A general class of estimators for the population mean using multi-phase sampling with the non-respondents

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1 Hacettepe Journal of Mathematcs Statstcs Volume A general class of estmators for the populaton mean usng mult-phase samplng wth the non-respondents Saba Raz Gancarlo Dana Javd Shabbr Abstract Ths paper addresses the problem of non-response when estmatng the populaton mean of the varable of nterest A general class of estmators s suggested for the unknown populaton mean of the study varable Two vectors of the auxlary nformaton under mult-phase samplng scheme are used n presence of non response The asymptotc varance of the proposed class s determned compared wth some other exstng estmators theoretcally numercally It s shown that the proposed class of estmators s more effcent than [13] [17] classes of estmators Keywords: Auxlary nformaton Mult-phase samplng Non-response Asymptotc varance Effcency 000 AMS Classfcaton: 6D05 1 Introducton Whle collectng nformaton through the sample surveys there may arse several problems one of the common problems s non-response Ths happens especally n the surveys conducted through mals When research partcpants can not be approached drectly they may refuse to acknowledge survey questonnares sent to them through mals The estmates obtaned from such ncomplete surveys are often based The data obtaned from respondents group dffers from that of non-respondents group thus affects data relablty [6] suggested nonrespondents sub samplng scheme to hle ths problem In ths scheme ntally the nformaton s collected from the respondent s group through mal survey then non-respondents are re-contacted by usng sub samplng process complete Department of Statstcs Quad--Azam Unversty Islamabad Pakstan Emal: sabaqau@gmalcom Correspondng Author Department of Statstcal Scences Unversty of Padova Italy Emal: dana@statunpdt Department of Statstcs Quad--Azam Unversty Islamabad Pakstan Emal: javdshabbr@gmalcom

2 nformaton s retreved through personal ntervews The man objectve n samplng theory s to estmate the unknown populaton parameter of nterest the nformaton about populaton characterstc of the auxlary varables may or may not be avalable Of course the use of the auxlary nformaton ncreases the effcency of the estmates of the populaton parameter Many authors have proposed dfferent estmators to estmate the populaton mean usng the auxlary nformaton wth or wthout consderng non-response see [19] [7] [14] [1] [9] [13] [0] [8] [4] [16 17] [10] [3] references cted theren When nformaton about the populaton characterstc of the auxlary varable X s unavalable n such stuaton t s estmated from a large frst phase sample drawn from the populaton then a smaller second phase sample s taken from the frst phase sample to estmate the populaton parameter of the varable of nterest Y [10] has consdered regresson-cum-rato estmator usng sngle auxlary varable n two phase samplng scheme n presence of non-response [17] have proposed regresson-cum-rato estmators consderng dfferent stuatons usng two auxlary varables n two-phase samplng scheme n presence of non-response Mult-phase samplng s very useful scheme n a stuaton when the varable of nterest s very expensve connected wth other cheaper auxlary varables Lmted lterature s avalable about ths technque see for nstance [18] [11] [1] [] [15] Ths paper s nspred by the prevous studes the man objectve s to propose a general class of estmators to estmate the unknown populaton mean Ȳ under mult-phase samplng scheme usng a vector X = x 1 x k t wth unknown populaton mean vector X = X 1 Xk t another vector Z = z 1 z k t wth known or unknown populaton mean vector Z = Z 1 Z k t n the presence of non-response Ths research wll also explore that the optmal estmators proposed n the generalzed class are regresson type estmators Notatons Background Let us assume that P be a fnte populaton of N dstnct unts Let Y X be the study the auxlary varables havng values y x = 1 N Let X s correlated wth Y s used to estmate the unknown populaton mean Ȳ When the mean of the auxlary varable X s avalable the rato product regresson estmators are used to ncrease the effcency of the estmates of Ȳ When X s unknown two-phase samplng scheme s used at frst phase only X s estmated the second phase s devoted for the estmaton of Ȳ Let a large sample s of sze m 1 m 1 < N s drawn by smple rom samplng wthout replacement SRSWOR to collect nformaton on the auxlary varable X It s assumed that all m 1 unts provde complete nformaton on X In the second phase a smaller sample s of sze m from m 1 unts m < m 1 s drawn by SRSWOR for obtanng nformaton of the study varable Y Suppose that non-response s present n second phase n ths stuaton a subset s 1 of sze m supples nformaton on Y the remanng m = m m unts are nonrespondents Therefore followng the famlar technque of [6] a sub-sample s r of sze r = m /b b > 1 s selected from the m non-response unts r would be an nteger otherwse must be rounded Assumng that all r selected unts show

3 full response on second call Consequently the whole populaton s sad to be stratfed nto two strata P 1 P P 1 s the stratum of respondents of sze N 1 that would gve response on frst call at second phase as P s the stratum of non-respondents of sze N whch would not respond on frst call at second phase but wll cooperate on the second call Obvously N 1 N are not known n advance Now we can defne a dummy varable u = y x z n the followng presentaton ū = ū = d 1 ū 1 + d ū r m1 =1 u m =1 ū = u ū 1 = m 1 m u = y x z havng th value Also Smlarly we have Ū 1 = The varance of ū s gven by m =1 u m d 1 = m m d = m m Ū = D 1 Ū 1 + D Ū r =1 ū r = u r N1 =1 u N =1 Ū = u D 1 = N 1 N 1 N N D = N N 1 Varū = θ S u + ω S u = S u N Su = u Ū N Su N 1 = u Ū N 1 1 θ 1 = 1 1 θ = 1 ω = N b 1 m 1 N m N m N One can defne the covarance as Covū v = θ S uv + ω S uv = S uv N S uv = u Ūv V N 1 S uv = N u Ūv V u = y v = x z N 1 From now on we shall consder that the Asymptotc Varance AV of the consdered estmators obtaned by usng a frst order Taylor seres [1] [6] proposed an estmator ȳ for the populaton mean Ȳ when non response occurs 3 ȳ = d 1 ȳ 1 + d ȳ r [13] proposed the followng regresson estmator usng double samplng scheme when non-response occurs on Y X both 4 t OL = ȳ + ˆβ yx x x

4 ˆβ yx = s yx s a sample estmate of the populaton regresson coeffcent β yx = s x m =1 y x + b r =1 y x m ȳ x m 1 AVt OL = θ 1 S y + θ θ 1 S y m =1 x + b r S yx Sx wth s yx = s =1 x = x m x m 1 The asymptotc varance of t OL s obtaned by usng the Taylor lnearzaton s gven by } 1 ρ yx + ω {Sy + β yxsx β yxs yx ρ yx = S yx SyS x [10] proposed the followng regresson-cum-rato estmator usng [13] regresson estmator a x α + b a x β + b 5 t K = t OL a x + b a x + b a b are known constants α β are sutably chosen constants The mnmum asymptotc varance of t K s gven by mnavt K = θ 1 Sy + θ θ 1 Sy 1 ρ yx + ω Sy 1 ρ yx ρ yx = S yx S y S x [17] consdered two auxlary varables x z n four dfferent stuatons We consder only two of those stuatons whch are n our opnon the most nterestng related to our proposed class Stuaton I: X unknown Z known When the populaton mean of the frst auxlary varable x s unknown the populaton mean of the second auxlary varable z s known Also non-response occurs on both the study the auxlary varables The proposed estmator s gven by 6 t SK1 = [ ȳ + ˆβ ] Z yx x x Z + α z Z α s a sutably chosen constant The mnmum asymptotc varance of the estmator t SK1 s gven by mnavt SK1 = AVt OL M D M = {θ S zβ yz + ω S z β yz} β yx {θ θ 1 S zβ xz + ω S z β xz} D = θ S z + ω S z Stuaton II: X Z both unknown

5 When the populaton means X Z are unknown Also non-response occurs on the study the auxlary varables The estmator s gven by 7 t SK = [ ȳ + ˆβ ] yx x x z z + γ z z γ s a sutably chosen constant The mnmum asymptotc varance of t SK s gven by N = mnavt SK = AVt OL N D { } { } θ θ 1 Szβ yz + ω Sz β yz β yx θ θ 1 Szβ xz + ω Sz β xz It s mportant to note that our proposed general class s an extenson of regresson estmator We consder specfcally regresson estmators because our consderaton s to show that regresson estmators perform better than regresson-cum-rato estmators Therefore we expect that our proposed class of estmators perform better than [17] class 3 Mult-Phase Scheme By generalzng the mult-phase samplng scheme proposed by [] we construct a samplng desgn wth two vectors X = x 1 x k t Z = z 1 z k t of auxlary varables n such a way that at each th + 1 th phase the samples s s +1 of szes m m < N m +1 m +1 < m are drawn by SRSWOR At the th phase the varables x z are observed whle at last phase the auxlary varables as well as y are measured accordng to Table 1 Table 1 Suggested mult-phase samplng desgn 1 3 k k + 1 m 1 m m 3 m m k m k+1 X 1 Z 1 X 1 Z 1 X Z X Z X 3 Z 3 X 1 Z 1 X Z X k 1 Z k 1 X k Z k X k Z k Y

6 4 General Class of Estmators Usng ths mult phase samplng desgn we propose a general class of estmators for the populaton mean Ȳ when non response can occur on the study varable as well as on the auxlary vectors 41 t kk = g ȳ w t w = x 1 1 z1 1 x 1 z 1 xk k zk k xk+1 k z k+1 g s a functon satsfyng the followng regularty condtons C1: g : S R S R 4k+1 s a convex bounded set contanng the pont Ȳ W t wth W = E w = X1 X 1 Z 1 Z 1 X k X k Zk Z k t; C: t s contnuous bounded n S; C3: ts frst second partal dervatves exst are contnuous bounded n S; C4: g ȳ W t = ȳ In order to determne the mnmum asymptotc varance AV of the class t kk let us ndcate g 0 = g ȳ w t ȳ ȳ w t =Ȳ Wt k t g = g ȳ w t w ȳ w t =Ȳ Wt the frst partal dervatves of g wth respect to the component ȳ w = 1 k of the vector w Expng t kk at the pont Ȳ W t n a frst order Taylor s seres we get 4 t kk = ȳ + + =1 =1 η X x + φ Z z + =1 =1 ξ X x +1 ϕ Z z +1 frst k ndcates the general class second k for havng k auxlary varables n the estmator t kk Further consder ũ = δ u ū + δ u ū wth δ u = 1 δ u δ u s an ndcator functon takng value δ u = 1 when non-response occurs 0 otherwse u = x z = 1 k

7 41 Stuaton 1: X unknown Z known In ths stuaton t s assumed that the populaton mean vector X s unknown but the mean vector Z s known Snce X s unknown so t s necessary to mpose a constrant η = ξ for computatonal reasons t may be useful to set φ + ϕ = ψ We can wrte Eq4 as 43 t 1k = ȳ + =1 ξ x x +1 + =1 ϕ z We can wrte 43 n a generalzed vector form as 44 t 1k = ȳ + ν ν t ξ + Z z ψ ν = x 1 1 z1 ν = x 1 z 1 xk k 1 xk+1 ξ = ξ 1 ϕ 1 ξ k ϕ k t t z = z 1 1 zk k k ψ = ψ 1 ψ k t z +1 t zk k t z k+1 k + =1 ψ Z z The asymptotc varance of the proposed class t 1k 45 AVt 1k = Varȳ + ξ t Sxz ξ + ψ t S z z ψ ξ t Syxz + ψ t Sy z ξ t Sxz z ψ Varȳ = S y = θ k+1 S y + ω k+1 S y S xz = E [ [ ν ν ν ν t] Sxx Sxz = S xz Szz ] S yxz = E [ ȳ Ȳ ν ν ] = [ Syx S yz ] t S z z = E [ z Z z Z t] S y z = E [ ȳ Ȳ z Z ] = [ θ 1 S yz1 θ S yz θ k S yzk ] t S xz z = [ S x z S z z ] t The detals of above mentoned matrces are gven n Appendx A Mnmzng AVt 1k wrt ξ ψ leads to optmum vector { ψ = S z z S t xz z S 1 S } 1 { xz xz z Sy z S xz z S 1 S } yxz = ψ o say ξ = 1 S xz S yxz S { xz z S z z S t xz z S 1 S } 1 { xz xz z Sy z S xz z S 1 S } yxz = ξ o say

8 Snce t s not easy to express mnmum AV t 1k n close form n the followng sub sectons we can express the class t 1k n two sub classes t 1k1 t 1k wth two three phases 411 Two Phase Usng Eq43 let consder X 1 Z 1 auxlary varables under two-phase samplng scheme assumng X 1 unknown Z 1 known 46 t 1k1 = ȳ + ξ 1 x 1 1 x 1 + ϕ 1 z 1 1 z 1 + ψ 1 Z1 z 1 1 The asymptotc varance of t 1k1 can be wrtten as 47 AV t 1k1 = S y+ξ 1 S x 1 +ϕ 1 S z 1 +ψ 1 S z 1 ξ 1 Syx1 ϕ 1 Syz1 ψ 1 Syz1 +ξ 1 ϕ 1 Sx1z 1 Mnmzng Eq47 wrt ξ 1 ϕ 1 ψ one can get ξ 1 = x1 S z 1 1 x 1z 1 z1 Syx1 x1z 1 Syz1 = ξ1say o ϕ 1 = x1 S z 1 1 x 1z 1 x1 Syz1 x1z Syx1 1 = ϕ o 1say ψ 1 = 1 S z1 Syz1 = ψ1say o The mnmum asymptotc varance of t 1k1 s gven by [ 48 mnavt 1k1 = S y z1 yx1 + S x 1 yz1 S ] yx1 Syz1 Sx1z 1 S x 1 z1 x 1z 1 S yz 1 S z 1 [ ] ρ 49 mnavt 1k1 = S y yx1 + ρ yz 1 1 ρ yx1 ρ yz1 ρ x1z 1 1 ρ ρ yz1 x 1z 1 ρ yu 1 = 410 mnavt 1k1 = S y S yu 1 S y S u 1 u 1 = x 1 z 1 ρ x 1z 1 = S x 1z 1 S x 1 z1 [1 R yx1z1 ρ yz1 ] R yx1z 1 s a multple correlaton coeffcent 41 Three Phase In ths case we take X 1 X Z 1 Z auxlary varables usng three-phase samplng assumng populaton means X 1 X unknown Z 1 Z known t 1k = ȳ + ξ 1 x 1 1 x 1 + ξ x x 3 + ϕ 1 z 1 1 z ϕ z z 3 + ψ 1 Z1 z ψ Z z

9 The asymptotc varance of t 1k AVt 1k = y + ξ S x j + ϕ S j z j ξ j Syxj ψj S z j + The AV of t 1k s mnmum when ϕ j Syzj + ψ j Syzj ξ j ϕ j Sxjz j + ψ 1 ψ + ξ 1 ψ Sx1z + ϕ 1 ψ ξ = x S z 1 x z z Syx xz Syz = ξsay o ϕ = x S z 1 x z x Syz xz Syx = ϕ o say ξ 1 = E 1 A = ξ o 1say ϕ 1 = E 1 B = ϕ o 1say ψ 1 = E 1 C = ψ o 1say ψ = E 1 D = ψ o say The detals of A B C D E are gven n Appendx B The resultng mnmum asymptotc varance of t 1k s gven by 413 mnavt 1k = S y 1 R yx1z1 z1 z R yxz R yx1z 1 z 1 z R yxz are the multple correlaton coeffcents 4 Stuaton : X Z both unknown In ths stuaton t s assumed that the populaton mean vectors X Z are unknown From the expresson 43 we have 414 t k = ȳ + =1 ξ x x ϕ z We can wrte Eq414 n a generalzed vector form as 415 t k = ȳ + ν ν ξ The asymptotc varance of t k s 416 AVt k = Varȳ + ξ t Sxz ξ ξ t Syxz z +1 ξ S xz S yxz are defned earler n Stuaton 1 Mnmzaton of AVt k wrt g xz leads to optmum vector [ ] [ ξ o ξ o = ϕ o = S xx Szz t S xz xz 1 S zz Syx ] xz Syz S xx Szz t S xz xz 1 S xx Syz xz Syx

10 Replacng ξ o n Eq416 one can get the optmal estmator n the class whch attans the mnmum asymptotc varance bound gven by 417 mnavt k = S [ y 1 R ] yxz R yxz s a vector of the squared multple correlaton coeffcents of Y on X Z s explaned n detal n Appendx C 41 Two Phase Now suppose that both X 1 Z 1 have unknown means n ths case expresson 414 can be expressed as 418 t k1 = ȳ + ξ 1 x 1 1 x 1 + ϕ 1 z 1 1 z 1 The asymptotc varance of t k1 can be wrtten as 419 AVt k1 = S y + ξ 1 S x 1 + ϕ 1 S z 1 ξ 1 Syx1 ϕ 1 Syz1 + ξ 1 ϕ 1 Sx1z 1 Now mnmzng 419 wrt ξ 1 ϕ 1 we have ξ 1 = x1 S z 1 1 x 1z 1 z1 Syx1 x1z Syz1 1 = ξ1say o ϕ 1 = x1 S z 1 1 x 1z 1 x1 Syz1 x1z Syx1 1 = ϕ o 1say One can wrte mnmum AV of t k1 as [ 40 mnavt k1 = Varȳ z1 yx1 + S x 1 yz1 S ] yx1 Syz1 Sx1z 1 S x 1 z1 x 1z 1 41 mnav t k1 = S y 1 R yx1z1 R yx 1z 1 s explaned earler n the Secton Three Phase Now consder the same auxlary varables as taken earler n Secton 41 wth unknown means t k = ȳ + ξ 1 x 1 1 x 1 + ξ x x ϕ 1 z 1 1 z 1 + ϕ z z 3 The asymptotc varance of t k can be expressed as 43 AVt k = S y+ ξ S x j + ϕ S j z j ξ j Syxj + ϕ j Syzj + ξ j ϕ j Sxjz j

11 Now mnmzng 43 wrt ξ 1 ξ ϕ 1 ϕ we have ξ o x1 S z1 1 x1z1 z1 Syx1 x1z Syz1 1 1 ξ o = ξ o ϕ o = x S z 1 x z z Syx xz Syz 1 ϕ o x1 S z1 1 x1z1 x1 Syz1 x1z 1 Syx1 x S z 1 xz x Syz xz Syx By replacng ξ1 o ξ o ϕ o 1 ϕ o n 43 one can get the optmal estmator n the class whch attans the mnmum asymptotc varance bound gven by: mnavt k = S y z1 yx1 + S x 1 yz1 S yx1 Syz1 Sx1z 1 S x 44 1 z1 x 1z 1 z yx + S x yz S yx Syz Sxz S x z x z 45 mnavt k = S y 1 R yx1z1 R yxz R yx 1z 1 R yx z are the squared multple correlaton coeffcents From Eq410 Eq413 Eq41 Eq45 we can accomplsh that t1k1 t 1k t k1 t k are regresson type estmators n ther optmal cases 5 Numercal Comparson In order to llustrate the gan n effcency for the best estmator n the two stuatons dscussed n prevous secton we carred out a numercal study usng the Populaton Census Report of Salkot Dstrct 1998 Pakstan ths data s earler used by [5] Each varable s taken from rural localty The descrpton of varables s gven below Varable Y X 1 X Z 1 Z Descrpton Lteracy Rato Populaton of prmary but below matrc Populaton of matrc above Populaton of 18 years old above Populaton of women years old We compute the Percent Relatve Effcency PRE of the consdered estmators wth respect to the [6] estmator ȳ for dfferent values of b as PREt = Varȳ AVt 100 t = t OL t K t SK1 t SK t 1k1 t 1k t k1 t k

12 Table PREs of the consdered classes wth respect to ȳ for dfferent values of b m 1 m Stuaton b t OL t K t SK t SK t 1k t k t OL t K t SK t SK t 1k t k t OL t K t SK t SK t 1k t k t OL t K t SK t SK t 1k t k t OL t K t SK t SK t 1k t k t OL t K t SK t SK t 1k t k

13 Table 3 PREs of the proposed class wth respect to ȳ for dfferent values of b b m 1 m m 3 Stuaton t 1k t k t 1k t k t 1k t k t 1k t k t 1k t k t 1k t k In Table we assume 40% non-response rate of the total populaton N = 68 assumng last 107 unts as non-respondents for second phase the numercal results of the estmators t 1k1 t k1 wth dfferent combnatons of sample szes m 1 m are gven To compute PREs of the estmators t 1k t k we consder 40% 30% non-response rates of the total populaton at frst second phase assumng last 80 unts as non-respondents for thrd phase The results are provded n Table 3 for dfferent choces of sample szes m 1 m m 3 In Tables 3 t s seen that the PREs of all consdered estmators are hgher than ȳ due to ncluson of the auxlary nformaton Also the PREs of all estmators ncrease wth the ncrease of nverse samplng rate b In Table the estmators t OL t K wth sngle auxlary varable t SK1 t SK wth two auxlary varables perform almost smlar but the proposed estmators t 1k1 t k1 show hgher effcency It s also seen that the regresson estmators t 1k1 t k1 perform better than the regresson-cum-rato estmators t SK1 t SK When we consder more auxlary varables phases for t 1k t k n Table 3 t s observed that they have consderable ncrease n effcency Moreover t s observed that when we ncrease sample szes the PREs of all consdered estmators also ncrease whch confrms the large sample theory aspect It s also expected that the frst proposed class t 1k shows always hgher effcency than the second class t k because we are usng more auxlary nformaton n Stuaton 1 Hence we can conclude that the estmator t 1k s the best choce f Z s known of course t k s to use when Z s unknown

14 6 Conclusons In ths paper we propose a general class of estmators for the estmaton of populaton mean Ȳ For ths we consder mult phase samplng scheme when auxlary nformaton s avalable The effects of non-response on the study on the auxlary varables are dscussed n detal We determne the asymptotc varance for the proposed classes To compare the effcency of the suggested ones wth other exstng estmators n the lterature [6] estmator s used It s noted that the performance of the proposed estmator t 1k s better than the other consdered estmators From the numercal analyss one can draw the concluson that regresson type estmators t 1k t k always perform better f compared to regressoncum-rato estmators t SK1 t SK However our proposed generalzed class of estmators s more effcent n terms of asymptotc varance f compared to all prevous estmators avalable n the lterature Acknowledgement The authors wsh to thank anonymous referees for ther careful readng constructve suggestons whch led to mprovement over an earler verson of the paper Thanks to the Department of Statstcal Scences Unversty of Padova Italy the Hgher Educaton Commsson HEC Islamabad Pakstan for ther logstc fnancal support for ths research Appendx A The detals of matrces n Eq45 are descrbed here θ 1 Sz S z z = E [ 1 θ 1 S z1z θ 1 S z1z k z Z z Z t] θ 1 S z1z θ Sz θ S zz k = θ 1 S z1z k θ S zz k θ k Sz k S xz z = [ ] t S x z S z z 0 θ θ 1 S x1z θ 3 θ S xz 3 0 S x z = θ k θ k 1 S xk 1 z k θ θ 1 S z1z θ 3 θ S zz 3 0 S z z = θ k θ k 1 S zk 1 z k ] S xz = E [ [ ν ν ν ν t] Sxx Sxz = S xz Szz

15 Now usng the dummy varable u for the followng representatons S uu = dag S u S u = θ +1 θ S u + ω +1 S u S yu = [ Syu1 S yu Syuk ] t wth θ = S yu 1 1 θ +1 = m N = θ +1 θ S yu + ω +1 S yu u = x z 1 m +1 1 N S uv = dag S uv u = x v = z = 1 k ω +1 = N +1b 1 m +1 N b > 1 S uv = θ +1 θ S uv + ω +1 S uv Appendx B Mnmzng Eq41 the followng terms are obtaned A = S z 1 S z1 S z Syx1 S z 1 S z Syz1 Sx1z 1 z 1 S z1 Syz Sx1z z 1 Syx1 S z1z S z 1 Syx1 S z1z + S z 1 Syz1 Sx1z + S z 1 Syz1 Sx1z + S z 1 Syz Sx1z 1 + S yz1 Sx1z 1 S z1z S yz1 Sx1z 1 B = S x 1 S z1 S z Syz1 S z 1 S z Sx1z 1 Syx1 x 1 S z1 Syz x 1 S z1z Syz1 S z 1 S x1z Syz1 + S z 1 Sx1z Syx1 + S x 1 Syz1 + S z 1 Sx1z 1 Sx1z Syz + S x1z 1 S z1z Syx1 x1z 1 Sx1z Syz1 C = S x 1 z1 S z Syz1 x 1 z1 Syz x 1 S z1z Syz1 z 1 S x1z Syz1 S z x1z 1 Syz1 + S x 1z 1 Syz + S z 1 Sx1z Syx1 + S x 1 Syz1 x 1z 1 Syx1 x 1z 1 Sx1z Syz1 + S x 1z 1 Sx1z Syz1 D = S x 1 z1 S z1 Syz x 1 z1 Syz1 x 1 S z1 Syz1 z 1 S z1 Sx1z Syx1 S z 1 x1z 1 Syz + S x 1z 1 Syz1 + S z 1 Sx1z 1 Syx1 + S z 1 Sx1z 1 Sx1z Syz1 E = S x 1 z1 S z1 S z S x 1 z1 S z1z + S x 1 S z1 S z1z + S z 1 S z1 S x1z + S z 1 S z x1z 1 + S x 1z 1 S z1z + S z 1 Sx1z 1 Sx1z

16 Appendx C From 417 the vector of the squared multple correlaton coeffcent R yxz s explaned as R yxz = R yx 1z 1 + R yx z + + R yx κz κ R yx z = ρ yx + ρ yz ρ yx ρ yz ρ xz 1 ρ x z = 1 κ ρ yu = S yu S y S u u = x z ρ x z = S x z S x z References [1] Ahmed M S Generalzed chan estmators under mult phase samplng Journal of Appled Statstcs [] Dana G Perr P F Estmaton of fnte populaton mean usng mult-auxlary nformaton Metron LXV [3] Dana G Perr P F A class of estmators n two-phase samplng wth subsamplng the non-respondents Appled Mathematcs Computaton [4] Gupta S Shabbr J On mprovement n estmatng the populaton mean n smple rom samplng Journal of Appled Statstcs [5] Hanf M Ahmed Z Ahmad M Generalzed multvarate rato estmators usng multauxlary varables for mult-phase samplng Pakstan Journal of Statstcs [6] Hansen M H Hurwtz W N The problems of non-response n sample surveys Journal of Amercan Statstcal Assocaton [7] Jan R K Propertes of estmators n smple rom samplng usng auxlary varable Metron XLV [8] Kadlar C Cng H Improvement n estmatng the populaton mean n smple rom samplng Appled Mathematcs Letters [9] Khare B B Srvastava S Transformed rato type estmtors for the populaton mean n the presence of non-response Communcaton n Statstcs Theory Methods [10] Kumar S Rato cum regresson estmator for estmatng a populaton mean wth a sub samplng of non respondents Communcatons of the Korean Statstcal Socety [11] Mukerjee R Rao T J Vjayan K Regresson type estmators usng mult auxlary nformaton The Australan Journal of Statstcs [1] Nak V D Gupta PC A general class of estmators for estmatng populaton mean usng auxlary nformaton Metrka [13] Okafor F C Lee H Double samplng for rato regresson estmaton wth subsamplng the non-respondents Survey Methodology [14] Rao P S R S Rato estmaton wth sub samplng the non-respondents Survey Methodology [15] Raz S Shabbr J An mproved class of regresson estmators for the populaton mean under mult-phase samplng scheme n the presence of non-response Proceedngs of the 11th Islamc Countres Conference on Statstcal Scences

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