PROPERTIES OF THE FOUR-PARAMETER WEIBULL DISTRIBUTION AND ITS APPLICATIONS

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1 Pak. J. Statist. 207 Vol. 33(6), PROPERTIES OF THE FOUR-PARAMETER WEIBULL DISTRIBUTION AND ITS APPLICATIONS T.H.M. Abouelmagd &4, Saeed Al-mualim &2, M. Elgarhy 3, Ahmed Z. Afify 4 ad Muir Ahmad 5 Maagemet Iformatio System Departmet, Taibah Uiversity Saudi Arabia tabouelmagd@taibahu.edu.sa 2 Departmet of Statistics, Saa a Uiversity, Yeme. s_almoalim@hotmail.com 3 Vice Presidecy for Graduate Studies ad Scietific Research Jeddah Uiversity, Saudi Arabia ad Istitute of Statistical Studies ad Research, Cairo Uiversity Cairo, Egypt. m_elgarhy85@yahoo.com 4 Departmet of Statistics, Mathematics ad Isurace, Beha Uiversity, Egypt. ahmed.afify@fcom.bu.edu.eg 5 Natioal College of Busiess Admiistratio ad Ecoomics, Lahore, Pakista. drmuir@cbae.edu.pk ABSTRACT I this article, we study the so-called the Weibull Weibull distributio. Geeral explicit expressios for the quatile fuctio, expasio of its desity fuctio, ordiary ad icomplete momets, momets of the residual ad reversed residual lifes, order statistics ad Réyi ad q etropies are derived. The model parameters are estimated usig the maximum likelihood method. Simulatio results are provided to assess the accuracy ad performace of the maximum likelihood estimators. The usefuless ad flexibility of the Weibull Weibull model are illustrated usig real data sets. KEY WORDS Etropy, Maximum Likelihood, Momets, Order Statistics, Weibull Distributio, Weibull-G Family.. INTRODUCTION The Weibull (W) distributio is a very popular distributio for modelig lifetime data i reliability where the hazard rate fuctio is mootoe. However, i may applied areas such as lifetime aalysis, the two-parameter W distributio is iadequate whe the true hazard shape is of uimodal or bathtub shape. May geeralizatios of the W distributio have bee proposed i the statistical literature to hadle with bathtub shaped failure rates. Mudholkar ad Srivastava (993) ad Mudholkar et al. (996) pioeered expoetiated W (EW) distributio to aalyze bathtub failure data. Xie et al. (2002) proposed a threeparameter modified W distributio with a bathtub shaped hazard fuctio. Carrasco et al. (2008) suggested the geeralized modified W distributio. Cordeiro et al. (206) proposed the Kumaraswamy expoetial W distributio, amog others. 207 Pakista Joural of Statistics 449

2 450 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Recetly, ew geerated families of cotiuous distributios have attracted several statisticias to develop ew models. These families are obtaied by itroducig oe or more additioal shape parameter(s) to the baselie distributio. Some of the geerated families are: the beta-g (Eugee et al., 2002), gamma-g (Zografos ad Balakrishaa, 2009), Kumaraswamy-G (Cordeiro ad de Castro, 20), McDoald-G (Alexader et al., 202), trasformed-trasformer (Alzaatreh et al., 203), Kumaraswamy odd log-logistic (Alizadeh et al., 205), type half-logistic (Cordeiro et al., 206), Garhy geerated family (Elgarhy et al., 206), Kumaraswamy Weibull-G (Hassa ad Elgarhy, 206), additive Weibull-G (Hassa ad Hemeda, 206), type II half logistic-g ( Hassa et al. 207) ad Weibull-G (W-G) (Bourguigo et al., 204) families, amog others. The cumulative distributio fuctio (cdf ) of the W-G family is give by Gx ( ) F( x) exp, x 0,, 0, Gx ( ) where ad are two positive shape parameters. The cdf () provides a wider family of cotiuous distributios. The probability desity fuctio (pdf) correspodig to () is give by g( x) G( x) G( x) f( x) exp. Gx ( ) Gx ( ) (2) Recetly, may authors costructed geeralizatios based o the W-G family. For example, Tahir et al. (205a) itroduced the W Lomax, Merovci ad Elbatal (205) defied the W Rayleigh, Tahir et al. (206) proposed the W Pareto, Tahir et al. (205b) studied the W Dagum, Afify et al. (206b) pioeered the W Fréchet, Hassa et al. (206) itroduced the W quasi Lidley, Afify et al. (206a) proposed the W Burr XII distributios. Bourguigo et al. (204) defied the Weibull-Weibull (WW) distributio. However, they do ot ivestigate its several properties. Therefore, the mai objective of this paper is to study the WW distributio defied from the W-G family ad give a comprehesive accout of some of its mathematical properties. Further, we prove empirically that the WW distributio provides better fits tha at least three other competitive models i two applicatios. The WW distributio cotais several lifetime distributios as special cases (see Table ). We are motivated to study the WW distributio because (i) It cotais a umber of kow lifetime sub models listed i Table ; (ii) The WW distributio exhibits buthtab hazard rate which makes this distributio to be superior to other lifetime distributios, which exhibit oly mootoically icreasig/decreasig, or costat hazard rates. (iii) It is show i Sectio 3. that the WW distributio ca be viewed as a mixture of Weibull distributio itroduced by Weibull (95); ad (iv) The WW distributio outperforms several of the well-kow lifetime distributios with respect to two real data sets. ()

3 T.H.M. Abouelmagd et al. 45 The remaider of the paper is orgaized as follows: I Sectio 2, we defie the WW distributio ad provide its special models. I Sectio 3, we derive a very useful represetatio for the WW desity ad distributio fuctios. Further, we derive some mathematical properties of the proposed distributio. The maximum likelihood method is used to estimate the model parameters i Sectio 4. I Sectio 5, simulatio results to assess the performace of the proposed maximum likelihood estimatio procedure are discussed. I Sectio 6, we prove imperially the importace of the WW distributio usig two real data sets. Fially, we give some cocludig remarks i Sectio THE WW DISTRIBUTION The cdf of the W distributio with scale parameter 0 ad shape parameter 0 is give (for x 0 ) by G ( x ;, ) exp. (3) The correspodig pdf of (3) is give by x g( x;, ) x exp x. (4) The radom variable X is said to have a WW distributio, deoted by X WW,,,, if its cdf is give (for x 0 ) by x 0 F( x;,,, ) exp exp( x ), (5) where, ad are shape parameters ad is a scale parameter. The correspodig pdf of X is f ( x;,,, ) x exp x exp exp x x. (6) The hazard rate fuctio (hrf), reversed hazard rate fuctio ad cumulative hazard rate fuctio of X are, respectively, give by ad h( x;,,, ) x exp x exp x, x exp x exp exp x x ( x;,,, ) exp expx ( ;,,, ) l exp H x exp x.

4 452 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Plots of the WW desity for some selected parameter values are displayed i Figure. Figure 2 displays some possible shapes of the hrf of the WW model for selected parameter values. The plots i Figure reveal that the pdf of the WW distributio ca be reversed J-shape, right skewed, left skewed or cocave dow. It ca be see, from Figure 2, that the hrf ca be decreasig, icreasig or bathtub failure rate shapes. The WW model is a very flexible distributio that approaches differet distributios whe its parameters are chaged. Table lists the special sub-models of the WW distributio. Table Special Models of the WW Distributio S# Model Author Weibull expoetial Ogutude et al. (205) 2 Weibull Rayleigh 2 Merovci ad Elbatal (205) 3 Expoetial Weibull New 4 Expoetial Expoetial New 5 Expoetial Rayleigh 2 New 6 Rayleigh Weibull 2 New 7 Rayleigh Expoetial 2 New 8 Rayleigh Rayleigh 2 2 New 3. STATISTICAL PROPERTIES This sectio discusses some importat statistical properties of the WW distributio. Let Z be a radom variable havig the W distributio with cdf (3) ad pdf (4). The, the rth ordiary ad icomplete momets of Z are give by r/ r/ / rz, r / ad rz t r t t w x 0 respectively, where,, /,, w t x e dx is the lower icomplete gamma fuctio. 3. Useful Expasios Now, we derive a useful mixture represetatio for the pdf ad cdf of the WW distributio. The pdf (6) ca be rewritte as x exp x exp x f ( x) exp x exp. exp exp x x Usig the expoetial series, we ca write

5 f(x) T.H.M. Abouelmagd et al x Figure : Some Possible Shapes for the pdf of the WW Distributio Figure 2: Some Possible Shapes for the hrf of the WW Distributio exp x k k ( ) exp x exp. (7) k exp x k0 k! exp x Usig (7), the WW desity fuctio reduces to k k k exp ( ) x f ( x) x exp x. 0! k k k exp x k (8)

6 454 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Usig the geeralized biomial series, we ca write ( k ) ( k) j exp x exp. j! ( ) The, equatio (8) reduces to j0 k x k ( ) ( ) k j f ( x) x exp x exp x. k! j! ( ) k, j0 k k j k Cosider the geeralized biomial expasio, for b 0 is real o iteger ad z, b ib i z ( ) z. (9) i0 i Applyig expasio (9) to the last equatio gives ks k ( ) ( k) j k j f ( x) x exp ( s ) x. k, j, s0 k! j! ( k ) s or, equivaletly, we ca write f ( x) s g s x, (0) s0 ks k k j ( ) ( ( k) j) where s ad g k, j0 k! j!( s ) ( k ) s x s the W distributio with shape parameter ad scale parameter s j is the pdf of. Thus, the WW desity fuctio ca be expressed as a liear combiatio of W desities. The, several of its structural properties ca be obtaied from equatio (0) ad those properties of the W distributio. By itegratig equatio (0), the cdf of X ca be give i the mixture form F( x) s G s x, s0 where G s x is the W cdf with with shape parameter ad scale parameter s 3.2 Quatile Fuctio The quatile fuctio, say follows. Q( u) F ( u) of X ca be obtaied by ivertig (5) as Qu ( ) u exp e.

7 T.H.M. Abouelmagd et al. 455 After some simplificatios, it reduces to / / Q( u) l l( u), where u is a uiform radom variable o the uit iterval 0,. I particuler, the media ca be derived from () by settig u Momets The rth momet of X ca be obtaied from (0) as r r E (X ) s x gs x dx. The, we have s0 s0 r r/ s r ( ) /. r s The th icomplete momet of X ca be expressed, based o (0), as t t t x f x dx s x g x dx. Hece, we have s0 s / r, / t s s s t. s0 The th momet of the residual lifetime is defied (for t 0 ad,2,... ) by m ( t) E[( X t) X t] x t f ( x) dx, Rt () where Rt () is the reliability fuctio. The, we have d d / d / m( t) t s( s ), t, Rt () s0d0 d y x a where, a x y e dy is the upper icomplete gamma fuctio. t The th momet of the reversed residual life is defied (for t 0,,2,... ) by t Mt E t X X t Mt 0t x f xdx. F t ()

8 456 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Usig equatio (0), we ca write d d d / d / Mt t s s, t Ft d0s0 d. The mea iactivity time (MIT) of X follows from the above equatio with. 3.4 Order Statistics Let X: X2: X: be the order statistics of a radom sample of size followig the WW distributio as give i (5) ad (6), respectively. The, the pdf of the kth order statistic, X k :, deoted by fk : ( x ), is defied by f( x) k v k vk fk : ( x) F( x), (2) B( k, ) v k v0 where B (.,.) is the beta fuctio. Hece, we ca write m exp x vk m vk F( x) ( ) exp. m 0 m exp x Usig (6), we have vk exp m vk x x x m f ( x) F( x) ( ) exp m0 expx m expx exp. expx Applyig the expoetial series ad the geeralized biomial expasio, we have f ( x) F( x) vk mrs r r ( ) m ( r ) j vk m, r, j, s0 r! j! ( r ) m r j x exp ( s ) x. s By isertig the last equatio i (2), we obtai f k : vmrs r r k ( ) m ( r ) j k v k ( x) m, r, j, s0 v0 r! j! B( k, k ) ( r ) v m r j x exp ( s ) x. s

9 T.H.M. Abouelmagd et al. 457 where The, The pdf of X k : reduces to k: s ( s) s0 f ( x) g x, (3) vmrs r r k ( ) m ( r ) j k v k r j s m, r, j0 v0 r! j! s B( k, k ) ( r ) v m s ad, as before, g x s is the W pdf with shape parameter ad scale parameter s. So, the desity fuctio of the WW order statistics is a liear combiatio of W desities. Based o equatio (3), we ca obtai some structural properties of X k : from those W properties. For example, the qth momet of X k : is give by q q/ k : s /. E X s q s0 3.5 Réyi ad q-etropies The Réyi etropy (Réyi, 96) of X is defied by I( X ) log f ( x) dx, 0,. Usig the pdf (6) ad the expoetial series, we ca write ( k) k k ( ) ( ) ( ) exp x f ( x) x exp x. ( k) k 0 k! exp x Applyig the geeralized biomial expasio ad after some algebra, the above equatio reduces to k j k ( ) ( ) ( ) f ( x) ( ) x exp ( j) x. k, i, j0 k! i! ( ) The, I ( X) reduces to k i ( k) i ( ) k j where ( ) ( ) I( X ) log d j ( j), (4) j 0 d j k j k k ( k) i ( ) ( ) ( k) i. ki, 0 k! i! ( k) j

10 458 Properties of the Four-Parameter Weibull Distributio ad its Applicatios The q-etropy is defied (for q0 ad q ) by Hq( X ) log Jq, q q where J f ( x) dx, follows from (4) as J ( q) I ( X ). q 4. MAXIMUM LIKELIHOOD ESTIMATION The maximum likelihood estimates (MLEs) of the ukow parameters for the WW distributio are determied based o complete samples. Let X,..., X be a radom T sample of size from the this distributio with vector of parameters (,,, ). The total log-likelihood fuctio for ca be expressed as ad xi i i q q xi i ( ) l l l l ( ) l x ( ) l e xi e. The score vector elemets come out as i i xi x i e xi x i i xi U x x e e, i i e i i xi l xi e xi l i i l i i i xi i U x x x e i i xi i i x l x e e, U e i xi x xi xi xi U l e e l e. i i The MLEs ˆ of ca be determied by maximizig (for a give x ) either directly by usig the Mathcad, R (optim fuctio), SAS (PROC NLMIXED), Ox program (sub-routie MaxBFGS) or by solvig the above oliear system obtaied by differetiatig this equatio ad equatig its four compoets to zero.

11 T.H.M. Abouelmagd et al SIMULATION STUDY I this sectio, a extesive umerical ivestigatio is carried out to assess o the fiite sample behavior of the MLEs of,, ad. We evaluate the performace of MLEs through their biases ad mea square errors (MSEs) for sample sizes =0, 30, 50 ad 00. All results are obtaied from 3000 Mote Carlo replicatios. The meas, MSEs ad biases for the differet estimators will be reported from these experimets. Table 2 presets the meas of the MLEs of the parameters of the WW distributio ad the correspodig biases ad MSEs. It ca be verified that the estimates are stable ad quite close the true parameter values for all sample sizes. Further, the MSEs decrease whe the sample size icreases i all cases Table 2 The Parameter Estimatio from WW Distributio usig MLE Iit MLE Bias MSE Iit MLE Bias MSE DATA ANALYSIS I this sectio, we use two real data sets to illustrate the importace ad flexibility of the WW distributio. We compare the fits of the WW model with some models amely: the beta Weibull (BW) (Lee et al., 2007), Mcdoald Weibull (McW) (Cordeiro et al., 204) ad expoetiated Weibull (EW) (Mudholkar ad Srivastava, 993) dsitributios. The maximized log-likelihood ( 2 ), Akaike iformatio criterio (AIC), the corrected Akaike iformatio criterio (CAIC), Bayesia iformatio criterio (BIC), Haa-Qui iformatio criterio (HQIC), Aderso-Darlig A * ad Cramér-Vo Mises ( W *) statistics are used for model selectio.

12 460 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Table 2 The Parameter Estimatio from WW Distributio usig MLE (Cotiued) Iit MLE Bias MSE Iit MLE Bias MSE Example : The data have bee obtaied from Nicholas ad Padgett (2006). The data represet tesile stregth of 00 observatios of carbo fibers ad they are: 3.7, 3., 4.42, 3.28, 3.75, 2.96, 3.39, 3.3, 3.5, 2.8,.4, 2.76, 3.9,.59, 2.7, 3.5,.84,.6,.57,.89, 2.74, 3.27, 2.4, 3.09, 2.43, 2.53, 2.8, 3.3, 2.35, 2.77, 2.68, 4.9,.57, 2.00,.7, 2.7, 0.39, 2.79,.08, 2.88, 2.73, 2.87, 3.9,.87, 2.95, 2.67, 4.20, 2.85, 2.55, 2.7, 2.97, 3.68, 0.8,.22, 5.08,.69, 3.68, 4.70, 2.03, 2.82, 2.50,.47, 3.22, 3.5, 2.97, 2.93, 3.33, 2.56, 2.59, 2.83,.36,.84, 5.56,.2, 2.48,.25, 2.48, 2.03,.6, 2.05, 3.60, 3.,.69, 4.90, 3.39, 3.22, 2.55, 3.56, 2.38,.92, 0.98,.59,.73,.7,.8, 4.38, 0.85,.80, 2.2, For the data i Example, Table 3 gives the MLEs of the fitted models ad their stadard errors (SEs) i parethesis. The values of goodess-of-fit statistics are listed i Table 4. It is oted, from Table 4, that the WW distributio provides a better fit tha other competitive fitted models. It has the smallest values for goodess-of-fit statistics amog all fitted models. Plots of the histogram, fitted desities ad estimated cdfs are show i Figures 3 ad 4, respectively. These figures supported the coclusio draw from the umerical values i Table 4.

13 T.H.M. Abouelmagd et al. 46 Example 2: The secod data set is obtaied from Tahir et al. (205) ad represets failure times of 84 Aircraft Widshield. The data are: 0.040,.866, 2.385, 3.443, 0.30,.876, 2.48, 3.467, 0.309,.899, 2.60, 3.478, 0.557,.9, 2.625, 3.578, 0.943,.92, 2.632, 3.595,.070,.94, 2.646, 3.699,.24,.98, 2.66, 3.779,.248, 2.00, 2.688, 3.924,.28, 2.038, 2.82,3, 4.035,.28, 2.085, 2.890, 4.2,.303, 2.089, 2.902, 4.67,.432, 2.097, 2.934, 4.240,.480, 2.35, 2.962, 4.255,.505, 2.54, 2.964, 4.278,.506, 2.90, 3.000, 4.305,.568, 2.94, 3.03, 4.376,.65, 2.223, 3.4, 4.449,.69, 2.224, 3.7, 4.485,.652, 2.229, 3.66, 4.570,.652, 2.300, 3.344, 4.602,.757, 2.324, 3.376, Table 5 lists the MLEs of the fitted models ad their SEs i parethesis. The values of goodess-of-fit statistics are preseted i Table 6. Table 3 The MLEs ad SEs of the Model Parameters for First Data Set Model Estimates (SEs) WW (,,, ) (.582) (0.236) (0.073) (0.086) BW ( ab,,, ) (0.96) (0.9) (0.) (0.077) McW ( a, b,,, c) (0.96) (0.254) (0.3) (0.085) (6.993) EW (,, a) (0.03) (0.057) (0.662) Table 4 Goodess-of-Fit Statistics for First Data Set Model 2 AIC CAIC BIC HQIC A * W * WW BW McW EW

14 462 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Table 5 The MLEs ad SEs for Secod Data Set Model Estimates (SEs) WW (,,, ) (.44) (0.298) (0.069) (0.3) BW ( ab,,, ) (2.77) (0.278) (0.278) (0.84) McW ( a, b,,, c) (5.329) (0.605) (0.48) (0.424) EW (,, a) (0.34) (0.063) (0.827).525 (38.539) Figure 3: Estimated Desities of the Fitted Models for Data Set Figure 4: Estimated cdfs of for Data Set

15 T.H.M. Abouelmagd et al. 463 Table 6 Goodess-of-Fit Statistics for Secod Data Set Model 2 AIC CAIC BIC HQIC A * W * WW BW McW EW It is observed, from Table 6, that the WW distributio gives a better fit tha other fitted models. Plots of the histogram, fitted desities ad estimated cdfs are displayed i Figures 5 ad 6, respectively. Figure 5: Estimated pdfs for Data Set 2

16 464 Properties of the Four-Parameter Weibull Distributio ad its Applicatios Figure 6: Estimated cdfs for Data Set 2 7. CONCLUSIONS I this paper, we study a four-parameter model, amed the Weibull Weibull (WW) distributio. The WW model is motivated by the wide use of the Weibull distributio i practice. The WW pdf ca be expressed as a mixture of Weibull desities. We derive explicit expressios for the quatile fuctio, ordiary ad icomplete momets, momets of the residual ad reversed residual fuctio, order statistics ad Réyi ad q- etropies. The maximum likelihood estimatio method is used to estimate the model parameters. We provide some simulatio results to assess the performace of the proposed model. The practical importace of the WW distributio is demostrated by meas of two real data sets. ACKNOWLEDGMENTS The authors would like to thak the Editor ad the five aoymous referees for carefully readig the article ad providig valuable commets which greatly improved the paper. REFERENCES. Afify, A.Z., Cordeiro, G.M., Ortega, E.M.M., Yousof, H.M. ad Butt, N.S. (206a). The Four-Parameter Burr XII distributio: properties, Regressio Model ad applicatios. Commuicatios i Statistics-Theory ad Methods, forthcomig. 2. Afify, A.Z., Yousof, H.M., Cordeiro, G.M., Ortega, E.M.M. ad Nofal, Z.M., (206b). The Weibull Fréchet distributio ad its applicatios. Joural of Applied Statistics, 43, Alexader, C., Cordeiro, G.M., Ortega, E.M., Sarabia, J.M. (202). Geeralized beta geerated distributios. Computatioal Statistics ad Data Aalysis, 56(6),

17 T.H.M. Abouelmagd et al Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G.M., Ortega, E.M.M., Pescim, R.R. (205). Kumaraswamy odd log-logistic family of distributios: Properties ad applicatios. Hacettepe Uiversity Bulleti of Natural Scieces ad Egieerig Series B: Mathematics ad Statistics. Forthcomig, available at DOI: /HJMS Alzaatreh, A., Lee, C. ad Famoye, F. (203). A ew method for geeratig families of cotiuous distributios. Metro, 7(), Bourguigo, M., Silva, R.B. ad Cordeiro, G.M. (204). The Weibull-G family of probability distributios. Joural of Data Sciece, 2(), Carrasco, J.M., Ortega, E.M. ad Cordeiro, G.M. (2008). A Geeralized Modified Weibull Distributio for Lifetime Modelig. Computatioal Statistics & Data Aalysis, 53(2), Cordeiro, G.M. ad de Castro, M. (20). A ew family of Geeralized Distributios. Joural of Statistical Computatio ad Simulatio, 8(7), Cordeiro, G.M., Alizadeh, M. ad Diiz Mariho, P.R. (206). The type I halflogistic family of distributios. Joural of Statistical Computatio ad Simulatio, 86(4), Cordeiro, G.M., Hashimoto, E.M. ad Ortega, E.M.M. (204). The McDoald Weibull model. Statistics: A Joural of Theoretical ad Applied Statistics, 48(2) Cordeiro, G.M., Saboor, A., Kha, M.N., Ozel, G. ad Pascoa, M.A. (206). The Kumaraswamy expoetial-weibull Distributio: Theory ad applicatios. Hacettepe Joural of Mathematics ad Statistics, 45(4), Elgarhy, M., Hassa, A.S. ad Rashed, M. (206). Garhy-Geerated Family of Distributios with Applicatio, Mathematical Theory ad Modelig, 6, Eugee, N., Lee C. ad Famoye, F. (2002). Beta-ormal distributio ad its applicatios. Commuicatio i Statistics-Theory Methods, 3, Hassa, A.S. ad Elgarhy, M., (206). Kumaraswamy Weibull-Geerated Family of Distributios with Applicatios. Advaces ad Applicatios i Statistics, 48, Hassa, A.S. ad Hemeda, S.E. (206). The Additive Weibull-g Family of Probability Distributios. Iteratioal Jourals of Mathematics ad its Applicatios, 4, Hassa, A.S., Elbatal, I. ad Hemeda, S.E. (206). Weibull Quasi Lidley Distributio ad its Statistical Properties with Applicatios to Lifetime Data. Iteratioal Joural of Applied Mathematics ad Statistics, 55, Hassa, A.S., Elgarhy, M. ad Shakil, M. (207). Type II Half Logistic Family of Distributios with Applicatios. Pakista Joural of Statistics ad Operatio Research, 3, Lee, C., Famoye, F. ad Olumolade, O. (2007). Beta-Weibull distributio: some properties ad applicatios to cesored data. Joural of Moder Applied Statistical Methods, 6(), Merovci, F. ad Elbatal, I. (205). Weibull rayleigh distributio: Theory ad applicatios. Applied Mathematics & Iformatio Scieces, 9(4), Mudholkar, G.S. ad Srivastava, D.K. (993). Expoetiated Weibull family for aalysig bathtub failure rate data. IEEE Trasactios o Reliability, 42,

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ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,