Mixture Ratio Estimators Using Multi-Auxiliary Variables and Attributes for Two-Phase Sampling

Transcription

1 Opn Journal of Sascs Publshd Onln Ocobr 04 n Scs hp://scrporg/ournal/os hp://ddoorg/0436/os Mur ao Esmaors Usng Mul-Aular Varabls and Arbus for To-Phas Samplng Paul Mang Waru John Kung u Jams Kahr Dparmn of Sascs and Acuaral Scnc Knaa Unvrs Narob Kna Emal: Warupaul@ahoocom ohnkungu08@ahoocom cvd 5 Jul 04; rvsd 7 Augus 04; accpd Spmbr 04 Coprgh 04 b auhors and Scnfc sarch Publshng Inc Ths ork s lcnsd undr h Crav Commons Arbuon Inrnaonal Lcns (CC BY hp://cravcommonsorg/lcnss/b/40/ Absrac In hs papr hav proposd hr classs of mur rao smaors for smang populaon man b usng nformaon on aular varabls and arbus smulanousl n o-phas samplng undr full paral and no nformaon cass and analzd h proprs of h smaors A smulad sud as carrd ou o compar h prformanc of h proposd smaors h h sng smaors of fn populaon man I has bn found ha h mur rao smaor n full nformaon cas usng mulpl aular varabls and arbus s mor ffcn han man pr un rao smaor usng on aular varabl and on arbu rao smaor usng mulpl aular varabl and mulpl aular arbus and mur rao smaors n boh paral and no nformaon cas n o-phas samplng A mur rao smaor n paral nformaon cas s mor ffcn han mur rao smaors n no nformaon cas Kords ao Esmaor Mulpl Aular Varabls Mulpl Aular Arbus To-Phas Samplng B-Sral Corrlaon Coffcn Inroducon Th hsor of usng aular nformaon n surv samplng s as old as hsor of h surv samplng Th ork of Nman [] ma b rfrrd o as h nal orks hr aular nformaon has bn usd Cochran [] usd aular nformaon n sngl-phas samplng o dvlop h rao smaor for smaon of populaon man In h rao smaor h sud varabl and h aular varabl had a hgh posv corrlaon and h rgrsson ln as passng hrough h orgn Hansn and Hurz [3] also suggsd h us of aular Ho o c hs papr: Waru PM Kung u J and Kahr J (04 Mur ao Esmaors Usng Mul-Aular Varabls and Arbus for To-Phas Samplng Opn Journal of Sascs hp://ddoorg/0436/os

2 P M Waru al nformaon n slcng h sampl h varng probabls If rgrsson ln s sll lnar bu dos no pass hrough h orgn h rgrsson smaor s usd Wason [4] usd h rgrsson smaor of laf ara on laf gh o sma h avrag ara of h lavs on a plan Olkn [5] as h frs prson o us nformaon on mor han on aular varabl hch as posvl corrlad h h varabl undr sud usng a lnar combnaon of rao smaor basd on ach aular varabl a [6] suggsd a mhod of usng mul-aular nformaon n sampl surv Sngh [7] proposd a rao-cum-produc smaor and s mul-varabl prsson Th concp of doubl samplng as frs proposd b Nman [] n samplng human populaons hn h man of aular varabl as unknon I as lar ndd o mul-phas b obson [8] Ahmad [9] proposd a gnralzd mulvara rao and rgrsson smaors for mul-phas samplng hl Zahoor Muhhamad and Munr [0] suggsd a gnralzd rgrsson-cum-rao smaor for o-phas samplng usng mulpl aular varabls Jha Sharma and Grovr [] proposd a faml of smaors usng nformaon on aular arbu Th usd knon nformaon of populaon proporon possssng an arbu (hghl corrlad h sud varabl Y Th arbu ar normall usd hn h aular varabls ar no avalabl g amoun of mlk producd and a parcular brd of co or amoun of ld of ha and a parcular var of ha Jha Sharma and Grovr [] usd h nformaon on aular arbus n rao smaor n smang populaon man of h varabl of nrs usng knon arbus such as coffcn of varaon coffcn kuross and pon b-sral corrlaon coffcn Th smaor prformd br han h usual sampl man and Nak and Gupa [] smaor Jha Sharma and Grovr [] also usd h aular arbu n rgrsson produc and rao p ponnal smaor follong h ork of Bahl and Tua [3] Hanf Ha and Shahbaz [4] proposd a gnral faml of smaors usng mulpl aular arbu n sngl- and doubl-phas samplng Th smaor had a smallr MSE compard o ha of Jha Sharma and Grovr [] Th also ndd hr ork o rao smaor hch as gnralzaon of Nak and Gupa [] smaor n sngl- and doubl-phas samplngs h full nformaon paral nformaon and no nformaon Kung u and Odongo [5] and [6] proposd rao-cum-produc smaors usng mulpl aular arbus n sngland o-phas samplng Mon Shahbaz and Hanf [7] proposd a class of mur rao and rgrsson smaors for sngl-phas samplng for smang populaon man b usng nformaon on aular varabls and arbus smulanousl In hs papr ll nd h mur rao smaor proposd b Mon Shahbaz and Hanf [7] n sngl-phas samplng o o-phas samplng undr full paral and no nformaon cas srags nroducd b Samuddn and Hanf [8] and also ncorpora Arora and Bans [9] approach n rng don h man suard rror Prlmnars Noaon and Assumpon Consdr a populaon of N uns L Y b h varabl for hch an o sma h populaon man and X X Xk ar k aular varabls For o-phas samplng dsgn l n and n ( n < n ar sampl h szs for frs and scond phas rspcvl ( and ( dno h aular varabls form frs and scond phas sampls rspcvl and dno h varabl of nrs from scond phas X and C h dno h populaon mans and coffcn of varaon of aular varabls rspcvl and dnos h populaon corrlaon coffcn of Y and X Furhr l θ θ ( θ < θ n N n N hr and ( ( Y + X + ( ( ( ( ( + ( ( ( and X k ar samplng rror and ar vr small W assum ha ( ( ( ( ( (0 E E E 0 ( 777

3 P M Waru al Consdr a sampl of sz n dran b smpl random samplng hou rplacmn from a populaon of sz N L and dnos h obsrvaons on varabl and r rspcvl for h h un hr k+ k+ In dfnng h arbus assum compl dchoom so ha L N A and a h h f un of populaon possss aular arbu ( 0 ohrs n b h oal numbr of uns n h populaon and sampl rspcvl pos- A a sssng arbu L P and p ( b h corrspondng proporon of uns possssng a spcfc N n arbus and s h man of h man varabl a scond phas L p ( and p ( dno h h aular arbu form frs and scond phas sampls rspcvl and dno h varabl of nrs from scond phas Th man of man varabl of nrs a scond phas ll b dnod b Also l us dfn hr z L Also v z and ( ( hn X ( hn ( ( p P p P (3 ( ( ( ( ar samplng rror and ar vr small W assum ha ( ( ( ( ( E E E 0 ( ( X ( ( v Thrfor v + Smlarl X X X X ( ( ( ( ( ( ( ( ( ( ( v + ( ( + X X X W shall ak v o rm of ordr ( ( ( ( + X X Smlarl ( ( X + X n as ( ( v hnc X Th coffcn of varaon and corrlaon coffcn ar gvn b u ( + P ( ( (4 P S S S S C C C Y X P SS Sz S S z Pb and Pb SS SS SS z (5 Thn for smpl random samplng hou rplacmn for boh frs and scond phass r b usng phas s opraon of pcaons as: 778

4 P M Waru al ( θ ( ( ( ( ( ( ( ( θ θ θ θ ( ( θ ( ( θ Pb ( ( ( ( ( ( θ ( ( ( ( ( ( θ θ θ ( ( ( ( ( ( ( ( ( ( θ θ PC E θ XXC C ( ( ( ( ( ( ( ( ( ( (( ( ( ( ( ( ( ( ; θ θ (( ( ( ( ( ( ( θ θ XC C ;( ( ( ( ( ( ( ( ; θ θ ( ( ( ( ( ( ( ; θ θ E Y C E P C E X C E YX C C E YP C C E YX C C E X C E E θ PPC C E θ θ YX C C Pb E PPC C E X E XXC C E PPC C A A ( C ( T (7 Ad A A ( Arora and La [ 9] Th follong noaons ll b usd n drvng h man suar rrors of proposd smaors : Drmnan of populaon corrlaon mar of varabls and r : Drmnan of h mnor of p corrspondng o h h lmn of : Dnos h mulpl coffcn of drmnaon of on r and r r r : Dnos h mulpl coffcn of drmnaon of on and : Drmnan of populaon corrlaon mar of varabls r and r r : Drmnan of populaon corrlaon mar of varabls and : Drmnan of h corrlaon mar of r and r : Drmnan of h corrlaon mar of and : Drmnan of h mnor corrspondng o r r ( and : Drmnan of h mnor corrspondng o of h corrlaon mar of of h corrlaon mar of Man pr Un n To-Phas Samplng ( (6 (8 and (9 Th sampl man usng smpl random samplng hou rplacmn n o-phas samplng s gvn b s gvn b n n (0 779

5 P M Waru al hl s varanc s gvn ( θ Var ( Y C 3 ao Esmaor Usng Aular Varabl n To-Phas Samplng L n n b h sampl man of h aular varabl n o-phas samplng Th rao smaor hn nformaon on on aular varabls s avalabl for populaon (full nformaon cas s: V α X Th man suar rror of V can b rn as: ( V θy ( C α C αcc ( MSE + (3 C hr α and ar h opmum valu and h corrlaon coffcn rspcvl C 4 ao Esmaor Usng Mulpl Aular Varabls n To-Phas Samplng Th rao smaor b Ha [9] hn nformaon on k aular varabls s avalabl for populaon (full nformaon cas s: Th opmum valus of unknon consans ar β β βk X X X (4 k MV ( ( ( k Th man suar rror of MV can b rn as: + C k β ( C k ( MV θy C ( k MSE (5 (6 5 ao Esmaor Usng Aular Arbu n To-Phas Samplng In ordr o hav an sma of h populaon man Y of h sud varabl assumng h knoldg of h populaon proporon P Nak and Gupa [] dfnd rao smaor of populaon man hn h pror nformaon of populaon proporon of uns possssng h sam arbu s varabl Nak and Gupa [] proposd h follong smaor: P p A Th MSE of A up o h frs ordr of appromaon ar gvn rspcvl b α ( A θy ( C α CP αcc P Pb (7 MSE + (8 C hr α and Pb ar h opmum valu and h b-sral corrlaon coffcn rspcvl C 6 ao Esmaor Usng Mulpl Aular Arbus n To-Phas Samplng Th rao smaors b Hanf Ha and Shahbaz [4] for o-phas samplng usng nformaon on mulpl 780

6 P M Waru al aular arbus s gvn b Th opmum valus of unknon consans ar γ γ γ P P P (9 MA p ( p ( p ( C + γ ( C Th MSE of h MA up o h frs ordr of appromaon s gvn b In gnral hs smaors hav a bas of ordr h uan bas s s of ordr 3 Mhodolog n ( MA θy C ( Pb MSE (0 ( n Snc h sandard rror of h smas s of ordr n and bcoms nglgbl as n bcoms larg 3 Mur ao Esmaors Usng Mul-Aular Varabl and Arbus for To-Phas Samplng (Full Informaon Cas If sma a sud varabl hn nformaon on all aular varabls s avalabl from populaon s ulzd n h form of hr mans B akng h advanag of mur rao smaors chnu for o-phas samplng a gnralzd smaor for smang populaon man of sud varabl Y h h us of mul aular varabls and arbus ar suggsd as: MVAF( 30 α α αk βk+ βk+ β X X X P P P (30 k k+ k+ ( ( ( p k ( p k ( p + k+ ( Usng (0 (3 and (4 n (30 and gnorng h scond and hghr rms for ach panson of produc and afr smplfcaon r + Y Y Y (3 MVAF( 30 Th man suard rror of MVAF( 30 s gvn b k h+ k ( ( α β X k+ P k h+ k ( ( MSE ( MVAF( 30 E( MVAF( 30 Y E α Y β Y X k+ P W dffrna h Euaon (3 parall h rspc o hn ua o zro usng (6 (7 and (9 g (3 α ( k and β ( k+ k+ α + C ( (33 C C + β ( C Usng normal uaon ha s usd o fnd h opmum valus gvn (3 can r (34 78

7 P M Waru al k h+ k ( ( MSE ( MVAF( 30 E α Y β Y X k+ P Takng pcaon and usng (6 n (35 g (35 k k+ h Y Y MSE ( MVAF( 30 θy C α XCC P β CC Pb X k+ P Subsung h opmum (33 and (34 n (36 g (36 k k h C + C MSE( MVAF( 30 θy C + ( CC ( + CC Pb C C Usng (8 g (37 k k h + MSE( MVAF( 30 θy C + ( + ( Pb k+ (38 MSE ( ( MVAF( 30 θy C ( ( ( ( θ MVA 30 Y C ( MSE 3 Mur ao Esmaor Usng Mul-Aular Varabl and Arbus n To-Phas Samplng (Paral Informaon Cas (39 (30 In hs cas suppos hav no nformaon on all s and aular varabls bu onl for r and g aular varabls from populaon Consdrng mur rao chnu of smang chnu h populaon man of sud varabl Y can b smad for o-phas samplng usng mul-aular varabls and arbus as: MVAP( 3 γ k+ ( k + Pk + α β α β αs βs αr+ αr+ αk ( X ( X ( r X ( s ( r s ( + + k ( ( ( ( ( r ( r ( s ( + s+ ( k λk γk λk γh λh γh γh γ p p( k P p( k h P p( p( p( + + h h+ h+ p ( k + p ( p k + ( k+ p ( p k ( h p ( p h ( p h ( p + + h+ ( (3 Usng (0 (3 and (4 n (3 and gnorng h scond and hghr rms for ach panson of produc and afr smplfcaon r + Y Y + Y + Y MVAP( 3 r ( r ( s+ r k ( ( k+ h ( ( ( α β α γ X X X P + k+ h h+ ( ( ( Yλ Yγ P h+ P Man suard rror of MVAP( 3 smaor s gvn b (3 78

8 P M Waru al MVAP( 3 EE + Y Y + Y + Y r ( r ( s+ r k ( ( k+ h ( ( ( α β α γ X X X P k+ h h+ ( ( ( Yλ + Yγ P h+ P W dffrna h Euaon (33 h rspc o α ( r β ( r α ( + + γ ( + + λ ( + + ( r r k k k h k k h o zro and us (6 (7 and (9 Th opmum valus ar as follos + C m α ( C m ( r ; C + p γ ( C m k+ k+ h; + C r β ( ( r ; C C λ C ( + k+ k+ h; + C s α ( r+ r+ k; C s + C γ ( ψ h+ h+ C p Usng normal uaon ha ar usd o fnd h opmum valus gvn (33 can r MSE ( MVAP( 3 EE + Y Y + Y r ( r ( s+ r k ( ( ( α β α X X X k+ h ( k+ h ( h+ m ( ( ( + Yγ Yλ + Yγ P P h+ P Takng pcaon and usng (6 n (35 g MSE + ( MVAP( 3 ( θ θ γ PC C Pb θ λ PC C Pb + ( θ θ γ PC C Pb (33 γ h+ h+ and ua Y Y Y θ + θ θ α θ β + θ θ α X X X r r r+ s k Y C ( X CC ( X CC X CC r+ Y Y Y P P P + k + k h k+ k+ h+ Usng h opmum valu (34 n (36 g (34 (35 (36 783

9 P M Waru al MSE ( MVAP( 3 Y C + r r + + θ ( θ θ ( θ ( + + r+ s k + k h + + r+ k+ ( θ θ ( ( θ θ ( θ + θ θ γ + k h + + ( Pb ( ( Pb k+ h + Pb (37 MSE MSE r r + + ( MVAP( 3 Y C θ + ( θ θ ( θ ( ( MVAP( 3 r+ s k + k h ( θ θ ( + ( θ θ ( r+ k+ θ + θ θ γ + k h + + ( Pb ( ( Pb k+ h+ r r s k k h + + Y C ( θ θ + ( + ( ( + Pb r+ k + r + k h + ( γ ( ( Pb + θ + + h+ Pb k+ ( MSE ( ( θ θ ( MVAP( 3 Y C ( + ( ( ( r + k h θ r + + ( ( + k ( ( + ( Pb Pb (38 (39 (30 784

10 P M Waru al Usng (8 n (3 g Smplfng (3 g MSE ( ( ( ( Y C MVAP 3 ( θ θ + θ ( ( ( ( ( ( ( ( ( MVAP 3 θ θ ( + θ MSE Y C ( ( ( ( ( ( ( MVAP 3 θ ( + θ MSE Y C (3 (3 (33 33 Mur ao Esmaor Usng Mul-Aular Varabl and Arbus n To-Phas Samplng (No Informaon Cas If sma a sud varabl hn nformaon on all aular varabls s unavalabl from populaon s ulzd n h form of hr mans B akng h advanag of mur rao chnu for o-phas samplng a gnralzd smaor for smang populaon man of sud varabl Y h h us of mul aular varabls and arbus ar suggsd as: MVAN( 3 α α αs γk+ γk+ γ ( ( ( p( p ( p k k k ( + + ( ( ( p k ( k + p ( p k+ ( (34 Usng (0 (3 and (4 n (34 and gnorng h scond and hghr rms for ach panson of produc and afr smplfcaon r k ( k+ h ( ( ( MVAN( 3 + Yα + Yγ (35 X P Man suard rror of MVAN( 3 smaor s gvn b MSE k ( k+ h ( ( ( ( MVAN( 3 E + Yα + Yγ X P W dffrna h Euaon (37 parall h rspc o α ( k and hn ua o zro usng (6 (7 and (9 g (36 β ( k+ k+ + C k α ( (37 C C + β ( (38 C Usng normal uaon ha ar usd o fnd h opmum valus gvn (36 can r k ( k+ h ( ( ( MSE ( MVAN( 3 E + Yα + Yγ X k+ P Takng pcaon and usng (6 n (39 g (39 k k+ h ( ( Y C MVAN 3 θ + ( θ θ α CC ( + θ θ β CC Pb k+ MSE (

11 P M Waru al k k+ h MSE( MVAN( 3 θy C θ + ( θ θ ( + ( θ θ ( Pb ( ( ( MSE( MVAN( 3 θy C ( θ θ + ( + θ ( θ + ( ( MSE θ θ ( ( Y C MVAN 3 + ( ( + θ ( ( ( Usng (8 n (334 g Smplfng (335 g MSE ( ( ( Y C MVAN 3 ( θ θ + θ ( ( ( ( Y C ( ( MVAN 3 θ θ ( + θ MSE ( ( ( ( ( MVAN 3 θ ( + θ MSE Y C (33 (33 (333 (334 (335 ( Bas and Conssnc of Mur ao Esmaors Ths mur rao smaors usng mulpl aular varabls and arbus n o-phas samplng ar basd Hovr hs bass ar nglgbl for larg sampls ha s ( n 30 I s asl shon ha h mur rao smaors ar conssn smaors usng mulpl aular varabls snc h ar lnar combnaons of conssn smaors follos ha h ar also conssn 4 Smulaon sul and Dscusson W carrd ou daa smulaon prmns o compar h prformanc of mur rao smaors usng mulpl aular varabls and arbus n o-phas samplng h rao smaor usng on aular varabl and on aular arbu or rao smaor usng mulpl aular varabl or mulpl aular arbus n o-phas samplng smaors for fn populaon All h rsuls r oband afr carrng ou o hundrd smulaons and akng hr avrag Sud varabl N 800 n 96 n 34 man 50 sandard dvaon 6 For rao smaor h aular varabl s posvl corrlad h h sud varabl and h ln passs hrough h orgn N 500 n 96 n 34 man 9 sandard dvaon 67 N 500 n 96 n 34 man 38 sandard dvaon 86 Corrlaon coffcns For rao smaor h aular arbus s posvl corrlad h h sud varabl and h ln passs hrough h orgn N 500 n 96 n 34 man

12 P M Waru al N 500 n 96 man 0486 Corrlaon coffcns In ordr o valua h ffcnc gan could achv b usng h proposd smaors hav calculad h varanc of man pr un and h man suard rror of all smaors hav consdrd W hav hn calculad prcn rlav ffcnc of ach smaor n rlaon o varanc of man pr un W hav hn compard h prcn rlav ffcnc of ach smaor h smaor h h hghs prcn rlav ffcnc s consdrd o b h mor ffcn han h ohr smaors Th prcn rlav ffcnc s calculad usng h follong formula ( Y ˆ ( ˆ Var ff 00 (40 MSE Tabl shos prcn rlav ffcnc of proposd smaor h rspc o man pr un smaor for sngl-phas samplng I s vr clar from Tabl ha our proposd mur rao smaor usng mulpl aular varabls and mulpl aular arbus smulanousl s h mos ffcn compard o rao smaor usng on aular varabl and on aular arbu or rao smaor usng mulpl aular varabl or mulpl aular arbus n o-phas samplng Tabl compars h ffcnc of full nformaon cas and paral cas o no nformaon cas and full o paral nformaon cas of proposd mur rao smaors I s obsrvd ha h full nformaon cas and paral nformaon cas ar mor ffcn han no nformaon cas bcaus h hav hghr prcn rlav ffcnc han no nformaon cas In addon h full nformaon cas s mor ffcn han h paral nformaon cas bcaus has a hghr prcn rlav ffcnc han paral nformaon cas 5 Concluson Th proposd mur rao smaor undr full nformaon cas s rcommndd for smang h fn populaon man snc s h mos ffcn smaor compard o man pr un rao smaor usng on aular varabl rao smaor usng on aular arbu rao smaor usng mulpl aular varabl and rao smaor usng mulpl aular arbus n o-phas samplng In cas som aular varabls or arbus ar unknon rcommnd mur rao smaor undr paral nformaon cas snc s mor ( Y ˆ Tabl lav ffcnc of sng and proposd smaor h rspc o man pr un smaor for o-phas samplng Esmaor lav prcn ffcnc h rspc o man pr un n o-phas samplng var ( 00 V 46 A 85 MV MA 33 (proposd MVAF( Tabl Comparsons of full paral and no nformaon cass for proposd rao-cum-produc smaor usng mulpl aular varabls Populaon Prcn rlav ffcnc of full and paral o no nformaon Prcn rlav ffcnc of full o paral n formaon cas Esmaor MVAN( 3 MVAP( 3 MVAF( 30 MVAP( 3 MVAF( 30 lav prcn ffcnc

13 P M Waru al ffcn han h mur rao smaor undr no nformaon cas and f all ar unknon rcommnd h mur rao smaor undr no nformaon cas o sma fn populaon man frncs [] Nman J (938 Conrbuon o h Thor of Samplng Human Populaons Journal of h Amrcan Sascal Assocaon hp://ddoorg/0080/ [] Cochran WG (940 Th Esmaon of h Ylds of h Cral Eprmns b Samplng for h ao of Gran o Toal Produc Journal of Agrculural Scnc hp://ddoorg/007/s [3] Hansn MH and Hurz WN (943 On h Thor of Samplng from Fn Populaons Annals of Mahmacal Sascs hp://ddoorg/04/aoms/ [4] Wason DJ (937 Th Esmaon of Laf Aras Journal of Agrculural Scnc hp://ddoorg/007/s x [5] Olkn I (958 Mulvara ao Esmaon for Fn Populaon Bomrka hp://ddoorg/0093/bom/45-54 [6] a D (965 On a Mhod of Usng Mul-Aular Informaon n Sampl Survs Journals of h Amrcan Sascal Assocaon hp://ddoorg/0080/ [7] Sngh MP (967 ao-cum-produc Mhod of Esmaon Mrka 34-4 hp://ddoorg/0007/bf06348 [8] obson DS (95 Mulpl Samplng of Arbus Journal of h Amrcan Sascal Assocaon hp://ddoorg/0080/ [9] Ahmad Z (008 Gnralzd Mulvara ao and grsson Esmaors for Mul-Phas Samplng Unpublshd PhD Thss Naonal Collg of Busnss Admnsraon and Economcs Lahor [0] Zahoor A Muhhamad H and Munr A (009 Gnralzd grsson-cum-ao Esmaors for To Phas Samplng Usng Mulpl Aular Varabls Paksan Journal of Sascs [] Jha HS Sharma MK and Grovr LK (006 A Faml of Esmaor of Populaon Man Usng Informaon on Aular Arbus Paksan Journal of Sascs [] Nak VD and Gupa PC (996 A No on Esmaon of Man h Knon Populaon of Aular Characr Journal of h Indan Soc of Agrculural Sascs [3] Bahl S and Tua K (99 ao and Produc Tp Esmaor Informaon and Opmzaon Scncs [4] Hanf M Ha IU and Shahbaz MQ (009 On a N Faml of Esmaor Usng Mulpl Aular Arbus World Appld Scnc Journal 49-4 [5] Kung u J and Odongo L (04 ao-cum-produc Esmaor Usng Mulpl Aular Arbus n Sngl Phas Samplng Opn Journal of Sascs hp://ddoorg/0436/os [6] Kung u J and Odongo L (04 ao-cum-produc Esmaor Usng Mulpl Aular Arbus n To-Phas Samplng Opn Journal of Sascs hp://ddoorg/0436/os [7] Mon M Shahbaz Q and Hanf M (0 Mur ao and grsson Esmaors Usng Mul-Aular Varabl and Arbus n Sngl Phas Samplng World Appld Scncs Journal [8] Smuddn M and Hanf M (007 Esmaon of Populaon Man n Sngl and To Phas Samplng h or hou Addonal Informaon Paksan Journal of Sascs [9] Arora S and Lal B (989 N Mahmacal Sascs Saa Prakashan N Dlh 788

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