A NEW FAMILY OF GENERALIZED GAMMA DISTRIBUTION AND ITS APPLICATION

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1 Joural of Mathmatics ad Statistics 10 (2): , 2014 ISSN: Scic Publicatios doi: /jmssp Publishd Oli 10 (2) 2014 ( a 1 y whr, Γ ( a, b) λ = y dy is th uppr icomplt gamma α-1 -( λx) g x = λx for x>0; α,b, λ > 0 (1) b Γ α fuctio. Corrspodig Author: Wiai Bodhisuwa, Dpartmt of Statistics, Faculty of Scic, Kastsart Uivrsity, Bagkok, 10900, P.O.Box 1086, Chatuchak, Bagkok, 10903, Thailad Tl: Fax: Scic Publicatios A NEW FAMILY OF GENERALIZED GAMMA DISTRIBUTION AND ITS APPLICATION Satsayamo Suksagrakcharo ad Wiai Bodhisuwa Dpartmt of Statistics, Faculty of Scic, Kastsart Uivrsity, Bagkok, 10900, Thailad Rcivd ; Rvisd ; Accptd ABSTRACT Th mixtur distributio is dfid as o of th most importat ways to obtai w probability distributios i applid probability ad svral rsarch aras. Accordig to th prvious raso, w hav b lookig for mor flxibl altrativ to th liftim data. Thrfor, w itroducd a w mixd distributio, amly th Mixtur Gralizd Gamma (MGG) distributio, which is obtaid by mixig btw gralizd gamma distributio ad lgth biasd gralizd gamma distributio is itroducd. Th MGG distributio is capabl of modlig bathtub-shapd hazard rat, which cotais spcial submodls, amly, th xpotial, lgth biasd xpotial, gralizd gamma, lgth biasd gamma ad lgth biasd gralizd gamma distributios. W prst som usful proprtis of th MGG distributio such as ma, variac, skwss, kurtosis ad hazard rat. Paramtr stimatios ar also implmtd usig maximum liklihood mthod. Th applicatio of th MGG distributio is illustratd by ral data st. Th rsults dmostrat that MGG distributio ca provid th fittd valus mor cosistt ad flxibl framwork tha a umbr of distributio iclud importat liftim data; th gralizd gamma, lgth biasd gralizd gamma ad th thr paramtrs Wibull distributios. Kywords: Gralizd Gamma Distributio, Lgth Biasd Gralizd Gamma Distributio, Mixtur Distributio, Hazard Rat, Lif Tim Data Aalysis 1. INTRODUCTION Th family of th gamma distributio is vry famous distributio i th litratur for aalyzig skwd data such as Rsti t al. (2013). Th Gralizd Gamma (GG) distributio was itroducd by Stacy (1962) ad was icludd spcial sub-modls such as th xpotial, Wibull, gamma ad Rayligh distributios, amog othr distributios. Th GG distributio is appropriatd for modlig data with dissimilar typs of hazard rat: I th figur of bathtub ad uimodal. This typical is practical for stimatig idividual hazard rat ad both rlativ hazards ad rlativ tims by Cox (2008). Its Probability Dsity Fuctio (PDF) is giv by Equatio 1: 211 Whr: α ad = Shap paramtrs ad λ = A scal paramtr Γ (α) = Th gamma fuctio, dfid by a -1 -y a y dy 0 Γ = As wll as th Cumulativ Distributio Fuctio (CDF) of GG distributio, dotd as G(x), ca b xprssd as follows Equatio 2-4: ( ( ) ) Γ α, λx G x =1- (2)

2 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 Furthrmor, som usful mathmatical proprtis such as ma ad th rth momt ar giv as follows: 1 Γ α E g X = λγ α Ad: r Γ α r Eg ( X ) = r λ Γ α Scic Publicatios (3) r = 1, 2, 3, (4) Rctly, Ahmd t al. (2013a) prstd a Lgth biasd Gralizd Gamma (LGG) distributio which obtaid pdf as Equatio 5: λ α -( λx) gl ( x ) = ( λx) for x > 0; α,, λ > 0 (5) By (5), it is simpl to show that th cdf of LGG distributio is giv by Equatio 6: 1, λx G L ( x ) =1- (6) From (5), w ca provid som hlpful mathmatical proprtis; such as ma ad th rth momt of th LGG distributio, rspctivly ar giv by Equatio 7 ad 8: 2 EL ( X ) = λ Ad: r1 r EL ( X ) = r = 1, 2, 3 r λ (7) (8) Morovr, th cocpt of lgth biasd distributio foud i various applicatios i liftim ara such as family history disas ad survival vts. Th study of huma 212 familis ad wildlif populatios wr th subjct of a articl dvlopd by Patil ad Rao (1978). Patill t al. (1986) prstd a list of th most commo forms of th wight fuctio usful i scitific ad statistical litratur as wll as som basic thorms for wightd distributios ad lgth biasd as spcial cas thy arrivd at th coclusio. For xampl, Nauwog ad Bodhisuwa (2014) prstd th lgth biasd Bta Parto distributio. Howvr, LGG distributio simultaously provids grat flxibility i modlig data i practic. O such class of distributios was gratd from th logit of th two-compot mixtur modl, which xtds th origial family of distributios with th lgth biasd distributios, provid powrful ad popular tools for gratig flxibl distributios with attractiv statistical ad probabilistic proprtis. Th mixtur distributio is dfid as o of th most crucial ways to obtai w probability distributios i applid probability ad svral rsarch aras. Accordig to th formr raso. W hav b lookig for a mor flxibl altrativ to th Gralizd Gamma (GG) distributio. Nadarajah ad Gupta (2007) usd th GG distributio with applicatio to drought data. Th Cox t al. (2007) offrd a paramtric survival aalysis ad taxoomy of th GG distributio. Alkari (2012) obtaid a class of distributios gralizs svral distributios with ay propr cotiuous liftim distributio by compoudig trucatd logarithmic distributio with dcrasig hazard rat. Sattayatham ad Talagtam (2012) foud th ifiit mixtur Logormal distributios for rducig th problm of th umbr of compots ad fittig of trucatd ad/or csord data. Rctly, Thr ar may rsarchrs hav applid i various fild such as Mahsh t al. (2014) proposd a gralizd rgrssio ural twork for th diagosis of th hpatitis B virus dias ad Biswas t al. (2014) usd th tworks of th prst day commuicatio systms, frqutly flood or watr loggig, sudd failur of o or fw ods i gralizd ral tim multigraphs. Th purpos of this study is to ivstigat th proprtis of a w mixtur gralizd gamma distributio, which was obtaid by mixig th GG distributio with th LGG distributio ad is mor flxibl i fittig liftim data. Sctio 2 itroducs th Mixtur Gralizd Gamma (MGG) distributio ad is cocrd with mixtur of th GG distributio with th LGG distributio. It cotais as wll-kow liftim spcial sub-modls. Usful mathmatical proprtis of th MGG distributio icludig th rth momt, ma, variac, skwss, kurtosis ad hazard rat. I additio, sctio 3 th paramtrs of th MGG

3 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 distributio ar stimatd by Maximum Liklihood Estimatio (MLE) ad ar prstd th compariso aalysis amog th GG, LGG, MGG ad th thr paramtrs Wibull distributios basd o ral data st. Fially, coclusio is icludd i sctio MATERIALS AND METHODS 2.1. Mixtur Gralizd Gamma Distributio I this sctio w proposd a w mixtur distributio to crat xtsivly flxibl distributio ad cosidrd som spcial cass. Dfiitio 1 Lt g(x) ad g L (x) ar th pdf ad lgth biasd pdf of th radom variabl (r.v.) X rspctivly, whr x > 0 ad 0 p 1 th th mixtur lgth biasd distributio of X producd by th mixtur btw g(x) ad g L (x) i th form of pg (x)(1-p)g L (x). Thorm 1 Lt X~MGG(α,, λ, p). Th pdf ad cdf rspctivly ar giv by Equatio 9: p ( 1-p) λx f ( x ) = λ( λx) Scic Publicatios α-1 -( λx) For x > 0; α,, λ > 0.; 0 p 1 ad Equatio 10: ( ) ( 1-p),( λx) pγ α, λx F x = (9) (10) If X is distributd as MGG distributio with α,, λ ad mixig p paramtrs ad if its pdf, is obtai by rplacmt (1) ad (5) i Dfiitio 1, (9) calld th two-compot mixtur distributio, ca b followd as: λ α-1 -( λx) f x =p λx ( 1-p) Γ α λ p ( 1-p) λx ( λx) = λ( λx) α -( λx) α-1 -( λx ) 213 Lt F (x) is th cdf for a gralizd class of distributio for dfid by dfiitio 2, is gratd by applyig to th MGG distributio Equatio 11: x F x = pg t 1-p g t dt L 0 x L (11) =p g t dt 1-p g t dt 0 0 ( x) =pg x 1-p G L x By substitut (2) ad (6) ito (11), w th obtai: 1 Γ α,x Γ( α,x) F( x ) =p 1- ( 1-p) 1- ( ) ( 1-p),( λx) pγ α, λx = I Fig. 1, w prst som graphs of MGG distributio, for diffrt valus of α, similarly i Fig. 2, for. W cosidr som wll-kow spcial sub-modls of th MGG distributio i th followig corollaris. Corollary 1 If p = 0 th th MGG distributio rducs to th LGG distributio with paramtrs α, ad λ is dfid by Equatio 12: λ f ( x ) = ( λx) α -( λx ) (12) Substitutig p = 0 ito (9), w obtaid (12) which is itroducd by Ahmd t al. (2013b). Corollary 2 If α = = 1 ad p = 0, th th MGG distributio dducs to lgth biasd xpotial distributio Ahmd t al. (2013a) ad its pdf is giv by: 2 -λx f ( x ) =λ x

4 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 Fig. 1. Th pdf of MGG distributio for diffrt valus of α Fig. 2. Th pdf of MGG distributio for diffrt valus of Scic Publicatios 214

5 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 Substitutig α = = 1 ito (12) rducs to Corollary 3 Scic Publicatios 2 -λx f ( x ) =λ x If = 1 ad p = 0 th th MGG distributio rducs to lgth biasd gamma distributio which prstd by Ahmd t al. (2013b) as follows: α1 λ f ( x ) = x Γ α1 α -λx Rplacig = 1 i (12), w hav: Corollary 4 α1 λ f ( x ) = x Γ α1 α -λx If p = 1, th th MGG distributio drivd to GG distributio ad its pdf is dfid by Stacy (1962) Equatio 13: λ f ( x ) = λx Γ α α-1 -( λx) Rplacig p = 1 i (9) may b xprssd as (1). Corollary 5 (13) If α = = 1 ad p = 1, th th MGG distributio rducs to xpotial distributio ad its pdf ca b writt as: -λx f ( x ) =λ Rplacig α = = 1 i (13) w obtai: -λx f ( x ) =λ 2.2. Momts of th MGG Distributio I this sctio, w will cosidr th rth momt of r.v. X~MGG(α,,λ,p). Th MGG distributio prsts various proprtis icludig: Th rth momt, ma, 215 variac, cofficit of kurtosis, cofficit of skwss ad hazard rat ar providd as follows: Dfiitio 2 E g (X r ) ad E L (X r ) ar th rth momts of origial distributio ad lgth biasd distributio of th r.v. X rspctivly. If 0 p 1, th th rth momts of th mixtur distributio is dfi by: Thorm 2 r r r g L E X =pe X 1-p E X x > 0, r = 1,2,3 Lt X~MGG(α,,λ,p), th rth momt of r.v. X is writt Equatio 14: r r1 p ( 1-p) r 1 E( X ) = r λ whr, x > 0, r = 1, 2, 3,, 0 p 1. (14) If X~MGG (α,,λ,p) from Dfiitio 2, by substitut (4) ad (8), th th rth momt is giv by: r r1 r E( X ) =p r ( 1-p) λ r λ r r1 p ( 1-p) 1 = r λ From (14), it is straightforward to ma, th scod four momts ad variac rspctivly as: 2 p ( 1-p) 1 E( X ) = λ 2 3 p ( 1-p) 2 1 E( X ) = 2 λ

6 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 W st: 3 4 p ( 1-p) 3 1 E( X ) = 3 λ 4 5 p ( 1-p) 4 1 E( X ) = 4 λ 2 3 p ( 1-p) 1 Var ( X ) = 2 λ Γ α Scic Publicatios 2 2 p ( 1-p) - Γ α i i1 p ( 1-p) ω( α,,p,i ) = Not that, ω (α,,p,i) is dfid wh i I ad lt, 2 W= ω( α,,p,2) -ω ( α,,p,1) cosqutly, th cofficit of skwss (α 3 ) i (15) ad th cofficit of kurtosis (α 4 ) i (16) ca b writt as Equatio 15 ad 16: α = 3 3 ω α,,p,3-3ω α,,p,2 ω α,,p,1 2ω α,,p,1 3 W α 4= ω α,,p,4-4ω α,,p,3 ω α,,p,1 6ω α,,p,2 ω α,,p,1-3ω α,,p,1 W (15) (16) W illustrat activitis of ma ad variac i Tabl 1 that ar icrasig fuctios of α. Also, Tabl 2 show skwss i (15) ad kurtosis i (16) for diffrt valus α ad p ar idpdt of paramtr α. Morovr, w discovr that both th skwss ad kurtosis ar icrasig fuctios of p xcpt ar both dcrasig fuctios of α Hazard Rat Hazard rat (or failur rat) ar xpasivly apply i svral filds. For xampl; Wahyudi t al. (2011) offrd th trivariat hazard rat fuctio of trivariat liftim distributio. By dfiitio, th hazard rat of a r.v. X with pdf f(x) ad cdf F(x) ca b writt by: f x h ( x ) = 1-F x Usig (9) ad (10), th hazard rat of th MGG distributio may b xprssd as Equatio 17: p ( 1-p) λx λ ( λx ) h( x ) = 1 Γ α,x pγ( α,x ) ( 1-p) Γ α α-1 -( λx) p ( 1-p) λx λ( λx) = 1 pγ( α,x) ( 1-p),x α-1 -( λx) (17) Wh substitutig diffrt valus of paramtrs i (17) th w gt som hazard rat of th MGG distributio which it prst i Fig. 3: Wh p = 0 th th hazard rat of th MGG distributio rducs to th hazard rat of th LGG distributio Wh p = 1 th th hazard rat of th MGG distributio dducs to th hazard rat of th GG distributio Wh α = = p = 1 th th hazard rat of th MGG distributio drivd to th hazard rat of th xpotial distributio 2.4. Limit Bhaviour Th limit of pdf of MGG as x is 0 ad th limit as x 1/λ is giv by:

7 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 Fig. 3. Plot of th hazard rats of th MGG distributio for diffrt valus of paramtrs Tabl 1. Ma ad variac of MGG distributio for various valus of α,, λ ad p p = 0.2 p = 0.5 p = α λ Ma Variac Ma Variac Ma Variac Scic Publicatios 217

8 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 Tabl 2. Skwss ad kurtosis of MGG distributio for various valus of α, ad p p = 0.2 p = 0.5 p = α Skwss Kurtosis Skwss Kurtosis Skwss Kurtosis λ, p = 0 p ( 1-p) λ lim f ( x ) =, 0 < p < 1 1 x 1 λ Γ α λ, p = 1 Γ( α ) It is straightforward to dmostrat th abov from th pdf of MGG i (9) as: p ( 1-p) λx lim f ( x ) = lim λ( λx) 1 1 x x 1 λ λ Γ α Scic Publicatios p ( 1-p) λ =. 3. RESULTS 3.1. Paramtrs Estimatio α-1 -( λx) Th stimatio of paramtrs for th MGG distributio will b discussd via th MLE mthod procdur. Th liklihood fuctio of th MGG (α,, λ, p) is giv by: 218 p 1-p λx i α-1 -( λxi ) L( x;θ ) = λ( λxi ) i=1 Γ α From which w calculat approximatly th logliklihood fuctio Equatio 18: logl( θ ) = log ( λ ) ( α-1) log( λxi )-λ xi i=1 i=1 p ( 1-p) λxi log i=1 (18) Th first ordr coditios for fidig th optimal valus of th paramtrs wr obtaid by diffrtiatig (18) with rspct to α,, λ ad p w gt th followig diffrtial Equatio 19-22: logl θ = α - i=1 log ( λxi ) i=1 2 2 pγ ( α) Γ α ( 1-p) λxγ i ( α) Γ α Γ α pγ α ( 1-p) λxγ i ( α) logl θ = log λx -λ x log x - 1 ( i ) ( i i ) i=1 i=1 ( 1-p) λxγ i ( α) Γ α ( 1-p) λx Γ α pγ α i=1 2 i (19) (20)

9 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 Tabl 3. Maximum liklihood stimats ad K-S distacs with thir associatd p-valus for th four mixtur distributios fittd to dprssiv coditio data Distributios Maximum liklihood stimats K-S statistic p-valu MGGD ˆα = , ˆ = , ˆλ = , ˆp = LGGD ˆα = , ˆ = , ˆλ = GGD ˆα = , ˆ = , ˆλ = Wibull ˆα = 318, ˆ = , ˆλ = logl( θ ) = ( α-1) log x -λ x λ λ i=1 2 Ad: Scic Publicatios -1 i i i=1 i=1 ( 1-p) xγ i ( α) pγ α ( 1-p) λxγ i ( α) 1 (21) -λxγ i α (22) i=1 logl( θ ) = p 1-p λxγ i α p Ths four drivativ quatios caot b solvd aalytically, as thy d to rly o Nwto-Raphso: Th Nwto-Raphso mthod is a powrful tchiqu for solvig quatios umrically. I practic ˆα, ˆ, ˆλ ad ˆp ar th solutio of th stimatig quatios obtaid by diffrtiatig th liklihood i trms of α,,λ ad p solvig i (19)-(22) to zro. Thrfor, ˆα, ˆ, ˆλ ad ˆp ca b obtaid by solvig th rsultig quatios simultaously usig a umrical procdur with th Nwto-Raphso mthod Applicatios of th MGG Distributio For o applicatio of th MGG distributio, w usd a ral data st. This was th flood rats data from th Floyd Rivr locatd i Jams, Iowa, USA for th yars from Akist t al. (2008). Th maximum liklihood mthod provids paramtrs stimatio. By comparig ths fittig distributio i Tabl 3 basd o th p-valu of this compariso, th rsults hav show that th MGG distributio 219 providd a bttr fit tha th GG, LGG ad th thr paramtrs Wibull distributios. Sic, Mahdi ad Gupta (2013) prstd th thr paramtrs Wibull distributio obtaid th pdf as: 1 x α λ x α fw ( x) = for x > 0; α,, λ > 0 λ λ 4. DISCUSSION Th MGG distributio is sigificac of mixtur distributio mthod which is a w family of GG distributio. I this study, th MGG distributio foud that it provids a cosidrably bttr fit tha th LGG ad GG distributios which ar som sub-modls of th MGG distributio. Idicatig that MGG distributio maks th approach modratly usful for liftim data. Basd o p-valus of th MGG distributio is bttr tha LGG, GG ad thr paramtrs Wibull distributios. As wll as, th rsarch by Kamaruzzama t al. (2012) fit th two compot mixtur ormal distributio by usig data sts o logarithmic stock rturs of Bursa, Malaysia idics bttr tha a ormal distributio. Furthmor, Cordiro t al. (2012) suggstd th Kumaraswamy gralizd half-ormal distributio usig th flood rats data of th Floyd Rivr, locatd i Jams, Iowa, USA provids a bttr fit tha sub-modls of it. I additio, Fato ad Lluka 2014 graliz th Parto distributio ca b usd quit ffctivly to provid bttr fits tha th Parto distributio. 5. CONCLUSION This study offrs th MGG distributio which is obtaid by mixig GG distributio with LGG

10 S. Suksagrakcharo ad W. Bodhisuwa / Joural of Mathmatics ad Statistics 10 (2): , 2014 distributio. W showd that th LGG, GG, Gamma, lgth biasd xpotial ad xpotial distributios ar sub-modls of this w mixd distributio. W hav drivd svral proprtis of th MGG distributio which icluds ma, variac, skwss, kurtosis ad hazard rat. Additioally, paramtrs stimatio ar also implmtd usig MLE mthod ad th usfulss of this distributio is illustratd by ral data st. Basd o p-valus of goodss of fit tst, w foud that th MGG distributio provids highst p-valus wh w compard with LGG, GG ad thr paramtrs Wibull distributios as show i Tabl 3. Accordig to th classical statistics, th MGG distributio is th bst fit for ths data. I coclusio, it is blivd that th MGG distributio may attract widr applicatio i ral liftim data from divrs disciplis. I th futur rsarch w should b cosidrd i paramtr stimatio usig Baysia or othr approachs. Scic Publicatios 6. ACKNOWLEDGEMENT Th rsarchrs wish to thak School of Scic Uivrsity of Phayao. Also, th authors thak Dr. Chookait Pudprommarat, th ditor ad rfrs for thir commts that aidd i improvig this articl 7. REFERENCES Ahmd, A., K.A. Mir ad J.A. Rshi, 2013a. O w mthod of stimatio of paramtrs of siz-biasd gralizd gamma distributio ad its structural proprtis. IOSR J. Mathm., 5: DOI: / Ahmd, A., K.A. Mir ad J.A. Rshi, 2013b. Structural proprtis of siz-biasd gamma distributio. IOSR J. Mathm., 5: DOI: / Akist, A., F. Famoy ad C. L, Th bta- Parto distributio. Statistics, 42: DOI: / Alkari, S.H., Nw family of logarithmic liftim distributios. J. Math. Stat., 8: DOI: /jmssp Biswas, S.S., B. Alam ad M.N. Doja, A rfimt of dijkstraâ s algorithm for xtractio of shortst paths i gralizd ral timmultigraphs. J. Comput. Sci., 10: DOI: /jcssp Cox, C., Th gralizd F distributio: A um brlla for paramtric survival aalysis. Stat. Md., 27: DOI: /sim Cox, C., H. Chu, M.F. Schidr ad A. Muoz, Paramtric survival aalysis ad taxoomy of hazard fuctios for th gralizd gamma distributio. Stat. Md., 26: DOI: /sim.2836 Cordiro, G.M., R.R. Pscim ad E.M.M. Ortga, Th kumaraswamy gralizd half-ormal distributio for skwd positiv data. J. Data Sci., 10: Mahdi, T. ad A.K. Gupta, A gralizatio of th gamma distributio. J. Data Sci., 11: Mahsh, C., E. Kaa ad M.S. Saravaa, Gralizd rgrssio ural twork basd xprt systm for hpatitis b diagosis. J. Comput. Sci., 10: DOI: /jcssp Nadarajah, S. ad A.K. Gupta, A gralizd gamma distributio with applicatio to drought data. Mathm. Comput. Simulatio, 74: 1-7. DOI: /j.matcom Nauwog, N. ad W. Bodhisuwa, Lgth biasd bta-parto distributio ad its structural proprtis with applicatio. J. Math. Stat., 10: DOI: /jmssp Patil, G.P. ad C.R. Rao, Wightd distributios ad siz-biasd samplig with applicatios to wildlif populatios ad huma familis. Biomtrics, 34: DOI: / Patill, G.P., C.R. Rao ad M.V. Rataparkhi, O discrt wightd distributios ad thir us i modl choic for obsrvd data. Commu. Statisit- Thory Math., 15: DOI: / Rsti, Y., N. Ismail ad S.H. Jamaa, Estimatio of claim cost data usig zro adjustd gamma ad ivrs Gaussia rgrssio modls. J. Math. Stat., 9: DOI: /jmssp Sattayatham, P. ad T. Talagtam, Fittig of fiit mixtur distributios to motor isurac claims. J. Math. Stat., 8: DOI: /jmssp Stacy, E.W., A gralizatio of th gamma distri butio. Aals Mathm. Statist., 33: DOI: /aoms/ Wahyudi, I., I. Purhadi ad Sutiko, Th dvlopmt of paramtr stimatio o hazard rat of trivariat wibull distributio. Am. J. Biostat., 2: DOI: /amjbsp Kamaruzzama, Z.A., Z. Isa ad M.T. Ismail, Mixturs of ormal distributios: Applicatio to bursa malaysia stock markt idics. World Applid Sci. J., 16: