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1 R ASHI :43-48 (06 ESIA IO OF HE SALLES LOCA IO OF WO EGAIVE EXPOEIAL POPULAIOS Partha Pal ad Uttam Badyopadhyay Dpartmt of Statsts aulaa Azad Collg Kolkata Dpartmt of Statsts Uvrsty of Calutt a ABSRAC K ywords : Expotal dstrbuto; Loato ad sal paramtr; axmum lklhood stmator; Bas; a squar rror; Ptma Closss Itroduto W propos a stmator of th smallst loato of two gatv xpotal populatos h proposd stmator s ompard wth th xstg maxmum lklhood stmator through dffrt prforma rtra I support of ths om- parsos w also provd som umral omputatos Estmato of th smallst (largst loato of sv- ral xpotal populatos has always b a tr- stg problm h problm of stmatg th largst ma of K ormal populatos was osdrd by Kuo ad ukhopadhyay (990 ukhopadhyay t al (993 Saxa ad og (969 ad og (970 amog oth- rs A rlatd problm of stmatg th largst ompo- t of a multomal paramtr has rtly b o- sdrd by Alam ad Fg (997 Kuo ad ukhopadhyay (990a osdrd th pot stmato problm of th largst loato of K gatv xpotal populatos But most of ths works ar for fxd- wdth trval stmato basd o squtal or mult- stag samplg I ths papr w osdr th pot stmato of th smallst loato of two gatv xpotal populatos th fa of ukow sal paramtr(s I th la- guag of rlablty ad lf tstg ths amouts to th stmato of th mmum guarat tm wh th as- sumd dstrbutos ar all xpotals wth ukow falur rat(s h xpotal dstrbuto has b wdly usd for dsrbg th dstrbuto of falur tms of omplx qupmt vauum tubs ad othr small ompots Applatos of smpl xpotal modls ar ord wth amal tumour systms ad aut lukama dstruto of tumour lls wth lasr rgy ad th aalyss of survval data wth oomtat formato O a s Zl (996 ths otxt Lt E(µ = b two dpdt gatv x- potal populatos ad ( X X b a sampl from th th populato Our objtv s to stmat µ ( = m ( µ µ Frst w osdr th as of qual s al paramtr = = say W propos a st- mator of wth a slght modfato ovr th xstg R ASHI : (06 43 Bas Dstrbutos maxmum lklhood (L stmator sto 3 h suggstd stmator may b usful to stmat th m- mum guarat tm of a srs systm omprsg of two ompots havg dpdt xpotal lf ds- trbutos I sto 4 th proposd stmator s om- pard wth th L stmator both asymptotally ad also trms of thr small sampl prforma xt sto 5 w osdr th as of uqual sal pa- ramtrs Som umral omputatos for th as of qual sal paramtr ar also provdd to justfy our fdgs S lf-tm s o-gatv t s qut raso- abl to assum µ > 0 = Also X j >µ j= ( = Lt = m {X J= } ( j S = (X j - = ad = m( It s wll-kow that ~ E > µ = ( S ~ X ( ad th varabls ar all dpdt ow th dstrbuto futo (df of F (y =- xp ( y 0 f y > µ s gv by f y < µ = (3 If G(y s th df of th usg (3 w gt
2 Estmato of two gatv xpotal populatos { G(y = P ( > y = xp xp µ ( = m ( µ µ (y( f y µ f µ < y µ ( ( ( y f y > µ (4 ( µ = max ( µ µ ( ad s th sampl sz orrspodg to th p opulato havg loato µ = Clarly ( Lt ad mum ( ( = w h µ µ ( = = + = ( + ( = ( s usually stmatd by ˆ lklhood m( = ˆ ( m ( (5 s a stmator of (L ˆ stmator - If ˆ th max of w gt whh s th L stmator of Lt us dot ths stmator by 3 Proposd Estmator dus I ths sto w propos a stmator whh r- th bas ( SE of B as ( = { } h bas ad ma squar rror a b asly obtad as SE( ( ( (3 ( ( (3 ( ( ( ( (33 W s that B as ( 0 ad h ovrst- m ats ( o rdu th bas w propos th follow- g modfd stmator ˆ ˆ ad ˆ ˆ S S SS f ad also dpdt H ar dpdt ˆ E( E( E( ˆ E ˆ Hr E ˆ ˆ E P ( ( P( P( ( hus E h t a b show that E ( f (34 a d ˆ ar (35 (36 (37 mplyg that ( (38 R ASHI : (06 44
3 Pal ad Badyopadhyay E Bas 0 (39 a d thus u drstmats ( ow to fd th SE of w s that ˆ V ˆ ˆ ˆ E E ˆ E E E E (30 E ( ad E a b asly obtad by us ˆ - g th dstrbuto of ad ˆ Also aftr smpl ma- pulatos w ad E gt P ( ( E EE E P ( ( ( (3 (3 4 Comparsos 4 Small Sampl Comparsos S Bas Bas( 0 ( s mallr absolut bas tha that of to has s losr t ha hs fat howvr a b provd by usg Ptma s losss proprty whh says: A stmator s losr to tha aothr stmator f P I our as ad (4 ˆ ˆ ( hus ' ' ˆ ˆ ' (4 ow aftr som rout stps t a b show that QP ˆ P P ˆ (43 Q Q P hus SE R ASHI : (06 ( ( ( Q 4 ( ( ( ( ( 3 ( (44 ot that It s asy to vrfy that for > v (45
4 Estmato of two gatv xpotal populatos S lt ( x P X R ASHI : (06 v v x/v v v x P Xv whh a b wrtt as ad x P U x U ~ X v v (46 V ~ dpdtly U V U ow s x 0 ad ~ F V x x PU V x PU V 0 P F (45 ad (47 mply that w gt (47 P (48 a d h h s Ptma losr tha SE of s ot asy to ompar algbra- ally wth that of W shall howvr osdr som asymptot ad umral omparsos th latr s- tos 4 Asymptot Comparsos It a b asly s that Bas Bas SE ad SE a ll td to zro as H ad ar osstt Also w s that lm Bas( (49 46 ad that lm lm Bas( 0 (40 A ga as (4 SE SE( (49 ad (40 mply lm Bas( Bas( (4 ( 4 ad (4 ma that s at last asymptot- ally a bttr stmator of 5 Cas of Uqual Sal Hr w osdr ˆ ad th stmators ˆ ˆ t ha S ˆ f h as th as of qual sal w Bas Bas s that ( 0 ( ( (5 ( 0 ( (5 SE
5 Pal ad Badyopadhyay SE ( ( (53 SE R ASHI : (06 ad Hr ( ( ( (54 ( Bas f ( ( ( Bas( 0 ( ad P P ˆ ˆ ˆ P ˆ Q QP P (55 Q 4 ( ( ( ( (56 47 As sto 4 t a b smlarly show that a d h s Ptma losr tha Aga lm ad lm Bas SE mplyg that s tha 6 umral Rsults Hr w wrt Bas( SE( also P asymptotally B Bas B Bas SE SE From th tabl 6 w obsrv that : ( Both B as ad Bas bttr dras wth th ras sampl szs ad ras wth th - ras as s vdt from thr thortal xprs- sos ( h ma squar rrors too dras wth th ras sampl szs ad ras wth th ras ( As has b alrady algbraally stablshd B as( s always lss tha B as( absolutly But what s mor mportat s that SE( s lss tha SE( 7 Coludg Rmarks Comparso of th proposd stmator wth aothr ˆ xstg stmator U m s lft udo baus as th as of proposd stmator or th L stmator w aot hav xat mathmatal xprssos for th prforma rtra of Howvr U w a gt a far da of th prforma of ovr U through smulato studs h as of qual sampl szs follows asly from th abov study as a partular as
6 Estmato of two gatv xpotal populatos abl 6: Calulatos of B B ( ad basd o 0000 smulatos B B ( 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 05 ( ( ( ( ( ( ( REFERECES Alam K ad Z Fg (997 : Estmatg probablty of ourr of th most lkly multomal vt J Statst Pla Ifr 59: Balakrsha ad Basu AP h Expotal Dstrbuto: hory thods ad Applatos Gordo ad Brah Publshrs Kuo L ad ukhopadhyay (990: ult-stag pot ad trval stmato of th largst ma of K ormal populatos ad th assoatd s- od ordr proprts trka 37: Kuo L ad ukhopadhyay (990a: Pot stmato of th largst loato of K gatv xpotal populatos Squtal Aalyss 9: ukhopadhyay Chattopadhyay S ad Sahu SK (993: Furthr dvlopmts stmato of th largst ma of K ormal populatos trka 40: Saxa KL ad L og (969 : Itrval stmato of th largst ma of K ormal populatos wth kow varas J Amr Statst Asso 64: L og (970 : ult-stag trval stmato of th largst of K ormal mas J Royal Statst So B 3: 7-77 Zl (966: Applato of xpotal modls to problms ar rsarh J Royal Statst So A 9: Zha PW (966 : Ivara of axmum lklhood stmators A ath Statst 37: 744 R ASHI : (06 48
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