Mean Estimation with Imputation in Two- Phase Sampling

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Rosamond McKenzie
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1 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: a smao w Impuao wo- Pas amplg aa g au Kalpaa aav aa Paa * fo amacal ccs () aasal Uvsaasal ajasa * pam of amacs a ascs. H.. Gou al Uvs aga (.P.) Absac: ssg aa s a poblm cou almos v aa collco acv bu paculal sampl suv. mssg aa auall occus sampl suvs w som o all samplg us fus o uabl o pacpa suv o w aa fo spcfc ms o a qusoa compl fo a ows coopag u a mssg. Impuao s a moolog wc uss avalabl aa as a ool fo placm of mssg obsvaos. Impuao mos us o fll o sposs a la u f coos o suabl fc. s acl suggss som mpuao mos a scusss pops of ma smaos. umcal su s pfom ov wo populaos usg pssos of bas a m.s. a ffcc compa w sg smaos. I. Iouco I lau sval mpuao cqus a scb som of m a b ov os. ub (6) ass cocps: OA (obsv a aom) A (mssg a aom) a P (paamc sbuo). H f a f pobabl of obsv mssgss pa gv obsv a uobsv aa os o p o valu of uobsv aa aa a A. obsv aa a obsv a aom (OA) f fo ac possbl valu of mssg aa a paam φ cooal pobabl of obsv pa of mssg aa gv mssg aa a obsv aa s sam fo all possbl valus of obsv aa. Hz a asu (6) av sgus mag of A a A a v c wa. I wa follows A (mssg compll a aom) s us. l a ub () f ff classs of mssgss. f ms us scussg mssgss lau. aa mssg o a obsv a aom (OA) f mssgss o s o a fuco of. Pas ao wa f ms mssgss o aa a o OA. aa mssg o a mssg a aom (A) f mssgss o s o a fuco of. Pas ao wa f ms mssgss o aa a o A. aa a ssg ompll a aom (A) f mssgss o s ula o o. I o wos AOA A. If aa a A o a las A mssg aa mcasm s cos goabl. Ows mssg aa mcasm s cos o-goabl. a ff was a mas o cool o-spos. O wa of alg w poblm of o-spos s o ma mo ffos o collc fomao b ag a sub-sampl of us o spog a fs amp. Ao wa of alg w poblm of o-spos s o sma pobabl of spog fomas of bg a om a a spcf po of m a wgg suls w vs of s pobabl. A cqu o al w poblm of o-spos was vlop b Has a Huwz (46). assum a populao s v o wo classs a spos class wo spo fs amp a a o-spos class wo o. A qusoa coas ma qusos a w call ms. m o-spos occus subsaal fomao abou o-spo s usuall avalabl fom o ms o qusoa. a mpuao mos lau us slco of s ms as aula vaabl assgg valus o o-spo fo m. ao a (5) g a Ho (000) Am al. (006) a ula a au (00) av gv applcaos of vaous mpuao pocus. vaabl s of ma s a b a aula vaabl cola w a populao ma of aula vaabl s uow. A lag plma smpl aom sampl (wou placm) of us s aw Ω... o sma a a scoa sampl of sz ( ) aw as a sub-sampl fom populao of sampl o sma populao ma of ma vaabl. sampl coas spog us a o-spog us. Usg cocp of pos-safcao sampl ma b v o wo goups: spog ( ) a o-spog ( ). sampl ma b cos as saf o wo classs aml a spos class a o-spos class pocu s ow as pos-safcao. uam (4) avocas a pos-safcao pocu s as pcs as saf samplg u popooal allocao f sampl sz s lag oug. smao poblm sampl suvs sup of pos-safcao u o-spos suao s su u o ula a ub (004 a 00). ula al. (00) av also gv cocp of ulzao of (populao ma of o-spos goup of ) mpuao fo mssg obsvaos of aula fomao u o o-spos. ow ma b cos populao as wo ps of vuals l as umb of spos ( ) a o-spos ( ) us oal us of populao wll comps a spcvl suc a. populao popoos of us a goups a pss as / a / suc a. Fu l a b populao mas of a spcvl. Fo v u valu s obsv 56 Pag

2 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: avalabl. Howv fo us s a mssg a mpu valus a o b v. valu of aula vaa s us as a souc of mpuao fo mssg aa w. s s o assum a fo sampl aa : a ow. followg oaos a us s pap: s { } : sampl ma of a spcvl ; : sampl ma of a spcvl ; : populao ma squas of a spcvl; : coffc of vaao of a spcvl; : olao offc populao bw a spcvl. Fu cos fw mo smbolc psaos: ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) Q. U s sup as scb abov cas of smpl aom samplg wou placm a assumg s ow som wll ow mpuao mos a gv blow:.. a o of Impuao: f f as Fo (.) f Usg abov mpuao-bas smao of populao ma s: m (.) mma.: bas a ma squa o s gv b: () 0 (.).. ao o of Impuao: Fo a f as: U s mpuao-bas smao s: w () m (.4) m f w bˆ ˆ (.5) b f a A s gv b: () ( ) ( ) (.6) mma.: bas a ma squa o of (.) A A () ( ) [ ] A w (.).. ompoms o of Impuao: g a Ho (000) popos ompoms mpuao pocu as gv blow: ( α ) ( α ) ( α ) b ˆ b ˆ f (.) f w α s a suabl cos cosa suc a sula vaac of smao s opmum. mpuao-bas smao fo s cas smao of populao ma s α ( α ) OP (.0) mma.: bas ma squa o a mmum ma squa o a α of s gv b OP () ( ) ( α ) ( ) OP (.) 56 Pag

3 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag () z OP α (.) () m O (.).4. Am s os: (A) f f (.4) U s mo po smao of s: (.5) mma.4: bas vaac a mmum vaac a of s gv b: () 0 (.6) () (.) () m (.) () f f (.) u s sup po smao of s : (.0) mma.5: bas ma squa o a mmum ma squa o a of s gv b () (.) () (.) () m (.) () f f (.4) U s mo po smao of s : (.5) mma.6: bas ma squa o a mmum ma squa o a of s gv b () (.6)

4 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag () (.) () m (.) () 0 f f (.) u s po smao of populao ma s: 0 (.0) mma.: bas ma squa o a mmum ma squa o a of 0 s gv b () 0 (.) () 0 (.) () m 0 (.) II. ag ampl Appomaos: ; ; a wc mpls suls ; ; a. ow b usg cocp of wo-pas samplg a mcasm of A fo gv a [s ao a (5)] w av: 0 ; mlal 0 ; ; ; ; ) / ( ; ; ; ; III. Popos ff Impuao os vj os avalabl obsvao fo j mpuao. suggs followg mpuao mos: () f f (.) w a a suabl cos cosas suc a vaac sula smao s mmum. U s (.)

5 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag () f f (.) w s suabl cos cosa suc a vaac sula smao s mmum. U s (.4) () f f (.5) w s suabl cos cosa suc a vaac sula smao s mmum. U s po smao of populao ma s (.6) (4) 0 f f (.) w s suabl cos cosa suc a vaac sula smao s mmum. U s po smao of populao ma s 0 (.) I. as a.. of Popos os: (.) a (.) o bas a ma squa o (...) of a smao u a gv samplg sg. pops of smaos a v followg oms spcvl. om 4.: () smao ms of s a : (4.) Poof: () smao s ubas.. 0 (4.) Poof: 0 () vaac of s (4.) Poof: [ ] [ ]

6 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag (4) mmum vaac of s (4.4) Poof: ffag (4.) w spc o a qua o zo 0 a 0 Af placg valu of a (4.) w oba om 4.: (5) smao ms of s a : [ ( )] (4.5) Poof: { }... [ ( )] (6) bas of smao s (4.6) Poof: [ ( ) ] () a squa o of s (4.) Poof: [ ( ) ] () mmum m.s.. of s w (4.) Poof: ffag (4.) w spc o qua o zo 0

7 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag Af placg valu of (4.) w oba om 4.: () smao ms of s a : [ )]...(4.) Poof: { }... [ )] (0) bas of smao s (4.0) Poof: [ ) ] () a squa o of s (4.) Poof: [ ) ] () mmum m.s.. of s w (4.) Poof: ffag (4.) w spc o qua o zo 0 Af placg valu of (4.) w oba om 4.4: () smao 0 ms of s a : 0 (4.) Poof: 0

8 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag { }... (4) bas of smao 0 s 0 (4.4) Poof: 0 0 (5) a squa o of 0 s 0 (4.5) Poof: 0 0 (6) mmum m.s.. of 0 s 0 w (4.6) Poof: ffag (4.5) w spc o qua o zo 0 0 Af placg valu of (4.5) w oba 0. ompasos I s sco w v coos u wc suggs smaos a supo o Am al. (006). () m m s b a f 0 >

9 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: Pag Q ± Q Q w > Q > () m m s b a f 0 > 0 > 0 > ± w () m m s b a f 0 > 0 > ± w (4) m m 0 s b a 0 f 0 0 >

10 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: w > ± Q Q Q ( )( ) ( )( ) ( )( ) Q ( )( ) > ( )( ) ( )( ) I. umcal Illusaos cos wo populaos A a fs o s afcal populao of sz 00 [souc ula al. (00a)] a ao o s fom Am al. (006) w followg paams: abl 6. Paams of Populaos A a Populao A fo populao A a fo populao spcvl. bas a.. of suggs smaos (usg pssos of bas a m.s.. of co 5) a o sg smaos w Am al. (006) mos a gv abl 6. a 6. fo populao A a spcvl. abl 6. as a fo Populao A a smaos Populao A Populao as as smaos abl 6. as a fo Populao A a fo Am al. (006) Populao A Populao as as A OP samplg ffcc of suggs smaos ov Am al. (006) s f as: Op[ ( ) ] ; 0 (6.) Op ffcc fo populao A a populao a gv abl 6.4 abl 6.4 ffcc fo Populao A a ov Am al. (006) ffcc Populao A Populao Pag

11 Iaoal Joual of o gg sac (IJ) ol. Issu.5 p-oc. 0 pp-56-5 I: II. scusso a of wo-pas samplg s us wl cosg aula populao ma s uow a umbs of avalabl obsvaos a cos as aom vaabl. om sags a suggs co a smao of populao ma v. Pops of v smaos l bas a m.s. a scuss co 4. opmum valu of paams of suggs smaos s oba as wll sam sco. Am al. (006) smaos a cos fo compaso pupos a wo populaos A a cos fo umcal su fs o fom ula al. (00) a ao o s Am al. (006). samplg ffcc of suggs smao ov Am al. (006) s oba a suggs sag s fou v clos w Am al. (006) w s o ow. III. oclusos popos smaos a usful w som obsvaos a mssg samplg a populao ma of aula fomao s uow. Fo populao A popos smaos a fou o b mo ffc a sg smaos. Fo populao popos smaos a a fou o b mo ffc a 0 sg smaos. F [] Am.. Al- O. Al-aw Z. a Abu-a. (006): smao of a populao ma usg ff mpuao mos ascs aso [] Has. H. a Huwz.. (46): poblm of o-spos ampl uvs. Joual of Amca ascal Assocao [] Hja. F. a asu. (6): sgusg ssg a aom a ssg compll a aom. Amca asca [4] Kalo G. Kaspz. a aos. (): Issus of o-spos a mpuao uv of Icom a Pogam Pacpao. u opcs uv amplg (. Kws. Pla a J..K. ao s.) Acamc Pss w o. [5] l.j.a. a ub.. (): ascal Aalss w ssg aa Jo l a os. [6] H. acou. a aal.. (4): pms w vaac smao fom suv aa w mpu valus. Joual of offcal ascs 0-4. [] H. acou. a aal.. (5): aac smao psc of mpu aa fo galz smao ssm. Poc. of Amca as. Assoc. (ocal uv sac os co) 4-. [] ao J.. K. a.. (5): aac smao u wo-pas samplg w applcao o mpuao fo mssg aa omca [] ub.. (6): Ifc a mssg aa. omca [0] a I. G. (): A psoal vw of o c appoac o auomac a mpuao. Joual Impuao Pocus. uv oolog [] ula. (00): F- smao u wo-pas samplg o [] ula. a ub J. (00): smao mal suvs u P samplg scm. I. oc. Ag. a [] ula. a ub J. (004): O smao u pos-saf wo pas o-spos samplg scm mal suv. Iaoal Joual of aagm a sms [4] ula. a ub J. (006): O ama saa pos-safcao. ascs aso [5] ula. a au.. (00): smao of ma w mpuao of mssg aa usg faco-p smao ascs aso -4. [6] ula. au.. Paa. a ajpu.. (00): smao of ma w mpuao of mssg aa usg faco- p smao wo-pas samplg ascs aso [] ula. au.. au.. (00a):Ulzao of o-spos aula populao ma mpuao fo mssg obsvaos Joual of labl a ascal us -40. [] ula. au.. au.. a Paa. (0): a combao bas mpuao mo fo mssg aa sampl Iaoal Joual of o gg sac [] uam P.. uam.. uam. a Aso. (4): amplg o of uvs w Applcaos Iowa a Uvs Pss I..A.. Publcao w l. [0] g. a Ho. (000): ompoms mpuao suv samplg a [] au.. aav K. a Paa. (0): smao of ma psc of mssg aa u wo-pas samplg scm Joual of labl a ascal us [] au.. aav K. a Paa. (0): om mpuao mos oubl samplg scm fo smao of populao ma Iaoal Joual of o gg sac pp Pag

SOME IMPUTATION METHODS IN DOUBLE SAMPLING SCHEME FOR ESTIMATION OF POPULATION MEAN

SOME IMPUTATION METHODS IN DOUBLE SAMPLING SCHEME FOR ESTIMATION OF POPULATION MEAN aoal Joual of Mod Egg Rsach (JMER) www.jm.com ol. ssu. Ja-F 0 pp-00-07 N: 9- OME MPUTATON METHOD N DOUBLE AMPLNG HEME FOR ETMATON OF POPULATON MEAN ABTRAT Nada gh Thaku Kalpaa adav fo Mahmacal ccs (M)