but wth poor ft to the data. Ths applcaton suggests that LWD has the ablty to adequately ft bmodal data. Concluson A four-parameter LWD was proposed as an extenson of the Webull dstrbuton and a member of T-Webull{Y} famly defned by Almhedat et al. (25). The LWD s found to be unmodal or bmodal and reduces to some exstng dstrbutons that are nown n the lterature. Varous propertes of the LWD are nvestgated, ncludng the hazard functon, the quantle functon, and the regons of unmodalty and bmodalty. Expressons for the moments, the Shannon s entropy, and the mean devatons are derved. The parameters are estmated by the method of maxmum lelhood. The LWD s ftted to three real data sets to llustrate the applcaton of the dstrbuton. The frst data set s the exceedances of flood peas of Wheaton Rver, the second s the strength of.5 cm glass fbers, and the thrd s the percentage of the body fat of 22 Australan Athletes. In fttng these data sets, dfferent dstrbutons are compared wth the LWD based on goodness of ft statstcs. The two most compettve dstrbutons to the LWD are the GWD (used n the flood data set) and the EWED (used n the glass fbers data set). The results show that the LWD outperformed these two dstrbutons n fttng both data sets. LWD has an advantage over several other dstrbutons due to the flexblty of ths dstrbuton and ts ablty to model dfferent shapes n real lfe data sets, ncludng unmodal and bmodal cases. References Abramowtz, M., & Stegun, I. (972). Handboo of mathematcal functons wth formulas, graphs, and mathematcal tables. New Yor: Dover Publcatons. 87

32 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Ansete, A., Famoye, F., & Lee, C. (28). The beta-pareto dstrbuton. Statstcs, 42(6), do:.8/ Al-Aqtash, R., Famoye, F., & Lee, C. (24). On generatng a new famly of dstrbutons usng the logt functon. Journal of Probablty and Statstcal Scence, 3(), Aljarrah, M. A., Lee, C., & Famoye, F. (24). On generatng T- famly of dstrbutons usng quantle functons. Journal of Statstcal Dstrbutons and Applcatons, (2), -7. do:.86/ Almhedat, M., Famoye, F., & Lee, C. (25). Some generalzed famles of Webull dstrbuton: Propertes and applcatons. Internatonal Journal of Statstcs and Probablty, 4(3), do:.5539/jsp.v4n3p8 Alshawarbeh, E., Lee, C., & Famoye, F. (22). The beta-cauchy dstrbuton. Journal of Probablty and Statstcal Scence, (), Alzaatreh, A., Famoye, F., & Lee, C. (22). Gamma-Pareto dstrbuton and ts applcatons. Journal of Modern Appled Statstcal Methods, (), Retreved from Alzaatreh, A., Famoye, F., & Lee, C. (23). Webull-Pareto dstrbuton and ts applcatons. Communcatons n Statstcs Theory and Methods, 42(9), Alzaatreh, A., Lee, C., & Famoye, F. (23). A new method for generatng famles of contnuous dstrbutons. METRON, 7(), do:.7/s y Alzaatreh, A., Famoye, F., & Lee, C. (24a). The gamma-normal dstrbuton: Propertes and applcatons. Journal of Computatonal Statstcs & Data Analyss, 69(), do:.6/j.csda Alzaatreh, A., Famoye, F., & Lee, C. (24b). T-normal famly of dstrbutons: A new approach to generalze the normal dstrbuton. Journal of Statstcal Dstrbutons and Applcatons, (6), -8. do:.86/ Alzaghal, A., Famoye, F., & Lee, C. (23). Exponentated T- famly of dstrbutons wth some applcatons. Internatonal Journal of Statstcs and Probablty, 2(3), do:.5539/jsp.v2n3p3 Barreto-Souza, W., Cordero, G. M., & Smas, A. B. (2). Some results for beta Fréchet dstrbuton. Communcaton n Statstcs Theory and Methods, 4(5), do:.8/

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34 A GENERALIZATION OF THE WEIBULL DISTRIBUTION Mahmoud, E. (2). The beta generalzed Pareto dstrbuton wth applcaton to lfetme data. Mathematcs and Computers n Smulaton, 8(), do:.6/j.matcom Moors, J. J. (988). A quantle alternatve for urtoss. Journal of the Royal Statstcal Socety. Seres D (The Statstcan), 37(), do:.237/ Mudholar, G. S., & Srvastava, D. K. (993). Exponentated Webull famly for analyzng bathtub falure-rate data. IEEE Transactons on Relablty, 42(2), do:.9/ Nadarajah, S., & Kotz, S. (24). The beta Gumbel dstrbuton. Mathematcal Problems n Engneerng, 24(4), do:.55/s Nadarajah, S., & Kotz, S. (26). The beta exponental dstrbuton. Relablty Engneerng & System Safety, 9(6), do:.6/j.ress Shannon, C. E. (948). A mathematcal theory of communcaton. Bell System Techncal Journal, 27, Smth, R. L., & Naylor, J. C. (987). A comparson of maxmum lelhood and Bayesan estmators for the three-parameter Webull dstrbuton. Journal of the Royal Statstcal Socety. Seres C (Appled Statstcs), 36(3), do:.237/ Stenbrecher, G., & Shaw, W. T. (28). Quantle mechancs. European Journal of Appled Mathematcs, 9(2), do:.7/s Webull, W. (939). Statstcal theory of the strength of materals. Ingenörs Vetensaps Aademens, Handlngar, 5, -45. Webull, W. (95). Statstcal dstrbuton functons of wde applcablty. Journal of Appled Mechancs, 8(3), e, M., & La, C. (995). Relablty analyss usng an addtve Webull model wth bathtub-shaped falure rate functon. Relablty Engneerng & System Safety, 52(), do:.6/95-832(95)49-2 e, M., Tang, Y., & Goh, T. (22). A modfed Webull extenson wth bathtub falure rate functon. Relablty Engneerng & System Safety, 76(3), do:.6/s95-832(2)