A MODIFIED CHI-SQUARED GOODNESS-OF-FIT TEST FOR THE KUMARASWAMY GENERALIZED INVERSE WEIBULL DISTRIBUTION AND ITS APPLICATIONS

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1 Jourl of Sscs: Advces Theory d Applcos Volume 6 Number 06 Pges Avlble hp://scefcdvces.co. DOI: hp://dx.do.org/0.864/js_ A MODIFIED CHI-SQUARED GOODNESS-OF-FIT TEST FOR THE KUMARASWAMY GENERALIZED INVERSE WEIBULL DISTRIBUTION AND ITS APPLICATIONS HAFIDA GOUAL d NACIRA SEDDIK-AMEUR Lborory of Probbly d Sscs Uversy of Bdj Mokhr Ab Alger e-ml: hfd-06@homl.fr crseddk@yhoo.fr Absrc I hs pper we hve proposed d sudy ew fve prmeer geerlzed verse Webull model h s bsed o he cumulve dsrbuo fuco of Kumrswmy [4] dsrbuo. The mporce of hs model les s bly o model moooe d o-moooe flure re fucos whch re que commo o lfeme d lyss d relbly. We prese ew goodess-of-f es proposed by Bgdovcus d Nkul [] for he Kumrswmy geerlzed verse Webull model he cse of cesored d. The mehod of mxmum lkelhood s used o esme he model prmeers. We llusre he model d he proposed es by pplcos o wo rel d ses. 00 Mhemcs Subjec Clssfco: 60E5 6E0 6F 6G0 6N05. Keywords d phrses: cesored d geerlzed verse Webull dsrbuo formo mrx Kumrswmy dsrbuo mxmum lkelhood esmo Nkul Ro Robso ssc. Receved December Scefc Advces Publshers

2 76 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR. Iroduco I he survvl d relbly lyss he observed d re frequely complee (cesorg mes). Therefore s ecessry o cosder es of he cesored flure mes d. I 0 Bgdovcus d Nkul developed he de of Akrs [] of comprg observed d expeced umbers of flures me ervls. The choce of rdom groupg ervls used re cosdered s d fucos roducg modfed ch-squred es Y whch s well dped o cesored flure me d. To gve more objecve ssessmes of he seleced model of f he Y ssc s bsed o he mxmum lkelhood esmo (MLE) d he Fsher mrx whch mesure he formo bou he prmeers coed he model chose. I he rece yers severl crero sscs d her pplcos re cosdered such s Hghgh d Nkul [3] Gervlle-Réche e l. [7] Voov e l. [3] Bgdovcus e l. [4] Nkul d Tr [4] Goul d Seddk-Ameur [9]. We roduce hs pper he geerlzed verse Webull (GIW) model bsed o he cumulve dsrbuo fuco of Kumrswmy [4] d clled Kumrswmy geerlzed verse Webull (Kum-GIW) dsrbuo. Tll ow he Kumrswmy dsrbuo hs bee fdg dffere res of pplcos such s survvl lyss relbly bologcl d hydrologcl d Sulo e l. [9] Psco e l. [6] Cordero e l. [5] Ndrjh e l. [8] Cordero e l. [6] Shhbz e l. [30]. O he oher hd he geerlzed verse Webull (GIW) dsrbuo proposed by de Gusmão e l. [7] s flexble model whch c descrbe d predc he flure mes of much rel sysems. The mporce of he ew Kum-GIW dsrbuo les s bly o model mooocy o-mooocy d shpe flure re fucos whch re que commo o lfeme d lyss d relbly. The

3 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 77 purpose of hs pper s o cosruc d lyze he geerlzed Nkul- Ro-Robso goodess-of-f ssc es Y (Bgdovcus d Nkul [] Bgdovcus d Nkul [3]) for he Kumrswmy geerlzed verse Webull (Kum-GIW) dsrbuo boh of complee d cesored d. We roduce he Kum-GIW dsrbuo d we dscuss he shpes of he hzrd re fuco Seco. I Seco 3 we clcule he mxmum lkelhood esmors of he Kum-GIW prmeers he cse of cesored d. We defe Seco 4 he ew goodess-of-f ssc es Y d he vldo of our ew model s vesged Seco 5. Flly he mporce of he proposed model s llusred by wo rel d ses Seco 6.. Kumrswmy Geerlzed Iverse Webull Dsrbuo The geerlzed verse Webull (GIW) dsrbuo s proposed by Gusmão e l. [7]. Ths dsrbuo rses s rcble lfeme model curl sceces lfe esg d relbly. The rdom vrble T follows he geerlzed verse Webull dsrbuo f s cumulve dsrbuo fuco s gve by FGIW ( ) exp α T θ γ θ ( α γ) > 0 > 0. Is probbly desy fuco s ( ) ( ) + exp α f GIW θ γα γ > 0 θ > 0. We propose hs pper exeso of he geerlzed verse Webull dsrbuo bsed o he fmly of geerlzed Kumrswmy dsrbuos (deoed Kum-G) roduced by Cordero d de Csro [6] d Ndrjh e l. [8]. The Kumrswmy (Kum) dsrbuo s o very commo o sscs d hs bee lle explored he

4 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR 78 sscl lerure. Is cumulve dsrbuo fuco s ( ) ( ) b x x F for 0 < < x d where 0 > d 0 > b re shpe prmeers d s desy fuco hs smple form ( ) ( ). b x bx x f A rdom vrble T follows Kumrswmy geerlzed verse Webull (Kum-GIW) dsrbuo f he cumulve dsrbuo fuco s gve s ( ) ( ) 0 0 exp > γ α θ > α γ θ T b b F where > > > b re he shpe prmeers 0 > α s he scle prmeer d 0 > γ s he shf prmeer. Is survvl fuco s ( ) ( ). 0 0 exp > γ α θ > α γ θ T b b S The probbly desy fuco hzrd d cumulve hzrd fucos of he Kum-GIW dsrbuo re ( ) ( ) exp exp + α γ α γ γα θ b b f ( ) 0 > γ α θ T b ( ) ( ) ( ) θ θ θ λ S f ( ) 0 0 exp exp > θ > α γ α γ γα + b ( ) ( ) 0 0 exp l > γ α θ > α γ θ Λ T b b respecvely.

5 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 79 Where he cumulve dsrbuo fuco of geerlzed b Kumrswmy dsrbuo s FKum ( θ ) [ { G() } ] ( G( ) s he cumulve dsrbuo fuco of he geerlzed verse Webull dsrbuo). The grphs of hzrd re fucos re show below (Fgure ) for vrous choces of he prmeers. The flexbly of he Kum-GIW dsrbuos s show her hzrd fuco plos (Fgure ) wh eresg properes: If 0 < d < γ < 4 he s hzrd re hs cup shpe ( -shpe) form. If >. d 0 < γ he s hzrd re s decresg o 0 whch c chrcerze he sysems h mproves (bur- phse or youh bur f morly). If γ 4 he he hzrd re s cresg hs suo chrcerzes sysem whch deerores (phse obsolescece or geg).

6 80 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR

7 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 8 Fgure. Dffere shpes of hzrd re fuco of he Kumrswmy geerlzed verse Webull (Kum-GIW) dsrbuo. 3. Esmg he Prmeers of he Kum-GIW Dsrbuo wh Cesored D Le T be rdom vrble dsrbued wh he vecor of prmeers T θ ( b α γ). Suppose h T re flure mes o-egve d depede d he probbly desy fuco of T belogs o prmerc fmly (Kum-GIW our cse). The cesorg mes C re lso o-egve d ssumed o be rdom smple. Suppose h he d coss of depede observos m( T C ) for. The rgh cesorg s o formve ( C does o deped o θ ). So hs cse we ob he followg expressos of he lkelhood fucos:

8 8 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR δ δ L( θ) f ( θ) S ( θ) δ λ ( θ) S( θ) δ { }. T C The log-lkelhood fuco for he Kum-GIW model s gve by α θ γ ( ) δ [ l + l b + l γ + l + l α ( + ) l( ) l l exp ] l α α + exp γ γ b l( θ) m l + m l b + m l γ + m l + m l α ( + ) l( ) mγα F F ( ) l exp l α α + b exp + γ γ b F C where F d C re he ses of complee d cesored observos respecvely m s he umber of flures. The mxmum lkelhood esmos (MLE s) θ b α γ of ( ) he prmeers re he soluo of he o-ler sysem of score T ( ). fucos ( θ) ( θ) ( θ) ( θ) ( θ) 0 (For more dels see he b α γ 5 Appedx). 4. Goodess-of-f Tes for Rgh Cesored D For esg he goodess-of-f of prmerc fmly of survvl dsrbuo from rgh cesored d Hbb d Thoms [] Hollder d Pe [] cosdered url modfcos of he Nkul-Ro- Robso (NRR) ssc for d whou covres. Also Hjor [3] Hollder d Pe [] cosdered goodess-of-f for prmerc Cox models Gry d Perce [0] Akrs [] d Zhg [33] for ler regresso models. T

9 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 83 Bgdovcus d Nkul [3] gve ch-squred ype goodess-of-fes for cesored d wh possbly me depede covres d cosdered rdom groupg ervls s d fucos. These ess re bsed o mxmum lkelhood esmos for ugrouped d d rdom groupg ervls re cosdered s d fucos. We dp hs es for Kumrswmy geerlzed verse Webull model. Le us cosder he hypohess H0 : s F ( x) F { F ( x θ) x R θ Θ R } 0 0 T s where θ { θ θs} Θ R s ukow s-dmesol vecor prmeer d F 0 s kow dsrbuo fuco. Le us cosder fe me ervl oly sy [ 0 τ] d dvde o k > s smller ervls I ( ] where j j j 0 < 0 < < k < k +. I hs cse he esmed j s gve by ( ( () ˆ )) ( ) j Λ Ej Λ X l θ + θ X( ) j k k l where θ s he mxmum lkelhood esmor of he prmeer θ Λ s he verse of he cumulve hzrd fuco Λ X () s he -h eleme he ordered sscs ( X () X ( )) d E ( + ) Λ ( θ ) + ( X() ); l l Λ θ j re rdom d fucos such s he k ervls chose hve equl expeced umbers of flures e j. The es s bsed o he vecor j j Z T ( Z Zk ) Z j ( U j e j ) j k; U j represe he umbers of observed flures hese ervls.

10 84 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR Uder he hypohess H 0 for prmerc models wh survvl fucos d hzrd res bsoluely couous d o formve cesorg Bgdovcus e l. [4] (Theorems d ) showed h he lm dsrbuo of he ssc es ( ) T Uj ej Y Z Z k + Q U j where s geerlzed verse of he esmed covrce mrx of he vecor Z d T A + A C G CA G I CA C T Q W G W Aj Uj Uj δ X : I k ˆ s ll s s ll ll lj lj j j ( T ) [ ] W W W G g g C C A ( ) l λ( ) l ( θ λ θ ) Clj l ll δ λ θ δ θ θ θ X : Ij k W C A Z l l s l lj j j j s ch-squre wh r rk( ) degrees of freedom. The hypohess s rejeced wh pproxme sgfcce level α f Y > χ α ( r) where χ α ( r) s he qule of ch-squre wh r degrees of freedom. For more dels see Bgdovcus d Nkul [] Bgdovcus d Nkul [3]. j j l T l

11 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT Vldy of he Kum-GIW Model Cesorg Cse The choce of j; j k he cse of Kum-GIW model s obed s follows: j bl exp γα Ej + b l ( exp{ γα X ()}) ( + ) l l k X ( ). To ob he explc form of he modfed ch-squred Y ssc for he Kum-GIW model we mus clcule he mrx C [ c lj] 55 d he Fsher formo mrx ll. (For more dels see he 55 Appedx). 5.. The mrx C The elemes of he esmed mrx C wh C δ : I l λ( θ) l m d j k θ l α α γ M ( ) γ θ C j δ M ( ) : I θ j j C j δ b : Ij γαα γα M ( θ) C3 j δ α M ( ) : I θ j

12 86 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR α α α α l( ) γ l l M ( ) γ θ C4 j δ M ( ) : I θ j α α γ M ( ) θ C5 j δ. γ M ( ) : I θ j So we c clcule he esmed mrx w defed s k w l CljAj Zj l 5 j k. j The we ob he ssc Y for he model of Kumrswmy geerlzed verse Webull s follow: Y k k ( Uj e ) T j + W ll CljCl j Aj W l l 5. U j j j 6. Smulo Sudy 6.. Kum-GIW s prmeers esmo I order o evlue he performce of he mxmu lkelhood mehod of esmg he Kum-GIW prmeers complee d cesored cse we use he R sscl sofwre d Brzl-Borme (BB) lgorhms (Rv e l. [8]). We gve he me smuled mxmum lkelhood esmors vlues b α γ d her me squre errors Tble. Seve smple szes ( d 500) re cosdered. The d were smuled N wh he followg vlues of he prmeers:.5 b.5 α.5 γ 3.

13 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 87 Tble. Me smuled mxmum lkelhood esmors vlue d her me squre errors N SME b SME α SME SME γ SME From Tble we c oe h he mxmum lkelhood mehod of esmg he Kum-GIW prmeers s well perform. The smuled verge bsolue errors of MLE θ ( b α γ ) versus her rue vlue c be fd Tble. For ech smuled smple we compue he verge bsolue of errors s he ol of bsolue dffere of he MLE s gs he rue vlue of he umber of rus N 0000 mes. T

14 Tble. Smule verge bsolue errors of MLE θ versus her rue vlue θ (.5 b.5 α.5 γ 3) T whe d s compleed ( p 0) d he rgh cesorg probbly p 0% 0.5 b b α α γ γ p 0% p 0% p 0% p 0% p 0% p 0% p 0% p 0% p 0% p 0%

15 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT Tble show h ll smuled MLE coverge fser h d hece cofrm he heorem esblshed by Fsher [8] so we demosre h he MLE of Kum-GIW dsrbuo s - cosse he cses of complee d cesored d. 6.. Smuled dsrbuo of Y ssc for Kum-GIW model We es he hypohess H 0 h he ssc Y follows chsqured dsrbuo wh k degrees of freedom. For h we ru 0000 smulos of smples d (smple szes d 500) from Kum-GIW dsrbuo. We group Tble 3 he umber of rejeco s cses of he hypohess H 0 whe Y > χ α ( k ) wh α ( α % α 5% α 0% ) sgfcce levels. Tble 3. Smuled levels of sgfcce for Y ( θ ) es gs her heorecl vlues ( α ) N 0000 α 0. 0 α α Oe c observe h heorecl levels of he ch-squred dsrbuos wh k degrees of freedom mches wh he smuled levels of sgfcce for Y es. Hece we c sy h he es proposed c djus d from Kumrswmy geerlzed verse Webull dsrbuos ccepble mer. For demosrg h he Y ssc follows he lm; chsqured dsrbuo wh k degrees of freedom; we compue N 0000 mes he smuled dsrbuo of Y ( θ ) uder he ull hypohess H 0

16 90 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR wh dffere vlues of prmeers Kum -GIW ( b α γ); for exmple θ (.5 b.5 α.5 γ 3 ) θ (.5 b.5 α.5 γ 3 ) θ (.5 b.5 α.5.5 γ 3 ) θ (.5 b.5 α.5 γ 3 ) θ (.5 b.5 α 0.5 γ 3 ) ( ) θ.5 b.5 α.5 γ.5 d r 3 ervls versus he ch-squred dsrbuo wh k r degree of freedom. Ther hsogrms re represeed Fgure versus he ch-squred dsrbuo wh k degree of freedom. As c see from Fgure oe c observe h he sscl dsrbuo of Y wh dffere vlues of prmeers d dffere umbers k of groupg cells; he lm follows ch-squred wh r k degrees of freedom wh he sscl errors of smulo. We ob he sme resuls for dffere vlue of prmeers d dffere umber of equprobble groupg ervls. I s mes h he lmg dsrbuo of he geerlzed ch-squred Y ssc s dsrbuo free. Ths resul lso esblshed he heorem of Nkul [7 8 9].

17 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 9

18 9 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR

19 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 93 Fgure. Smuled dsrbuo of he Y ssc uder he ull hypohess H 0 wh dffere prmeers of θ versus he ch-squred dsrbuo wh degrees of freedom wh 50 N Applco o rel d To show he pplco of he Kumrswmy geerlzed verse Webull model o survvl d relbly d he cse of cesored d we use wo rel lfe d ses by usg Y ssc.

20 94 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR 6.4. Survvl lyss d cse Pke [7] gve some d from lborory vesgo whch he vgs of rs were ped wh he crcoge DMBA d he umber of dys T ul crcom ppered ws recorded. The d below re for group of 9 rs (Group Pke s pper); he wo observos wh sersks re cesorg mes * 44*. These d re lyzed by Nkul d Tr [5]; where hey suggesed h he probbly plos for wo prmeers Webull dsrbuos he he d re bes fg wh geerlzed Brbum- Suders logsc d he geerlzed Brbum-Suders cuchy dsrbuos. We cosder he hypohese h hese d follow Kum-GIW dsrbuo. Choosg k 6 groupg ervls we fd he vlues of he mxmum lkelhood esmors prmeers: 0.57 b.3884 α γ The vlues of j he frequecy vecor Z re gve he followg ble: j j U J e j Z j The resul of he elemes of he Y ssc re W [ ] d he esmed Fsher s formo mrx I 55 s

21 I d G

22 96 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR By usg he NRR ssc for fve prmeers Kumrswmy geerlzed verse Webull dsrbuo we ob he vlue Y Sce Y.9589 < χ ( 6). 596 (he crcl vlue) we c coclude h he Kumrswmy geerlzed verse Webull (Kum-GIW) model s cocordce wh he ppered me of he crcom of Pke Relbly d (cycles-o-flure of cyldrcl seel specmes) Cycles-o-flure of cyldrcl seel specmes of 4% C el for 30 m 870 C; fer surfce fsh wh corse emery esed by Oo s Rory Uform Bedg Teser 000rpm wh ± 35.6kg/mm. Ths d s obed from Lwless [6] The MLE s of he prmeers geerlzed verse Webull (Kum-GIW) model re b α γ of he Kumrswmy b 0.87 α γ If we choose k 8 ervls he followg ble gves he resuls j j U J e j Z j

23 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 97 The sscl fereces for esg he ull hypoheses h cue cycles-o-flure of cyldrcl seel specmes belog he Kum-GIW model re gve s follows: W [ ] d he esmed Fsher's formo mrx s

24 I

25 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 99 We ob X d Q so he vlue of he ssc es s Y Sce Y 5.09 < χ ( 8) (he crcl vlue) he hypohess H 0 of he Kumrswmy geerlzed verse Webull (Kum-GIW) dsrbuo s well cceped he sgfcce level α Refereces [] M. G. Akrs Perso-ype goodess-of-f ess: The uvre cse Jourl of he Amerc Sscl Assoco 83 (988) -30. [] V. Bgdovcus d M. Nkul Ch-squred ess for geerl compose hypoheses from cesored smples Compes Redus Mhemque 349(3) (0) 9-3. [3] V. Bgdovcus d M. Nkul Ch-squred goodess-of-f es for rgh cesored d Ierol Jourl of Appled Mhemcs d Sscs 4(SI-A) (0) [4] V. Bgdovcus e l. No-prmerc Tess for Cesored D Wley New York 03. [5] G. M. Cordero E. M. M. Oreg d S. Ndrjh The Kumrswmy Webull dsrbuo wh pplco o flure d J. Frkl Is. 349 (00) [6] G. M. Cordero d M. de Csro A ew fmly of geerlzed dsrbuos Jourl of Sscl Compuo d Smulo 8 (0) [7] Felpe R. S. de Gusmão M. Edw M. Oreg Guss d M. Cordero The geerlzed verse Webull dsrbuo S. Ppers 5(3) (009) [8] R. A. Fsher d ohers O propery coecg he χ mesure of dscrepcy wh he mehod of mxmum lkelhood Jourl of he Royl Sscl Socey 87 (98) [9] H. Goul d N. Seddk-Ameur Ch-squred ype es for he AFT-geerlzed verse Webull dsrbuo Commucos Sscs-Theory d Mehods 43(3) (04) [0] Rober J. Gry d Dold A. Perce Goodess-of-f ess for cesored survvl d A. Ss. 3() (985) [] M. G. Hbb d D. R. Thoms Ch-squre goodess-f-f ess for rdomly cesored d The Als of Sscs 4() (986) [] M. Hollder d E. Pe Ch-squre goodess-of-f es for rdomly cesored d JASA 87(48) (99)

26 300 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR [3] N. L. Hjor Goodess of f ess models for lfe hsory d bsed o cumulve hzrd res A. Ss. 8 (990) -58. [4] P. Kumrswmy Geerlzed probbly desy-fuco for double-bouded rdom-processes Jourl of Hydrology 46 (980) [5] J. F. Lwless Sscl Models d Mehods for Lfeme D (Secod Edo) Wley New York 003. [6] J. F. Lwless Relbly Lfe Tesg d he Predco of Servce Lves: For Egeers d Scess by Sm C. Suders Sprger Seres Sscs 75() (007) 60. [7] L. Gervlle-Réche M. S. Nkul R. Thr e l. O relbly pproch d sscl ferece demogrphy Jourl of Relbly d Sscl Sudes 5 (0) [8] S. Ndrjh G. M. Cordero d E. M. M. Oreg Geerl resuls for he Kumrswmy-G dsrbuo Jourl of Sscl Compuo d Smulo 8(7) (0) [9] M. S. Nkul L. Gervlle-Réche d X. Q. Tr O ch-squred goodess-of-f es for ormly I V. Couller L. Gervlle-Réche H. Huber-Crol N. Lmos d M. Mesbh Edors Sscl Models d Mehods for Relbly d Survvl Alyss Wley Ole Lbrry New York (03) 3-7. [0] M. S. Nkul Ch-squred es for ormly proceedgs of he Ierol Vlus Coferece o Probbly Theory d Mhemcl Sscs (973) 9-. [] M. S. Nkul Ch-squred es for couous dsrbuos wh shf d scl prmeers Theory of Probbly d s Applcos 8 (973b) [] M. S. Nkul O ch-squred es for couous dsrbuos Theory of Probbly d s Applcos 9 (973c) [3] M. Nkul d F. Hghgh A ch-squred es for he geerlzed power webull fmly for he hed-d-eck ccer cesored d Jourl of Mhemcl Sceces 33(3) (006) [4] M. S. Nkul d X. Q. Tr O ch-squred esg ccelered rls Ierol Jourl of Performbly Egeerg 0() (04) [5] M. S. Nkul d X. Q. Tr Ch-squred goodess-of-f ess for geerlzed Brbum-Suders models for rgh cesored d d s relbly pplcos Jourl of Relbly: Theory & Applco Elecroc Jourl of Ierol Groups o Relbly 8(#) (03) 7-0. [6] M. A. R. Psco E. M. M. Oreg d G. M. Cordero The Kumrswmy geerlzed gmm dsrbuo wh pplco survvl lyss Sscl Mehodology 8 (0) [7] M. C. Pke A mehod of lyss of cer clss of expermes crcogeess Bomercs (966) 4-6.

27 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 30 [8] V. Rv d D. Glber Pul BB: A R pckge for solvg lrge sysem of oler equos d for opmzg hgh-dmesol oler objecve fuco Jourl of Sscl Sofwre 3(04) (009) -6. [9] H. Sulo J. Leo d M. Bourgugo The Kumrswmy brbum-sders dsrbuo J. S. Theory Prc. 6 (0) [30] M. Q. Shhbz S. Shhbz d N. S. Bu The Kumrswmy verse-webull dsrbuo Pks J. Ss. Oper. Res. 8 (0) [3] V. Voov M. S. Nkul d N. Blkrsh Ch-Squred Goodess of F Tess wh Applcos Acdemc Press New York 03. [3] V. Voov N. Py N. Shpkov d Y. Voov Goodess-of-f ess for he power geerlzed Webull probbly dsrbuo Commucos Sscs- Smulo d Compuo 4(5) (03) [33] B. Zhg A ch-squred goodess-of-f es for logsc regresso models bsed o csecorol d Bomerk 86(3) (999) g

28 30 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR Appedx The score fucos re l ( θ) l( θ) r α γ M ( θ) γα + ( ) r b M ( θ) F F l ( θ) b l( θ) b r b + F + b C α γ M ( θ) M ( θ) l( M ( θ) ) + l( M ( θ) ) C l ( θ) α l( θ) α r α rγα γα M ( θ) + ( b ) M ( θ) F F + b C γα M ( θ) M ( θ) l ( θ) l( θ) r F F l( ) rγα ( l( α) l( )) + γα [ l( α) l( )] M ( θ) ( b ) M ( θ) F γα [ l( α) l( )] M ( θ) + b M ( θ) C l ( θ) γ l( θ) r rα γ γ F + ( b ) F α M ( θ) + b M ( θ) C α M ( θ) M ( θ) α where M ( θ) exp γ.

29 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 303 Fsher formo mrx The elemes of he Fsher formo mrx l λ( θ) l λ( θ) ll whll δ θ θ l l re α α γ M ( ) γ θ δ M ( ) θ α α γ M ( ) γ θ δ b M ( ) θ α α γ M ( ) γ θ γαα γα M ( θ) 3 δ M ( ) ( ) θ α M θ α α γ M ( ) γ θ 4 δ M ( ) θ α α α α l( ) γ l l M ( ) γ θ M ( θ) α α α α γ M ( ) M ( ) γ θ γ θ 5 δ M ( ) ( ) θ γ M θ

30 304 HAFIDA GOUAL d NACIRA SEDDIK-AMEUR δ b γαα γα M ( θ) 3 δ b α M ( ) θ α α α α l( ) γ l l M ( ) γ θ 4 δ b M ( ) θ α α γ M ( ) θ 5 δ b γ M ( ) θ γαα γα M ( θ) 33 δ α M ( ) θ γαα γα M ( θ) 34 δ α M ( ) θ α α α α l( ) γ l l M ( ) γ θ M ( θ) α α ( ) ( ) γ M θ γαα γα M θ 35 δ M ( ) ( ) M α θ γ θ α α α α l( ) γ l l M ( ) γ θ 44 δ M ( ) θ

31 A MODIFIED CHI-SQUARED GOODNESS-OF-FIT 305 α α α α l( ) γ l l M ( ) γ θ 45 δ M ( ) θ α α γ M ( ) θ γ M ( θ) α α γ M ( ) θ 55 δ γ M ( ) θ.