IMPUTATION USING REGRESSION ESTIMATORS FOR ESTIMATING POPULATION MEAN IN TWO-PHASE SAMPLING

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1 Joural of Rlal ad asal uds; I (Pr: , (Ol:9- ol., Issu (0: - IPUAIO UIG RGRIO IAOR FOR IAIG POPUAIO A I WO-PHA APIG ardra gh hakur, Kalpaa adav ad harad Pahak r for ahmaal s (, Baashal Uvrs, Rajasha, Ida, P-00 parm of ahmas ad ass, r. H.. Gour ral Uvrs, agar (.P., Ida, P al: sharadpahaksas@ahoo.om. (Rvd a 0, 0 Asra hs papr prss h smao of ma prs of mssg daa udr wo-phas samplg dsg usg rgrsso smaors as a ool for mpuao whl h sz of rspodg ( R ad o-rspodg R group s osdrd as a radom varal. h as ad ma squard rror of suggsd smaors ar drvd h form of populao paramrs usg h op of larg sampl approxmao. umral sud s prformd ovr wo populaos usg h xprssos of as ad ma squard rror ad ff ompard wh xsg smaors. K Words: smao, ssg daa, Rgrsso smaors, Bas, a squard rror (, wo-phas samplg, RWOR, arg sampl approxmaos.. Iroduo A qusoar oas ma qusos ha w all ms. Wh m orspos ours, susaal formao aou h o-rspod s usuall avalal from ohr ms o h qusoar. a mpuao mhods lraur ar usd slo of hs ms as auxlar varal assgg valus o h h orspod for m. Rao ad r (99, gh ad Hor (000, Ahmd al. (00 ad hukla ad hakur (008 hav gv applaos of varous mpuao produrs for ma ad vara smao. hukla al. (009a hav gv h op of ulzao of (populao ma of o-rspos group of mpuao for mssg osrvaos of auxlar formao du o o-rspos. hukla al. (0 hav dsussd o h lar omao asd mpuao mhod for mssg daa for auxlar formao h sampl. hukla al. (0 hav gv h op of us of h mxur of,, mpuao for mssg osrvaos of auxlar formao du o o-rspos. hakur al. (0 hav prsd h smao of ma prs of mssg daa udr wo-phas samplg shm. hakur al. (0 ad hav gv som mpuao mhods doul samplg shm for smao of populao ma ad hukla al. (009 proposd h smao of ma wh mpuao of mssg daa usg faor- p smaor wophas samplg. hukla al. (0 hav dsussd o a rasformd smaor for smao of populao ma wh mssg daa sampl-survs. hukla al. (0a a smaor for ma smao prs of masurm rror. hukla

2 Joural of Rlal ad asal uds, mr 0, ol. ( al. (0 hav dsussd o smao of populao ma usg wo auxlar sours sampl survs. hakur al. (0a advoad o ma smao wh mpuao wo-phas samplg. hukla al. (0 dsussd o smao of ma usg mprovd rao-um-produ p smaor wh mpuao for mssg daa. om ohr usful ad rsg oruos ar du o Bhusha al. (008, Barj ad war (0, gh al. (0. h varal s of ma rs ad a auxlar varal orrlad wh ad h populao ma of auxlar varal s ukow. A larg prlmar smpl radom sampl (whou rplam of us s draw from h Ω,,..., o sma ad a sodar sampl of sz ( populao draw as a su-sampl of h sampl o sma h populao ma of ma varal. h sampl oas rspodg us ad ( orspodg us. Usg h op of pos-srafao, sampl ma dvdd o wo groups: rspodg ( R ad o-rspodg ( R. h sampl ma osdrd as srafd o wo lasss aml a rspos lass ad o-rspos lass, ad h h produr s kow as pos-srafao. ukham al. (98 advoad ha pos-srafao hqu s as prs as h srafd samplg hqu udr proporoal alloao f h sampl sz s larg ough. ow ma osdr h populao has wo ps of dvduals lk as umr of rspods ( R ad o-rspods ( R, hus h oal us of h populao wll omprs ad, rspvl, suh ha. h populao proporos of R ad R groups ar xprssd as W / ad W / suh ha W W. Furhr, l ad h populao mas of ad rspvl. For vr u R, h valu s osrvd avalal. Howvr, for h us R, h s ar mssg ad mpud valus ar o drvd. h h valu x of auxlar vara s usd as a sour of mpuao for mssg daa wh R. hs s o assum ha for sampl, h daa x { x : } ar kow. s h followg oaos ar usd hs hapr: x, : h sampl ma of ad rspvl ; x, : h sampl ma of ad rspvl R ;, : h populao ma squars of ad rspvl;, : h off of varao of ad rspvl; : orrlao off populao w ad rspvl. Furhr, osdr fw mor smol rprsaos: ( ( W ( W, W ( (,

3 Impuao Usg Rgrsso smaors ( ( ( ( ( ( Q.. arg sampl approxmao ( ; x ( ; x ( ad x (, whh x x x mpls h rsuls ; ; ad. ow usg h op of wo-phas samplg ad h mhasm of AR, for gv, ad [s Rao ad r (99] w hav: 0 ( ad ( ( ( 0 Also, ; mlarl, ( ; ( ; ( ( ( ( x ( / ( ; ; ( ; ; ;. Proposd dffr mpuao mhods j dos h h avalal osrvao for h j h mpuao. W suggs h followg mpuao mhods: f R ( (. [ ] a x x f R whr a s a osa, suh ha h vara of h smaor s mmum. Udr hs, h po smaor of s ( x a x (. ;

4 Joural of Rlal ad asal uds, mr 0, ol. ( ( (. x x ( W f f R R whr s a osa, suh ha h vara of h smaor s mmum. Udr hs, h po smaor of s ( x x (. f R ( x x f R ( W whr s a osa, suh ha h vara of h smaor s mmum. (. Udr hs, h po smaor of s ( x x (.. Bas ad ma squard rror ( of proposd mhods B(. ad (. do h as ad ma squard rror ( of a smaor udr a gv samplg dsg. h proprs of smaors ar drvd h followg horms rspvl. horm. ( h smaor rms of,, ad s : ( a( (. Proof: a xx ( ( a( s a uasd smaor,.. [ ] 0 Proof: B( 0 B (. ( h vara of upo frs ordr of approxmao ould wr as ( ( a a (. Proof: ( [ a( ] [ ] a a a ( a(

5 Impuao Usg Rgrsso smaors a a ( h mmum vara of h s, wh a (. Proof: B dffrag (. wh rsp o a ad qua o zro 0 da d a Afr rplag valu of a (., w oad horm. ( h smaor rms of s ad,, : (. Proof: x x ( h smaor s uasd,.. 0 B (. Proof: 0 B (7 h vara of s (.7 Proof: (8 h mmum vara of h s (.8 Proof: B dffrag (.7 wh rsp o ad qua o zro 0 d d Afr rplag valu of (.7, w oad

6 Joural of Rlal ad asal uds, mr 0, ol. ( horm. (9 h smaor rms of s ad,, : (.9 Proof: x x (0 h as smaor s 0 B (.0 Proof: 0 B ( h vara of s (. Proof: ( h mmum vara of h s (. Proof: B dffrag (. wh rsp o ad qua o zro 0 d d Afr rplag valu of (., w oad. omparso I hs so w drvd h odos udr whh h suggsd smaors ar supror o h Ahmd al. (00. ( m m

7 Impuao Usg Rgrsso smaors 7 s r ha, f 0 > 0 > 0 > ± whr ad ( m m s r ha, f 0 > 0 > ± whr ( m m

8 8 Joural of Rlal ad asal uds, mr 0, ol. ( s r ha, f 0 > Q ± Q Q whr > Q >. umral Illusraos W osdrd wo populaos A ad B, frs o s h arfal populao of sz 00 [sour hukla al. (009a] ad aohr o s from Ahmd al. (00 wh h followg paramrs: Populao A B al. Paramrs of Populaos A ad B 0, 0, for populao A ad 000, 00, 0 for populao B rspvl. h h as ad of suggsd smaors (usg h xprssos of as ad of o ad Ahmd al. (00 mhods ar gv al. ad. for populao A ad B rspvl. smaors Populao A Populao B Bas Bas al. Bas ad for Populao A ad B

9 Impuao Usg Rgrsso smaors 9 smaors Populao A Populao B Bas Bas r RA OP al. Bas ad for Populao A ad B for Ahmd al. (00 h samplg ff of suggsd smaors ovr Ahmd al. (00 s dfd as: [ ( ] [ ( ] Op ;,, (. Op h ff for populao A ad populao B ar gv al. ff Populao A Populao B al. ff for Populao A ad B ovr Ahmd al. (00 7. susso ad olusos: h form rgrsso p smaors ar usd as a sour of mpuao udr h sup of wo-phas samplg udr h assumpo ha, h auxlar populao ma s ukow ad h szs of h rspod ad o-rspod group ar osdrd as radom varal. om srags ar suggsd o ad h smaor of populao ma drvd. Proprs of drvd smaors lk as ad ar dsussd h o. h suggsd smaors ar uasd ad h opmum valu of paramrs of suggsd smaors s oad as wll sam so. Ohr xsg smaors ar osdrd for omparso purpos ad wo populaos A ad B usd for umral sud frs o from hukla al. (009 ad aohr o s from Ahmd al. (00. h samplg ff of suggsd smaor ovr Ahmd al. (00 s oad ad suggsd srag s foud vr los wh Ahmd al. (00 wh s o kow. h proposd smaors ar usful wh som osrvaos ar mssg h samplg ad populao ma of

10 0 Joural of Rlal ad asal uds, mr 0, ol. ( auxlar formao s ukow. For populao A proposd smaors s foud o mor ff ha h xsg smaors ad ad rsuls ar also vr los wh Ahmd smaors. For populao B proposd smaors,, ar foud o mor ff ha h xsg smaors. Akowldgm Auhors ar hakful o h doral Board of JR ad rfrs for hr valual suggsos ad for rommdg h mausrp for pulao. Rfrs. Ahmd,.., Al-, O., Al-Raw, Z. ad Au-ah, W. (00. smao of a populao ma usg dffr mpuao mhods, ass raso, 7 (, p Barj, J. ad war, (0. Improvd rao p smaor usg jakkf mhod of smao, Joural of Rlal ad asal uds, (, p. -.. Bhusha,., Pad, Aad Kaara,. (008. A lass of smaors doul samplg usg wo auxlar varals, Joural of Rlal ad asal uds, (, p Rao, J.. K. ad r, R. R. (99. ara smao udr wo-phas samplg wh applao o mpuao for mssg daa, Bomra, 8, p hukla,. ad hakur,.. (008. smao of ma wh mpuao of mssg daa usg faor-p smaor, ass raso, 9 (, p hukla,., Pahak,. ad hakur,.. (0. A rasformd smaor for smao of populao ma wh mssg daa sampl-survs, Joural of urr grg Rsarh, (. 7. hukla,., Pahak,. ad hakur,.. (0a. A smaor for ma smao prs of masurm rror, Rsarh & Rvws: A Joural of ass, (. 8. hukla,., Pahak,. ad hakur,.. (0. smao of populao ma usg wo auxlar sours sampl survs, ass rasow srs, (, p hukla,., hakur,.. ad hakur,.. (0. Ulzao of mxur of, ad mpuao for mssg daa pos-srafao, Afra Joural of ahmas ad ompur Rsarh, (, p hukla,., hakur,.., Pahak,. ad Rajpu.. (009. smao of ma wh mpuao of mssg daa usg faor- p smaor wophas samplg, ass raso, 0 (, p hukla,., hakur,.., Pahak,. ad adav, K. (0. smao of ma usg mprovd rao-um-produ p smaor wh mpuao for mssg daa, Iraoal Joural of ahmas ad ompuaoal hods & holog,,.. hukla,., hakur,.., hakur,.. (009a. Ulzao of o-rspos auxlar populao ma mpuao for mssg osrvaos, Joural of Rlal ad asal uds, (, p. 8-0.

11 Impuao Usg Rgrsso smaors. hukla,., hakur,.., hakur,.. ad Pahak,. (0. ar omao asd mpuao mhod for mssg daa sampl, Iraoal Joural of odr grg Rsarh, (, p gh,. ad Hor,. (000: ompromsd mpuao surv samplg, rka,, p gh, R., alk,., haudhar,. K., rma, H. K. ad Adwara, A. A. (0. A gral faml of rao-p smaors ssma samplg, Joural of Rlal ad asal uds, (, p ukham, P.., ukham, B.., ukham,. ad Ashok,. (98. amplg hor of urvs wh Applaos, Iowa a Uvrs Prss, I..A.. Pulao, w lh. 7. hakur,.., adav, K. ad Pahak,. (0. smao of ma prs of mssg daa udr wo-phas samplg shm, Joural of Rlal ad asal uds, (, p hakur,.., adav, K. ad Pahak,. (0. om mpuao mhods doul samplg shm for smao of populao ma, Iraoal Joural of odr grg Rsarh, (, p hakur,.., adav, K. ad Pahak,. (0a. a smao wh mpuao wo-phas samplg, Iraoal Joural of odr grg Rsarh, (, p. -7.

Almost unbiased exponential estimator for the finite population mean

Almost unbiased exponential estimator for the finite population mean 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