EXTENDED POWER LINDLEY DISTRIBUTION: A NEW STATISTICAL MODEL FOR NON-MONOTONE SURVIVAL DATA

Transcription

1 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( ETENDED POWER LINDLEY DISTRIBUTION: A NEW STATISTICAL MODEL FOR NON-MONOTONE SURVIVAL DATA Said Hofa Alkai Depatmet of Quatitative Aalysis, Kig Saud Uivesity, Riyadh, Saudi Aabia ABSTRACT: A ew statistical model fo o-mootoe suvival data is poposed with some of its statistical popeties as a etesio of powe Lidley distibutio. These iclude the desity ad hazad ate fuctios with thei behavio, momets, momet geeatig fuctio, skewess, kutosis measues, ad quatile fuctio. Maimum likelihood estimatio of the paametes ad thei estimated asymptotic distibutio ad cofidece itevals ae deived. Réyi etopy as a measue of the ucetaity i the model is deived. A applicatio of the model to a eal data set is peseted ad compaed with the fit attaied by othe well-kow eistig distibutios. KEYWORDS: Eteded powe Lidley distibutio; powe Lidley distibutio; omootoe suvival data INTRODUCTION The modelig ad aalysis of lifetimes is a impotat aspect of statistical wok i a wide vaiety of scietific ad techological fields, such as public health, actuaial sciece, biomedical studies, demogaphy, ad idustial eliability. The failue behavio of ay system ca be cosideed as a adom vaiable due to the vaiatios fom oe system to aothe esultig fom the atue of the system. Theefoe, it seems logical to fid a statistical model fo the failue of the system. I othe applicatios, suvival data ae categoized by thei hazad ate, e.g., the umbe of deaths pe uit i a peiod of time. Suvival data ae categoized by thei hazad ate which ca be mootoe (o-iceasig ad o-deceasig) o omootoe (bathtub ad upside-dow bathtub, o uimodal). Fo modelig of such suvival data, may models have bee poposed based o hazad ate type. Amog these, Weibull distibutio has bee used etesively i suvival studies, but it does ot fit data with a o-mootoe hazad ate shape. A oe-paamete distibutio was itoduced by Lidley (958) as a alteative model fo data with a o-mootoe hazad ate shape. This model becomes the well-kow Lidley distibutio. Popeties ad applicatios of Lidley distibutio i eliability aalysis wee studied by Ghitay et al. (008) showig that this distibutio may povide a bette fit tha the epoetial distibutio. Some eseaches have poposed ew classes of distibutios based o modificatios of the Lidley distibutio, icludig thei popeties ad applicatios. Recetly, seveal authos icludig Zakezadeh ad Dolati (009), Nadaajah et al. (0), Elbatal et al. (03), Ashou ad Eltehiwy (04), ad Oluyede ad Yag (05) poposed ad geealized Lidley distibutio with its mathematical popeties ad applicatios. A discete vesio of Lidley distibutio has bee suggested by Deiz ad Ojeda (0) with applicatios i cout data elated to isuace. A ew etesio of Lidley distibutio, called eteded Lidley (EL) distibutio, which offes a moe fleible model fo lifetime data, was itoduced by Bakouch et al. (0). Shake et al. (03) poposed a two-paamete Lidley distibutio fo modelig waitig ad suvival time data. ISSN (Pit), ISSN (Olie) 9

2 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( The powe Lidley (PL) distibutio with its ifeece was poposed by Ghitay et al. (03) ad geealized by Liyaage ad Paaai (04). Estimatio of the eliability of a stessstegth system fom powe Lidley distibutio was discussed by Ghitay et al. (05). The ivese Lidley distibutio with applicatio to head ad eck cace data was itoduced by Shama et al. (05). I this pape we itoduce a ew type of Lidley distibutio with thee paametes as a etesio to powe Lidley distibutio. The Lidley distibutio has bee poposed by Lidley (958) i the cotet of Bayes theoem as a coute eample of fiducial statistics with the followig pobability desity fuctio (pdf) y f ( y ; ) ( y ) e ;, y 0. () Ghitay et al. (008) discussed Lidley distibutio ad its applicatios etesively ad showed that the Lidley distibutio fits bette tha the epoetial distibutio based o the waitig time at a bak fo sevice. Shake et al. (03) poposed two-paamete etesios of Lidley distibutio with the followig pdf z f ( z ; ) ( z ) e ;,, z 0. () This gives bette fittig tha the oigial Lidley distibutio. Aothe two-paamete Lidley distibutio was itoduced by Ghitay et al. (03) amed powe Lidley distibutio usig the tasfomatio Y, with pdf f e ( ; ) ( ) ;,, 0. whee Y is a adom vaiable havig pdf (). Usig the tasfomatio Z, whee Z has the pdf (), we itoduce a moe fleible distibutio with thee paametes called eteded powe Lidley distibutio (EPL), which gives us a bette pefomace i fittig o-mootoic suvival data. The aim of this pape is to itoduce a ew Lidley distibutio with its mathematical popeties. These iclude the shapes of the desity ad hazad ate fuctios, the momets, momet geeatig fuctio ad some associated measues, the quatile fuctio, ad stochastic odeigs. Maimum likelihood estimatio of the model paametes ad thei asymptotic stadad distibutio ad cofidece iteval ae deived. Réyi etopy as a measue of the ucetaity i the model is deived. Fially, applicatio of the model to a eal data set is peseted ad compaed to the fit attaied by some othe well-kow Lidley distibutios. ISSN (Pit), ISSN (Olie) 0

3 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( The eteded powe Lidley distibutio A eteded powe Lidley distibutio with paametes,, ad is defied by its pobability desity fuctio ad cumulative distibutio fuctio accodig to the followig defiitio. Defiitio: Let Z be a adom vaiable havig pdf (), the the adom vaiable said to follow a EPL distibutio pdf Z is f( ;,, ) ( ) e ;,,, 0 (3) ad cumulative distibutio fuctio (cdf) F( ;,, ) e ;,,, 0. (4) Remak: The pdf (3) ca be show as a mitue of two distibutios as follows: f ( ;,, ) pf ( ) ( p)f ( ) whee p, f ( ) e, 0 ad f ( ) e, 0. We see that the EPL is a two-compoet mitue of Weibull distibutio (with shape ad scale ), ad a geealized gamma distibutio (with shape paametes, ad scale ), with miig popotio p / ( ). We use EPL (,, ) to deote the adom vaiable havig eteded powe Lidley distibutio with paametes,, with cdf ad pdf i (3) ad (4), espectively. The behavio of f ( ) at 0 ad, espectively, ae give by f, if 0 (0), if, f ( ) 0. 0, if, The deivative of f ( ) is obtaied fom (3) as f e ( ) ( ), 0, whee ISSN (Pit), ISSN (Olie)

4 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( ( y ) ay by c, y, with a, b ( ), c. Clealy, f ( ) ad ( y ) have the same sig. ( y ) is deceasig quadatic fuctio b (uimodal with maimum value at the poit y ) with (0) c ad ( ). If a, i.e., the case of Lidley distibutio, f() is deceasig (uimodal) if with f (0) ad f( )=0. Fig. shows the pdf of the EPL distibutio fo some values of,, ad displayig the behavio of f ( ). Fig.. Plots of the pobability desity fuctio of the EPL distibutio fo diffeet values of,, ad. ISSN (Pit), ISSN (Olie)

5 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Suvival ad hazad fuctios The suvival ad hazad ate fuctios of the EPL distibutio ae give espectively by s( ) F ( ) e ;,,, 0, (5) ad f ( ) ( ) h( ) ;,,, 0. s ( ) (6) The behavio of h ( ) i (6) of the EPL (,, ) distibutio at 0 ad, espectively, ae give by, if 0 0, if, h(0), if, h( ), if,, if. 0, if, It ca be see that whe, i.e., i the case of Lidley distibutio, h ( ) is iceasig fo all with h(0) ad h ( ). Fig. shows h ( ) of the EPL distibutio fo some choices of,, ad. Fig.. Plots of the hazad ate fuctio of the EPL distibutio fo diffeet values of,, ad. ISSN (Pit), ISSN (Olie) 3

6 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Momets, momet geeatig fuctio, ad associated measues Theoem. Let be a adom vaiable that follows the EPL distibutio with pdf as i (3), th the the ow momet (about the oigi) is give by ( ) E ( ), ( ) (7) ad the momet geeatig fuctio (mgf) is give by M t [ ( ) ] ( t), ( ) (8) ( ) a whee a e d. 0 Poof: ( E ) f ( ) d. Fo EPL (,, ), we have ( ) 0 e d e d e d 0 0. Lettig y, we have dy. y y y e y e dy 0 0 Usig the a e d a ( a ) ( a ), a 0, the above epessio is educed to [ ( ) ], ( ) The mgf of a cotiuous adom vaiable, whe it eists, is give by t M ( t ) e f ( ) d. ISSN (Pit), ISSN (Olie) 4

7 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Fo EPL (,, ), we have M t e e d t ( ) ( ). 0 Usig the seies epasio, e t t, the above epessio is educed to 0 M t e d e d t ( ) Usig the same ideas above, we ed up with M t [ ( ) ] ( t), ( ) ( ) whee a a e d 0. Theefoe, the mea ad the vaiace of the EPL distibutio, espectively, ae [ ( ) ], ad ( ) ( )[ ( ) ] [ ( ) ] 4 ( ) The skewess ad kutosis measues ca be obtaied fom the epessios skewess cutosis, 4 upo substitutig fo the ow momets i (7). Quatile fuctio Theoem. Let be a adom vaiable with pdf as i (3), the quatile fuctio, say Q( p ) is / ( )( p) Q ( p) W, ( ) e ISSN (Pit), ISSN (Olie) 5

8 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( whee,, 0, p (0,), ad W (.) is the egative Lambet W fuctio. Poof: We have get Q ( p) F ( p), p (0,) which implies F ( Q ( p)) p, by substitutio we ( Q( p)) [ ( Q( p)) ]e ( )( p), multiply both sides by ( ) e ad ise them to, we have the Lambet equatio, ( Q( p)) ( ) [ ( Q( p)) ]e ( )( p) e. Hece we have the egative Lambet W fuctio of the eal agumet W p e Q p ( ) (( )( ) ) [ ( ( )) ] solvig this equatio fo QP ( ), the poof is complete. ( ) ( )( pe ) i.e., Special cases of the EIL distibutio The EPL distibutio cotais some well-kow distibutios as sub-models, descibed below i bief some eamples. Lidley distibutio The oigial Lidley distibutio show by Lidley (958) is a special case of the EPL distibutio;. Usig (3) ad (4), the pdf ad cdf ae give by f( ; ) ( ) e ;, 0 ad F( ; ) e ;, 0. ( ) The associated hazad ate fuctio usig (6) is give by h( ), 0. Usig (7), th ( ) the ow momet (about the oigi) is give by. Substitute i the geeal ( ) fom of mgf i (8), we have the Lidley mgf, M () t t. The mea ad the ( ) vaiace of Lidley distibutio ae the give, espectively, by ( ) 4 ad ( ) ( ) Two-paamete Lidley distibutio. The two-paamete Lidley distibutio poposed by Shake et al. (03) is a special case of the EPL distibutio;. Usig (3) ad (4), the pdf ad cdf ae give by 6 ISSN (Pit), ISSN (Olie)

9 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( f( ;, ) ( ) e ;,, 0 ad F( ;, ) e ;,, 0. ( ) The associated hazad ate fuctio usig (6) is give by h( ) ; 0. Usig th ( ) (7), the ow momet (about the oigi) is give by. Substitute i the ( ) geeal fom of mgf i (8), we have the Lidley mgf, M () t t. The mea ( ) ad the vaiace of Lidley distibutio ae the give, espectively, by ( ) 4 ad. ( ) ( ) Powe Lidley distibutio The two-paamete Lidley distibutio poposed by Ghitay et al. (03 is a special case of the EPL distibutio;. Usig (3) ad (4), the pdf ad cdf ae give by F( ;, ) e ;,, 0. f( ;, ) ( ) e ;,, 0 ad ( ) The associated hazad ate fuctio usig (6) is give by h( ) ; 0. Usig th ( ) ) (7), the ow momet (about the oigi) is give by. Substitute i ( ) the geeal fom of mgf i (8), we have the Lidley mgf, t [ ( ) ] M () t. The mea ad the vaiace of Lidley distibutio ae ( ) ( ) the give, espectively, by ) ad ( ) ( )( ) ( ). 4 ( ) ISSN (Pit), ISSN (Olie) 7

10 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Weibull distibutio A two-paamete Weibull distibutio is a special case of the EPL distibutio; 0. Usig (3) ad (4), the pdf ad cdf ae give by f( ;, ) e ;, 0 ad F( ;, ) e ;, 0. The associated hazad ate fuctio usig (6) is give by h( ) ; 0. Usig (7), the th ow momet (about the oigi) is give by. Substitute i the geeal fom t of mgf i (8), we have the Lidley mgf, M () t. The mea ad the ( ) vaiace of Lidley distibutio ae the give, espectively, by ad. Stochastic odeigs Stochastic odeigs of positive cotiuous adom vaiables is a impotat tool to judge the compaative behavio. A adom vaiable is said to be smalle tha a adom vaiable Y i the followig cotests: (i) Stochastic ode ( Y ) if F ( ) F ( ) ; st Y (ii) Hazad ate ode ( Y ) if h ( ) h ( ) ; h Y (iii) Mea esidual life ode ( Y ) if m ( ) m ( ) ; ml Y (iv) Likelihood atio ode ( Y ) if f ( ) / f ( ) deceases i. l Y The followig implicatios (Shaked ad Shathikuma, 994) ae well kow that Y Y Y l h ml st Y The followig theoem shows that the EPL distibutio is odeed with espect to likelihood atio odeig. ISSN (Pit), ISSN (Olie) 8

11 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Theoem 3. Let PL(,, ) ad Y PL(,, ). If ad (o if ad ), the Y ad hece Y, Y ad Y. h ml st Poof: fo we have l f f Y ( ) ( ) e ( ) ;, 0, ad f ( ) log log log log( ) fy ( ) log( ) ( ). Thus d f ( ) log ( ) d f ( ) Y ( ) ( )( ) Case (i) : If ad, the d f ( ) log 0. This meas that l Y d f ( ) Y ad hece Y, Y ad Y. h ml st Case (ii) : If ad, the d f ( ) log 0. This meas that l Y ad hece d f ( ) Y Y, Y ad Y. h ml st Estimatio ad ifeece Let,..., be a adom sample, with obseved values,..., fom EPL (,, ) distibutio. Let (,, ) be the 3 paamete vecto. The log likelihood fuctio is give by ISSN (Pit), ISSN (Olie) 9

12 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( i i i i i i l [log log log( )] log( ) ( ) log. The the scoe fuctio is give by, U ( ) ( l/, l/, l/ ) T ae l i i l i i i,, l log log log. i i i i i i i i i The maimum likelihood estimatio (MLE) of say is obtaied by solvig the oliea system U (; ) 0. This oliea system of equatios does ot have a closed fom. Fo iteval estimatio ad hypothesis tests o the model paametes, we equie the obseved ifomatio mati I I I I ( ) I I I I I I whee the elemets of I ae the secod patial deivatives of U ( ). Ude stadad egula coditios fo lage sample appoimatio (Co ad Hikley, 974) that fulfilled fo the poposed model, the distibutio of is appoimately (, ( ) N ), 3 J whee J ( ) E[I ( )]. Wheeve the paametes ae i the iteio of the paamete space but ot o the bouday, the asymptotic distibutio of ( ) J ( ) lim I ( ) is N J 3 (0, ( ) ), whee is the uit ifomatio mati ad p is the umbe of paametes of the distibutio. The asymptotic multivaiate omal (, ( ) N ) 3 I distibutio of ca be used to appoimate the cofidece iteval fo the paametes ad fo the hazad ate ad suvival fuctios. A 00( ) asymptotic cofidece iteval fo paamete i is give by ii ( i Z I, Z I ), ii i whee ii I is the ( ii, ) diagoal elemet of / of the stadad omal distibutio. I ( ) fo,...,3 i ad Z is the quatile ISSN (Pit), ISSN (Olie) 30

13 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Réyi etopy Etopy is a measue of vaiatio of the ucetaity i the distibutio of ay adom vaiable. It povides impotat tools to idicate vaiety i distibutios at paticula momets i time ad to aalyze evolutioay pocesses ove time. Fo a give pobability distibutio, Réyi (96) gave a epessio of the etopy fuctio, so-called Réyi etopy, defied by Re( ) log f ( ) d whee 0 ad 0. Fo the EPL distibutio i (3), we have Re( ) log ( ) e d. 0 Now usig the fact j ( z) z, we have j 0 j j j ( ) Re( ) log e d j 0 j 0 ( ) (/ ) / j log ( / ) / ( j / ) j 0 j Applicatios I this sectio, we demostate the applicability of the EIL model fo eal data. Bade ad Piest (98) obtaied tesile stegth measuemets o 000-cabo fibe impegated tows at fou diffeet gauge legths. These cabo fibe mico-composite specimes wee tested ude tesile load util beakage, ad the beakig stess was ecoded (i gigapascals, GPa). At the gauge legth of 50 mm, ¼ 30 obseved beakig stesses wee ecoded. The data ae listed i Table. The data wee ecetly used as a illustative eample fo powe Lidley distibutio by Ghitay et al. (03). Table : Cabo fibes tesile stegth ISSN (Pit), ISSN (Olie) 3

14 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Fo this data, we fit the poposed EPL, EPL (,, ), as well as the sub models that wee itoduced i Sectio 6. The epectatio maimizatio (EM) algoithm is used to estimate the model paametes. The MLEs of the paametes, the Kolmogoov Smiov statistics (K-S) with its espective p-value, the maimized log likelihood fo the above distibutios, as well as ou poposed model ae give i Table. They idicate that the EPL distibutio (poposed model) fits the data bette tha the othe distibutios. The EPL (,, ) takes the smallest K-S test statistic value ad the lagest value of its coespodig p-value. I additio, it takes the lagest log likelihood. The fitted desities ad the empiical distibutio vesus the fitted cumulative distibutios of all models fo this data ae show i Figs. 3 ad 4, espectively. Table : Paamete estimates, K-S statistic, p-value, ad logl of the cabo fibes tesile stegth Dist. ˆ ˆ ˆ K-S p-value log L EPL (,, ) EPL (,, ) EPL (,0, ) EPL (,,) EPL (,,) Fig. 3. Plot of the fitted desities of the models i Table. ISSN (Pit), ISSN (Olie) 3

15 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Fig. 4. Plot of the fitted CDFs of the models i Table. CONCLUDING REMARKS I this pape, a ew thee-paamete distibutio called the eteded powe Lidley distibutio is itoduced ad studied i detail. This model has moe fleibility tha othe types of Lidley distibutios (oe ad two paametes) due to the shape of its desity as well as its hazad ate fuctios. The desity of the ew distibutio ca be epessed as a two-compoet Weibull desity fuctios ad a geealized gamma desity fuctio. Maimum likelihood estimates of the paametes ad its asymptotic cofidece itevals fo model paametes ae show. Applicatio of the poposed distibutio to eal data shows bette fit tha may othe well-kow distibutios, such as Lidley, two-paamete Lidley, Weibull, ad powe Lidley. REFERENCES Ashou, S. ad Eltehiwy, M., 04. Epoetiated powe Lidley distibutio. Joual of Advaced Reseach, pepit, ISSN (Pit), ISSN (Olie) 33

16 Euopea Joual of Statistics ad Pobability Vol.3, No.3, pp.9-34, Septembe 05 Published by Euopea Cete fo Reseach Taiig ad Developmet UK ( Bade, M. ad Piest, A., 98. Statistical aspects of fibe ad budle stegth i hybid composites. I: Hayashi, T., Kawata, S., ad Umekawa S. (Eds.), Pogess i Sciece ad Egieeig Composites, ICCM-IV, Tokyo, Bakouch, H., Al-Zahai, B., Al-Shomai, A., Machi, V. ad Louzad F., 0. A eteded Lidley distibutio. Joual of the Koea Statistical Society 4, Co, D. ad Hikley, D., 974. Theoetical Statistics. Chapma ad Hall, Lodo. Deiz, E., ad Ojeda, E., 0. The discete Lidley distibutio: Popeties ad applicatios. Joual of Statistical Computatio ad Simulatio 8, Elbatal, I., Meovei, F. ad Elgahy, M., 03. A ew geealized Lidley distibutio. Mathematical Theoy ad Modelig, 3(3), Ghitay, M., Al-Mutaii, D. ad Aboukhamsee, S., 05. Estimatio of the eliability of a stess-stegth system fom powe Lidley distibutio. Commuicatios i Statistics: Simulatio & Computatio 44(), Ghitay, M., Al-Mutaii, D., Balakisha, N. ad Al-Eezi, I., 03. Powe Lidley distibutio ad associated ifeece. Computatioal Statistics ad Data Aalysis 64, Ghitay, M., Atieh, B. ad Nadadajah, S., 008. Lidley distibutio ad its applicatios. Mathematics ad Computes i Simulatio 78, Lidley, D., 958. Fiducial distibutios ad Bayes theoem. Joual of the Royal Statistical Society 0(), Liyaage, G. ad Paaai, M., 04. The geealized powe Lidley distibutio with its applicatios. Asia Joual of Mathematics ad Applicatios, 04, -3. Nadaajah, S., Bakouch, H. ad Tahmasbi, R., 0. A geealized Lidley distibutio. Sakhya B: Applied ad Itediscipliay Statistics, 73, Oluyede, B. ad Yag, T., 05. A ew class of geealized Lidley distibutio with applicatios. Joual of Statistical Computatio & Simulatio 85(0), Réyi, A., 96. O measue of etopy ad ifomatio. Poceedigs of the 4 th Bekeley Symposium o Mathematical Statistics ad Pobability, Uivesity of Califoia Pess, Bekeley, Shaked, M. ad Shathikuma, J., 994. Stochastic odes ad thei applicatios. Bosto: Academic Pess. Shake, R., Shama, S. ad Shake, R., 03. A two-paamete Lidley distibutio fo modelig waitig ad suvival time seies data. Applied Mathematics 4, Shama, V., Sigh, S., Sigh, U. ad Agiwal, V., 05. The ivese Lidley distibutio: A stess-stegth eliability model with applicatios to head ad eck cace data. Joual of Idustial & Poductio Egieeig 3(3), Zakezadeh, H. ad Dolati, A., 009. Geealized Lidley distibutio. Joual of Mathematical Etesio 3(), 3-5. ISSN (Pit), ISSN (Olie) 34