Rajs Sg, Pakaj aua, rmala Saa Scool of Sascs, DAVV, Idor (M.P., Ida Flor Smaradac Uvrs of Mco, USA Almos Ubasd Epoal Esmaor for F Populao Ma Publsd : Rajs Sg, Pakaj aua, rmala Saa, Flor Smaradac (Edors AUXILIARY IFORMATIO AD A PRIORI VALUES I OSTRUTIO OF IMPROVED ESTIMATORS Rassac Hg Prss, A Arbor, USA, 7 (ISB-: -59973-46-4 (ISB-3: 978--59973-46- pp. 4-53
Absrac I s papr av proposd a almos ubasd rao ad produc p poal smaor for f populao ma Y. 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. 4
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 smpl radom samplg ou rplacm (SRSWOR o sma populao ma of caracr udr sud. L (, b sampl ma smaor of ( Y, X populao mas of ad rspcvl. I ordr o av a surv sma of populao ma Y 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( = Y K (.3 MSE (.4 4 ( = Y K B( = Y K (.5 4
MSE (.6 4 ( = Y K r S =, ( ( Y S =, ( ( X S =, Y S = X, K = ρ, S ρ =, ( S S S = ( ( Y( X. 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 Y.. 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 Y. 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 = Y(, X( suc a =. E ( =E ( =. 43
44 ( E =, ( E =, ( E ρ =. Eprssg rms of s, av ( = p p Y (.3 Epadg rg ad sd of (.3 ad rag rms up o scod pors of s, av ( = 8 8 Y (.4 Takg pcaos of bo sds of (.4 ad subracg Y from bo sds, g bas of smaor, up o frs ordr of appromao as ( ( = K 4 Y ( B (.5 From (.4, av ( Y Y (.6 r = -. (.7 Squarg bo sds of (.7 ad akg pcaos, g MSE of smaor, up o frs ordr of appromao, as = K 4 Y 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 m.mse( Y ( ρ = (. c s sam as a of radoal lar rgrsso smaor. From (.7 ad (.9, av - = = 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 45
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 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 Y, rspcvl, as = p (3. d = p (3. d To oba bas ad MSE of d ad d, r suc a = Y(, = X(, = X( ( = E( = E( = E 46
ad E ( =, f E ( =, f E ( =, f E( fρ =, E(, = f ρ E ( =. f r f =, f =. Follog sadard procdur oba B ( d = Yf 3 ρ 8 (3.3 B ( d = Yf 3 ρ 8 (3.4 MSE ( = ρ d Y f f 3 4 (3.5 MSE ( = ρ d Y 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 47
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 Y. B dfo, s W s a lar var f W = W. (4. for =, R. (4. r (,, umbrs. = dos sascal cosas ad R dos s of ral To oba bas ad MSE of, usg oaos of sco 3 ad prssg rms of s, av = Y( p p (4.3 = Y[ 8 8 4 4 ( ( ( ( ] (4.4 r =. (4.5 Takg pcaos of bo sds of (4.4 ad subracg Y from bo sds, g bas of smaor, up o frs ordr f appromao as Bas ( = Yf 3 ρ (4.6 8 From (4.4, av 48
( Y (4.7 Squarg bo sds of (4.7 ad akg pcao, g MSE of smaor, up o frs ordr of appromao, as MSE c s mmum ( = Y f f K 3 (4.8 4 = K. (4.9 Tus mmum MSE of s gv b m.mse( [ f f ρ ] = Y 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 49
= 8K = K 4K = K 4K Us of s 's(,, = rmovs bas up o rms of ordr ( o a (4.. (4.4 5. Emprcal sud T daa for mprcal sud ar ak from o aural populao daa ss cosdrd b ocra (977 ad Rao (983. Populao I: ocra (977 =.477, =.445, ρ =. 887. Populao II: Rao (983 =.46, =.8, ρ =. 736. I abl (5., valus of scalar s ( =,, ar lsd. Tabl (5.: Valus of s ( =,, Scalars Populao I II -.65 -.93.4985 8.6.779 3.3 Usg s valus of s ( =,, gv abl 5., o ca rduc bas o ordr o ( - smaor a (.. 5
I abl 5., Prc rlav ffcc (PRE of,, ad ( opmum cas ar compud rspc o. Tabl 5.: PRE of dffr smaors of Y rspc o. Esmaors PRE (., Populao I Populao II 7.75 3.55 47.7 6.8 (opmum 468.97 98.4 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(,, = ar lsd. 5
Tabl 5.3: Valus of 's( =,, Scalars Populao I Populao II.659.45.88.73.5.687 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 Y rspc o. Esmaors PRE (., Populao I Populao II 8.7 74.68 d 4.4 3.64 d 38.7 6. 5
Rfrcs Bal, S. ad Tuja, R.K. (99: Rao ad produc p poal smaor. Iformao ad opmzao sccs, (, 59-63. ocra, W. G. (977: Samplg cqus. Trd do Wl ad Sos, York. Hdroglou, M. A. ad Sardal,.E. (998: Us of aular formao for o-pas samplg. Surv Modolog, 4(, ad. Mur, M.. (967: Samplg Tor ad Mods. Sascal Publsg Soc, alcua, Ida. Rao, T. J. (983: A class of ubasd produc smaors. Tc. Rp. o. 583, Sa. Ma. Ida Sascal Isu, alcua, Ida. Sg, R. ad Sg, J. (6: Sparao of bas smaors of populao ma usg aular formao. Jour. Rajasa Acad. P. Scc, 5, 3, 37-33. Sg, S. ad Sg, R. (993: A mod: Almos sparao of bas prcpas sampl survs.. Jour. Id. Sa. Assoc., 3, 99-5. Sl, R. G. D. ad Torr, J. H. (96: Prcpls ad procdurs of sascs, Mc Gra Hll, York. 53