Transmuted Generalized Lindley Distribution

Transcription

1 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 Tsmuted Geelzed Ldley Dstbuto M. Elghy, M.Rshed d A.W.Shwk 3, Buydh colleges, Deptmet of Memtcl Sttstcs, KSA.,, 3 Isttute of Sttstcl Studes d Resech, Deptmet of Memtcl Sttstcs, Co Uvesty E-ml: m_elghy85@yhoo.com, e-ml: m.shed3030@gml.com, e-ml: Ahmed3484@hotml. Com. Abstct:The Ldley dstbuto s oe of e mpott fo studyg stess-steg elblty modelg. Besdes, some eseches hve poposed ew clssesof dstbutos bsed o modfctos of e qus Ldley dstbuto. I s ppe, ew geelzed veso of s dstbuto whch s clled e tsmutedgeelzed Ldley () dstbuto s toduced. A compehesve memtcl tetmet of e dstbuto s povded. We deve e momet d momet geetg fucto s dstbuto. Moeove, we dscuss e mxmum lkelhood estmto of s dstbuto. Keywods: Geelzed Ldley dstbuto; Suvvl fucto; Momets. Mxmum lkelhood estmto. - Itoducto d Motvto Geelzed Ldley dstbuto w pmetes pobblty desty fucto (p.d.f) s gve by x d s toduced by Ndjh et l. (0) ts x x g( x,, ) x e e ; x 0, 0,. (.) The cumultve dstbuto fucto (cdf) of GLD s obted s x x G( x,, ) e, x 0, 0,. (.).. Tsmutto Mp I s subsecto we demostte tsmuted pobblty dstbuto. Let F d F be e cumultve dstbuto fuctos, of two dstbutos w commo smple spce. The geel k tsmutto s gve (007) s defed s GR ( u) F ( F ( u)) d GR( u) F ( F ( u)). Note t e vese cumultve dstbuto fucto lso kow s qutle fucto s defed s F ( y) f x RF ( x) yfo y 0,. The fuctos G ( u R ) d G ( u R ) bo mp e ut tevl I 0, to tself, d ude sutble ssumptos e mutul veses d ey stsfy G Rj ( 0) 0 d G Rj ( 0). A qudtc Rk Tsmutto Mp (QRTM) s defed s G ( u) u u( u),, (.3) R fom whch t follows t e cdf's stsfy e eltoshp F ( x) ( ) F( x) F( x) (.4) whch o dffeetto yelds, f ( x ) f ( x ) ( ) F ( x ) (.5) whee f ( x ) d f ( x) e e coespodg pdfs ssocted w cdf F ( x ) d F ( x) espectvely. A extesve fomto bout e qudtc k tsmutto mp s gve Shw d Buckely. (007). Obseve t t 0 we hve e dstbuto of e bse dom vble. The followg Lemm poved ISSN: Pge 45

2 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 t e fucto f ( ) gve (.5) stsfes e popety of pobblty desty fucto. x Lemm: f ( ) gve (.5) s well defed pobblty desty fucto. x Poof: Rewtg f ( x ) s f ( x) ( ) ( ( ( ) f x F x we obseve t f ( x ) s oegtve. We eed to show t e tegto ove e suppot of e dom vble s equl oe. Cosde e cse whe e suppot of f ( ) s (, ). I s cse we hve x f ( x ) dx f ( x ) ( ) F ( x ) dx ( ) f ( x) dx f ( x) F( x) dx ( ) Smlly, oe cses whee e suppot of e dom vble s pt of el le follows. Hece f ( x ) s well-defed pobblty desty fucto. We cll f ( x ) e tsmuted pobblty desty of dom vble w bse desty f ( x ). Also ote t whe 0 e f ( x) f( x).ths poves e equed esult. My uos delg w e geelzto of some well- kow dstbutos. Ayl d Tsokos (009) defed e tsmuted geelzed exteme vlue dstbuto d ey studed some bsc memtcl chctestcs of tsmuted Gumbel pobblty dstbuto d t hs bee obseved t e tsmuted Gumbel c be used to model clmte dt. Also Ayl d Tsokos (0) peseted ew geelzto of Webull dstbuto clled e tsmuted Webull dstbuto. Recetly, Ayl (03) poposed d studed e vous stuctul popetes of e tsmuted Log- Logstc dstbuto, d Muhmmd kh d kg (03) toduced e tsmuted modfed Webull dstbuto whch exteds ecet developmet o tsmuted Webull dstbuto by Ayl et l. (0). Ad ey studed e memtcl popetes d mxmum lkelhood estmto of e ukow pmetes. Elbtl (03) peseted tsmuted modfed vese Webull dstbuto. Elbtl d Elghy (03) peseted tsmuted qus Ldley dstbuto. The est of e ppe s ogzed s follows. I Secto we demostte tsmuted pobblty dstbuto, e hzd te d elblty fuctos of dstbuto. I Secto 3 we studed e sttstcl popetes clude qutle fuctos, expso of desty fucto, momets, momet geetg fucto. The dstbuto of ode sttstcs s expessed Secto 4. Flly, I Secto5, we demostte e mxmum lkelhood estmtesof e ukow pmetes.. Tsmuted Qus Ldley Dstbuto I s secto we studed e tsmutedgeelzed Ldley ( ) dstbuto. Now usg (.) d (.) we hve e cdf of tsmutedgeelzed Ldley dstbuto x x x x F ( x,,, ) e e (.) whee scle pmete, shpe pmete d s e tsmuted pmete. The pobblty desty fucto (pdf) of e tsmutedgeelzed Ldley dstbuto s gve by x x x x x f ( x,,, ) x e e e x x x x e e x x x x e e (.) The elblty fucto of e tsmutedgeelzed Ldley dstbuto s deoted by R ( x ) lso ISSN: Pge 46

3 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 kow s e suvvo fucto d s defed s x x x x R ( x ) F ( x ) e e. (.3) It s mpott to ote t R ( x) F ( x). Oe of e chctestc elblty lyss s e hzd te d e evesed hzd tefuctos defed by f ( x) h ( x) F ( x) x x x x x x e e e,(.4) x x x x e e x x x x e e f ( ) x ( x ). (.5) F ( x) x x x x e e espectvely. It s mpott to ote t e uts fo h ( x ) s e pobblty of flue pe ut of tme, dstce o cycles. These flue tes e defed w dffeet choces of pmetes. The cumultve hzd fucto of e tsmuted geelzed Ldley dstbuto s deoted by H ( x ) d s defed s x x x x H ( x) l e e. (.6) It s mpott to ote t e uts fo H ( x ) s e cumultve pobblty of flue pe ut of tme, dstce o cycles. We c show t. Fo ll choce of pmetes e dstbuto hs e decesg pttes of cumultve stteous flue tes. Fgue : Plot of e cdf of dstbuto fo selected vlues of e pmetes. ISSN: Pge 47

4 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 Fgue : Plot of e pdf of dstbuto fo selected vlues of e pmetes. Fgue 3: Plot of e suvvl fucto of dstbuto fo selected vlues of e pmetes. ISSN: Pge 48

5 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 Fgue 4: Plot of e hzd te fucto of dstbuto fo selected vlues of e pmetes. Fgue 5: Plot of e evesed hzd te fucto of dstbuto fo selected vlues of e pmetes. Specl Cses of e Dstbuto The tsmuted geelzed Ldley s vey flexble model t ppoches to dffeet dstbutos whe ts pmetes e chged. The dstbuto cots s specl- models e followg well kow dstbutos. If X s dom vble w cdf (.), e we hve e followg cses: - If e Equto (.) gves tsmuted Ldley dstbuto. Tht toduced by Fto (03). - If 0 d e Equto (.) gves Ldley dstbuto. Tht toduced by Ldley (958). 3- If 0 (03). e Equto (.) gves geelzed Ldley dstbuto. Tht toduced by Ndjh et l. ISSN: Pge 49

6 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy Sttstcl Popetes Ths secto s devoted to studyg sttstcl popetes of e ( ) dstbuto, specfclly qutle fucto, momets d momet geetg fucto. 3.. Qutle Fucto The q qutle x of e tsmuted geelzed Ldley dstbuto c be obted fom (.) s q x x 4q q q e (3.) We smulte e dstbuto by solvg e ole equto x x 4u q q e whee u hs e ufom U (0,) dstbuto. 3.. Expso of Desty Fucto I s secto epesetto of pdf fo tsmuted Kumswmy qus Ldley dstbuto wll be peseted. The memtcl elto gve below wll be useful s subsecto. It s well-kow t, f 0 s el o tege d z, e geelzed boml eoem s wtte s follows z ( ) z. (3.) 0 The, by pplyg e boml eoem (3.) (.), e pobblty desty fucto of dstbuto becomes () x x f ( x ) x e 0 x () x x x e e 0 M j j () x N j j () x f ( x ) x x e x x e (3.3) whee j, 0 j0 j d N j j., j0k 0 j If s tege e dex e pevous sum stops t Momets I s subsecto we dscuss e momet fo dstbuto. Momets e ecessy d mpott y sttstcl lyss, especlly pplctos. It c be used to study e most mpott fetues d chctestcs of dstbuto (e.g., tedecy, dspeso, skewess d kutoss). ISSN: Pge 50

7 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 Theoem (3.). If X hs ( x,,, ), e e momet of X s gve by e followg N j j j j M j j ( x ) j j Poof: Let X be dom vble w desty fucto (3.3). The dstbuto s gve by ) ( ) ( ( ) 0 x E X x f x dx 0 0 (3.4) ody momet of e ( ) M j j x N j j x x x e dx x x e dx e M j j ( x ) j j N j j j j Ths completes e poof. Bsed o e fst fou momets of e dstbuto, e mesues of skewess A () d kutoss k () of e TKQL dstbuto c obted s ( ) 3 ( ) ( ) ( ) A ( ), (3.5) d ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) 6 ( ) ( ) 3 ( ) k ( ). (3.6) 3.4. Momet geetg fucto I s subsecto we deved e momet geetg fucto of dstbuto. Theoem (3.): If X hs dstbuto, e e momet geetg fucto M X (t) hs e followg fom M N j j M j j () t t t j j t t X j j (3.7) ISSN: Pge 5

8 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 Poof: We stt w e well kow defto of e momet geetg fucto gve by tx tx M () t E( e ) e f ( x) dx The, X 0 M j j ( t) x N j j ( t) x x x e dx x x e dx 0 0 M Ths completes e poof. 4. Dstbuto of e ode sttstcs N k k b k k () t j t j t b k k j t j t X k k I s secto, we deve closed fom expessos fo e pdfs of e ode sttstc of e dstbuto, lso, e mesues of skewess d kutoss of e dstbuto of e ode sttstc smple of sze fo dffeet choces of ; e peseted s secto. Let X, X,..., X be smple dom smple fom dstbuto w pdf d cdf gve by (.) d (.), espectvely. Let X, X,..., X deote e ode sttstcs obted fom s smple. We ow gve e pobblty desty fucto of X :, sy f : ( x,,, ) d e momets of X :,,,...,. Theefoe, e mesues of skewess d kutoss of e dstbuto of e X : e peseted. The pobblty desty fucto of X : s gve by f : ( x,,, ) F( x,,, ) F( x,,, ) f ( x,,, ) (4.) B (, ) whee F( x,,, ) d f ( x,,, ) e e cdf d pdf of e dstbuto gve by (.), (.), espectvely, d B (.,.) s e bet fucto, sce 0 F( x,,, ), fo x 0, by usg e boml sees expso of (,,, ) F x, gve by j j F( x,,, ) ( ) F( x,,, ), (4.) j 0 j we hve j j f : ( x,,, ) ( ) F( x,,, ) f ( x,,, ), (4.3) B(, ) j 0 j substtutg fom (.) d (.) to (4.3), we c expess e k ody momet of e ode sttstcs X : sy ( k E X : ) s le combto of e k momets of e dstbuto w dffeet shpe pmetes. Theefoe, e mesues of skewess d kutoss of e dstbuto of X : c be clculted. 5. Estmto d Ifeece I s secto, we deteme e mxmum lkelhood estmtes (MLEs) of e pmetes of e dstbuto fom complete smples oly. Let X,...,, X X be dom smple of sze fom ISSN: Pge 5

9 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 ( x,,, ).The lkelhood fucto fo e vecto of pmetes ( x,,, ) c be wtte s L f ( x, ) f ( x, ) x x x x e e x x e. (5.) Tkg e log-lkelhood fucto fo e vecto of pmetes ( x,,, ) we get l L l l l l x x x x x x le l e. (5.) The log-lkelhood c be mxmzed ee dectly o by solvg e ole lkelhood equtos obted by dffeettg (5.). The compoets of e scoe vecto e gve by x x x x e l e l L x x l e,(5.3) b x x e x x e l L x x e, (5.4) d x x x x e xe l L x x x e x x x x x x e e x e. (5.5) x x e We c fd e estmtes of e ukow pmetes by mxmum lkelhood meod by settg ese bove o-le equtos to zeo d solve em smulteously. Theefoe, we hve to use memtcl pckge to get e MLE of e ukow pmetes. Also, ll e secod ode devtves exst. Thus we hve e vese dspeso mtx s gve by ISSN: Pge 53

10 Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 ˆ ˆ ˆ N ˆ,. (5.6) ˆ ˆ E Whee L L L L L L,,,,,,. By solvg s vese dspeso mtx ese solutos wll yeld symptotc vce d covce's of ese ML estmtos fo ˆ,ˆ d ˆ. Usg (5.6), we ppoxmte 00( )% cofdece tevls fo,,, b d e detemed espectvely s ˆ z ˆ,ˆ z ˆ d ˆ z ˆ. whee z s e uppe 00 e pecetle of e stdd oml dstbuto. REFERENCES [] Ayll, G.R. d Tsokos, C.P. (0). Tsmuted Webull dstbuto: A geelzto of ewebull pobblty dstbuto, Euope Joul of Pue d Appled Memtcs, 4(), [] Ayll, G.R. d Tsokos, C.P. (009). O e tsmuted exteme vlue dstbuto w pplctos,nole Alyss: Theoy, Meods d Applctos, 7, [3] Ayll, G.R.(03).Tsmuted log-logstc dstbuto, J. Stt. Appl. Po., (), -0. [5] Dez, E.G. d Ojed, E.C. (0). The dscete Ldley dstbuto-popetes d pplctos, J.Stt. Comput. Smul, 8(), [4] Elbtl, I. (03). Tsmuted Modfed Ivese Webull Dstbuto. Itetol Joul of Memtcl Achve. 4(8), 7-9. [6] Elbtl,I. d Elghy,M. (03). Tsmuted Qus Ldley Dstbuto: A Geelzto of e Qus Ldley Dstbuto. It. J. Pue Appl. Sc. Techol., 8(), [7] Fto Meovc (03). Tsmuted Ldley dstbuto. It. J. Ope Poblems Compt. M, 6():63-7. [8] Ghty, M.E., Ateh, B. d Ndjh, S. (0). Ldley dstbuto d ts pplctos, M.Comput. Smul, 78(4), [9] Kh, M.S. d Kg, R. (03). Tsmuted modfed Webull dstbuto: A geelzto of emodfed Webull pobblty dstbuto, Euope Joul of Pue d Appled Memtcs, 6(), [0] Ldley, D.. (958). Fducl dstbutos d Byes' eoem, J. Royl Stt. Soc. Sees B,0, [] Ndjh, S., Bkouch, H.S. d Thmsb, R. (0). A geelzed Ldley dstbuto. Skhy B, 73, [] Shw, W. d Buckley, I. (007). The lchemy of pobblty dstbutos: beyod Gm- Chleexpsos d skew- kutotcoml dstbuto fom k tsmutto mp. Xvpeept, Xv, ISSN: Pge 54