REPLICATION VARIANCE ESTIMATION UNDER TWO-PHASE SAMPLING IN THE PRESENCE OF NON-RESPONSE

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1 STATISTICA, anno LXXIV, n. 3, 2014 REPLICATION VARIANCE ESTIMATION UNDER TWO-PHASE SAMPLING IN THE PRESENCE OF NON-RESPONSE Muqaddas Javed 1 Natonal College of Busness Admnstraton and Economcs, Lahore, Pakstan. Zahoor Ahmad Department of Statstcs, Unversty of Gujrat, Gujrat, Pakstan. Muhammad Hanf Natonal College of Busness Admnstraton and Economcs, Lahore, Pakstan. 1. Introducton Two phase samplng ntroduced by Neyman 1938 he gve the concept of use of auxlary nformaton to get maxmum response about the requred nformaton. He also gave the dea of the use of stratfcaton n the second phase. Hansan and Hurwtz 1946 suggest the survey statstcan to get maxmum response by recontactng wth expansve method to noespondent. A smple method of double samplng for stratfcaton s proposed and the classcal non-response theory s obtaned as a specal case by Rao Rato and regresson estmator of the populaton mean was proposed by Cochran 1977 for the study varable n whch nformaton on the auxlary varable s obtaned from all the sample unts. More mprovement n populaton mean n the presence of noesponse usng auxlary nformaton was suggested by Rao Sngh et al proposed a class of estmators based on general samplng desgns for populaton parameter utlzng auxlary nformaton of some other parameters. They also studed the general propertes for the suggested class and fnd the asymptotc lower bound to the mean square error of the estmators belongng to the class. They 1990 also proposed several unbased rato and product estmators wth ther expressons of asymptotc varances usng Jackknfe technque n two-phase samplng. Double expanson estmator DEE and reweghted expanson estmator REE are developed by Kott and Stukel 1997 for regresson estmator n two phase stratfed samplng. Khare and Srvastve 1997 proposed some mproved estmator of Rato and Regresson usng auxlary nformaton n the presence of non response.precse varance estmator s always a choce of survey statstcan; replcaton varance estmaton s a tool whch overcomes ths problem. The concept of the jackknfe was ntroduced by Quenoulle 1956 n connecton wth reducton 1 Correspondng Author. E-mal: muqaddas.javed@uog.edu.pk

2 248 M. Javed, Z. Ahmad and M. Hanf of bas for nonlnear estmaton. The possblty of usng ths technque for the purpose of varance estmaton was proposed by Tukey He 1958 suggested that each jackknfe replcate estmate mght be regarded as an ndependent and dentcally dstrbuted random varable, whch n turn suggests a very smple varance estmator. Jones 1958 modfed the jackknfe varance estmator for the populaton mean under two phase samplng for rato estmator. Durbn 1959 may have been the frst to use t n the context of fnte populatons. Rao and Shao 1992 proposed a consstent jackknfe varance estmator for reweghted expanson estmator n the context of hot deck mputaton. Bredt and Fuller 1993 suggested a replcaton varance estmator for multphase samples. Ther 1993 procedure s partcularly applcable f the frst phase and second phase prmary samplng unts are the same and f there are at least two second phase prmary samplng unts n each frst phase stratum. Rust and Rao 1995 provde comprehensve overvews oeplcaton varance estmaton topc. A jackknfe procedure has been suggested for the rato estmator by Rao and Stter 1995 and for the regresson estmator by Stter Kott 1990 dscuss varance estmaton n a stuaton the stratfed desgn for the second phase dffers from the stratfed desgn for the frst phase, and suggests a jackknfe estmator of varance for a partcular two phase estmator. Kott and Stukel 1997 concluded that the jackknfe varance estmator works well for the reweghted expanson estmator f the frst phase samplng rate s neglgble.fuller 1998 gve the estmaton of the varance of regresson estmator for a two phase sample and replcaton varance estmator s developed n second phase sample. Km and Stter 2003 developed a replcaton varance estmator for two phase samplng when the second phase sample s much smaller than the frst phase sample; Method s appled to DEE and REE. A consstent varance estmator whch s applcable n DEE and REE n two phase samplng was developed by Km, Navarro and Fuller 2006 when the second phase s stratfed. Wolter 2007 provde comprehensve overvews oeplcaton varance estmaton topc. Ramasubramanan et al proposed a two new jackknfe method as the counterparts of two exstng Bootstrap methods of varance estmaton under two phase samplng.farrell and Sngh 2010 dscuss the problem of estmatng the varance of varous estmators of the populaton mean n two phase samplng by jackknfng the two phase calbrated weghts. They 2010 also estmated the varance of the chaato and regresson type estmators due to Chand 1975 usng the jackknfe. Km and Yu 2011 developed a replcaton based bas adjusted varance estmator that extends the method of Km, Navarro and Fuller 2006 under two phase samplng. In case of non-response the sampler has an ncomplete sample data that affects the qualty of estmates of the unknown populaton parameters. The problem was frst consdered by Hansen and Hurwtz 1946, n whch the populaton was dvded nto groups of respondents and non-respondents and the sample sze to be attempted at the frst occason and the proporton of non-respondents to be repeated n the remanng sample was also determned. They constructed the estmate of the populaton mean by combnng the data from the two attempts and derved the expresson for the samplng varance of the estmates and optmum samplng fracton among the noespondents s also obtaned.

3 Replcaton varance estmaton n the presence of non-response 249 In ths paper, we propose some new replcaton varance estmators n the presence of non-response under two phase samplng. The estmators are proposed n two dfferent stuatons, Km and Yu 2011 methodology s used for the replcaton. The rest of the paper s organzed as follows. In secton 2, the four estmators are developed under two phase samplng n the presence of noesponse and ther replcaton varance estmators are developed n the Secton 3. In Secton 4, proposed method s extended for DEE-type estmators and REE-type estmators. In secton 5, a lmted smulaton study s presented. Concludng remarks are made n last secton of the paper. 2. Proposed Estmators of Mean under Two-phase Samplng n the Presence of Non-response In ths secton, we suggest two phase stratfed estmators for estmatng populaton mean of varable of nterest n the presence of noesponse. The estmators are proposed for two dfferent condtons. Estmaton of mean n the presence of non-response at frst phase Estmaton of mean n the presence of non-response at second phase 2.1. Estmaton of Mean n the Presence of Non-response at Frst Phase Sample Here, we suggest two phase stratfed estmators for estmatng populaton mean of the study varable when non-response s present on frst phase under the followng two dfferent stuatons. Estmaton by gnorng non-respondng unts Estmaton by re-contactng noespondng unts Estmaton by Ignorng Non-respondng Unts In ths secton, we smply assume the stuaton the frst phase sample of sze n s selected wth smple random samplng wthout replacement SRSWOR from a fnte populaton of sze N and selected frst phase sample s dvded nto two parts, respondent and non-respondent n r. Usng the nformaton obtaned from frst phase respondng sample, t s stratfed nto L strata for second phase samplng. We have h frst phase respondng sample elements and assume A rh1 be the set of ndces for the frst phase respondng sample elements n stratum h. A stratfed random sample of sze n wth sample sze n rh n stratum h s selected n the second phase samplng. Where n = L n rh and h/n s samplng rate for each stratum.

4 250 M. Javed, Z. Ahmad and M. Hanf The study varable y s observed n the second phase sample; the populaton means of y s estmated by t 1 = 1 L h n y rh, 1 A rh2 rh A rh2 s the set of ndces for the second phase sample elements that belong to stratum h. Usng proof 1 of Appendx, the varance of t 1 can be wrtten as V t 1 = E n 1 r 1 f 1 w h ȳ rh1 ȳ ] rh n 1 rh f 1 whs 2 2 rh1, 2 ȳ 1 = =1 y r/ be the mean of the frst phase sample and the consstent estmator of the varance of t 1 can be derved by replacng ȳ h1 and s 2 h1 by ther estmates ȳ rh2 = rh A rh2 y r and s 2 rh2 = n rh 1 1 A rh2 y r ȳ rh2 2 respectvely n 2. So the consstent estmator of 1 s ˆV t 1 = n 1 r 1 f 1 w h ȳ rh2 ȳ f 1 = N, ȳ 2 = L w hȳ h2 and w h = h. rh n 1 rh f 1 whs 2 2 rh2, Estmaton by Re-contactng non Respondng Unts Here, we develop an estmator by the theory of Hansan and Hurwtz 1946 to re-contact the non-respondng unts wth a dfferent method. In ths stuaton the frst phase sample unts s stratfed for both respondng unts and nonrespondng unts n r nto L strata for second phase samplng.we have h frst phase respondng sample elements and assume A rh1 be the set of ndces for the frst phase respondng sample elements n stratum h and we also have n rh frst phase non-respondng sample elements n stratum h. In the second phase samplng, a stratfed random sample of sze n 1 s selected from respondng unts wth sample sze n 1h n stratum h, and a stratfed random sample of sze n 2 s selected from re-contacted frst phase non-respondng unts wth sample sze n 2h n stratum h, n 1 = L n 1h, n 2 = L n 2h and a fxed samplng rate for each frst phase respondng stratum at second phase s n 1h /h and frst phase non-respondng stratum at second phase s n 2h /n rh. The study varable y s observed n the second phase sample, the populaton mean of y s estmated by t 2 = w 1 h n y rh + w 2 A rh2 1h n r A rh2 n rh n 2h y rh, 4 w 1 = n 1, w 2 = n 2 and A n r rh2 s the set of ndces for the second phase respondng sample elements that belong to stratum h, A rh2 s the set of ndces

5 Replcaton varance estmaton n the presence of non-response 251 for the second phase re-contactng sample elements that belong to stratum h. Usng the result gven n the proof 2 of Appendx, the varance of estmator t 2 can be wrtten as ˆV t 2 = n 1 1 f 1 n 1 r + n 21 f 2 n 1 r w h ȳ rh2 ȳ r2 2 +w1 2 wh ȳ rh2 ȳ r2 2 +w h 1 2h f 1 n 1 whs 2 2 rh2 rh f 2 n 1 wh 2 s 2 rh2, 5 rh 2.2. Estmaton of Mean n the Presence of Non-response at Second Phase Sample In ths secton we agan suggest two new estmators for the populaton mean n two phase stratfed desgn when non-response s present on second phase under the followng stuatons. Estmaton by gnorng non-respondng unts Estmaton by re-contactng noespondng unts Estmaton by Ignorng Non-respondng Unts In ths secton, we assume the stuaton the frst phase sample of sze n s selected wth SRSWOR from a fnte populaton of sze N. Usng the nformaton obtaned from frst phase sample, t s stratfed nto L strata for second phase samplng. We have n h frst phase sample elements and assume A h1 be the set of ndces for the frst phase sample elements n stratum h. A stratfed random sample of sze n wth sample sze n h n stratum h s selected n the second phase samplng of whch n rh respond and n rh non-respondent. Where n = L n h = L n rh + n rh and a fxed samplng rate for each stratum s n rh /n h. The study varable y s observed n the second phase sample; the populatons mean of y s estmated by t 3 = 1 n n h n A rh2 rh y h, 6 A rh2 s the set of ndces for the second phase sample elements that belong to stratum h. The varance of t 3 can be wrtten as V t 3 = E n 1 1 f1 wh ȳ rh1 ȳ ] rh n 1 h f 1 wh 2 s 2 rh1. 7 A consstent estmator of the varance of t 3 can be derved by replacng ȳ h1 and s 2 h1 wth ther estmates ȳ rh2 = rh A rh2 y r and

6 252 M. Javed, Z. Ahmad and M. Hanf s 2 rh2 = n rh 1 1 A rh2 y r ȳ rh2 2 respectvely n 7. So the consstent estmator of 6 s ˆV t 3 = n 1 1 f 1 wh ȳ rh2 ȳ f 1 = n N, ȳ 2 = L w hȳ rh2 and w h = n h n. rh n 1 h f 1 wh 2 s 2 rh2, Estmaton by re-contactng noespondng unts In ths secton, we assume the stuaton the frst phase sample of sze n s selected wth SRSWOR from a fnte populaton of sze N. Usng the nformaton obtaned from frst phase sample, t s stratfed nto L strata for second phase samplng. We have n h frst phase sample elements and assume A h1 be the set of ndces for the frst phase respondng sample elements n stratum h. A stratfed random sample of sze n wth sample sze n h n stratum h s selected n the second phase samplng of whch n rh respond and n rh non-respondent. Where n = L n h = L n rh + n rh and a fxed samplng rate for each stratum respondng unt s n rh /n h and non-respondng unt s n rh /n h. The study varable y s observed n the second phase sample, the populaton mean of y s estmated by t 4 = 1 n w n h h1 A rh2 n rh y rh + 1 n w n h h1 n rh A rh2 y rh, 9 wh1 = n h n and A rh2 s the set of ndces for the second phase respondng sample elements that belong to stratum h and A rh2 s the set of ndces for the second phase re-contactng sample elements that belong to stratum h. The varance of t 4 can be wrtten as ˆV t 4 = + w wh1 2 ] n 1 h + wh1n 2 1 rh n 1 h w f1 wh 2 s 2 rh2 w wh1 2 ] n 1 h + wh1n 2 1 rh n 1 h w f1 wh 2 s 2 rh w n 1 1 f 1 w h ȳ rh2 ȳ ȳ rh2 ȳ 2 2], f1 = n N, ȳ 2 = L w hȳ rh2, w = n2, ȳ n 2 rh2 = rh ȳ rh2 = rh A rh2 y r. A rh2 y r, w h = n h n,

7 Replcaton varance estmaton n the presence of non-response Replcaton Varance Estmaton of Proposed Estmators In ths secton we proposed some replcaton varance estmators for the estmators developed n secton 2, the proposed method of jackknfe replcaton varance estmaton of Km and Yu 2011 by successvely deletng unts from the entre frst phase sample s adopted. In ths method frst phase samplng rate s not necessarly neglgble and does not requre addtonal replcates for bas correcton n the varance estmaton. The full jackknfe varance estmator of 1 s t k 1 = 1 L n r n rhȳk rh2 ˆV J1 = c k t k 1 t 1 2, 11 and k s the ndex of the unts deleted n the jackknfe replcates, A 1 s the set of ndces for the frst phase sample elements and c k s a factor assocated wth replcate k determned by the replcaton method { 1 n nr 1 rh = 1 h 1 f k A rh1 1 1 h f k / A rh1 n r and { n rh2 = h 1 1 n hȳrh2 y rk f k A rh2 ȳ rh2 f k / A rh2. ȳ k The full jackknfe varance estmator of 4 s ˆV J2 = c k t k 2 t 2 2, 12 t k 2 = w1 L n r n rhȳk rh2 + w2 L n r n rhȳk rh2 { 1 n n nr 1 rh = 1 h 1 f k A rh1 r 1 1 h f k / A rh1, { 1 n rh = n r 1 1 n rh 1 f k A rh1 n r 1 1 n rh f k / A rh1, n r and { n rh2 = 1h 1 1 n 1hȳrh2 y rk f k A rh2 ȳ rh2 f k / A rh2 ȳ k { n rh2 = 2h 1 1 n 2hȳ rh2 y rk f k A rh2 ȳ rh2 f k / A rh2. ȳ k Smlarly the full jackknfe varance estmator of 6 s ˆV J3 = c k t k 3 t 3 2, 13

8 254 M. Javed, Z. Ahmad and M. Hanf t k 3 = 1 L n n hȳk h2 1 n n h = { n 1 1 n h 1 f k A h1 n 1 1 n h f k / A h1, { and ȳ k n h2 = rh 1 1 n rhȳh2 y k f k A rh2 ȳ h2 f k / A rh2. Also the full jackknfe varance estmator of 9 s ˆV J4 = c k t k 4 t 4 2, 14 t k 4 = 1 L n u h1n hȳk rh2 + 1 L n u h1n hȳk rh2 ȳ k 1 n n h = { n 1 1 n h 1 f k A h1 n 1 1 n h f k / A h1, { n rh2 = rh 1 1 n rhȳrh2 y rk f k A h2 ȳ rh2 f k / A h2, { n rh2 = rh 1 1 n rhȳ rh2 y rk f k A h2 ȳ rh2 f k / A h2, ȳ k and u h1 = n 1 1 n h 1]. 4. Extenson of Proposed Estmators Here, we consder some extensons of the proposed replcaton method by DEE and REE under the condtons defned n secton 2 for proposed estmators DEE and REE n the Presence of Non-response at Frst Phase Sample The DEE and REE type estmators were developed for the two stuatons whch was dscussed n the secton Estmaton by Ignorng Non-respondng Unts Let w r = π 1 1 π 1 = Pr A rh1 s the frst phase samplng wght for respondng unts and wr = π 1 2 π 2 = Pr A rh2 A rh1 be the nverse of the condtonal probablty of respondng unts n the second phase, then we develop the followng DEE and REE. The proposed DEE for ths stuaton s t 1DEE = 1 N A rh2 w r w ry, 15

9 Replcaton varance estmaton n the presence of non-response 255 the replcaton varance estmator of 15 s developed by the Km and Yu 2011 method whch s as ˆV J1DEE = 2 t k 1DEE 1DEE t, 16 t k 1DEE = 1 N L c k Ah1 wk r w 1 r A h2 r w 1 r Ah2 wk r y and c k s a factor assocated wth replcate k determned by the replcaton method. The proposed REE estmator for ths stuaton s t 1REE = 1 N wh1 ȳ h2, 17 ȳ h2 = A rh2 w r wr 1 A rh2 w r wr y and w h1 = A rh1 w r. The replcaton varance estmator of 17 s as ˆV J1REE = c k t k 1REE t 1REE 2, 18 and t k w k h1 ȳ k h2 1REE = 1 L N k w h1 = A rh2 r, ȳk h2 = A rh2 r w r 1 A rh2 r w r y Estmaton by Re-contactng Non-respondng Unts Let we consder w r and wr as gven n Stuaton-I, further we let w r = π 1 r1 π r1 = Pr A rh1 s the frst phase samplng weght for re-contacted respondng unts and w r = π 1 r2 π r2 = Pr A rh2 A rh1 be the nverse of the condtonal probablty n the second phase of re-contacted respondng unts, then we develop the followng DEE and REE. The proposed DEE for ths stuaton s t 2DEE = 1 w r w N ry r + w r w r y r 19 A rh2 A rh2 and replcaton varance estmator of 19 s as ˆV J2DEE = c k t k 2DEE t 2DEE 2, 20 t k 2DEE = 1 N A rh1 r w 1 r A rh2 r w 1 r A rh2 r y r

10 256 M. Javed, Z. Ahmad and M. Hanf + w A rh1 k r w 1 r w A rh2 k r w 1 r A rh2 r y r and c k s a factor assocated wth replcate k determned by the replcaton method. The proposed REE for ths stuaton s t 2REE = 1 N wh1 ȳ rh2 + w h1 ȳ rh2, 21 wh1 = w r, ȳ Arh1 rh2 = w rw 1 Arh2 r A rh2 w r wr y r, w h1= A rh1 w r and ȳ rh2 = w r w r 1 w r w A rh2 A rh2 The replcaton varance estmator of 21 s ˆV J2REE = c k t k 2REE = 1 L k N w h1 ȳ k rh2 + h1 ȳ k rh2 ] r y r. t k 2REE t 2REE 2, 22 1 ȳ k rh2 = A rh2 r w r A rh2 r w r y r, h1 = A rh2 k w h1 = A rh2 r and ȳ k rh2 = A rh2 r w r 1 A rh2 r r, w r y r DEE and REE n the presence of non-response at second phase In ths secton we agan develop two new estmators of DEE and REE for the two stuatons dscussed n secton Estmaton by gnorng non-respondng unts Let w = π 1 π = Pr A h1 s the frst phase samplng weght and w = π 1 r π r = Pr A rh2 A h1, be the nverse of the condtonal probablty n the second phase respondng unts, then we developed DEE and REE by gnorng second phase non-respondng unts. The proposed DEE estmator for ths stuaton s t 3DEE = 1 N and replcaton varance estmator of 23 s ˆV J3DEE = c k A h2 w w y 23 t k 3DEE t 3DEE 2, 24

11 Replcaton varance estmaton n the presence of non-response 257 t k 3DEE = 1 N L Ah1 wk w 1 A rh2 w 1 wk Arh2 y and c k s a factor assocated wth replcate k determned by the replcaton method. The proposed REE estmator for ths stuaton s t 3REE = 1 N wh1 ȳ rh2, 25 wh1 = w Ah1 and ȳ h2 = w w 1 Arh2 A rh2 w w y r. The replcaton varance estmator of 25 s ˆV J3REE = c k t k 3REE t 3REE 2, 26 and t k 3REE = 1 L k N w h1 ȳ k rh2, 1 ȳ k rh2 = A rh2 w A rh2 w y r, h1 = A h Estmaton by re-contactng noespondng unts Consder w and w as gven n Stuaton-I, further we let w r = π 1 r π r = Pr A r h2 A h1 be the nverse of the condtonal probablty n the second phase re-contacted unts, then by re-contactng the non-respondng unts we develop DEE and REE. The proposed DEE estmator for ths stuaton s t 4DEE = 1 w w y r + w w r y r 27 N A h2 A h2 and replcaton varance estmator of 27 s ˆV J4DEE t k 4DEE = 1 N = c k t k 4DEE t 4DEE 2, 28 A h1 + A h2 A h1 w 1 A h2 w 1 w 1 w 1 y r A h2 y r A h2 and c k s a factor assocated wth replcate k determned by the replcaton method. ]

12 258 M. Javed, Z. Ahmad and M. Hanf The proposed REE estmator for ths stuaton s t 4REE = 1 N wh1 ȳ rh2 + ȳ rh2, 29 wh1 = w Ah1, ȳ rh2 = w Ah2 w 1 A h2 w w y r, ȳ r h2 = 1 A h2 w w r A h2 w w r y r. The replcaton varance estmator of 29 s ˆV J4REE = c k t k 4REE t 4REE 2, 30 and t k 4REE = 1 L k N w h1 ȳ k rh2 + ȳk r h2 1 ȳ k r h2 = A h2 w r 1 rh2 = A h2 w ȳ k A h2 w r y r, h1 = A h1, A h2 w y r. 5. Smulaton Study We conducted a smulaton study, to analyze the performance of the estmators proposed n Secton 2 to Secton 4. A fnte artfcal populaton of sze N = 1000 was generated wth four varables y, z, q, u, the populaton elements are ndependently generated from z exp1+2; q χ 2 1+2; u Unf 1, 2, 3 and y = β 0 + β 1 z + β 2 q + ε β 0, β 1, β 2 are 0, 1, 1 respectvely and ε N 0, 1 as dscussed by Km and Yu To obtan unequal probablty samples for ths smulaton study, we used SRSWOR, wth selecton probabltes proportonal to the measure of sze varable. The varable z was used as a sze measure for the unequal probablty samplng n the frst phase samplng and q was used as a sze measure for the unequal probablty samplng n the second phase samplng, the stratfcaton was defned usng varable u, t s assumed that the varable z, q, u, y and ε are mutually ndependent. A frst phase sample of sze n = 200 was drawn from the populaton of For frst stuaton of secton 2.1 t s assumed that 30% non-response at frst phase. Frst phase respondng sample of sze was stratfed usng u and stratfed random sample of sze n was taken at second phase. In stuaton-ii frst phase sampled n unts are stratfed usng u for second phase samplng for both assumed respondng and re-contactng elements of frst phase sample and stratfed random sample of sze n = n r + n r was taken. When non-response s present on second phase, then for stuaton-i dscussed n secton 2.2 a sample of sze n = 200 wth SRSWOR was drawn. Frst phase sampled n unts are stratfed nto L stratum. It s assumed that 30% non-responses exst n second phase sample, so gnorng non-respondng unts a second phase

13 Replcaton varance estmaton n the presence of non-response 259 stratfed sample of sze n rh was taken from hth stratum. For stuaton-ii,the estmator s developed by re-contactng second phase non-respondng unts and a stratfed random sample of sze n = L n rh + n rh s used for both assumed respondng and re-contactng elements of second phase sample. We generate S=4000 ndependent samples for smulaton from the fnte populaton generated above. the parameter of nterest s the populaton mean of the varable y. From generated samples, we compute two phase stratfed estmators, DEE and REE wth ther replcaton varances. Relatve bas RB and coeffcent of varaton CV were computed for each estmator and ther replcaton varance wth the followng method. 1. The percent relatve bas of the mean estmators wth respect to the populaton parameter P RB = E t T T 100, The percent relatve bas of the jackknfe varance estmators wth respect to the estmated true mean squared error s estmated by P RB = E V J t MSE true MSE true 100, 32 E V J t = 1 S S =1 V J t and MSE true = 1 S S =1 t T The percent coeffcent of varaton of the jackknfe varance wth respect to the estmated true MSE s estmated by 1 S S =1 P CV = V J t MSE true MSE true Results of proposed estmators wth percent relatve bas and percent coeffcent of varaton are gven n Table 1. Table 2 shows the percent relatve bas and percent coeffcent of varaton of the replcaton varance estmators. 6. Concludng Remarks Table 1 shows that estmators are very close values to populaton parameter wth small relatve bas. Table 2 shows the RB and CV of the replcaton varance of the proposed estmators. The result shows that the varances are wth smaller bas. The man purpose of ths study was to show that the replcaton varance estmator works well n two phase stratfed desgn when non-response s present. Expanson estmators are used n the estmaton strategy and smulaton study supported these results. ACKNOWLEDGEMENTS The authors are hghly thankful to Dr Govanna Galatà Executve Edtor Support and to the referee for ther valuable suggestons regardng mprovement of the paper.

14 260 M. Javed, Z. Ahmad and M. Hanf REFERENCES B. B. Khare, S. Srvastava Transformed rato type estmators for populaton mean n presence of non-response, Commu. Statst. Theory Meth., 26, pp H. L. Jones The jackknfe method, Proceedngs of the IBM scentfc computaton and symposum on statstcs, pp H. P. Sngh, L. N. Upadhyaya, R. Iachan On applyng the jackknfe technque to the rato and product estmators n two-phase samplng, Vkram Math. J., X, pp J. Bredt, W.A. Fuller 1993.Regresson weghtng for multplcatve samples, Sankhya, 55, pp J. Durbn A note on the applcaton of Quenoulle s method of bas reducton to the estmaton of ratos, Bometrka, 46, pp J. K. Km, A. Navarro, W. A. Fuller Replcaton varance estmaton for two phase stratfed samplng, Journal of the Amercan statstcal assocaton, 101, pp J. K. Km, R. R. Stter Effcent replcaton varance estmaton for two phase samplng, Statstca Snca, 13, pp J. K. Km, C. L. Yu Replcaton varance estmaton under two-phase samplng, Survey methodology, 37, pp J. Neyman Contrbuton to the theory of samplng human populatons, Journal of the Amercan Statstcal Assocaton, 33, pp J. N. K. Rao On double samplng for stratfcaton and analytcal surveys, Bometrka, 60, pp J. N. K. Rao, J. Shao Jackknfe varance estmaton wth survey data under hot deck mputaton, Bometrka, 79, pp J. N. K. Rao, R. R. Stter Varance estmaton under two-phase samplng wth applcaton to mputaton for mssng data, Bometrka, 82, pp J. Tukey Bas and Confdence n not qute large samples, Ann. Math. Statst., 29, pp K. F. Rust, J. N. K. Rao Varance estmaton under two phase samplng wth applcaton to mputaton for mssng data, Bometrka, 82, pp K. Wolter Introducton to Varance Estmaton. 2 nd Edton, New York: Sprnger. L. Chand Some rato type estmators based on two or more auxlary varables, PhD Thess, Iowa State Unversty, Ames, Iowa, USA.

15 Replcaton varance estmaton n the presence of non-response 261 M. H. Hansen, W. N. Hurwts The problem of non-response n sample surveys, Journal of the Amercan Statstcal Assocaton, 41, pp M. H. Quenoulle Notes on bas n estmaton, Bometrka, 43, pp P. J. Farrell, S. Sngh Some Contrbutons to Jackknfng Two-phase Samplng Estmators, Survey Methodology, 36, pp P. S. Kott Varance estmaton when a frst-phase area sample s restratfed, Survey Methodology, 16, pp P. S. Kott, D. M. Stukel Can the jackknfe be used wth a two phase sample, Survey Methodology, 23, pp P. S. R. S. Rao Rato estmaton wth subsamplng the non-respondents, Survey Meth., 12, pp R. R. Stter Varance estmaton for the regresson estmator n twophase samplng, Journal of the Amercan Statstcal Assocaton, 92, pp V. Ramasubramanan, A. Ra, R. Sng Jackknfe varance estmaton under two phase samplng, Model Asssted Statstcs and Applcaton, 2, pp W. G. Cochran Samplng Technques, New York: John Wley & Sons, lnc. W. A. Fuller Replcaton varance estmaton for two phase samples, Sta- tstca snca, 8, pp SUMMARY Replcaton Varance Estmaton under Two-phase Samplng n the Presence of Non-response Km and Yu 2011 dscussed replcaton varance estmator for two-phase stratfed samplng. In ths paper estmators for mean have been proposed n two-phase stratfed samplng for dfferent stuaton of exstence of non-response at frst phase and second phase. The expressons of varances of these estmators have been derved. Furthermore, replcaton-based jackknfe varance estmators of these varances have also been derved. Smulaton study has been conducted to nvestgate the performance of the suggested estmators. Keywords: Two-phase stratfed samplng; Jackknfe varance estmator; Non-response

16 262 M. Javed, Z. Ahmad and M. Hanf Appendx Proof of 2: As we know n the case of two phase samplng, n general the varance of an estmator can be expressed as Consderng E 1 V 2 t 1 ] from above Eq. 34 or V t 1 = V 1 E 2 t 1 ] + E 1 V 2 t 1 ] E 1 V 2 t 1 ] = E 1 V 2 E 1 V 2 t 1 ] = E 1 = E 1 V 2 L A rh2 nrh ] h ȳ rh 2 1 n rh ] h n y rh rh 1 ] s 2 rh1, 35 h s 2 rh1 = h 1 1 y ȳ rh1 2 and ȳ rh1 = n 1 rh y r. A rh1 A rh1 Now consderng V 1 E 2 t 1 ] fromeq V 1 E 2 t 1 ] = V 1 E 2 = V 1 ȳ ] h ȳ rh or = N N S2 As f 1 = N, S2 = N 1 1 N V 1 E 2 t 1 ] = n 1 r 1 f 1 S 2 36 =1 y ȳ N 2. Usng Eq. 35 and Eq. 36 n Eq. 34, we get Then V t 1 = n 1 r 1 f 1 S 2 + E n 1 r S 2 = E V t 1 = 1 f 1 E n 1 r n 1 r L nrh 2 1 n rh 1 ] s 2 rh1. h w h ȳ rh1 ȳ srh1 ] 2. w h ȳ ] rh1 ȳ s 2 h1

17 Replcaton varance estmaton n the presence of non-response 263 or = E n 1 r f 1 V t 1 = E +E L n 1 r 1 f 1 nrh w h s 2 rh1+ n 1 r 1 f n rh 1 ] s 2 rh1 h w h ȳ rh1 ȳ n 1 r rh w 2 hs 2 rh1 w h s 2 rh1 w h ȳ rh1 ȳ 1 2 n 1 r f 1 rh w 2 hs 2 rh1 ]. n 1 rh w2 hs 2 rh1 ] w h s 2 rh1 Solvng above equaton we can obtaequred result gven n Eq.2. Proof of 5: Estmator gven n 4 can be wrtten as t 2 = w 1 t 1 + w 2 t 2, 37 t 1 = 1 L h A rh2 n y rh, t 2 = 1 L n rh 1h n r A rh2 n y rh, w 1 = n 1 2h and w 2 = n 2. n r As t 1 and t 2 are ndependent, the varance expresson of Eq. 37 can be wrtten as V t 2 = w 2 1V t 1 + w 2 2V t Consderng V t 1 from Eq. 38, we can wrte V t 1 = V 1 E 2 t 1] + E 1 V 2 t 1]. 39 or Now consderng E 1 V 2 t 1] from Eq. 39, we can wrte ] E 1 V 2 t 1 1] = E 1 V 2 h ȳ rh L 2 E 1 V 2 t nrh 1 1] = E n 1 ] s 2 rh1, 40 1h h s 2 rh1 = h 1 1 A rh1 y r ȳ rh1 2 andȳ rh1 = n 1 rh A rh1 y r. Now consderng V 1 E 2 t 1] from Eq. 39, we can wrte ] V 1 E 2 t 1 1] = V 1 E 2 h ȳ rh

18 264 M. Javed, Z. Ahmad and M. Hanf or = V 1 ȳ V 1 E 2 t 1] = n 1 r 1 f 1 S 2 41 f 1 = nr N S2 = N 1 1 N =1 y ȳ N 2. Usng Eq. 40 and Eq. 41 n Eq. 39, we get ˆV t 1 = n 1 r 1 f 1 w h ȳ rh2 ȳ r2 2 + From Eq. 38, now we consder 1h n 1 rh f 1 whs 2 2 rh2. 42 or or V t 2 = V 1 E 2 t 2] + E 1 V 2 t 2] 43 Now consderng E 1 V 2 t 2] from Eq. 43, we can wrte E 1 V 2 t 1 2] = E 1 V 2 E 1 V 2 t 2] = E L n rh n r ] n rh ȳ rh n r 2 1 n 2h 1 ] s 2 rh1 n rh Now consderng V 1 E 2 t 2] from Eq. 43, we can wrte V 1 E 2 t 1 2] = V 1 E 2 = V 1 ȳ] ] n rh ȳ rh n r 44 V 1 E 2 t 2] = n 1 r 1 f 2 S 2 45 Usng Eq. 44 and Eq. 45 n Eq. 43, we get ˆV t 2 = n 1 r 1 f 2 wh ȳ rh2 ȳ r h n 1 rh f 2 wh 2 s 2 rh2 46 Usng Eq. 42 and Eq. 46 n Eq. 37, we get ˆV t 2 = w1 2 n 1 r 1 f 1 w h ȳ rh2 ȳ r h n 1 rh f 1 whs 2 2 rh2 + w 2 2 n 1 r 1 f 2 wh ȳ rh2 ȳ r h n 1 rh f 2 wh 2 s 2 rh2 Solvng Eq. 47 we can obtaequred result gven n Eq.5. 47

19 Replcaton varance estmaton n the presence of non-response 265 TABLE 1 RB and CV of Proposed Estmators Estmator Parameter Estmated Mean RB CV t t t t t 1DEE t 2DEE t 3DEE t 4DEE t 1REE t 2REE t 3REE t 4REE TABLE 2 RB and CV of Replcaton Varance Estmators Estmator RB CV V J t V J t V J t V J t V J t 1DEE V J t 2DEE V J t 3DEE V J t 4DEE V J t 1REE V J t 2REE V J t 3REE V J t 4REE