Kummer Beta -Weibull Geometric Distribution. A New Generalization of Beta -Weibull Geometric Distribution
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2 Althogh y sttos hv t th lwys oos fo vlog sttos whh o fll fo fttg sf l wol sos. Th otvt shs skg vlog w o fll sttos. As slt y w sttos thqs hv vlo st. Bt stto s vy vstl s to lyz fft tys of t sts. My of th ft g sttos ot t sly tsfo to th st t stto. Glz t sttos hv wly st sttsts os thos hv vlo vos lsss of ths sttos. Th thos [] oos gl lss of sttos fo o vl f fo th logt of th t o vl y loyg two ts whos ol s to to skwss to vy tl wght. Followg th thos [] who f th t ol BN stto th y shs follow th s st to vlo w sttos sh s th thos [] who to th t Gl stto BG th thos [3] who oos t-otl stto. A tso of th t-gt tho ws oos y th tho [4 5] y sg th swy stto of th tho [6] s gto st of t stto. Th swy sttos W s vy sl to th t stto t hs los f fo. My shs follow th swy oh to t vlo w sttos sh s th thos [7] who to th swy glz hlf-ol -GHN stto. t stto ws oos y th thos [8] o th t tvl wth ltv stto fto olty sty fto. Th s thoology of th thos [] th thos [5] s to ostt w lss of t glz BG sttos. My shs follow th oh of t to t vlo w sttos sh s th thos [9] who vlo th BG-Wll GBW. As slt of sg t oh w stto s f whh s ll Bt-Wll Got BWG stto. My ots of BWG stto llt. Fst th BWG sty fto wll to. So ots ot gtg fto wll ot. Th lklhoo stts wll ostt. Foth l llstto wll st.. Mol Dvto W s th s thoology of th thos [] th thos [5] to ostt w lss of t glz BG sttos. Fo ty t f G th BG fly of ltv sttos s f y F G 59
3 wh > > -<< F ; ; F ; ; Wll got stto whh s: t t t t!. Tkg G to th f of wh G > s sl t > > > sh ts. A o vl X hs Bt-Wll Got BWG stto f t hs th fllowg f s fg. : F Cltv DsttoF F F.5 F F X Fg.. ltv stto fto of BWG stto 6
4 lso w ot th f of BWG stto y fftt s follow: F th ; f 3 wh > > s sl t > > > sh ts-<< F ; ;! ; ; t t t t F s fg.. t ot tht qto 3 s olty sty fto wh ; f fo ll vls of X ; f. Fg.. olty sty fto of BWG stto X Polty Dsty Fto f f f f f f
5 Th fl t fto osog to qto 3 wll s fg.3 h 3 Fl Rt Fto h H H 5 H X Fg. 3. fl t fto of BWG stto Usg 3 so sl ss ot s follows s fg.4: Wh th t-wll got BWG stto s ot. f ; B whh ws to y th thos []. Wh th t otl got BEG stto s ot. f ; B whh ws ot y th tho []. 6
6 3 Wh th glz otl got GEG stto ot. f ; B whh ws to y th thos []. 4 Wh th WG sty fto ot s follow f ; whh ws oos y th thos [3]. 5 Wh w ot th EG stto s follow f ; whh ws to y th thos [4]. 6 Wh to to th BWG stto s to th totl BE stto. Fg. 4. ltoshs of th BWG s-ols 63
7 3. Mots Mot Gtg Fto ths sto w v th th ot fo o vl X hvg th f 3 s follows: f E ; lt th X E 64
8 { } { } { } { } 4 y l o-tg tg so w hv th followg two ss: Cs : Wh s l o-tg w hv [ { }]! 5 lyg 5 to 4 gvs E X! { } { } { }! { } { } { } { }! Cs : Wh s tg t show tht [ { }] { } 6 lyg 6 to 4 gvs E { } { } { } { } { } X { } { } { } { } 65
9 W ot th ot gtg fto of BWG fo gl fol of gf s follows: M t t E t f ; t { } { } { } t { } { } lt th M t { } { } t { } 7 y l o-tg tg so w hv th followg two ss: Cs : Wh s l o-tg lyg 5 to 7 th gf of o vl X hvg th BWG stto s: M { } { } t! { } t { }! { } { } t { } 66
10 { } { } { } t { }! Cs : Wh s tg lyg 6 to 7 th gf of o vl X hvg th BWG stto s: M X t { } { } t { } { } { } t { } { } t { } 4. M-lklhoo Estto Lt... o sl of sz fo BWG stto 3 th th θ lklhoo fto fo th vto of ts T wll : L θ f 8 θ th th log-lklhoo fto fo th vto of ts T 8 wtt s: l L l θ [ l l l l ] l l l 9 wh θ s tsfo osvto. Dfftt 9 wth st to T w hv l L θ l l l l 67
11 l l l l L θ L l l l l θ L l l l θ L l θ A L l θ qlg wth zo gvs l l l l l l l 68
12 l l l l l wh. Eqto solv lly sg sttstl kgs. Th osv foto t θ fo ts wtt s follows: E θ wh [ ] [ ] l l l l l 69
13 l l l l l l l l 7
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15 5. Nl llstto W wll gt sls of sz 5 fo th BWG stto fo t vls of th ts sg oos o gto th th lklhoo stts fo h sl wll ot log wth th ss sq o skwss ktoss of thos stts fo fft sl szs. tl w lst th ss sq o skwss ktoss fo th MLE's of th ts fo o sls of sz 5. Th tl shows tht th sq o th s fo th stts of th ts s s th sl szs s. W s sttstl kg ll th. Tl. M Bss M Sq Eos Skwss toss fo th stts of BWG stto fo fft vls of ts.9.8 C N M Bs MSE Skwss toss C D Colsos W to w stto whh s ll k t-wll got stto wth s t. Th w stto y os glzto of t-wll got stto. W ov thtl ttt of th stto lg ts sty fto ots ot gtg fto th stll so of ts ots to vstgt. Th stto of th ts s oh y th tho of lklhoo th osv foto t s llt. W ho tht BWG stto ht o fft shs of fl t fto. Also w ho tht th w ol y ttt w ltos sttsts. Rfs [] N. Eg C. L F. Foy. Bt-ol stto ts ltos. Cotos Sttsts Thoy MthosVol. 3. 7
16 [] S. Njh S. otz. Th t Gl stto. Mthtl Pols EggVol. 4. [3] S. Njh S. otz. Th t otl stto. Rllty Egg Syst SftyVol [4] M.C.Jos. swy s stto: A t-ty stto wth so ttlty vtgs. Sttstl Mthoology Vol [5] G. M. Coo M. Csto. A w fly of glz sttos. Jol of Sttstl Cotto SltoVol. 8. [6] P. swy A glz olty sty fto fo ol o o osss. Jol of Hyology Vol [7] G. M. Coo R. R. Ps E. M. Otg. Th swy glz hlf-ol stto fo skw ostv t. Jol of Dt SVol.. [8]. W. Ng S. otz. -g -t vt ltvt sttos. Rsh Rot of Dtt of Sttsts Th Uvsty of Hog og Hog og Vol [9] R. R. Ps G. M. Coo C. G. B. Déto E. M. M. Otg S. Njh. Th w lss of t glz sttos. Sttsts Otos Rsh TstosVol. 36. [] G. M. Coo G. O. Slv E. M. Otg. Th t-wll got stto. A Jol of Thotl Al Sttsts Vol. 47. [] H. B. Th t otl-got stto. Cotos Sttsts - Slto CottoVol.4. [] R.B.Slv W. Bto-Soz G.M. Coo. A w stto wth sg sg s-ow tht fl t. Cottol Sttsts Dt AlyssVol. 54. [3] W. Bto-Soz A.L. Mos G. M. Coo. Th Wll-got stto. Jol of Sttstl Cotto SltoVol [4] C.D. L M. X.Stohst gg fo llty Sg USA 6. 73
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