A NEW MODIFIED GENERALIZED ODD LOG-LOGISTIC DISTRIBUTION WITH THREE PARAMETERS

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1 A NEW MODIFIED GENERALIZED ODD LOG-LOGISTIC DISTRIBUTION WITH THREE PARAMETERS Arbër Qoshja 1 & Markela Muça 1. Departmet of Appled Mathematcs, Faculty of Natural Scece, Traa, Albaa. Departmet of Appled Mathematcs, Faculty of Natural Scece, Traa, Albaa e-mal: qoshjaa@gmal.com Abstract Statstcal dstrbutos are very useful descrbg ad predctg real-world pheomea. Numerous exteded dstrbutos have bee extesvely used over the last decades for modelg data may appled sceces such as medce, egeerg ad face. Recet developmets focus o defg ew famles that exted wellkow dstrbutos ad at the same tme provde great flexblty modelg data practce. I ths paper, we have troduced a ew three-parameter expoetal dstrbuto called the geeralzed odd log-logstcexpoetal dstrbuto by usg the geerator defed by Cordero et al (017). Ths model exteds the odd loglogstc-expoetal ad expoetal dstrbutos. Several of ts structural propertes are dscussed detal. These clude shape of the probablty desty fucto, hazard rate fucto, quatle fucto order statstcs, ad momets. The method of maxmum lkelhood s adopted to estmate the model parameters. The applcablty of the ew models s llustrated by usg real data. The goodess-of-fts of the expoetal, beta expoetal, Kumaraswamy expoetal ad the geeralzed odd log-logstc-expoetal dstrbutos have bee compared through the AIC, AICC, BIC ad KS statstcs ad foud that the geeralzed odd log-logstcexpoetal dstrbuto fts well the data. Key Word: Expoetal dstrbuto, odd log logc dstrbuto, maxmum lkelhood estmato, Mote Carlo. Itroducto I 017, Cordero et al. troduced a geerator of cotuous dstrbuto called the geeralzed odd log-logstc famly of dstrbutos wth pdf ad cdf gve by: g( x, ξ) G( x, ξ) 1 [1 G( x, ξ) ] 1 f( x,,, ξ) G( x, ξ) [1 G( x, ξ) ] Gx (, ξ) F( x,,, ξ) G( x, ξ) [1 G( x, ξ ) ] () respectvely. (1) The am of ths paper s to cosder expoetal dstrbuto wth three parameters called the geeralzed odd log-logstc-expoetal dstrbuto. I ths artcle we cosder a ew way of expoetal dstrbuto wth three parameters by replacg x G( x, ) 1 e, x 0, 0 equato (), called the geeralzed odd log-logstc-expoetal dstrbuto. Defto 1. A radom varable s sad to have the geeralzed odd log-logstc-expoetal dstrbuto f t has the desty: 49

2 x x 1 x e 1 1 e [1 1 e ] f( x,,, ) x x 1 e [1 1 e ] The cumulatve dstrbuto fucto assocated wth Equato (5) s gve by x 1 e Fx (,,, ) x x 1 e 1 1 e (3) (4) Fgure 1, ad 3 llustrates some of the possble shapes of the pdf, cdf ad hazard fucto of the geeralzed odd log-logstc-expoetal dstrbuto for selected values of the parameters,, ad respectvely. Fgure 1. The pdf of the geeralzed odd log-logstc-expoetal dstrbuto for dfferet values of parameters alpha, theta ad lambda. Fgure. The cdf of the geeralzed odd log-logstc-expoetal dstrbuto for dfferet values of parameters alpha, theta ad lambda. 50

3 Relablty Aalyss.1 Survval fucto The relablty fucto (survval fucto) of the geeralzed odd log-logstc-expoetal dstrbuto s gve by R( x,,, ) 1 F( x,,, ) 1 1 e x x x 1 e 1 1 e (5). Hazard Rate Fucto The hazard rate fucto (falure rate) of a lfe- tme radom varable the geeralzed odd log-logstcexpoetal dstrbuto wth three parameters s gve by f( x,,, ) hx (,,, ) 1 F( x,,, ) 1 x x e 1 e x x x e e e [1 1 ] 1 [1 1 ] (6) Fgure 3. The hazard rate fucto of the geeralzed odd log-logstc-expoetal dstrbuto for dfferet values of parameters alpha, theta ad lambda..3 Quatles The quatle of ay dstrbuto s gve by solvg the equato F( x ) p, 0 p 1. The quatle of the geeralzed odd log-logstc-expoetal dstrbuto s gve by p 51

4 (7) 3. Order statstcs of the geeralzed odd log-logstc-expoetal dstrbuto Order statstcs has a mportat role qualty cotrol ad relablty aalyss, ad also hydrologcal ad extreme value aalyss. It s ofte used to detfy the stuatos ad parameter estmato. Here we assume that 1,,..., 1, s a radom sample from expoetal dstrbuto wth pdf ad cdf gve (3) ad (4) respectvely. Let (1), (),..., ( 1), () be the ordered values of the precedg sample o-decreasg order of magtude. The,..., 1 max (, th order statstcs of the geeralzed odd log-logstc-expoetal dstrbuto, () = 1, ) s gve by f x F x f x ( ) = [ ( )] 1 ( ) ( ) 1 x 1 e = x x 1 e 1 1 e x x 1 x e 1 1 e [1 1 e ] x x 1 e [1 1 e ] (8) The smallest order statstc, = (1) m (, 1,..., 1, ) has the pdf 5

5 f x F x f x 1 ( ) = [1 ( )] ( ) (1) x 1 e = 1 x x 1 e 1 1 e x x 1 x e 1 1 e [1 1 e ] x x 1 e [1 1 e ] (9) 1 Geerally the dstrbuto of the dstrbuto s as follows th r order statstcs wth the geeralzed odd log-logstc-expoetal! f x F x F x f x ( r 1)!( r)! r1 r ( ) = [ ( )] [1 ( )] ( ) ( r ) x! 1 e x x e e = ( 1)!( )!(1 ) r r r x 1 e 1 x x 1 e 1 1 e x x 1 x e 1 e [1 1 e ] 1 x x 1 e [1 1 e ] r1 (10) 3.3. Useful expasos Based o geeralzed bomal expasos, the pdf (1) of ca be expressed as (for more detals see Cordero 016) l jkl ( 1) ( 1) j 1 l f ( x) 1 ( ), j, l0 k0 k G x j l k (11) 53

6 By usg the same methodology, the pdf of the geeralzed odd log-logstc expoetal dstrbutos has the form l jk l ( 1) ( 1) j 1 l k x f ( x) 1 1 e, j, l0 k 0 j l k l ( 1) ( 1) 1 j jk l k j l x ( 1) e, j, l0 k 0 t0 t j l k Theorem. Let be a radom varable wth pdf (6). The expectato s gve by: l jk l1 k ( 1) ( 1) j 1 l 1 Ex ( ) ( 1), j, l0 k 0 t0 t j l k j E( ) xf ( x) dx 0 k ( 1) ( 1) j 1 l j x e dx, j, l0 k 0 t0 t j l k 0 l jk l1 k ( 1) ( 1) j 1l 1 ( 1), j, l0 k 0 t0 t j l k j l jk l x Proof. ( 1) 4. Estmato I ths secto, we defe the maxmum lkelhood estmato ad Newto Raphso procedure to estmate the parametrc values Maxmum lkelhood estmato I ths subsecto, terest s to defe the parameter estmato of the geeralzed odd log-logstcexpoetal dstrbuto by maxmum lkelhood estmato. dstrbuto s Let 1,,..., be..d radom varables of sze. The the lkelhood fucto for ths L(.) = f ( x,,, ) 1 x 1 x x e e e [1 1 ] 1 1 x x 1 e [1 1 e ] (1) the sample log-lkelhood fucto 54

7 ( ;,, ) = l L(.) = l l l x 1 x x e e ( 1) l 1 ( 1) l x x l 1 e [1 1 e ] 1 The maxmum lkelhood estmates ca be obtaed as the smultaeous solutos of the followg o-lear equatos: ( ;,, ) ( ;,, ) ( ;,, ) 0, 0, 0. The exact soluto for ukow parameters s ot possble aalytcally so the estmates are obtaed by solvg olear equatos smultaeously. The soluto of olear system s easer by teratve techques commo as Newto Raphso approach. By provdg tal guess of the parameters, Newto Raphso used these tal values to calculate parameter estmates. Asymptotcally these estmates of parameters approaches to ormalty ad the z-score are approxmately stadard ormal, whch ca be used to fd the 100(1 ) two sded cofdece terval for the parameters. 4. Maxmum product spacg estmates The maxmum product spacg (MPS) method has bee proposed by Cheg et al (1983). Ths method s based o a dea that the dffereces (Spacg) of the cosecutve pots should be detcally dstrbuted. The geometrc mea of the dffereces s gve as GM 1 = 1D (13) =1 where, the dfferece D s defed as x () x ( 1) D = f x,,, dx; = 1,,, 1. (14) where, Fx ( (0),,, ) = 0 ad Fx ( ( 1),,, ) =1. The MPS estmators ˆ, ˆ PS PS ad ˆ PS, of,, are obtaed by maxmzg the geometrc mea (GM) of the dffereces. Substtutg pdf of the geeralzed odd log-logstc-expoetal dstrbuto ad takg logarthm of the above expresso, we wll have 1 1 LogGM = log F( x,,, ) F( x,,, ) ( ) ( 1) 1 =1 (15) 55

8 The MPS estmators o-lear equatos: ˆ, 垐, of,, ca be obtaed as the smultaeous soluto of the followg PS PS PS 1 LogGM 1 F( x( ),,, ) F( x( 1),,, ) = = 0 1 =1 F( x( ),,, ) F( x( 1),,, ) 1 LogGM 1 F( x( ),,, ) F( x( 1),,, ) = = 0 1 =1 F( x( ),,, ) F( x( 1),,, ) 1 LogGM 1 F( x( ),,, ) F( x( 1),,, ) = = 0 1 =1 F( x( ),,, ) F( x( 1),,, ) 4.3 Least square estmates Let x(1), x(),, x( ) be the ordered sample of sze draw the geeralzed odd log-logstcexpoetal dstrbuto pdf. The, the expectato of the emprcal cumulatve dstrbuto fucto s defed as F = ; = 1,, E ( ), 1 (16) The least square estmates (LSEs) ˆ, 垐, of,, are obtaed by mmzg LS LS LS Z,, = F x(),,, =1 1 (17) Therefore, ˆ, 垐, of,, ca be obtaed as the soluto of the followg system of equatos: LS LS LS Z,, = F ( x( ),,, ) F x( ),,, = 0 =1 1 Z,, = F ( x( ),,, ) F x( ),,, = 0 =1 1 Z,, = F ( x( ),,, ) F x( ),,, = 0 =1 1 56

9 5. Smulato algorthms Sce the probablty tegral trasformato caot be appled explctly, we, therefore eed to follow the followg steps for geeratg a sample of sze from the geeralzed odd log-logstc-expoetal dstrbuto GOLLE (,, ) : 0 1. Set,,, ad tal value x.. GeerateU Uform 0,1. 3. Update 0 x by usg the Newto s formula 0 0 x = x R( x,,, ) where, Rx 0 (,,, ) = 0 F x,,, U f 0 x,,, log-logstc-expoetal dstrbuto, respectvely., F (.) ad (.) f are cdf ad pdf of the geeralzed odd 0 4. If x x, (very small, > 0 tolerace lmt), the store x= x as a sample from GOLLE (,, ). 5. If 0 > x x, the, set x 0 = x ad go to step 3. x, x,, x 6. Repeat steps 3-5, tmes for 1 respectvely. 6. Applcato Now we use a real data set to show that the geeralzed odd log-logstc-expoetal dstrbuto (GOLEE) ca be a better model tha the beta-expoetal, Kumaraswamy-expoetal ad expoetal dstrbuto. We cosder a data set of the lfe of fatgue fracture of Kevlar 373/epoxy that are subject to costat pressure at the 90% stress level utl all had faled, so we have complete data wth the exact tmes of falure. These data are: 0.051,0.0886,0.0891,0.501,0.3113,0.3451,0.4763,0.5650,0.5671,0.6566,0.6748,0.6751,0.6753,0.7696,0.8375, ,0.845,0.8645,0.8851,0.9113,0.910,0.9836,1.0483,1.0596,1.0773,1.1733,1.570,1.766,1.985,1.311, ,1.3551,1.4595,1.4880,1.578,1.5733,1.7083,1.763,1.7460,1.7630,1.7746,1.875,1.8375,1.8503,1.8808, ,1.8881,1.9316,1.9558,.0048,.0408,.0903,.1093,.1330,.100,.460,.878,.303,.3470,.3513,.4951,.560,.9911,3.056,3.678,3.4045,3.4846,3.7433,3.7455,3.9143,4.8073,5.4005,5.4435,5.595,6.5541,

10 Table 1.. Estmated parameters of the GOLLE, BE, KWE ad expoetal dstrbuto for data set. Model ML Estmate Stadard Error Log- Lkelhood LSE PS Estmator Geeralzed odd log-logstc exp.dst. Alpha=.636 theta= lambda= Kumaraswamy Expoetal Beta Expoetal a=1.556 b=.448 Lambda=0.38 a=1.679 b=1.508 Lambda= Expoetal Lambda= I order to compare the two dstrbuto models, we cosder crtera lke Kolmogorov-Smrov (K-S) statstcs, l, AIC (Akake formato crtero), ad CAIC (corrected Akake formato crtero). Table 1 shows the MLEs uder both dstrbutos, Table shows the values of KS, l, AIC, AICC, ad BIC values for the data set. The better dstrbuto correspods to smaller KS, l, AIC ad CAIC values. The values Table dcate that the GOLLE leads to a better ft tha the beta expoetal, Kumaraswamy expoetal ad expoetal dstrbuto. Table. Crtera for comparso. Model K-S l AIC CAIC BIC GOLLE Beta-E Kw-E Exp The P-P plots, ftted dstrbuto fucto ad desty fuctos of the cosdered models are plotted Fgures 4 ad 5, respectvely, for the data set. 58

11 Fgure 4. The P-P plots for the real data set Fgure 5. Ftted pdf s plots of the cosdered dstrbuto for the real data set 7. Cocluso I ths artcle, we propose a ew model, the so-called geeralzed odd log-logstc-expoetal dstrbuto whch exteds the expoetal dstrbuto the aalyss of data wth real support. A obvous reaso for geeralzg a stadard dstrbuto s that the geeralzed form provdes larger flexblty modelg real data. We study shape of the probablty desty fucto, hazard rate fucto, quatle fucto order statstcs, ad momets. The estmato of parameters s approached by the method of maxmum lkelhood, maxmum product spacg ad least square estmators. The goodess-of-fts of the expoetal, beta expoetal, Kumaraswamy expoetal ad the geeralzed odd log-logstc-expoetal dstrbutos have bee compared through the AIC, AICC, BIC ad KS statstcs ad foud that the geeralzed odd log-logstc-expoetal dstrbuto fts well the data. Fally, t s cocluded that the geeralzed odd log-logstc-expoetal dstrbuto ca be qut effectvely used to model the real problems ad so we ca recommed the use of the geeralzed odd log-logstcexpoetal dstrbuto varous felds of scece. Referece [1] Marshall AN, Olk I. A ew method for addg a parameter to a famly of dstrbutos wth applcato to the expoetal ad Webull famles. Bometrka. 1997;84: [] Eugee N, Lee C, Famoye F. Beta-ormal dstrbuto ad ts applcatos. Commu Stat, Theory Methods. 00;31:

12 [3] Cordero GM, de CastroM. A ew famly of geeralzed dstrbutos. J Statst Comput Smul. 011;81: [4] Alexader C, Cordero GM, Ortega EMM, Saraba JM. Geeralzed beta-geerated dstrbutos. Comput Stat Data Aal. 01;56: [5] Alzaatreh A, Lee C, Famoye F. A ew method for geeratg famles of cotuous dstrbutos. Metro.013;71: [6] Bourgugo M, Slva RB, Cordero GM. The Webull-G famly of probablty dstrbutos. J Data Sc. 014;1: JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 3 [7] Cordero GM, Alzadeh M, Ortega EMM. The expoetated half-logstc famly of dstrbutos: propertes ad applcatos. J Probab Stat. 014;0:1 1. do: /014/ [8] Tahr MH, Cordero GM, Alzaatreh A, Zubar M, Masoor M. The Logstc- famly of dstrbutos ad ts applcatos. Commu Stat Theory Methods. 014; [9] Cordero GM, Ortega EMM, Popovć B, Pescm RR. The Lomax geerator of dstrbutos: propertes, mfcato process ad regresso model. Appl Math Comput. 014;47: [10] Cordero, Gauss Moutho, Morad Alzadeh, Gamze Ozel, Bstoo Hosse, Edw Moses Marcos Ortega, ad Emrah Altu. "The geeralzed odd log-logstc famly of dstrbutos: propertes, regresso models ad applcatos." Joural of Statstcal Computato ad Smulato 87, o. 5 (017):