DISCRIMINATING BETWEEN WEIBULL AND LOG-LOGISTIC DISTRIBUTIONS

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1 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 DISCRIMINATING BTN IBU AND OG-OGISTIC DISTRIBUTIONS lsayed A lsherpey Sahar A N Ibrahm Noha U M M Radwa 3 Proessor Departmet o Mathematcal Statstcs Isttute o Statstcal Studes ad Research Caro Uversty gypt ecture Departmet o Mathematcal Statstcs Isttute o Statstcal Studes ad Research Caro Uversty gypt Postgraduate Studet Departmet o Mathematcal Statstcs Isttute o Statstcal Studes ad Research Caro Uversty gypt 3 Abstract: Webull ad log-logstc dstrbutos are two popular dstrbutos or aalyzg letme data I ths paper t s assumed that the data are comg ether rom Webull or log-logstc dstrbutos The mamzed lkelhood rato test to dscrmate betwee the two dstrbutos s used The asymptotc dstrbutos o the logarthm o the rato o the mamzed lkelhood are obtaed These asymptotc results are used to estmate the probablty o correct selecto ad the mmum sample sze eeded to dscrmate betwee the two dstrbutos Two real data le are aalyzed to see how the proposed method works practce Keywords: Asymptotc dstrbuto; Webull dstrbuto; og-logstc dstrbuto; Mamzed lkelhood rato statstc; Probablty o correct selecto I INTRODUCTION Choosg the correct or best-ttg dstrbuto or a gve data set s a mportat ssue Most o the tmes dstrbuto uctos may provde a smlar data t but selectg the correct or more early correct model s desrable The problem o choosg the correct model has bee attempted by may researchers Co 96 dscussed the eect o choosg the wrog model Co 96 tackled the problem o dscrmatg betwee the log-ormal ad the epoetal dstrbutos based o the lkelhood ucto ad derved the asymptotc dstrbuto o the lkelhood rato statstc Jackso 968 derved asymptotc results or the case log-ormal versus gamma The case log-ormal versus Webull was addressed by Dumoceau ad Atle 973 They proposed a certa test ad provded ts crtcal values Perera 977 developed aother two tests to dscrmate betwee log-ormal ad Webull dstrbutos Ba ad gelhardt 980 covered the case Webull versus gamma Che 980 made sgcat cotrbuto dscrmato problem whe usg small sample sze Kappema 98 studed the probablty o correct selecto or the pars Webull versus log-ormal Webull versus gamma ad gamma versus log-ormal Frth 988 dscussed the problem o dscrmatg betwee the log-ormal ad gamma dstrbutos Fear ad Nebezahl 99 used the mamum lkelhood rato method dscrmatg betwee the Webull ad gamma dstrbutos Wes 999 dscussed the eect o choosg the wrog model through a real data eample ad by usg log-ormal ad gamma models Gupta ad Kudu 003 cosdered the lkelhood rato statstc or dscrmatg betwee Webull ad geeralzed epoetal dstrbutos Gupta ad Kudu 004 dscussed the problem o dscrmatg betwee the gamma ad geeralzed epoetal dstrbutos by usg mamzed lkelhood rato test Pascual 005 dscussed the eect o msspeccato o the mamum lkelhood estmates whe dscrmatg betwee the log-ormal ad gamma dstrbuto uctos Kudu ad Maglck 005 used the rato o the mamzed lkelhoods dscrmatg betwee log-ormal ad gamma dstrbutos Dey ad Kudu 009 cosdered the problem o dscrmato amog Webull log-ormal ad geeralzed epoetal dstrbutos They used the mamzed lkelhood test to choose the best tted model Dey ad Kudu 00 used the mamzed lkelhood rato test the dscrmato problem betwee log-ormal ad log-logstc dstrbutos Some procedures or selectg betwee dstrbutros or data o ot oly complete but also cesored have bee pad atteto by some authors Sswad ad Queseberry 98 whe selectg amog Webull log-ormal ad gamma dstrbutos compared the scale varat scale shape varat ad mamzed lkelhood ucto tests or complete Copyrght to IJIRST wwwjrsetcom 3358

2 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 data ad scale varat ad mamzed lkelhood ucto tests or Type-I cesored data Km ad Yum 008 compared the rato o mamzed lkelhoods ad scale varat tests or dscrmatg betwee the Webull ad logormal dstrbutos or complete Type-Ι ad Type II cesored data Dey ad Kudu 0 cosdered the mamzed lkelhood rato test choosg betwee Webull ad log-ormal dstrbutos or type-ii cesored data Webull ad log-logstc dstrbutos are two popular dstrbutos or aalyzg letme data I ths paper the problem o dscrmatg betwee these two dstrbuto uctos s cosdered A hypothess testg method s used whch t s assumed that a data are comg ether rom Webull or log-logstc dstrbuto The rato o the mamzed lkelhood test s used to dscrmate betwee them The asymptotc dstrbutos o the logarthm o the rato o the mamzed lkelhood are obtaed through two theorems These asymptotc results are used to estmate the probablty o correct selecto rom whch the mmum sample sze eeded to dscrmate betwee the two dstrbuto uctos or a user speced probablty o correct selecto s obtaed Two real data le are aalyzed to see how the proposed method works practce Fgures ad 3 show the dverse shape o the probablty desty ucto pd ad cumulatve dstrbuto ucto cd respectvely o Webull dstrbuto at η= ad β = Whle Fgures ad 4 show that o loglogstc dstrbuto at ε = ad σ = From these gures the closeess o the two pd ad cd uctos ca be easly vsualzed However some o the characterstcs o Webull ad log-logstc dstrbutos ca be qute deret Ths ca be show whe cosderg ther hazard uctos gve Fgures 5 ad 6 respectvely Thereore the data are comg rom ay oe o them may be t s modeled by the other oe I addto the sample sze s ot very large the problem o choosg the correct dstrbuto becomes more dcult but t s stll very mportat to make the best decso based o the data at had The rest o the paper s orgazed as ollows I Secto the test statstc s preseted I Secto 3 the asymptotc dstrbutos o the test statstc uder ull hypotheses s obtaed The mmum sample sze eeded to dscrmate betwee Webull ad log-logstc dstrbutos at a user speced protecto level ad tolerace level s determed I Secto 4 Two real le data sets are aalyzed Secto 5 Fally a cocluso s gve Secto 6 Fg Desty uctos o the Webull dstrbuto at η= ad β = Copyrght to IJIRST wwwjrsetcom 3359

3 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 Fg Desty uctos o the log-logstc dstrbuto at ε = ad σ = Fg 3 Cumulatve dstrbuto uctos o the Webull at η= ad β = Fg 4 Cumulatve dstrbuto uctos o the log-logstc at ε = ad σ = Copyrght to IJIRST wwwjrsetcom 3360

4 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 Copyrght to IJIRST wwwjrsetcom 336 Fg 5 Hazard uctos o the Webull at η= ad β = Fg 6 Hazard uctos o the log-logstc at ε = ad σ = II TH TST STATISTIC Suppose are depedet ad detcally dstrbuted radom varables rom ay oe o the Webull or loglogstc dstrbuto uctos A Webull dstrbuto deoted by ηβ wth scale parameter η > 0 ad shape parameter β > 0 has probablty desty ucto ; e >0 > 0 A log-logstc dstrbuto deoted by εσ wth scale parameter ε > 0 ad shape parameter σ > 0 has probablty desty ucto / / / ; > 0 > 0 The lkelhood uctos o ηβ ad εσ dstrbutos are gve as ep / respectvely The rato o the mamzed lkelhood RM s deed as

5 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 RM= ˆ ˆ ˆ ˆ Where ˆ ˆ ad ˆ ˆ o RM ca be obtaed as ollows T l RMl Where ˆ are the mamum lkelhood estmators o ad ˆ ˆ l ˆ ˆ ˆ ˆ l ˆ ˆ respectvely The logarthm ˆ ˆ ˆ l ˆ l l / ˆ 3 ˆ ˆ ˆ ˆ The Webull dstrbuto s chose : T >0 Otherwse the log-logstc dstrbuto s the preerred model The eact dstrbutos o T gve quato 3 uder the respectve paret dstrbutos are dcult to obta thereore the asymptotc dstrbutos o T uder the ull hypotheses must be obtaed It s kow that also the probablty o correct selecto PCS depeds o the paret dstrbuto That s the data are orgally comg rom Webull dstrbuto the probablty o correct selecto PCS s gve as PCS = P T > 0 / data ollow Webull dstrbuto Also the data are orgally comg rom log-logstc dstrbuto the probablty o correct selecto PCS s gve as PCS = P T < 0 /data ollow log-logstc dstrbuto Thereore the asymptotc dstrbutos o T ca be used to compute the appromate PCS III ASYMPTOTIC DISTRIBUTIONS OF TH TST STATISTIC UNDR NU HYPOTHSS I ths secto the asymptotc dstrbuto o T statstc uder ull hypothess s obtaed two deret cases Case : The data are comg rom a Webull dstrbuto ad the alteratve they are rom a log-logstc dstrbuto Case : The data are comg rom a log-logstc dstrbuto ad the alteratve they are rom a Webull dstrbuto The results are obtaed through two theorems The ollowg deto ad lemmas are eeded to ollow up the et theorems The proo o lemmas ollows usg smlar argumets as that o Gupta ad Kudu 003 Deto For ay Borel measurable ucto h [hu] ad [hu] deote mea ad varace o hu uder the assumpto that U ollows Smlarly dee [hu] ad [hu] as mea ad varace o hu uder the assumpto that U ollows Also g ad h are two Borel measurable uctos dee alog the same le : [guhu]= [guhu] [gu] [hu] ad smlarly [guhu]= [guhu] [gu] [hu] where U ollows ad respectvely I the ollowg as deote the almost sure covergece emma Uder the assumpto that the data are rom η β ad as as as where we have: Copyrght to IJIRST wwwjrsetcom 336

6 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 l ; ma l ; as as where l ; ma l ; et us deote T l T T s asymptotcally equvalet to T T emma Uder the assumpto that the data are rom εσ ad as l as as where ; ma l ; as l et us deote T l ; as where ma l ; T T s asymptotcally equvalet to T T we have Theorem Uder the assumpto that the data are rom a Webull dstrbuto the dstrbuto o T s appromately ormally dstrbuted wth mea T ad varace T where ad T AM l l l l[ ] 3 Copyrght to IJIRST wwwjrsetcom 3363

7 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 Copyrght to IJIRST wwwjrsetcom ] l[ 4 ] l[ l 4 l ] l[ 4 l A T Wth Proo et us assume that data pots are obtaed rom η β wth scale parameter η ad shape parameter β Now to obta ad as deed emma ad dee h l l l l l ] ; [l ] l[ l l l l 33 Where Deretatg quato33 wth respect to σ the equatg by zero we get: 0 l l l l l h 34 Also deretatg quato33 wth respect to equatg wth zero we get 35 Usg quato 35 quato 34 we get:

8 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 Copyrght to IJIRST wwwjrsetcom l l 36 ad ca be obtaed by solvg quatos 35 ad 36 From these equatos t s clear that ad are both uctos o η ad β To obta T ad T deote lm AM T ad lm A T Thereore or large we get ; l ; l AM T ] l[ l l l l l l l l The the result o quato 3 s obtaed Also or large we have ; l ; l A T l 4 l l 4 l l 4 l

9 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 The the result o quato 3 s obtaed Usg the cetral lmt theorem ad usg o emma oe ca easly shows that T T s asymptotcally ormally dstrbuted wth mea T ad varace T gve quatos 3 ad 3 respectvely Theorem Uder the assumpto that the data are rom log-logstc dstrbuto the dstrbuto o T s appromately ormally dstrbuted wth mea T ad varace T where ad T AM l l T A 4 Wth Y l Y 4 l Y l Y Y Y Y 37 l l Y Y l Y l Y 4 Y l Y 38 Proo et us assume that data pots are obtaed rom εσ wth scale parameter ε ad shape parameter σ Now to obta ad as deed emma ad dee g [l ; ] [l l l ] l l l Y l Y 39 Where Y Deretatg quato 39 wth respect to ad equatg by zero we get Y 30 Also deretatg quato 39 wth respect to ad equatg by zero we get: Copyrght to IJIRST wwwjrsetcom 3366

10 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology l Y Y 0 Y ISO 397: 007 Certed Orgazato ol Issue 8 August 03 ad ca be obtaed by solvg quatos 30 ad 3 From these equatos t s clear that both uctos o ad Now to obta T ad T deote lm T AM ad lm T A Thereore or large we get T AM l ; l l l l l l Y ; l Y l l The the result o quato 37 s obtaed Also or large we have T A [l ; l ; ] Y ad 3 are l 4 l l 4 l l 4 l The the result o quato 38 s obtaed Usg the cetral lmt theorem ad usg o emma oe ca easly shows that T * T * s asymptotcally ormally dstrbuted wth mea T ad varace T gve quatos 37 ad 38 respectvely I the et secto the mmum sample sze requred to dscrmate betwee Webull ad log-logstc dstrbuto s obtaed I order to do that AM A AM ad A are computed umercally usg quatos ad 38 wth Copyrght to IJIRST wwwjrsetcom 3367

11 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 Mathcad program Wthout lose o geeralty we take = ad = whle = ad = The results are gve Tables ad Table I Deret values o AM A ad at = ad = AM A Table II Deret values o AM A ad at =ad = AM A I DTRMINATION OF SAMP SI I ths secto a method to determe the mmum sample sze eeded to dscrmate betwee Webull ad loglogstc dstrbutos s proposed The same argumets as that gve Gupta ad Kudu 003 are ollowed It s kow that two dstrbuto uctos are very close oe eeds a very large sample sze to dscrmate betwee them Whle they are qute deret the oe may ot eed very large sample sze to dscrmate betwee them Also rom a practcal pot o vew oe may ot eed to deretate betwee two so closed dstrbuto uctos Thereore t s epected that the user wll specy beore had the mmum dstace D* that he does ot wat to make the dscrmato betwee two dstrbuto uctos ther dstace s less tha t Ths mmum dstace s called tolerace lmt Here the Kolmogrov-Smrov K-S dstace s used to measure the closeess betwee the Webull ad log-logstc dstrbutos Where the Kolmogrov-Smrov K S dstace betwee two dstrbuto uctos say F ad G s deed as supf G Also t s epected that the user wll specy beorehad the probablty o correct selecto PCS to acheve a certa protecto level P* Wth the help o K S dstace ad PCS the requred sample sze s obtaed as ollows Cosderg Case where t s assumed that the data are comg rom ηβ Copyrght to IJIRST wwwjrsetcom 3368

12 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 the rom Theorem T s appromately ormally dstrbuted wth mea T ad varace T I ths case the PCS s gve by PCS = PrT > 0 AM A Where Ф s the dstrbuto ucto o the stadard ormal radom varable AM ad A are gve quatos 3 ad 3 respectvely Thereore to determe the mmum sample sze requred to acheve at least P* protecto level solve or the equato AM P* A e P* A AM 4 Here * P s the 00 P* percetle pot o a stadard ormal dstrbuto The values o are obtaed ad reported Table 3 where For P*=07 η= ad = wth ad Copyrght to IJIRST wwwjrsetcom 3369 as gve Table Also the K-S dstaces betwee β ad are obtaed ad preseted Table 3 Cosderg Case where t s assumed that the data are comg rom ηβ the rom Theorem T s appromately ormally dstrbuted wth mea T ad varace T I ths case the PCS s gve by PCS = PrT < 0 AM A Where AM ad A are gve quatos 37 ad 38 respectvely Thereore to determe the mmum sample sze requred to acheve at least P* protecto level solve or the equato AM P * A e P* AM A 4 The values o are obtaed ad reported Table 4 where P*= 07 ε= ad σ= wth ad as gve Table Also the K-S dstaces betwee σ ad are obtaed ad preseted Table 4 From Tables 3 ad 4 t ca be see that or a gve PCS e p*=07 as β ad σ crease the sample sze decreases Also as the K-S dstace betwee the two dstrbutos creases the sample sze decreases as epected To resume oe kows the rage o the shape parameter o the ull dstrbuto ad or a gve PCS that acheves a certa protecto level P* the the mmum sample sze ca be obtaed by takg the mamum obtaed rom quatos 4 ad 4 But uortuately practce the shape parameter may be completely ukow; thereore the K-S dstaces ca replace the ukow parameters to take the decso That s or a gve protecto level P* ad a gve pre-speced tolerace lmt D* the mmum sample sze ca be obtaed by takg the mamum obtaed rom

13 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 quatos 4 4 For eample suppose that or a gve P*= 07 ad or β= 05 ad σ= 05 the rom Tables 3 ad 4 the mmum sample sze requred to dscrmate betwee Webull ad log-logstc dstrbutos s ma33 84 =33 O the other had β ad σ are ukow ad suppose that the practtoer wats to dscrmate betwee a Webull ad a log-logstc dstrbuto uctos oly whe the dstace betwee them s greater tha or equal to 080 e D* 080 ad wth P* = 07 The rom Tables 3 ad 4 t s clear that D* 080 β 05 ad σ 3 Also whe the ull dstrbuto s Webull the or the tolerace lmt D* 080 oe eeds =33 to meet the PCS P*= 07 Smlarly whe the ull dstrbuto s log-logstc the oe eeds =9 to meet the same protecto level Fally the mmum sample sze requred to dscrmate betwee Webull ad log-logstc dstrbutos wth P*= 07 ad D* 080 s ma33 9 = 33 Table 3 The sample sze ad the K-S dstace or P*=07 η= = whe the ull hypothess s Webull dstrbuto β K-S Table 4 The sample sze ad the K-S dstace or P*=07 ε= σ= whe the ull hypothess s log-logstc dstrbuto σ K-S Notce that Tables 3 ad 4 are obtaed or the protecto level 07 but or other protecto levels the tables ca be easly moded For eample we eed a sample sze correspodg to protecto level P*=09 the all the etres correspodg to the row o must be multpled by 0 9 / 07 DATA ANAYSIS For llustratve purposes two real data sets to dscrmate betwee the Webull ad log-logstc dstrbuto uctos are aalyzed Data Set : The rst data set Gupta ad kudu003 represet the alure tmes o 30 ar codtos o a arplae hours: Whe the Webull dstrbuto s used the Ms o the deret parameters are: =0087 ad = Also l[ ] = Smlarly whe the log-logstc dstrbuto s used the Ms o the deret parameters are: =05 ad =66693 Also l[ ]= Cosequetly T= Thereore by usg the mamum lkelhood rato test to dscrmate betwee Webull ad log-logstc dstrbutos the Webull model s chose or ths data set Data Set : The secod data set Gupta ad kudu 003 represet the umber o mllo revolutos beore alure or each o 3 ball beargs the le test ad they are: Whe the Webull model s used the Ms o ad are: Copyrght to IJIRST wwwjrsetcom 3370 = 007 ad = 0490 Also l[ ]= Smlarly the log-logstc model s used the Ms o ε ad σ parameters are: = Also l[ = ad ]= Cosequetly T= -038 Thereore by usg the mamum lkelhood rato test to dscrmate betwee Webull ad log-logstc dstrbutos the log-logstc model s chose or ths data set

14 ISSN: Iteratoal Joural o Iovatve Research Scece geerg ad Techology ISO 397: 007 Certed Orgazato ol Issue 8 August 03 I CONCUSION I ths paper we cosder the problem o dscrmatg betwee Webull ad log-logstc dstrbuto uctos It s assumed that a data are comg ether rom Webull or log-logstc dstrbuto The mamzed lkelhood rato test to dscrmate betwee them s used The asymptotc dstrbutos o the logarthm o the rato o the mamzed lkelhood are obtaed These asymptotc results are used to estmate the probablty o correct selecto The mmum sample sze eeded to dscrmate betwee the two dstrbuto uctos or a user speced probablty o correct selecto ad a tolerace lmt based o the dstace betwee the two dstrbutos s calculated Two real data le are aalyzed to see how the proposed method works practce RFRNCS [] Ba J ad gelhardt M Probablty o Correct Selecto o Webull versus Gamma based o kelhood Rato Commucatos Statstcs Theory ad Methods ol 9 pp [] Che W W O the Tests o Separate Famles o Hypotheses wth Small Sample Sze Joural o Statstcal Computatos ad Smulatos vol [3] Co D R Tests o Separate Famles o Hypotheses Proceedg o the Fourth Berkely Symposum Mathematcal Statstcs ad Probablty Berkely Uversty o Calora Press pp [4] Co D R Further Results o Tests o Separate Famles o Hypotheses Joural o the Royal Statstcal Socety Seres B ol 4 pp [5] Dey A K ad Kudu D K Dscrmatg amog the og-ormal Webull ad Geeralzed poetal Dstrbutos I Trasactos o Relablty ol 58 o 3 pp [6] Dey A K ad Kudu D K Dscrmatg betwee the og-normal ad og-ogstc Dstrbutos Commucatos Statstcs Theory ad Methods ol 39 pp [7] Dey A K ad Kudu D K Dscrmatg betwee the Webull ad og-normal Dstrbutos or Type-II Cesored Data Statstcs ol 46 o pp [8] Dumoceau R ad Atle C Dscrmatg betwee the og-normal ad Webull Dstrbuto Techometrcs ol54 pp [9] Fear D H ad Nebezahl O the Mamum lkelhood Rato Method o Decdg Betwee the Webull ad Gamma Dstrbutos Commucatos Statstcs Theory ad Methods ol [0] Frth D Multplcatve rrors: og-normal or Gamma? Joural o the Royal Statstcal Socety Seres B pp [] Gupta R D ad Kudu D K Dscrmatg betwee Webull ad Geeralzed poetal Dstrbutos Joural o Computatoal Statstcs ad Data Aalyss ol 43 pp [] Gupta R D ad Kudu D K Dscrmatg betwee the Gamma ad Geeralzed poetal Dstrbutos Joural o Statstcal Computato ad Smulato ol 74 pp [3] Jackso O A Y Some results o tests separate amles o hypotheses Bometrka ol 55 pp [4] Kappema R F O a Method or Selectg a Dstrbutoal Model Commucato Statstcs Theory ad Methods ol pp [5] Km J S ad Yum B-J Selecto betwee Webull ad log-ormal dstrbutos: a comparatve smulato study Computatoal Statstcs ad Data Aalyss ol 53 pp [6] Kudu D K ad Maglck A Dscrmatg betwee Webull ad og-normal Dstrbutos Naval Research ogstcs NR ol 5 Issue 6 pp [7] Kudu D K ad Maglck A Dscrmatg betwee the og-normal ad Gamma Dstrbutos Joural o the Appled Statstcal Sceces ol 4 pp [8] awless J F Statstcal Models ad Methods or etme Data Joh Wley ad Sos New York 98 [9] Pascual F G Mamum kelhood stmato uder Msspeced og-normal ad Webull Dstrbutos Commucatos Statstcs Smulato ad Computatos ol 34 pp [0] Perera B de B A Note o the Cosstecy ad o the Fte Sample Comparsos o Some Tests o Separate Famles o Hypotheses Bometrka ol 64 pp [] Sswad ad Queseberry C P Selectg amog Webull og-normal ad Gamma Dstrbutos usg Complete ad Cesored Samples Naval Research ogstcs Quarterly ol 9 4 pp [] Stephes M A DF statstcs or goodess o t ad some comparsos Joural o the Amerca Statstcal Assocato ol69 pp [3] Wes B Whe og-normal ad Gamma Models Gve Deret Results: A Case Study Amerca Statstca ol 53 pp Copyrght to IJIRST wwwjrsetcom 337