Almost unbiased exponential estimator for the finite population mean
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1 Almos ubasd poal smaor for f populao ma Rajs Sg, Pakaj aua, ad rmala Saa, Scool of Sascs, DAVV, Idor (M.P., Ida (rsgsa@aoo.com Flor Smaradac ar of Dparm of Mamacs, Uvrs of Mco, Gallup, USA (smarad@um.du Absrac I s papr av proposd a almos ubasd rao ad produc p poal smaor for f populao ma. I as b so a Bal ad Tuja (99 rao ad produc p poal smaors ar parcular mmbrs of proposd smaor. Emprcal sud s carrd o dmosra supror of proposd smaor. Kords: Aular formao, bas, ma-squard rror, poal smaor.. Iroduco I s ll ko a us of aular formao sampl survs rsuls subsaal mprovm prcso of smaors of populao ma. Rao, produc ad dffrc mods of smao ar good ampls s co. Rao mod of smao s qu ffcv r s a g posv corrlao b sud ad aular varabls. O or ad, f s corrlao s gav (g, produc mod of smao ca b mplod ffcvl. osdr a f populao us ( U, U,..., U for ac of c formao s avalabl o aular varabl. L a sampl of sz b dra
2 smpl radom samplg ou rplacm (SRSWOR o sma populao ma of caracr udr sud. L (, b sampl ma smaor of (, X populao mas of ad rspcvl. I ordr o av a surv sma of populao ma of sud caracr, assumg koldg of populao ma X of aular caracr, Bal ad Tuja (99 suggsd rao ad produc p poal smaor X p (. X X p (. X Up o frs ordr of appromao, bas ad ma-squard rror (MSE of ad ar rspcvl gv b B( K (.3 MSE ( K (.4 4 B( K (.5 MSE ( K (.6 4 r S, ( ( S, ( ( X S, S X, K ρ, S ρ, ( S S S ( ( ( X.
3 From (.3 ad (.5, s a smaors ad suggsd b Bal ad Tuja (99 ar basd smaor. I som applcaos bas s dsadvaagous. Follog Sg ad Sg (993 ad Sg ad Sg (6 av proposd almos ubasd smaors of.. Almos ubasd smaor Suppos X X, p, p X X suc a,, H, r H dos s of all possbl smaors for smag populao ma. B dfo, s H s a lar var f H (. for, R (. r (,, dos sascal cosas ad R dos s of ral umbrs. To oba bas ad MSE of, r (, X( suc a. E ( E (. E(, E(., E( ρ Eprssg rms of s, av ( p p (.3
4 Epadg rg ad sd of (.3 ad rag rms up o scod pors of s, av ( 8 8 (.4 Takg pcaos of bo sds of (.4 ad subracg from bo sds, g bas of smaor, up o frs ordr of appromao as ( ( K 4 ( B (.5 From (.4, av ( (.6 r -. (.7 Squarg bo sds of (.7 ad akg pcaos, g MSE of smaor, up o frs ordr of appromao, as K 4 MSE( (.8 c s mmum K. (.9 Pug s valu of K (. av opmum valu of smaor as (opmum. Tus mmum MSE of s gv b (.MSE( m ρ (. c s sam as a of radoal lar rgrsso smaor. From (.7 ad (.9, av
5 - K. (. From (. ad (., av ol o quaos r ukos. I s o possbl o fd uqu valus for s,,,. I ordr o g uqu valus of s, sall mpos lar rsrco B(. (. r B( dos bas smaor. Equaos (., (. ad (. ca b r mar form as B( B( K (.3 Usg (.3, g uqu valus of s(,, as 4K K K K K (.4 Us of s s (,, rmov bas up o rms of ordr o( - a (.. 3. To pas samplg W populao ma X of s o ko, s of smad from a prlmar larg sampl o c ol aular caracrsc s obsrvd. T valu of populao ma X of aular caracr s rplacd b s sma. Ts cqu s ko as doubl samplg or o-pas samplg. T o-pas samplg apps o b a porful ad cos ffcv (coomcal procdur for fdg rlabl sma frs pas sampl for uko
6 paramrs of aular varabl ad c as m rol o pla surv samplg, for sac, s; Hdroglou ad Sardal (998. W X s uko, s somms smad from a prlmar larg sampl of sz o c ol caracrsc s masurd. T a scod pas sampl of sz ( < s dra o c bo ad caracrscs ar masurd. L do sampl ma of basd o frs pas sampl of sz ; ad scod pas of sz. b sampl mas of ad rspcvl basd o I doubl (or o-pas samplg, suggs follog modfd poal rao ad produc smaors for, rspcvl, as p (3. d p (3. d To oba bas ad MSE of d ad d, r suc a ad (, X(, X( ( E( E( E E (, f E (, f E (, f E( fρ,
7 E(, f ρ E (. f r f, f. Follog sadard procdur oba B ( d f 3 ρ 8 (3.3 B ( d f 3 ρ 8 (3.4 MSE ( ρ d f f 3 4 (3.5 MSE ( ρ d f f 3 4 (3.6 r f 3. From (3.3 ad (3.4 obsrv a proposd smaors d ad d ar basd, c s a draback of a smaor s som applcaos. 4. Almos ubasd o-pas smaor Suppos, d ad d as dfd (3. ad (3. suc a, d, d W, r W dos s of all possbl smaors for smag populao ma. B dfo, s W s a lar var f W W. (4.
8 for, R. (4. r (,, dos sascal cosas ad R dos s of ral umbrs. To oba bas ad MSE of, usg oaos of sco 3 ad prssg rms of s, av ( p p (4.3 ( ( ( [ ( ] (4.4 r. (4.5 Takg pcaos of bo sds of (4.4 ad subracg from bo sds, g bas of smaor, up o frs ordr f appromao as ρ 3 8 f ( Bas (4.6 From (4.4, av ( (4.7 Squarg bo sds of (4.7 ad akg pcao, g MSE of smaor, up o frs ordr of appromao, as
9 MSE ( f f K c s mmum 3 (4.8 4 K. (4.9 Tus mmum MSE of s gv b m.mse( [ f f ρ ] 3 (4. c s sam as a of o-pas lar rgrsso smaor. From (4.5 ad (4.9, av K (4. From (4. ad (4., av ol o quaos r ukos. I s o possbl o fd uqu valus for 's(,, 's, sall mpos lar rsrco. I ordr o g uqu valus of B( d (4. r B ( d dos bas smaor. Equaos (4., (4. ad (4. ca b r mar form as B( d B( d K (4.3 Solvg (4.3, g uqu valus of 's(,, as 8K K 4K K 4K (4.4
10 Us of s 's(,, 5. Emprcal sud rmovs bas up o rms of ordr ( o a (4.. T daa for mprcal sud ar ak from o aural populao daa ss cosdrd b ocra (977 ad Rao (983. Populao I: ocra ( ,.445, ρ Populao II: Rao (983.46,.8, ρ I abl (5., valus of scalar s (,, ar lsd. Tabl (5.: Valus of s (,, Scalars Populao I II Usg s valus of s (,, gv abl 5., o ca rduc bas o ordr o ( - smaor a (.. I abl 5., Prc rlav ffcc (PRE of,, ad ( opmum cas ar compud rspc o. Tabl 5.: PRE of dffr smaors of rspc o.
11 Esmaors PRE (., Populao I Populao II (opmum Tabl 5. clarl sos a suggsd smaor s opmum codo s br a usual ubasd smaor, Bal ad Tuja (99 smaors ad. For purpos of llusrao for o-pas samplg, cosdr follog populaos: Populao III: Mur (967 : Oupu : umbr of orkrs.354,. 9484, ρ. 95, 8,, 8. Populao IV: Sl ad Torr(96.483,. 7493, ρ. 4996, 3,, 4. I abl 5.3 valus of scalars 's(,, Tabl 5.3: Valus of 's(,, ar lsd. Scalars Populao I Populao II
12 o ordr ( Usg s valus of 's(,, o smaor a 5.3. gv abl 5.3 o ca rduc bas I abl 5.4 prc rlav ffcc (PRE of, d, d ad ( opmum cas ar compud rspc o. Tabl 5.4: PRE of dffr smaors of rspc o. Esmaors PRE (., Populao I Populao II d d Rfrcs Bal, S. ad Tuja, R.K. (99: Rao ad produc p poal smaor. Iformao ad opmzao sccs, (, ocra (977:
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