A NEW GENERALIZATION OF THE EXPONENTIAL-GEOMETRIC DISTRIBUTION

Transcription

1 Jou of Sttstcs: Advcs Thoy d Actos Voum 7 Num Pgs 5-48 A NW GNRAIZATION OF TH PONNTIA-GOMTRIC DISTRIBUTION M. NASSAR d N. NADA Dtmt of Mthmtcs Fcuty of Scc A Shms Uvsty Ass Co 566 gyt -m: m_ss_999@yhoo.com Actos Svc Mg Otos OB -Fc gyt Astct A w fou-mt ftm dstuto s oosd. Th ots of th oosd dstuto dscussd cudg fomu fo th fu t fucto momts d mod vu. Th mt stmto s sd o th usu mmum hood och. Th fty of th w mod s usttd y ctos to dt sts. Mthmtcs Suct Cssfcto: 65 6F. ywods d hss: umod vu md momts toy mmum hood stmto. Rcvd Fuy 9 Sctfc Advcs Pushs

2 6 M. NASSAR d N. NADA. Itoducto Th coct of hzd (fu) t s vy w ow ty s t costtuts sustt t of y study dvotd to th gth of f of ogsms dvcs stuctus mts tc. Ths study my d to scfc dstuto ut mo oft th choc s md o th ss of how w th ctu osvtos of tms to fu to fttd y th dstuto. My ftm dstutos hv oosd th ttu d ttmts w md to dsc w dstutos wth scfc fu t oty y comoudg two w-ow dstutos. Admds d ous [] oosd th two-mt ot-gomtc (G) dstuto whch hts th dcsg fu t (DFR) oty. A tso of th G dstuto tht c usd modg stutos wh th outo suvv ccty dcss ov tm ws dscussd y Admds t. []. Ths tdd ot-gomtc (G) dstuto hs oth th DFR d th csg fu t (IFR) ots. Sv t. [8] toducd th so-cd gzd ot gomtc (GG) dstuto hvg DFR IFR d usd-dow thtu fu t ddg o ts mts. Such fu t cuvs c osvd fo m th cous of dss whos motty chs ft som ft od th dcs gduy. ouzd t. [] oosd w ftm dstuto wth csg fu t cd th commty ot gomtc dstuto. Sttg fom t cumutv dstuto fucto (Cdf) F ( ) ug t. [6] dfd css of gzd dstutos y F ( ) G( ) w ( w) dw () B( ) wth t mts > d ( Γ Γ B ) Γ( )

3 A NW GNRAIZATION OF TH PONNTIA- 7 s th t fucto. Ths css of gzd dstutos hs cvg cosd ttto ov th st ys tcu ft th wos of ug t. [6] d Jos [9]. Th oty dsty fucto (df) cosodg to () c wtt s g( ) [ F ( ) ] [ F ( ) ] f ( ) B( ) df ( ) wh f ( ) s th t dsty fucto. d Rcty ttmts -y Ndh d Gut [3] Ndh d otz [ 4] og t. [] Ast t. [3] Pscm t. [5] Souz t. [9] d oths -hv md to td fms of oty dstutos foowg th wo oosd y ug t. [6] d Jos [9]. I th ght of th ov dscusso t s tu to s wht s th most wdy usd dstuto otd fom () tht s so my d oms ty. Bcus th ot dstuto s th most wdy d sttstc dstuto d comoudg t wth ts dsct u th gomtc dstuto s oosd y Admds d ous [] w motvtd to toduc th t ot gomtc (BG) dstuto y tg F ( ) () to th Cdf of ot gomtc dstuto wth mts d gv s foows: g ( ) B( ) ( ) ( ) > ( ) >. () A dom v wth th df gv y () s sd to foow th t ot gomtc (BG) dstuto. Th BG gzs som ct dstutos such s th ot gomtc (G) dstuto

4 8 M. NASSAR d N. NADA toducd y Admds d ous [] th tdd ot gomtc (G) dstuto oosd y Admds t. [] d th gzd ot gomtc (GG) dstuto dfd y Sv t. [8]. Th md of th s dvotd to comhsv study of ths w dstuto. I Scto w gv th cumutv dstuto fucto (Cdf) of th BG dstuto. Aomt foms fo th md mod d th fu t fucto dvd Scto 3. ssos fo th momt gtg fucto d hc th m d vc dscussd Scto 4. Th m dvto out th m d th md ccutd Scto 5. Moov w dscuss Scto 6 stmto y th mmum hood mthod. Scto 7 s dvotd to dvg Sho s toy. ct ssos fo od sttstcs of th BG dstuto otd Scto 8. Fy Scto 9 ctos usg dt sts std foowd y th cocudg ms.. Dstuto Fucto Th G d G dstutos sc css of th BG df gv () wh. Moov th GG dstuto s otd fom () wh. Pots of th BG df to show ts fty to mod ftm dt tg fo smcty gv Fgu.

5 A NW GNRAIZATION OF TH PONNTIA- 9 Fgu. Pots of th BG df fo dfft vus of mts ( ). Fo z < d > o-tg w hv ( z) Γ( ) z. Γ( )! (3) If > s tg th sum (3) stos t. Th th cumutv dstuto fucto (Cdf) of th BG dstuto s gv y G( ) B( ) Γ ( ) Γ( )! >. (4)

6 3 M. NASSAR d N. NADA 3. Md Mod d Hzd Rt Fucto Thom. Th md of th BG dstuto s t th foowg omt ot ( ) B m. (5) ( ) B Poof. Dvg th md m fom th w-ow fct G ( m) d th Cdf (4) w ot B( ) Γ ( ) Γ( )! m m. m Sc th sum o th HS s covgt fo < m fst omto tg s m m B( ). Thus omt vu fo th md w th vu gv y (5). W ow vstgt th umodty oty y th foowg thom: Thom. Th BG dstuto s umod t th foowg omt ot:. (6) ( ) Poof. Th dvtv of quto () s ccutd d sttg Ths ds to g η. (7) g ( ( ) )

7 A NW GNRAIZATION OF TH PONNTIA- 3 gvg th omt mod (6). Th hzd (fu t) fucto c thus otd fom g( ) h( ) G( ). Pots of th hzd t fucto gv Fgu fo dfft vus of th mts. Fgu. Pots of hzd t fucto of BG fo dfft vus of mts ( ).

8 3 M. NASSAR d N. NADA Th dvtv of quto (6) s η ( ) ( ) ( ) ( ). Fo < < w ot η ( ) <. Hc usg Gs s thom [7] th hzd t fucto h ( ) s dcsg. Howv fo > w dscuss two css: th fst cs s wh < wth th fct tht ( ) ( ) < w cocud tht η ( ) >. Ag usg Gs s thom [7] th hzd t fucto h ( ) s csg. Th scod cs ss wh > th ot wh η ( ) s gv y c > c. (8) c ( ) Th scod dvtv of th fucto η ( ) t th ot s η ( ) ( ) ( ) ( ) ( ) 3 < 3 3. It s ovous tht th tv ( ) η ( ) > wh th tv ( ) η ( ). Ag usg Gs s thom [7] th hzd < t s usd-dow thtu. 4. Momt Gtg Fucto Fo dom v Y foowg th GG dstuto Sv t. [8] showd tht th momt gtg fucto (mgf) s

9 A NW GNRAIZATION OF TH PONNTIA- 33 () ( ) ty M t t B( α ) B( α ) t <. A dom v foowg th BG dstuto w ths cs hv th foowg mgf: () ( ) M t B ( ) t B( ) B( ) t <. (9) Th -th momt of th BG dstuto foows fom th fct I tcu th m s gv y ( ) M () t t. t ( ) ( ) B( ) [ ( ) ( ) ] () d wh ( ) Γ( ) s th dgmm fucto. I th sm m d th vc V ( ) ( ) ( ) c dvd wh ( ) ( ) [( ( ) ( ) ) ( ) ( ) ]. B( ) 5. Th M Dvto of th BG Th mout of sd outo s vdty msud to som tt y th totty of dvtos fom th m d md. Ths ow s th m dvto out th m d th m dvto out th md. t t ot-gomtc dom v wth m µ ( ) d md m.

10 34 M. NASSAR d N. NADA Th m dvto fom th m c dfd s D ( µ ) ( µ ) µ g( )d µ µ G( µ ) g( ) d G( )d µ Γ ( ) ( ) Γ B Γ! Γ( ) ( ) [ ( ) µ ].!! Th m dvto fom th md s so dfd y D( m) ( m ) m g( )d µ m m G( )d Γ ( ) Γ( ) µ m B( ) ( ) Γ! Γ( ) ( ) [ ( ) m ].!! 6. Mmum hood stmto I wht foows w sh dscuss th stmto of th mts ϑ ( ). Th -hood fucto of dom sm foowg th BG dstuto s Γ( ) Γ Γ ( ) ( )

11 A NW GNRAIZATION OF TH PONNTIA- 35. () Dffttg () wth sct to th mts ϑ w ot. () Th mmum hood stmts of th mts ϑ th souto of th qutos (). Fo tv stmto d hyothss tsts o th mod mts w qu th fomto mt. Th Fsh fomto mt T θ θ s whos mts c sy dvd usg th suts gv th Ad s foows:

12 M. NASSAR d N. NADA 36 [ ] B [ ]; B [ ] B [ ]; B

13 A NW GNRAIZATION OF TH PONNTIA- 37 ( F B B ); F B B F B B wh th hygomtc fucto s dfd s.! α α α α α γ α γ α z z F Th T θ ˆ ˆ ˆ ˆ ˆ of θ s umcy dtmd fom th souto of th o systm of qutos gv. Ud codtos tht fufd fo th mt θ th to of th mt sc ut ot o th oudy th symtotc dstuto of [ ] ( 4 ˆ ˆ ˆ ˆ N T

14 38 M. NASSAR d N. NADA T ( ) ). Th symtotc om ( ( ˆ ˆ T N ˆ ˆ ) ) 4 dstuto of θ ˆ ( ˆ ˆ ˆ ˆ ) T c usd to costuct cofdc gos fo som mts d fo th hzd d suvv fuctos. I fct ( γ)% symtotc cofdc tv (ACI) fo ch mt s gv y ACI ACI ( ˆ zγ / ˆ zγ / ); ( ˆ ˆ zγ / zγ / ); ACI ( ˆ zγ / 33 ˆ zγ / 33 ); ACI ( ˆ ˆ zγ / 44 zγ / 44 ); wh dots th -th dgo mt of ( αˆ ˆ ˆ ˆ fo 3 d z γ / s th γ / of th stdd om ) T dstuto. 7. Th toy Th toy of dom v s msu of uctty vto. Sho s [7] toy s dfd y sh ( g) g( ) g( ) d ( g). Susttutg th BG dstuto gv quto () th toy s gv y sh ( g) B( ) ( ) [ ( )] ( ) [ ( )] ( ) ( ) ( ) ( ) ( ) (3)

15 A NW GNRAIZATION OF TH PONNTIA- 39 wh ( ) [ ( )] B( ) F ( ) B( ) ( ) [ ( )] B( ) F ( ) B( ) d ( ) s gv quto (). (4) (5) 8. Od Sttstcs Od sttstcs y mott o f tstg ty d cmt ocy stutos wh ctto ds to dct th fu of futu tms sd o tms of fw y fus. Ths dctos oft sd o momts of od sttstcs. W ow dv ct sso fo th dsty fucto of th -th od sttstc : sy f : ( ) dom sm of sz fom th BG dstuto. W c wt ( ) ( ) [ G( ) ] g. B( ) f: Susttutg wth th Cdf gv quto (3) tg to ccout tht [ ] ( ) G C > wh C( ) c c d c Γ( ) B( ) Γ( )! ( )

16 M. NASSAR d N. NADA 4 w ot th foowg fom:. : g C B f Ths c wtt s comto of th fom : g w f (6) wh g s th df of th BG dstuto gv () d ( ). B B B C w Sv mthmtc ots of th BG od sttstcs such s th ody momts momt gtg fucto tc. c ccutd. Th fst momt of th -th od sttstcs dtmd s foows: ( ) [ ]. : B w 9. Acto to R Dt I ths scto w sh com th fts of () BG dstuto oosd ths tc; () GG dstuto toducd y ; < < > > G () BG (t gzd ot) dstuto dfd y ; > λ > λ λ λ B G

17 A NW GNRAIZATION OF TH PONNTIA- 4 d (v) Wu (W) dstuto gv y λ 3 > G ( ) > λ > to dt sts. A mott so fo choosg th two-mt Wu dstuto s tht t s th most commo fmy fo modg suvv dt wth umod thtu hzd t wh th BGgzg som w-ow dstutos such s ottd ot d t ot dstutos-hs sgfct sttos suvv dt wth mooto d o-mooto hzd ts. Th sts of dt fttd y usg th fou BG GG BG d W dstutos. Th mts of th fou dstutos stmtd y th mmum hood tchqu y usg Wofm Mthmtc softw. W sh cosd th dt (D) gv y th tms of succssv fus of th -codtog systm of ft of t s toducd Posch [6]. Pots of th df of fttd mods gv Fgu 3. It s vdt ths cs tht fou mods ovd ct ft to ths dt. Fgu 3. stmtd df of BG BG W d GG fo D.

18 4 M. NASSAR d N. NADA Th scod dt st shot-d -tm outcoms of costtducd movmt thy ft sto (D) hv vstgtd domzd cotod fsty t toducd y Dh t. [5]. Pots of th df of fttd mods gv Fgu 4. It s vdt ths cs tht th BG d BG mods ovd tt ft th th GG d W mods to ths dt. Fgu 4. stmtd df of BG BG W d GG fo D. Th thd dt st (D3) ws cosdd y Co [4] to ft th 3 g dt. Pots of th df of fttd mods gv Fgu 5. It s vdt ths cs tht th BG mod ovds tt ft th th GG whs th BG d W mods oo fts to ths dt.

19 A NW GNRAIZATION OF TH PONNTIA- 43 Fgu 5. stmtd df of BG BG W d GG fo D3. Th mmum hood stmts fo th th dt sts std ow: T. Mmum hood stmts fo D BG ˆ.9679 ˆ ˆ.573 ˆ BG ˆ ˆ ˆ.9677 λ ˆ GG ˆ.3354 ˆ.3833 ˆ W ˆ.49 λ ˆ T. Mmum hood stmts fo D BG ˆ.984 ˆ ˆ.46 ˆ.5844 BG ˆ.4847 ˆ.957 ˆ 3.45 λ ˆ GG ˆ.774 ˆ ˆ W ˆ.354 λ ˆ 83.36

20 44 M. NASSAR d N. NADA T 3. Mmum hood stmts fo D3 BG ˆ.836 ˆ.995 ˆ ˆ BG ˆ ˆ.8435 ˆ.65 λ ˆ GG ˆ.7 ˆ ˆ W ˆ.6 λ ˆ Th foowg t sts th vus of th -hood fuctos to fou mods fo th th dt sts: T 4. Th -hood fuctos fo th dt sts Dstuto D D D3 BG BG GG W I ddto to th s of th mts d th -hood fuctos fo th th dt sts th vus of th A fomto cto (AIC) d th Bys fomto cto (BIC) fo th fou dstutos dducd T 5. T 5. AIC d BIC vus fo D D d D3 AIC BG BG W GG D D D BIC BG BG W GG D D D

21 A NW GNRAIZATION OF TH PONNTIA- 45 It s c tht th BG yds th hghst vu of th -hood fucto fo th th dt sts d hc w c cocud tht th BG mod s tt th th oth dstutos to ft ths dt sts. Aso w c s fom th umc suts T 5 th AIC vu fo th BG mod s th smst vu mog thos vus d hc ou w mod c chos s th st mod.. Cocudg Rms W toduc w dstuto th so-cd t ot gomtc (BG) dstuto whch tds th gzd ot gomtc dstuto d study som of ts g mthmtc d sttstc ots. W ovd mthmtc ttmt of th w dstuto cudg sos fo th dstuto fucto momts momt gtg fucto omt vus fo th md d th mod d Sho toy. Th stmto of th mod mts s ochd y th mthod of mmum hood. Fy ctos of th BG dstuto to dt gv to show tht th w dstuto ovds cosstty tt ft th oth mods cudg th gzd ot gomtc dstuto v th ttu. W ho tht th t ot gomtc (BG) mod w ttct wd ctos sttstcs th th th gzd ot gomtc dstuto. Rfcs []. Admds d S. ous A ftm dstuto wth dcsg fu t Sttstcs d Poty tts 39 (998) []. Admds T. Dmtoouou d S. ous O gzto of th ot gomtc dstuto Sttstcs d Poty tts 73 (5) [3] A. Ast F. Fmoy d C. Th t-pto dstuto Sttstcs 4 (8)

22 46 M. NASSAR d N. NADA [4] C. Co Th coct gs dt ftm Dt Ayss 8 () [5] A.. Dh T. Asm R. Stoc. gg S. yds d B. Iddv Shotd -tm outcom of costt ducd movmt thy ft sto: A domzd cotod fsty t Cc Rhtto (8) [6] N. ug C. d F. Fmoy Bt om dstuto d ts ctos Commuctos Sttstcs- Thoy Mthods 3 () [7] R.. Gs Bthtu d td fu t chctztos Jou of th Amc Sttstc Assocto 75 (98) [8] I. S. Gdshty d I. M. Ryzh T of Itgs Ss d Poducts Acdmc Pss Odo Fod 98. [9] M. C. Jos Fms of dstutos sg fom dstutos of od sttstcs Tst 3 (4) -43. []. og C. d J. H. Ss O th ots of t gmm dstutos Jou of Mod Ad Sttstc Mthods 6 (7) [] F. ouzd M. Rom d V. G. Ccho Th commty ot gomtc dstuto: Mod ots d comso wth ts coutt Comutto Sttstcs d Dt Ayss 55 () [] S. Ndh d S. otz Th t Gum dstuto Mthmtcs d Poty fo gs (4) [3] S. Ndh d A.. Gut Th t Fécht dstuto F st Jou of Thotc Sttstcs 4 (4) 5-4. [4] S. Ndh d S. otz Th t ot dstuto Rty gg d Systm Sfty 9 (5) [5] R. R. Pscm C. G. B. Dmto G. M. Codo. M. M. Otg d M. R. Uo Th t gzd hf-om dstuto Comutto Sttstcs d Dt Ayss 54 () [6] F. Posch Thotc to of osvd dcsg fu t Tchomtcs 5(3) (963) [7] C.. Sho A mthmtc thoy of commucto B Systm Tchoy Jou 7 (948) [8] R. B. Sv W. B. Souz d G. M. Codo A w dstuto wth dcsg csg d usd-dow thtu fu t Comutto Sttstcs d Dt Ayss 54 () [9] W. B. Souz A. H. S. Stos d G. M. Codo Th t gzd ot dstuto Jou of Sttstc Comutto d Smuto 8 () 59-7.

23 A NW GNRAIZATION OF TH PONNTIA- 47 Ad Th mts of Fsh fomto mt gv Scto 6 c dvd y usg th foowg cttos: ( ) () ( ) [ ( ) ( ) ] ( ). B( ) () ( ). ( ) (3) ( ) [ ( ) ( ) ]. ( ) (4) ( ) [ ( ) ( ) ]. B( ) B( ) (5) ( ) ( )( ) ( ). (6) ( ) ( ) B ( ) B( ) F ( ). (7) ( ) ( ) B ( ) B( ) F ( ).

24 48 M. NASSAR d N. NADA (8) ( ) ( ) B ( ) B( ) F ( ). Th ov cttos d th cttos qutos (4) d (5) w comutd y usg th ss so hygomtc fucto dfd Scto 6 d th tg λ µ ρ ( ) th ( ) ( u) d B( µ λ) F ( λ ρ λ µ u) R λ > R µ > gv T of Itgs Ss d Poducts. 87 y Gdshty d Ryzh [8]. g