Order Statistics from Exponentiated Gamma. Distribution and Associated Inference
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1 It J otm Mth Scc Vo 4 9 o 7-9 Od Stttc fom Eottd Gmm Dtto d Aoctd Ifc A I Shw * d R A Bo G og of Edcto PO Bo 369 Jddh 438 Sd A G og of Edcto Dtmt of mthmtc PO Bo 469 Jddh 49 Sd A Atct Od tttc fom ottd gmm EG dtto codd Ect o fo th g d do momt of od tttc fom EG dtto dd Bd o th momt of od tttc th t d tmto BLUE fo th octo d c mt of EG dtto d T-II cog otd Th c d coc of th tmto o td Th mmm hood tmto MLE fo th h mt dd Ao tmto d o od tttc fo th h mt otd F como tw th tmto md d o mto td Mthmtc Sct cfcto 6F; 6F; 6F3; 6G5; 6G3 Kwod Od tttc; Eottd gmm dtto; Momt of od tttc; T-II cog; Mmm hood tmto; Bt d tmto Itodcto Od tttc t m -f cto og dt tg to f ttg td M tho h tdd od tttc d octd fc fo m Dd 98 Bh d oh 99 Aod t 99 d Dd d Ng 3 Bh d Pth 986 otd th t d tmto BLUE of th octo d c mt of hf ogtc dtto d o od tttc Mmm hood tmto MLE of th mt of B XII dtto w dd fo T-II cod m Wgo 993 * Pmt dd Fc of Eg t Sho PO Bo 6 E Md 78 o Egt
2 7 A I Shw d R A Bo Bh d h 998 h dcd tmto fo og-gmm mt d o od tttc Moo Rq 998 h dcd two t of tmto of octo d c mt d o od tttc fom B X dtto Gt d Kd 999 todcd gzd ot dtto d dd th MLE Rq d Ahh otd th BLUE of th octo d c mt of gzd ot dtto d o od tttc Th BLUE of th mt of o-mt d two-mt ogthmc dcg dtto d T-II cod m w dtmd St t Ao Mhmod t 3 h dd th ct o fo th g d do momt of od tttc fom W dtto d cctd th c d coc Bd o th momt th otd th BLUE of th octo d c mt of W dtto d T-II cod m MLE fom B X dtto d T I d II cog w dcd S d Pdgtt 5 Lt X X X dom m of z fom ottd gmm EG dtto wth ot dt fcto df f ; [ ] > > d th cmt dtto fcto cdf g F ; [ ] > > fo dt ot th dtto Shw d Bo 8c Ao chctzto fom EG dtto d o cod d B tmto o th EG dtto dcd Shw d Bo 8 Wh th h mt oth d g th df d cdf of gmm dtto wth h mt α d c mt β G I th w d th ct o fo th g d th do momt of od tttc fom EG dtto d o T-II cod m cto I cto 3 MLE fo th h mt otd Ao om tmto d o od tttc fo th h mt dd cto 4 I cto 5 w th momt of od tttc to ot th c coffct fo th BLUE of th octo d c mt Th c d coc of th tmto o g Nmc m d mto td fo th tmto codd F cto 6 omo tw th tmto d coco md Od Stttc d Momt Lt X X X th od tttc otd fom EG dtto Th g d do momt of od tttc dd foow
3 Od tttc fom ottd Gmm dtto 73 Sg momt Th df of th th od tttc X g Dd 98 f [ F ; ] [ F ; ] f ; > wh f d F g d ct d!!! f c wtt wh f d f ; 3 d 4 Not tht th coffct d ot ddt o Th oto m tht f wghtd g of EG dt th Th momt of th th od tttc dotd α d g α E X f d 5 Th ct ct o fo th g momt of od tttc fom EG dtto g th foowg thom Thom Fo d th α 6 Poof Fom 5 3 d w gt α d [ ] d
4 74 A I Shw d R A Bo Ug th om o w h d d α d Th w gt th t 6 g 4 hc th thom od If ot tg m th th to 6 com α 7 Th g momt of od tttc fom gmm dtto G c otd fom 7 ttg A c c fom 6 th ft g momt m d th cod g momt ct 3 3 α 8 d 4 4 α 9 Do momt Th ot df of X d X < g Dd 98 ] ; [ ] ; ; [ ] ; [ F F F F f ; ; > > f f wh f d F g d ct d!!!! f c wtt ; ; ] ; [ ] ; [ f f F F f
5 Od tttc fom ottd Gmm dtto 75 Th do momt of X d X dotd α < d g d d f X X E α Th ct ct o fo th do momt of od tttc fom th EG dtto g th foowg thom Thom Fo < d th α! 3 Poof Fom d w gt d d ] [ ] [ α ] [ d I 4 wh d I ] [ d! 5 Th fom 4 d 5 w h
6 76 A I Shw d R A Bo d ] [! α! d α Th tg th t qto gmm fcto hc th thom od If ot tg m th th to 3 com α! 6 Th do momt of od tttc fom gmm dtto G c otd fom 6 ttg A c c fom 3 th odct momt of X d X α! Mmm Lhood Etmto So T-II cod m X X X X wh X th th od tttc Th m otd d codd fom EG dtto wth df d cdf g ct d Th hood fcto LF th c c wtt
7 Od tttc fom ottd Gmm dtto 77 wh T ; V 3 T [ ] d V Th ogthm of th LF g L ; T V 3 Th MLE of dotd ˆM g ˆ ˆ M M /[ V V ] 33 Th qto mct fom o t m od g mc tto ch Nwto Rho mthod g Mthmtc 4 I th of comt m tht th tmto ˆ M 33 dc to ˆ M 34 [ ] 4 Etmto Bd o Od Stttc Lt U U U th od tttc of dom m of z fom th fom dtto U d V V V th od tttc of dom m of z fom th ot dtto wth m / E Th o mg of th fct tht X U h E oto d mootoc dcg fcto w h X X d V 4 Lt V V V T II cod m fom E oto d cod th cott π π dfd π Not tht π th m of Z wh Z th th od tttc of dom m of z fom th E dtto To tmt th h mt d o th oto fom E w df Rq 998 S V π V
8 78 A I Shw d R A Bo It foow tht fo / / 3 S dttd mot χ t wth dg of fdom Thfo tmto of dotd ˆ g ˆ S Fom 4 w ot tmto of fcto of od tttc th foowg ˆ 4 [ ] X X X X π d Sc th tmto ˆ / Y wh Y dttd mot χ t foow tht th c of ˆ Vˆ > Not tht c c wh th tmto ˆ 4 th m th MLE of 34 Th tmto ~ dfd ~ [ ] X X X X π 43 mot d tmto fo > wth c / fo > whch m th tht of ˆ B g 4 43 d 33 w h mtd d o Mot-o th of d m q o MSE of ˆ ~ d ˆM I T w h td th fo o choc of d Fom th t w o tht th fomc of ~ tt th of ˆ d ˆ M fo choc of d codd
9 Od tttc fom ottd Gmm dtto 79 T Smtd of d MSE of ˆ ~ d 5 ˆ ~ ˆ M ˆ ~ ˆ M Not Th ft t th mtd Th cod t th mtd MSE T otd 3 35 ˆ ~ ˆ M ˆ ~ ˆ M Not Th ft t th mtd Th cod t th mtd MSE ˆM
10 8 A I Shw d R A Bo A w c fom T th MSE of ~ th tht of ˆ d ˆ M fo ~ comt d cod m codd Ao w ot tht MSE of ˆ d ˆ M c c d dc c Thfo w c tht ~ tt th ˆ d ˆ M fo choc of d codd Nt w dc th tmto of th octo d c mt of th EG dtto It md tht th h mt ow
11 Od tttc fom ottd Gmm dtto 8 5 BLUE of Locto d Sc Pmt Th ct d ct o fo th ft d cod g momt of od tttc g 8 d 9 ct ow to t th m d c of od tttc fo m of z Fo th comtto of c d coc th odct momt α < w comtd ft Th dgo mt α α otd fom th ct o g 9 Nt th ct d ct o g 7 w d fo comtto of th odct momt of two od tttc Th w d to t th coc β α α α Th of m c d coc w comtd g Mthmtc 4 c th gmm fcto It mott to m h tht o comtto wh fo gmm dtto G Am Y Y Y to dom m of z fom th EG dtto wth df μ μ / σ μ μ / σ g ; μ σ [ ] > μ σ > σ σ Lt Y Y Y T-II cod m fom th o dom m Th X X X X wh X Y / μ σ odd m fom oto wth th tdd EG dtto df d cdf g d ct Th w c wt th t d tmto BLUE of μ d σ [ Dd 98 Aod t 99 d Dd d Ng 3] ˆ μ Y Y Y 5 ˆ σ Y Y Y 5 T T wh A V A A V ; T T A α α α α α V th of th coc mt V β Vc d coc of th tmto g V ˆ μ d σ V ˆ σ dσ d o ˆ μ ˆ σ d σ 53 d d T wh D A V A d d I T d 3 w h td th coffct of th BLUE of th octo d c mt ct fo 6 / to d 555 Th coffct of th BLUE td T d 3 chcd th codto d Th c d coc of th BLUE g 53 td T 4 tm of σ
12 8 A I Shw d R A Bo T offct fo th BLUE of μ
13 Od tttc fom ottd Gmm dtto 83 T otd
14 84 A I Shw d R A Bo T 3 offct fo th BLUE of σ
15 Od tttc fom ottd Gmm dtto 85 T 3 otd
16 86 A I Shw d R A Bo T 4 Vc d oc of th BLUE of μ d σ tm of d σ
17 Od tttc fom ottd Gmm dtto 87 T 4 otd d
18 88 A I Shw d R A Bo Fom th ct o 8 9 d 7 w comt th g th do momt of od tttc d th c fo 6 d 55 5 Th c coffct 5 d 5 qd fo th BLUE of μ d σ ct c otd d o th comtto of th m c d coc of od tttc fom EG dtto T d 3 od th coffct of th BLUE of μ d σ ct fo 6 / to d 555 I T 4 w comt th c d coc of th BLUE g 53 d w ot tht fo m Vμˆ c c d dc Othw V σˆ dc c d c dc Ao o ˆ μ ˆ σ dc c d dc Nmc tto I od to tt th f of th fc ocd dcd th cto w cod th m Em Lt cod th c wh th comot h f tm whch foow EG dtto wth So tht w o th foowg mtd od f tm fom EG μ σ Thfo th odd f tm H th fo th odd dt wth 5 5 α d V d o 8 9 d 7 Th coffct 5 d 5 td T d 3 ct Thfo th BLUE of μ d σ comtd to ˆ μ 99 d ˆ σ Th coodg c d coc of μˆ d σˆ comtd to V ˆ μ 756σ V ˆ σ 6647σ d o ˆ μ ˆ σ 85548σ Lt cod th t oto m ζ E Y μ 75 σ Hd w d & ζ Y th m m W wod h & ζ 96 d tdd o SE ζ & Th BLUE of ζ ˆ ζ ˆ μ 75 ˆ σ 95 Th tdd o of ζˆ comtd to SEζˆ 377 Thfo th BLUE fom tt th th m m th of th tdd o fo comt m Em od th o dt t wth 5 d 3 thfo th cod dt
19 Od tttc fom ottd Gmm dtto 89 Fom t d 3 th BLUE of μ d σ comtd to ˆ μ d ˆ σ Th coodg c d coc of μˆ d σˆ comtd to V ˆ μ 49σ V ˆ σ 337σ d o ˆ μ ˆ σ 5394σ Lt cod th t oto m ζ E Y μ 75 σ Hd w d & ζ Y th m m W wod h & ζ 46 d tdd o SE ζ & Th BLUE of ζ ˆ ζ ˆ μ 75 ˆ σ 836 Th tdd o of ζˆ comtd to SEζˆ Thfo th BLUE fom tt th th m m th of th tdd o fo cod m 6 oco d omo I th wo th t of tmto of d o om od od tttc fom EG dtto h otd I ddto th BLUE of μ d σ otd Fom T 3 d 4 w cocd th foowg ot A w c fom T th MSE of ~ th tht of ˆ d ˆ M fo ~ comt d cod m codd Ao w ot tht MSE of ˆ d ˆ M c c d dc c Th coffct of th BLUE td T d 3 chcd th codto d 3 Fom T 4 w ot tht fo m - V μˆ c c d dc - V σˆ dc c d c dc - o ˆ μ ˆ σ dc c d dc 4 Fom Em d w tht th tdd o of BLUE th tht of th m m I coco w c tht ~ fom tt th ˆ d ˆ M fo choc of d codd Ao th BLUE of μ d σ fom tt th th m m th of tdd o fo comt d cod m
20 9 A I Shw d R A Bo Rfc [] A I Shw d R A Bo B d o-b tmto o th ottd gmm dtto Ad Mthmtc Scc [] A I Shw d R A Bo t chctzto of th ottd gmm dtto Acctd fo cto JATA 8c [3] A I Shw d R A Bo hctzto fom ottd gmm dtto d o cod Acctd fo cto JSTA o 7 8 [4] B Aod N Bh d H N Ng A Ft o Od Stttc W Nw Yo 99 [5] D R Wgo Mmm hood tmto of B XII dtto mt d t II cog Mcocto R [6] H A Dd Od Stttc Scod Edto W Nw Yo 98 [7] H A Dd d H N Ng Od Stttc Thd Edto W Nw J 3 [8] J G S d W J Pdgtt Som ot of cd B t X dtto Jo of Stttc Pg d Ifc [9] K S St A hd d N Bh A ogthmc-dcg dtto Ifc fo o-mt d two-mt c Pocdg of th Sth Mtg of SAMS 4-6 [] M A W Mhmod K S St d S M Am Od tttc fom W dtto d octd fc omtto Stttc & Dt A [] M Z Rq d M Ahh Etmto of th octo d c mt of gzd ot dtto d o od tttc J Sttt omt Sm [] M Z Rq Od tttc fom th B t X mod omt Mth Ac [3] N Bh d A oh Od Stttc d Ifc Etmto Mthod Acdmc P S Dgo 99
21 Od tttc fom ottd Gmm dtto 9 [4] N Bh d P S h Log-gmm od tttc d tmto of mt I Bh N d Ro R Ed Hdoo of Stttc Vo 7 E Scc Amtdm [5] N Bh d S Pth Bt d tmto of octo d c mt of th hf ogtc dtto J Sttt omt Sm [6] R D Gt d D Kd Gzd ot dtto Nw Zd J Sttt Rcd J 8
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