Estimation of the Population Mean in Presence of Non-Response

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1 Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, DOI: 0.535/CKSS Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics, Uiversity of Jammu b School of Mathematics, Shri Mata Vaisho Devi Uiversity Abstract I this paper followig Sigh et al. 008, we propose a modified ratio-product type expoetial estimator to estimate the fiite populatio mea Ȳ of the study variable y i presece of o-respose i differet situatios viz. i populatio mea X is kow, ad ii populatio mea X is ukow. The expressios of biases ad mea squared error of the proposed estimators have bee obtaied uder large sample approximatio usig sigle as well as double samplig. Some realistic coditios have bee obtaied uder which the proposed estimator is more efficiet tha usual ubiased estimators, ratio estimators, product estimators ad expoetial ratio ad product estimators reported by Rao 986 ad Sigh et al. 00 are foud to be more efficiet i may situatios. Keywords: Study variable, auxiliary variable, bias, mea squared error, expoetial estimator, o-respose.. Itroductio The problem of o-respose i sample surveys is commo ad is more prevalet i mail surveys tha i persoal iterview surveys. Hase ad Hurwitz 946 have give a samplig pla that calls for takig a sub sample of o-respodets after the first mail attempt ad the eumeratig the sub sample by persoal iterview see Sriath, 97. I estimatig populatio parameters like the mea, total or ratio, sample survey experts sometimes use auxiliary iformatio to improve precisio of the estimators. Further, various authors like Cochra 977, Rao 986, 987, Khare ad Srivastava 993, 995, 997, Okafor ad Lee 000, Tabasum ad Kha 004, 006, Sigh ad Kumar 008, 009a, 009b, 00 ad Sigh et al. 00 studied the problem of o-respose uder double two-stage samplig. Cosider a fiite populatio of size N ad a radom sample of size draw without replacemet. I surveys o huma populatios, frequetly uits respod o the items uder examiatio i the first attempt while remaiig = uits do ot provide ay respose. Whe o-respose occurs i the iitial attempt, Hase ad Hurwitz 946 proposed a double samplig scheme to estimate the populatio mea: i a simple radom sample of size is selected ad the questioaire is mailed to the sampled uits; ii a sub sample of size r = /k, k from the o-respodig uits i the iitial attempt is cotacted through persoal iterviews. Correspodig author: Assistat Professor, Departmet of Statistics, Uiversity of Jammu, J & K, Idia. suilbhougal06@gmail.com

2 538 Suil Kumar, Sadeep Bhougal I this procedure the populatio is supposed to be cosistig of the respose stratum of size N ad the o-respose stratum of size N = N N. Let Ȳ = N i= y i/n ad S y = N i= y i Ȳ /N deote the populatio mea ad the populatio variace of the survey variable y. Let Ȳ = N i= y i/n ad S y = N i= y i Ȳ /N deote the mea ad variace of the respose group. Similarly, let Ȳ = N i= y i/n ad S y = N i= y i Ȳ /N deote the mea ad variace of the orespose group. The populatio mea ca be writte as Ȳ = W Ȳ + W Ȳ, where W = N /N ad W = N /N. Let ȳ = i= y i/ ad ȳ = i= y i/ deote the meas of the respodig uits ad the o-respodig uits. Further, let ȳ r = r i= y i /r deote the mea of the r = /k sub sampled uits. Thus, a ubiased estimator, due to Hase ad Hurwitz 946 of the populatio mea Ȳ of the study variable y is give by ȳ = w ȳ + w ȳ r,. where w = /, w = / are respodig ad o-respodig proportios i the sample. The variace of ȳ to terms of order, is give by { f Var ȳ = Ȳ Cy + W } k Cy,. where Cy = S y/ȳ, Cy = S y /Ȳ. Let x i i =,,..., N deote a auxiliary variable correlated with the study variable y i i =,,..., N. The populatio mea of the auxiliary variable x is X = N i= x i/n. Let X = N i= x i/n ad X = N i= x i/n deote the populatio meas of the respose ad o-respose groups or strata. Let x = i= x i / deote the mea of all the uits. Let x = i= x i/ ad x = i= x i/ deote the meas of the respodig uits ad o-respodig uits. Further, let x r = r i= x i /r deote the mea of the r = /k, k > sub-sampled uits. With this backgroud we defie a ubiased estimator of the populatio mea X as The variace of x is give by Var x = X { f x = w x + w x r..3 C x + W } k C x,.4 where C x = S x/ X, C x = S x / X, S x = N i= x i X /N ad S x = N i= x i X /N. I some situatios, there may ot be ay o-respose o the auxiliary variables. Family size, years of educatio, ad years of employmet are the above type of auxiliary variables, see Rao 986, p.0. Whe the populatio mea X of the auxiliary variable x is kow, Rao 986 suggested a ratio estimator for the populatio mea Ȳ of the study variable y as X t = ȳ x..5 Khare ad Srivastava 993 suggested a product estimator for the populatio mea Ȳ of the study variable y as x t = ȳ..6 X

3 Estimatio of the Populatio Mea i Presece of No-Respose 539 A expoetial ratio ad product type estimators for the populatio mea Ȳ of the study variable y are X t3 = x ȳ exp.7 X + x ad t 4 = ȳ exp x X..8 x + X The objective of this paper is to suggest a ratio-product type expoetial estimator for estimatig the fiite populatio mea i the presece of o-respose i differet situatios viz. i populatio mea X is kow, ad ii populatio mea X is ukow. The expressios of biases ad mea squared errors of the proposed estimators, up to the first order of approximatio, have bee obtaied. The results obtaied are depicted with the help of umerical illustratio.. Proposed Estimators I geeral, the liear regressio estimator is more efficiet tha the ratio product estimator except whe the regressio lie of y o x passes through the eighborhood of the origi, i which case the efficiecies of these estimators are almost equal. I additio, i may practical situatios the regressio lie does ot pass through the eighborhood of the origi. I these situatios, the ratio estimator does ot perform as good as the liear regressio estimator. Sigh et al. 008 proposed a ratio-product type expoetial estimator by followig Sigh ad Ruize Espejo 003 for estimatig the fiite populatio mea. Followig Sigh et al. 008, we propose followig class of ratio-product estimators for estimatig populatio mea Ȳ of the study variable y i presece of o-respose, as X t 5 {α = x ȳ exp X + x + α exp x } X,. x + X where α is a real costat to be determied such that the MSE of t 5 is miimum. For α = 0, the class of estimators respectively reduce to the estimator t4 ad t 3 respectively. To obtai the bias ad variace of t 5, we write such that ȳ = Ȳ + ϵ 0 ; x = X + ϵ, Eϵ 0 = Eϵ = 0 ad E { f ϵ0 = Varȳ = Ȳ Cy + W } k Cy, E { f ϵ = Var x = X C x + W } k C x, { f E ϵ 0 ϵ = Covȳ, x = Ȳ X ρ yx C y C x + W } k ρ yx C y C x,

4 540 Suil Kumar, Sadeep Bhougal where ρ yx = S yx /S x S y ; ρ yx = S yx /S x S y ; S yx = /N N i= y i Ȳx i X; ad S yx = /N N i= y i Ȳ x i X. Now, expressig t 5 i terms of ϵ s we have [ { } { }] X t 5 = Ȳ + ϵ X + ϵ X + e X 0 α exp + α exp [ = Ȳ + ϵ 0 α exp [ = Ȳ + ϵ 0 α exp X + X + ϵ ϵ + ϵ { ϵ + α exp + ϵ ϵ ] + ϵ } + α exp X + e + X { ϵ + ϵ }].. Expadig the right had side of. ad eglectig the terms ivolvig powers of ϵ s greater tha two, we have t 5 = Ȳ + ϵ 0 + ϵ αϵ + ϵ 4 + ϵ 0ϵ αϵ 0ϵ t 5 Ȳ = Ȳ ϵ 0 + ϵ αϵ + ϵ 4 + ϵ 0ϵ αϵ 0ϵ..3 Takig expectatios of both sides of.3, we get the bias of the estimator t 5 as B [ t Ȳ f 5 = 4 { } + αkyx C x + W k { } ] + αkyx C x,.4 where k yx = ρ yx C y /C x ; k yx = ρ yx C y /C x. Squarig both sides of.3 ad eglectig terms of ϵ s ivolvig power greater tha two, we have t 5 Ȳ = Ȳ { ϵ 0 + ϵ α } } = Ȳ {ϵ 0 + ϵ 4 + α α + ϵ 0 ϵ α 4 + α α t 5 Ȳ = Ȳ {ϵ 0 + ϵ } + ϵ 0 ϵ α..5 Takig expectatios of both sides of.5, we get the exact mea squared errormse of t 5 ad approximatio to the first degree of approximatio MSE of t 5, as MSE t 5 = Ȳ [ f + W k { Cy + C x { Cy + C x 4 + α α 4 + α α } + ρ yx C y C x α Miimizatio of.6 with respect to α yields its optimum value as α = A + B A } ] + ρ yx C y C x α..6 = α 0 say,.7

5 Estimatio of the Populatio Mea i Presece of No-Respose 54 where A = { f /}C x + {W k }/ C x, B = { f /}k yxc x + {W k }/ k yx C x. Substitute the optimum value of α from.7 i. yields the optimum estimator as X t 5opt {α = x x } X ȳ 0 exp + α X + x 0 exp..8 x + X The exact MSE of the optimum estimator t 5opt is give by MSE t 5opt = mi MSE t 3. Efficiecy Comparisos 5 = Ȳ [ f + W k { Cy + B B } yx A A k C x { Cy + B B } ] yx A A k C x..9 I this sectio, the coditios for which the proposed estimator t 5 is better tha the usual ubiased estimator ȳ, t, t, t 3 ad t 4 have bee obtaied. The MSE s of these estimators to the first degree of approximatio are derived as [ f Var ȳ = Ȳ Cy + W ] k Cy, 3. MSE [ t f {C = Ȳ y + } k yx C W k { x + C y + } ] k yx C x, 3. MSE [ t f {C = Ȳ y + } + k yx C W k { x + C y + } ] + k yx C x, 3.3 MSE { f t3 = Ȳ Cy + C } x 4kyx + W k 4 C y + C x 4kyx 4, 3.4 MSE { f t4 = Ȳ Cy + C } x + 4kyx + W k 4 C y + C x + 4kyx To compare the efficiecy of the proposed estimator t 5 with the existig estimators, from.9 ad , we have Var ȳ MSE f t 5opt = k yx C x + W k k yx C x 0, 3.6 MSE t MSE t f 5opt = kyx C x + W k kyx C x 0, 3.7 MSE t MSE t f 5opt = + kyx C x + W k + kyx C x 0, 3.8 MSE t3 MSE t f 5opt = kyx C x + W k kyx C x 0, 3.9 MSE t4 MSE t f 5opt = + kyx C x + W k + kyx C x

6 54 Suil Kumar, Sadeep Bhougal From , we coclude that the proposed estimator t 5 outperforms the usual ubiased estimator ȳ, t, t, t 3 ad t 4. If α does ot coicide with α 0, i.e. α = α 0, the from 3., 3., 3.3, 3.4, 3.5 ad.6, we evisaged that the suggested estimator t 5 is better tha i the usual ubiased estimator ȳ if either < α < + 4kyx or + 4kyx < α < ii the ratio estimator t if either or ad ad 3 < α < 4kyx ad 4kyx < α < 3 ad iii the product estimator t if either < α < 4kyx + 3 ad or 4kyx + 3 < α < ad < α < + 4kyx, + 4kyx < α <, iv the expoetial ratio type estimator t3 if { either < α < kyx ad < α < k yx, or k yx < α < ad k yx < α <, 3. 3 < α < 4kyx, 4kyx < α < 3 3., < α < 4kyx + 3, 4kyx + 3 < α < 3.3, 3.4 v the expoetial product type estimator t4 if { either 0 < α < + kyx ad 0 < α < + k yx, or + k yx < α < 0 ad + k yx < α < The proposed class of ratio product estimator t 5 is more efficiet tha ȳ, t, t, t 3 ad t 4 respectively, if 3.0, 3., 3., 3.3 ad 3.4 respectively hold true. 4. Empirical Study To see the performace of the suggested estimators of the populatio mea, we cosider a atural dataset cosidered by Khare ad Srivastava 993. The descriptio of the populatio is give below: A list of 70 villages i Idia alog their populatio i 98 ad cultivated areas i acres i the same year is cosidered Sigh ad Choudhary, 986. Here the cultivated area i acres is take as the mai study variable ad the populatio of the village is take as the auxiliary variable. The parameters of the populatio are as follows: Ȳ = 98.9, X = , C y = 0.654, C x = , Ȳ = 597.9, X = 00.4, C y = , C x = , ρ yx = 0.778, ρ yx = 0.445, R = , k yx = , k yx = 0.369, W = 0.0, N = 70, = 35.

7 Estimatio of the Populatio Mea i Presece of No-Respose 543 Table : Percet-relative efficiecypre of differet estimators PRE, ȳ /k /5 /4 /3 / PRE t, ȳ PRE t, ȳ PRE t3, ȳ PRE t4, ȳ PRE t5opt, ȳ We have computed the percet-relative efficieciespre s of various suggested estimators with respect to the usual ubiased estimator ȳ for various values of k, by usig the formulae PRE t i, ȳ = Var ȳ MSE 00; i =,, 3, 4 ad 5opt. It is to be evisaged from Table that the PRE s of the ratio type estimators t, t 3 ad t 5 icrease while the PRE s of the product type estimators t ad t 4 decrease as the value of k icreases. Further, it has bee observed that the estimator t 5 is the best amog ȳ, t, t, t 3 ad t 4. Thus, the suggested estimator t 5 is to be recommeded for its use i practice. 5. Double Two-Stage Samplig I may of the large scale sample surveys a multi-stage samplig desig is geerally used for selectio of a sample ad data are collected for several items. However, it is oted that iformatio i most cases are ot obtaied at the first attempt eve after some call-backs. For the estimate of populatio mea X of the auxiliary variable x, a large first phase sample of size is selected from a populatio of size N uits by simple radom samplig without replacemetsrswor. A smaller secod phase sample of size is selected from by SRSWOR ad the variable y uder study is measured o it; however, take a sub-sample of the o-respodets ad re-coduct them if there is o-respose i the secod phase sample. Let us assume that at the first phase, all the uits supplied iformatio o the auxiliary variable x. At the secod phase from sample, let uits supply iformatio o y ad refuse to respod. Usig Hase ad Hurwitz 946 approach to sub-samplig from the o-respodets a subsample of size m uits is selected at radom ad is eumerated by direct iterview, such that x m =, k > see, Tabasum ad Kha, 004. k Whe the populatio mea X of the auxiliary variable x is ukow, the two phase ratio ad product type estimator are x t 6 = ȳ, 5. t 7 = ȳ x x, by Khare ad Srivastava, 995; Okafor ad Lee, 000; Tabasum ad Kha, 004, by Khare ad Srivastava, 995; Okafor ad Lee, 000; Tabasum ad Kha, 004, where x deote the sample mea of x based o first phase sample of size. The two phase ratio ad product type expoetial estimator [ x t8 = x ] ȳ exp x + x, by Sigh et al.,

8 544 Suil Kumar, Sadeep Bhougal ad [ x t9 = x ] ȳ exp x + x, by Sigh et al., The MSE of the estimators t 6, t 7, t 8 ad t 9 is give by MSE [ t 6 = Ȳ {C y + } k yx C W k { x + C y + } k yx C x + N MSE [ t7 = Ȳ {C y + } +k yx C W k { x + C y + } +k yx C x + N MSE t8 = Ȳ { Cy + C } x 4k yx + W k 4 C y + C x 4k yx 4 + N MSE t9 = Ȳ { Cy + C } x +4k yx + W k 4 C y + C x +4k yx 4 + N 6. Suggested Estimator ] Cy, 5.5 ] Cy, 5.6 C y C y, We defie a class of ratio-product estimator for estimatig the populatio mea Ȳ of the study variable y i presece of o-respose, as x t0 [α = x ȳ exp x + x + α exp x x x + x ], 6. where α is a real costat to be determied such that MSE of t0 is miimum. For α =, t0 reduces to estimator t 8, ad for α = 0, t0 reduces to t 9, respectively. Let x = X + ϵ such that Eϵ = 0 ad E ϵ = C N x; Eϵ ϵ = C N x; Eϵ 0 ϵ = ρ yx C y C x. N Expressig 6. i terms of ϵ s, we get t 0 Ȳ { = Ȳ ϵ 0 + ϵ ϵ + α ϵ ϵ + α ϵ 0 ϵ α ϵ 0 ϵ + ϵ ϵ + ϵ 0ϵ 4 ϵ } 0ϵ. 6. Takig expectatios of both sides of 6., we get the bias of the proposed estimator t0, up to the first degree of approximatio, as B t0 = Ȳ 4 { C x 4 α } k yx + W k C { x 4 α k yx 4 }. 6.3 From 6., we have t 0 Ȳ { Ȳ ϵ 0 + ϵ ϵ + α ϵ ϵ }. 6.4

9 Estimatio of the Populatio Mea i Presece of No-Respose 545 Squarig both sides of 6.4 ad the takig expectatios, we get the MSE of the proposed estimator t0, up to the first order of approximatio as MSE [ t0 = Ȳ { Cy + α α } k yx C x + Cy N + W k {C y + α α } ] k yx C x, 6.5 which is miimum whe where A = α = A + B A = α 0 say, C x + W k C x ; B = Thus, the resultig miimum MSE of t0 is give by MSE t 0opt = Ȳ [ {C y + B + W k 7. Efficiecy Comparisos A { C y + B A B } C x k yx A B k yx A k yx C x + W k k yx C x. C x } + ] Cy. 6.6 N From 3., 5.5, 5.6, 5.7, 5.8 ad 6.6, we have Var ȳ MSE t0opt = k yx C x + W k kyx C x 0, 7. MSE t 6 MSE t 0opt = kyx C x + W k kyx C x 0, 7. MSE t7 MSE t 0opt = + kyx C x + W k + kyx C x 0, 7.3 MSE t8 MSE t 0opt = kyx C x + W k kyx C x 0, 7.4 MSE t9 MSE t 0opt = + kyx C x + W k + kyx C x From , it is evisaged that the proposed estimator t0 is more efficiet tha the estimators t 6, t 7, t 8 ad t 9 respectively. If α does ot coicide with α 0, i.e. α α 0, the from 7., 7., 7.3, 7.4, 7.5 ad 6.5, we evisaged that the suggested estimator t0 is better tha i the usual ubiased estimator ȳ if either < α < + 4kyx or + 4kyx < α < ad ad < α < + 4kyx, + 4kyx < α < 7.6,

10 546 Suil Kumar, Sadeep Bhougal ii the ratio estimator t 6 if either 0 < α < + k yx ad 0 < α < + k yx, or + kyx < α < 0 ad + kyx < α < 0, 7.7 iii the product estimator t7 if either < α < 4kyx + 3 ad < α < 4kyx + 3, or 4kyx + 3 < α < ad 4kyx + 3 < α < 7.8, iv the expoetial ratio type estimator t8 if either < α < k yx ad < α < k yx, or k yx < α < ad k yx < α <, 7.9 v the expoetial product type estimator t9 if either 0 < α < + k yx ad 0 < α < + k yx, or + kyx < α < 0 ad + kyx < α < The proposed class of ratio product estimator t 0 is more efficiet tha ȳ, t 6, t 7, t 8 ad t 9 respectively, if 7.6, 7.7, 7.8, 7.9 ad 7.0 respectively hold true. 8. Empirical Study To look closely the excellece of the suggested estimators, we cosider the followig data set: Source: Khare ad Srivastava 995, p.0. The populatio of 00 cosecutive trips after leavig 0 outlier values measured by two fuel meters for a small family car i ormal usage give by Lewisi et al. 99 has bee take ito cosideratio. The measuremets of turbie meter i ml. is cosidered as mai variable y ad the measuremets of displacemet meter i cm 3 is cosidered as auxiliary variable x. We treat 5% last values as o-respose uits. The values of the parameters are as follows: Ȳ = 3500., X = 60.84, C y = 0.594, C x = , Ȳ = , X = 59.96, C y = , C x = 0.568, ρ yx = 0.985, ρ yx = 0.995, W = 0.5, k yx = , k yx = 0.977, R = 3.487, N = 00, = 30, = 50. Here, we have calculated the percet relative efficieciespre s of differet suggested estimators with respect to usual ubiased estimator ȳ for differet values of k by usig the followig formulae PRE t i, ȳ = Var ȳ MSE 00; i = 6, 7, 8, 9 ad 0opt. Table exhibits that the PRE s of estimators t 6, t 9 ad t 0opt decreases as the value of k icreases, while the PRE s of estimators t7 ad t 8 decreases as the value of k icreases. Further, it is to be oted that the estimator t0opt is the best amog ȳ, t 6, t 7, t 8 ad t 9. Thus, the suggested estimator t 0opt is to be recommeded for its use i practice.

11 Estimatio of the Populatio Mea i Presece of No-Respose 547 Table : Percet relative efficiecy of the differet estimators of Ȳ with respect to ȳ. PRE, ȳ /k /5 /4 /3 / PRE t6, ȳ PRE t7, ȳ PRE t8, ȳ PRE t9, ȳ PRE t0opt, ȳ Coclusio The preset article cosiders the problem for estimatig the fiite populatio mea Ȳ of the study variable y i presece of o-respose i differet situatios viz. i populatio mea X is kow, ad ii populatio mea X is ukow. Usig the Hase ad Hurwitz 946 procedure of sub-samplig the o-respodets for both the cases where the populatio mea of the auxiliary character is kow ad ot kow i advace, a ratio-product type estimator respectively have bee proposed ad their properties are studied. The optimum mea squared error smse s of the proposed estimators are also obtaied. The relative performace of the proposed estimators is compared with the covetioal estimators. The proposed estimators are efficiet ad should work very well i practical surveys. Ackowledgemet Authors wish to thak the leared referees for their critical ad costructive commets regardig improvemet of the paper. Refereces Cochra, W. G Samplig Techiques, 3rd ed., Joh Wiley ad Sos, New York. Hase, M. H. ad Hurwitz, W. N The problem of o-respose i sample surveys, Joural of the America Statistical Associatio, 4, Khare, B. B. ad Srivastava, S Estimatio of populatio mea usig auxiliary character i presece of o-respose, The Natioal Academy of Scieces, Letters, Idia, 6, 4. Khare, B. B. ad Srivastava, S Study of covetioal ad alterative two phase samplig ratio, product ad regressio estimators i presece of o-respose, Proceedigs of the Natioal Academy of Scieces, Idia, 65A, Khare, B. B. ad Srivastava, S Trasformed ratio type estimators for the populatio mea i the presece of o-respose, Commuicatios i Statistics - Theory ad Methods, 6, Lewisi, P. A., Joes, P. W., Polak, J. W. ad Tillotso, H. T. 99. The problem of coversio i method compariso studies, Joural of the Royal Statistical Society. Series C Applied Statistics, 40, 05. Okafor, F. C. ad Lee, H Double samplig for ratio ad regressio estimatio with subsamplig the o-respodets, Survey Methodology, 6, Rao, P. S. R. S Ratio estimatio with sub samplig the o-respodets, Survey Methodology,, Rao, P. S. R. S Ratio ad regressio estimates with sub samplig the o- respodets. Paper preseted at a special cotributed sessio of the Iteratioal Statistical Associatio Meetig, Sept., 6, Tokyo, Japa.

12 548 Suil Kumar, Sadeep Bhougal Sigh, D. ad Choudhary, F. S Theory ad Aalysis of Sample Survey Desigs, Wiley Easter Limited, New Delhi, p.08. Sigh, H. P. ad Kumar, S A regressio approach to the estimatio of fiite populatio mea i presece of o-respose, Australia ad New Zealad Joural of Statistics, 50, Sigh, H. P. ad Kumar, S. 009a. A geeral class of estimators of the populatio mea i survey samplig usig auxiliary iformatio with sub samplig the o-respodets, The Korea Joural of Applied Statistics,, Sigh, H. P. ad Kumar, S. 009b. A geeral procedure of estimatig the populatio mea i the presece of o-respose uder double samplig usig auxiliary iformatio, SORT, 33, Sigh, H. P. ad Kumar, S. 00. Estimatio of mea i presece of o-respose usig two phase samplig scheme, Statistical Papers, 50, Sigh, H. P., Kumar, S. ad Kozak, M. 00. Improved estimatio of fiite populatio mea whe sub-samplig is employed to deal with o-respose, Commuicatio i Statistics - Theory ad Methods, 39, Sigh, H. P. ad Ruiz Espejo, M O liear regressio ad ratio-product estimatio of a fiite populatio mea, Joural of the Royal Statistical Society. Series D The Statisticia, 5, Sigh, R., Chauha, P. ad Sawa, N O liear combiatio of ratio ad product type expoetial estimator for estimatig the fiite populatio mea, Statistics i Trasitio - New Series, 9, Sriath, K. P. 97. Multiphase samplig i o-respose problems, Joural of the America Statistical Associatio, 66, Tabasum, R. ad Kha, I. A Double samplig for ratio estimatio with o- respose, Joural of the Idia Society of Agricultural Statistics, 58, Tabasum, R. ad Kha, I. A Double samplig ratio estimator for the populatio mea i presece of o-respose, Assam Statistical Review, 0, Received July 00; Accepted May 0