Marquette Uiversity e-publicatios@marquette Mathematics, Statistics ad Computer Sciece Faculty Research ad Publicatios Mathematics, Statistics ad Computer Sciece, Departmet of --06 A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios R. S. Meshkat Yazd Uiversity H. Torabi Yazd Uiversity Gholamhossei G. Hamedai Marquette Uiversity, gholamhoss.hamedai@marquette.edu Published versio. Pakista Joural of Statistics ad Operatio Research, Vol., No. (06): 0-. DOI. 06 Uiversity of the Pujab. Used with permissio.
A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios R.S. Meshkat Departmet of Statistics, Yazd Uiversity, 8975-74, Yazd, Ira meshkat@yazd.ac.ir H. Torabi Departmet of Statistics, Yazd Uiversity, 8975-74, Yazd, Ira htorabi@yazd.ac.ir G.G. Hamedai Departmet of Mathematics, Statistics ad Computer Sciece, Marquette Uiversity, Milwaukee, USA g.hamedai@mu.edu Abstract We prepare a ew method to geerate family of distributios. The, a family of uivariate distributios geerated by the Gamma radom variable is defied. The geeralized gamma-weibull (GGW) distributio is studied as a special case of this family. Certai mathematical properties of momets are provided. To estimate the model parameters, the maximum likelihood estimators ad the asymptotic distributio of the estimators are discussed. Certai characterizatios of GGW distributio are preseted. Fially, the usefuless of the ew distributio, as well as its effectiveess i compariso with other distributios, are show via a applicatio of a real data set. Keywords: Gamma-geerated distributio, Geeralized gamma-weibull distributio, Weibull distributio, Hazard fuctio, Maximum likelihood estimatio, Characterizatios.. Itroductio Recetly, some attempts have bee made to defie ew families of probability distributios that exted well-kow families of distributios ad at the same time provide great flexibility i modellig data i practice. A commo feature of these geeralized distributios is that they have more parameters. Zografos ad Balakrisha (009) ad Torabi ad Motazeri (00) defied the Type I ad II family of Gammageerated distributios, respectively. Alzaatre at al. (03) developed a ew method to geerate family of distributios ad called it the T X family of distributios. Barreto- Souza ad Simas (03) defied a class of distributios give by G( x ) e 0 F( x) e G( x) 0, where ( ) Gx is a cumulative distributio fuctio (cdf) ad is a costat. The cdf F is called exp-g distributio. They obtaied several mathematical properties of this class of distributios ad discussed the two special cases: exp-weibull ad exp-beta Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
R.S. Meshkat, H. Torabi, G.G. Hamedai distributios. Recetly, Javashiri et al. (03) itroduced exp-uiform (EU) distributio by takig Gx ( ) to be the cdf of the uiform distributio with parameters a ad b ad studied its properties ad applicatio. I this paper, we exted the idea of T class as follows: X family of distributios to itroduce our ew Let Ft () be the cdf of a radom variable T ad Gx ( ) be the cdf of a radom variable X defied o. We defie the cdf of T X family of distributios by F ( G ( x )) F (0) H ( x ), x. F() F(0) Note that H( x ) is a cdf ad for the special case F ( x ) e x, the distributio is simplified to the previous oe. Whe X is a cotiuous radom variable, the probability desity fuctio (pdf) ad hazard fuctio of this family are give, respectively, by g ( x ) f ( G ( x )) hx ( ), F() F(0) g ( x ) f ( G ( x )) rx ( ), F () F ( G ( x )) where g( x ) is pdf correspodig to Gx ( ). The rest of the paper is orgaized as follows. I sectio, the Gamma-Weibull distributio is itroduced. Properties of this distributio are obtaied i Sectio 3. I Sectio 4, the maximum likelihood estimatios are discussed. Characterizatios are preseted i sectio 5. The proposed model is applied to a data set i Sectio 6. Cocludig remarks are give i sectio 7.. The Geeralized Gamma-Weibull Distributio The Weibull distributio is a popular distributio for modellig lifetime data as well as modellig pheomeo with mootoe failure rates. The two-parameter Weibull desity fuctio is usually expressed as follows: x ( ;, ) exp, 0, g x x x where 0 is a scale parameter ad 0 is a shape parameter ad its cumulative distributio fuctio is x G ( x ;, ) exp, x 0. Cosiderig the followig Gamma desity fuctio x f x x x ( ) ( ;, ) exp, 0, 0 Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios the cdf of the GGW distributio is x exp ( ), H ( x ;,,, ), x 0, ( ), ad the correspodig pdf is x x exp exp x exp h( x ;,,, ), x 0, x x exp ( ), where a t a z t e dt z (, ) deotes the icomplete gamma fuctio,, 0 are scale parameters ad, 0 are shape parameters. A radom variable X which follows the GGW distributio with parameters,, ad is deoted by X ~ GGW (,,, ). Pak.j.stat.oper.res. Vol.XII No. 06 pp0-03
R.S. Meshkat, H. Torabi, G.G. Hamedai The associated survival ad hazard fuctios of GGW distributio are, respectively, give by x exp,, S( x), ( ), ad x exp x x exp exp rx ( ). x exp x exp,, 04 Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios Due to the complicated form of rx ( ), it is ot possible to derive its properties mathematically. We make some observatios via certai plots of rx ( ). For some values of the parameters, the plots of the desity ad the hazard rate fuctio are show i Figures ad, respectively. Some of the shape properties of the GGW distributio ca be summarized as follows: The distributio is right-skewed whe 0.5. As icreases the degree of right skewess icreases. The distributio is reverse "J" shaped whe,, also whe, 0.5. Figure shows that the hazard rate fuctio of the GGW distributio ca take mootoic, bathtub ad uimodal-bathtub shapes for differet parametric combiatios. 3. Momets I geeral, k -th o-cetral momet of a GGW distributio caot be easily evaluated. Cosiderig commo defiitio, k -th o-cetral momet ca be writte as x x exp exp k x exp k EX [ ]. x exp ( ), These measures for the GGW (,,0.5,0.7) distributio are calculated ad preseted i Table for various values of ad. Pak.j.stat.oper.res. Vol.XII No. 06 pp0-05
R.S. Meshkat, H. Torabi, G.G. Hamedai From Table, it ca be cocluded that for fixed, the mea, the secod momet ad the variace are icreasig fuctios of, while the skewess ad the kurtosis are decreasig fuctios of. Also, for fixed, the mea, the secod momet ad the variace are icreasig fuctios of, while the skewess ad the kurtosis are decreasig fuctios of. Table also shows that the GGW distributio is right skewed. Over-dispersio i a distributio is a situatio i which the variace exceeds the mea, uder-dispersio is the opposite, ad equi-dispersio occurs whe the variace is equal to the mea. From Table, the GGW distributio satisfies the over-dispersio property for almost all values of the parameters. 4. Parameter estimatio ad iferece Let X,, X be a radom sample, with observed values x,, x from GGW (,,, ). The log-likelihood fuctio for the vector of parameters (,,, ) T ca be writte as l x i x i l( ) log exp exp i i x i x i log exp log log x i i i log ( ) (, ) log( ). i l l l l The compoets of the score vector U ( ),,, T are give by 3,0, ( ) ( ),3 log, G l 0,0, x i log exp, i ( ), / e l, ( ), x i x i x i exp log l ( ) x i i e x i x i x i x i exp log log, i i 06 Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios x i x i x i x i x i exp x i exp l ( ), i i x i exp m where (.) is the digamma fuctio ad G (.) is the Meijer G-fuctio. The maximum pq likelihood estimatio (MLE) of, say ˆ, is obtaied by solvig the oliear system U ( ) 0. This oliear system of equatios does ot have a closed form solutio. Uder coditios that are fulfilled for parameters i the iterior of the parameter space but ot o the boudary, the asymptotic distributio of ( ˆ ) is N (, I ( )) 4 0 where I ( ) is the iverse of the Fisher iformatio matrix. A 00( ) % asymptotic cofidece iterval (ACI) for each parameter i is give by where AIC ( ) ˆ z Iˆ ( ˆ ) /, ˆ z Iˆ ( ˆ ) /, ˆ ( ˆ) i i i / i i / i I is the i -th diagoal elemet of I ( ˆ ). 5. Characterizatios The problem of characterizig a distributio is a importat problem i various fields ad has recetly attracted the attetio of may researchers. Cosequetly, various characterizatio results have bee reported i the literature. I desigig a stochastic model for a particular modellig problem, a ivestigator will be vitally iterested to kow if their model fits the requiremets of a specific uderlyig probability distributio. The ivestigator, therefore, will rely o the characterizatios of the selected distributio. These characterizatios have bee established i may differet directios. The preset work deals with the characterizatios of GGW distributio which are based o a simple relatioship betwee two trucated momets. Our characterizatio results preseted here will employ a iterestig result due to Glazel (987) (Theorem 5. below). The advatage of the characterizatios give here is that, cdf H eed ot have a closed form ad are give i terms of a itegral whose itegrad depeds o the solutio of a first order differetial equatio, which ca serve as a bridge betwee probability ad differetial equatio. We believe that other characterizatios of GGW distributio may ot be possible due to the structure of its cdf. Theorem 5.. Let,, P I a, b be a iterval for some a b ( a, b might as well be allowed). Let X : I be a cotiuous radom variable with the distributio fuctio F ad let q ad q be two real fuctios defied o I such that F be a give probability space ad let Eq X X x E q X X x ( x ), x H, Pak.j.stat.oper.res. Vol.XII No. 06 pp0-07
R.S. Meshkat, H. Torabi, G.G. Hamedai is defied with some real fuctio. Assume that q, qc ( I ), C ( I ) ad F is twice cotiuously differetiable ad strictly mootoe fuctio o the set I. Fially, assume that the equatio q q has o real solutio i the iterior of I. The F is uiquely determied by the fuctios q, q ad, particularly x u u q u q u F x C exp s u du, a q where the fuctio s is a solutio of the differetial equatio s q q costat, chose to make df. I ad C is a We like to metio that this kid of characterizatio based o the ratio of trucated momets is stable i the sese of weak covergece, i particular, let us assume that there is a sequece X of radom variables with distributio fuctios F such that the fuctios q,, q, ad ( ) satisfy the coditios of Theorem 5. ad let q q, q q for some cotiuously differetiable real fuctios q ad q. Let,,, fially, X be a radom variable with distributio F. Uder the coditio that q, X ad q, X are uiformly itegrable ad the family F is relatively compact, the sequece X coverges to X i distributio if ad oly if coverges to, where x E q X X x. E q X X x This stability theorem makes sure that the covergece of distributio fuctios is reflected by correspodig covergece of the fuctios q, q ad, respectively. It guaratees, for istace, the 'covergece' of characterizatio of the Wald distributio to that of the Levy-Smirov distributio if, as was poited out i Glazel ad Hamedai (00), Remark 5.. (a) I Theorem 5., the iterval I eed ot be closed sice the coditio is oly o the iterior of I. (b) Clearly, Theorem 5.ca be stated i terms of two fuctios q ad by takig q x, provided that the cdf F has a closed form, which will reduce the coditio give i Theorem 5. to E q X X x x. However, addig a extra fuctio will give a lot more flexibility, as far as its applicatio is cocered. 08 Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios Propositio 5.3. x Let X : 0, be a cotiuous radom variable ad let q x e q x q x exp e ad x / for 0, the fuctio defied i Theorem 5. has the form x / x exp e, x 0. Proof. Let X have desity hx ( ), the ad E ( ) / x. The X has pdf hx ( ) if ad oly if / e / H x q X x exp e x, x 0, K E / e x / H x q X X x exp e, x 0, K K,. where Fially x / x q x q x q x exp e 0 for x 0. Coversely, if is give as above, the ad hece x / x / x e exp e x q x s x, x 0, x qx qx x / exp e s x x / exp e l, x 0. e Now, i view of Theorem 5., X has desity hx ( ). Pak.j.stat.oper.res. Vol.XII No. 06 pp0-09
R.S. Meshkat, H. Torabi, G.G. Hamedai Corollary 5.4. Let X : 0, be a cotiuous radom variable ad let q x be as i Propositio 5.3. The pdf of X is hx ( ) if ad oly if there exist fuctios q ad defied i Theorem 5.satisfyig the differetial equatio Remark 5.5. (a) x / x / x e exp e s x, x 0. x / exp e The geeral solutio of the differetial equatio i Corollary 5.4 is x for x 0 x / exp e exp, x / x / x e e q x q x dx D, where D is a costat. Oe set of appropriate fuctios is give i D. Propositio 5.3 with (b) Clearly there are other triplets of fuctios,, Theorem 5.. We preseted oe such triplet i Propositio 5.3. q q satisfyig the coditios of 6. Applicatios I this sectio, the flexibility ad applicability of the proposed model is illustrated as compared to the alterative Gamma-Weibull (GW.a) distributio itroduced by Alzaatre at al. (04) ad Gamma-Weibull (GW.p) distributio preseted by Provost et al. (0). The Gamma-Weibull model is applied to a data set published i Suprawhardaa et al. (999) which cosists of time betwee failures (thousads of hours) of secodary reactor pumps. The data set cosists of 3 observatios. The TTT plot of this set of data i Figure 3 displays a bathtub-shaped hazard rate fuctio that idicates the appropriateess of the GGW distributio to fit the data set. I order to compare the models, the MLEs of the parameters, -log-likelihood, the Kolmogorov-Smirov test statistic (K-S), p-value, the Aderso-Darlig test statistic (AD), the Cramer-vo Misestest statistic (CM) ad Durbi-Watsotest statistic (DW) are give i Table for this data set. The CM ad DW test statistics are described i details i Che ad Balakrisha (995) ad Watso (96), respectively. I geeral, the smaller the values of K-S, AD, CM ad WA, the better the fit to the data. From the values of these statistics, we coclude that the GGW distributio provides a better fit to this data set tha the other models. 0 Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
A Geeralized Gamma-Weibull Distributio: Model, Properties ad Applicatios The fitted desities ad the empirical distributios versus the fitted cumulative distributio fuctios of NP, N ad SN models are displayed i Figure 4. These plots suggest that the GGW distributio is superior to the other distributios i terms of the model fittig. 7. Coclusios I this paper, a ew method to geerate family of distributios is proposed. The, a family of uivariate distributios geerated by the Gamma radom variable was defied. As a special case, the geeralized gamma-weibull (GGW) distributio is studied ad its parameter estimatios are cosidered. Fially, i order to show the usefuless of the ew distributio, a applicatio to a real data set is demostrated. The curret study cofirms that the proposed distributio, with better flexibility, ca be cosidered to be a great model for real data i compariso with the other competig distributios. Ackowledgemet The authors are highly grateful to the editor ad referees for their valuable commets ad suggestios for improvig the paper. Pak.j.stat.oper.res. Vol.XII No. 06 pp0-
R.S. Meshkat, H. Torabi, G.G. Hamedai Refereces. Alzaatreh, A., Lee, C. ad Famoye, F. (0). O the discrete aalogues of cotiuous distributios, Statistical Methodology, 9, 589-603.. Alzaatreh, A., Lee, C. ad Famoye, F. (03). A ew method for geeratig families of cotiuous distributios, Metro, 7(), 63-79. 3. Alzaatre, A., Famoyeb, F. ad Lee, C. (04). The gamma-ormal distributio: Properties ad applicatios, Computatioal Statistics ad Data Aalysis, 69, 67-80. 4. Barreto-Souza, W. ad Simas, A.B. (03). The exp-g family of probability distributios, Brazilia Joural of Probability ad Statistics, 7(), 84-09. 5. Che, G. ad Balakrisha, N. (995). A geeral purpose approximate goodessof-fit test, Joural of Quality Techology, 7, 54-6. 6. Glazel, W. (987). A characterizatio theorem based o trucated momets ad its applicatio to some distributio families, Mathematical Statistics ad Probability Theory (Bad Tatzmasdorf, 986)}, Vol. B, Reidel, Dordrecht, 75-84. 7. Glazel, W. ad Hamedai, G.G. (00). Characterizatios of uivariate cotiuous distributios, Studia Sci. Math. Hugar., 37, 83-8. 8. Javashiri, Z., Habibi Rad A. ad Hamedai, G. G. (03). Exp-uiform distributio: Properties ad characterizatios, Joural of Statistical Research of Ira, 0, 85-06. 9. Provost, S.B., Saboor, A. ad Ahmad, M. (0). The gamma-ormal distributio, Pakista Joural of Statistics, 7(), -3. 0. Suprawhardaa, S. Prayoto, M, ad Sagadji. (999). Total time o test plot aalysis for mechaical compoets of the RSG-GAS reactor, Atom Idoes, 5().. Torabi, H. ad Motazeri, H. N. (00). The Gamma-Uiform Distributio, Proceedig of the 0 th Iraia Statistical Coferece, 430-437.. Watso, G.S. (96). Goodess-of-fit tests o a circle, Biometrika, 48, 09-4. 3. Zografos, K. ad Balakrisha, N. (009). O families of beta- ad geeralized gamma-geerated distributios ad associated iferece, Statistical Methodology, 6, 344-36. Pak.j.stat.oper.res. Vol.XII No. 06 pp0-