J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) 49 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/070105 Dual to Ratio Estimators for Mea Estimatio i Successive Samplig usig Auxiliary Iformatio o Two Occasio azeema Beevi Departmet of Statistics, Uiversity of Calicut, Kerala - 673 635, Idia. Received: 14 Ja. 016, Revised: 30 Dec. 017, Accepted: Ja. 018 Published olie: 1 Mar. 018 Abstract: I this paper, cosider dual to ratio estimator for estimatig mea usig auxiliary iformatio o both occasios i successive samplig scheme. Dual to ratio estimators have bee developed by Srivekataramaa (1980) uder simple radom samplig. Usig this estimator uder successive samplig scheme, the bias ad mea squared error are obtaied upto the first order of approximatio ad show theoretically that the proposed estimator is more efficiet tha the Cochra s estimator usig o auxiliary variable ad simple mea per uit estimator. Optimum replacemet strategy is also discussed. Results have bee justified through empirical iterpretatio. Keywords: Auxiliary Iformatio, Dual to Ratio Estimator, Optimum Replacemet, Successive Samplig. 1 Itroductio ow a days, it is ofte see that sample surveys are ot limited to oe time iquiries. A survey carried out o a fiite populatio is subject to chage overtime if the value of study character of a fiite populatio is subject to chage (dyamic) overtime. A survey carried out o a sigle occasio will provide iformatio about the characteristics of the surveyed populatio for the give occasio oly ad ca ot give ay iformatio o the ature or the rate of chage of the characteristics over differet occasios ad the average value of the characteristics over all occasios or most recet occasio. A part of the sample is retaied beig replaced for the ext occasio ( or samplig o successive occasios, which is also called successive samplig or rotatio samplig). The successive method of samplig cosists of selectig sample uits o differet occasios such that some uits are commo with samples selected o previous occasios. If samplig o successive occasios is doe accordig to a specific rule, with replacemet of samplig uits, it is kow as successive samplig. Replacemet policy was examied by Jesse (194) who examied the problem of samplig o two occasios, without or with replacemet of part of the sample i which what fractio of the sample o the first occasio should be replaced i order that the estimator of Ȳ may have maximum precisio. Yates (1949) exteded Jesse s scheme to the situatio where the populatio mea of a character is estimated o each of (h > ) occasios from a rotatio sample desig. These results were geeralized by Patterso (1950) ad arai (1953), amog others. Rao ad Mudhdkar (1983) ad Das (198), used the iformatio collected o the previous occasios for improvig the curret estimate. Data regardig chagig properties of the populatio of cities or couties ad uemploymet statistics are collected regularly o a sample basis to estimate the chages from oe occasio to the ext or to estimate the average over a certai period. A importat aspect of cotiuous surveys is the structure of the sample o each occasio. To meet these requiremets, successive samplig provide a strog tool for geeratig the reliable estimates at differet occasios. Se (1971) developed estimators for the populatio mea o the curret occasio usig iformatio o two auxiliary variables available o previous occasio. Se (197, 1973) exteded his work for several auxiliary variables. Sigh et.al (1991) ad Sigh ad Sigh (001) used the auxiliary iformatio o curret occasio for estimatig the curret populatio mea i two occasios successive samplig ad Sigh (003) exteded his work for h occasios successive samplig. Feg ad Zou (1997) ad Biradar ad Sigh (001) used the auxiliary iformatio o both the occasios for Correspodig author e-mail: azeemathaj@gmail.com
50. Beevi : Dual to ratio estimators for mea estimatio i... estimatig the curret mea i successive samplig. I may situatios, iformatio o a auxiliary variate may be readily available o the first as well as the secod occasios; for example, toage (or seat capacity) of each vehicle or ship is kow i survey samplig of trasportatio ad umber of beds i hospital surveys. Most of the studies related to dual to ratio estimators have bee developed by Srivekataramaa (1980). He cosidered, the relatioship betwee the respose y ad the subsidiary variate x, is liear through the origi ad variace of y is proportioal to x. It is assumed that X is kow. Motivated with the above argumet ad utilizig the iformatio o a additio auxiliary variable is available o the both occasios, the dual to ratio estimator for estimatig the populatio mea o curret occasio i successive samplig has bee proposed. It has bee assumed that the additioal auxiliary variable over two occasios. The paper is spread over te sectios. Sample structure ad otatios have bee discussed i sectio ad sectio 3 respectively. I sectio 4, the proposed estimators have bee formulated. Properties of proposed icludig mea square error are derived uder sectio 5. I sectio 6, optimum replacemet policy is discussed. Sectio 7 cotais compariso of the proposed estimator with Cochra (1977) ad simple mea per uit whe there is o matchig from the previous occasio ad the estimator whe o additioal auxiliary iformatio has bee used. I Sectio 8 ad 9, the theoretical results are supported by a umerical iterpretatio ad give coclusio i Sectio 10. Selectio of the sample Cosider a fiite populatio U = (U 1,U...U ) which has bee sampled over two occasios. Let x ad y be the study variables o the first ad secod occasios respectively, further assumed that the iformatio o the auxiliary variable z, whose populatio mea is kow which is closely related (positively related) to x ad y o the first ad secod occasios respectively available o the first as well as o the secod occasio. For coveiece, it is assumed that the populatio uder cosideratio is large eough. Allowig SRSWOR (Simple Radom Samplig without Replacemet) desig i each occasios, the successive samplig scheme as follows is carried out: uits which costitutes the sample o the first occasio. A radom sub sample of m = λ (0 < λ < 1) uits is retaied (matched) for use o the secod occasio. I the secod occasio u = µ (= m ) (0 < µ < 1) uits are draw from the remaiig ( ) uits of the populatio. Where µ is the fractio of fresh sample o the curret occasio. So that the sample size o the secod occasio is also (= λ + µ). 3 Descriptio of otatios The followig otatios i this paper. X: The populatio mea of the study variable o the first occasio. Ȳ : The populatio mea of the study variable o the secod occasio. Z: The populatio mea of the auxiliary variable o both occasios. S y: Populatio mea square of y. z : The sample mea based o uits draw o the first occasio. z u : The sample mea based o u uits draw o the secod occasio. x : The sample mea based o uits draw o the first occasio. x m : The sample mea based o m uits observed o the secod occasio ad commo with the first occasio. ȳ u : The sample mea based o u uits draw afresh o the secod occasio. ȳ m : The sample mea based o m uits commo to both occasios ad observed o the first occasio. ρ yx : The correlatio coefficiet betwee the variables y o x. ρ xz : The correlatio coefficiet betwee the variables x o z. ρ yz : The correlatio coefficiet betwee the variables y o z. m : The sample uits observed o the secod occasio ad commo with the first occasio. u : The sample size of the sample draw afresh o the secod occasio. : Total sample size.
J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) / www.aturalspublishig.com/jourals.asp 51 4 Proposed Product Ratio Estimators i Successive samplig I this sectio some dual to ratio estimators usig oe auxiliary variable have bee proposed. To estimate the populatio mea Ȳ o the secod occasio, two differet estimators are suggested. The first estimator is dual to ratio estimator based o sample of size u (= µ) draw afresh o the secod occasio ad is give by: t u = ȳ u z u Z, where z u =(1+g) Z g z ad g=. The secod estimator is a chai dual to ratio estimator based o the sample of size m (= λ) commo with both the occasios ad is defied as, t m = ȳ m x m x z Z, where x m =(1+g) X g x, x =(1+g) X g x ad g=. Combiig the estimators t u ad t m, the fial estimator t dr as follows t dr = ψt u +(1 ψ)t m, where ψ is a ukow costat to be determied such that MSE(t dr ) is miimum ad prove theoretically that the estimator is more efficiet tha the proposed estimator by (i) Cochra (1977) whe o auxiliary variables are used at ay occasio.this classical differece estimator is a widely used estimator to estimate the populatio mea Ȳ, i successive samplig. It is give by y = φ ȳ u +(1 φ )ȳ m, where φ is a ukow costat to be determied such that V( ˆȲ) opt is miimum ad y u = y u is the sample mea of the umatched portio o the secod occasio ad ȳ m = ȳ m+ b( y 1 ȳ 1m ) is based o matched portio. The variace of this estimator is V( ˆȲ) opt =[1+ (1 ρyx )] S y. Similarly, the variace of the mea per uit estimator is give by V(ȳ)= S y. (4.1) (4.) (4.3) 4.1 Properties of t dr Sice t u ad t m both are biased estimators of t dr, therefore, resultig estimator t dr is also a biased estimator. The bias ad MSE up to the first order of approximatio are derived as usig large sample approximatio give below: y u = Ȳ(1+eȳu ), ȳ m = Ȳ(1+eȳm ), x m = X(1+e xm ), x = X(1+e x ), z = Z(1+e z ), z u = Z(1+e zu ) where eȳu,eȳm,e xm,e x,e z, z u are samplig errors ad are of very small quatities. We assume that E(eȳu )=E(eȳm ) = E(e xm )=E(e x )=E(e z )=E(e zu )=0. The for simple radom samplig without replacemet for both first ad secod ) occasios, we write) by usig occasio wise operatio of expectatio as: E(eȳ u )=( 1u 1 Sy,E(e ȳ m )=( 1 m 1 Sy ), E(e x m )=( 1 m 1 Sx,E(e x )= ) 1 S x, E(e z )= ) 1 S z, E(e ) yu e zu )=( 1u 1 S yz, ) S yx, E(eȳm e xm )=( 1 m 1 ( ) E(eȳm e x )= 1 m 1 S yx, E(eȳm e x )= ) 1 Syx, E(eȳm e z )= ) 1 Syz, E(e xm e x )= ) 1 S x, E(e xm e z )= ) 1 Sxz, E(e x e z )= ) 1 Sxz. Derive the bias of t u i lemma 4.1. Lemma 4.1The bias of ) t u deoted by B(t u ) is give by B(t u )= Ȳ( 1u 1 gρ yz S y S z
5. Beevi : Dual to ratio estimators for mea estimatio i... Proof.Expressig (4.1) i terms of e s ad get t u = Ȳ(1+eȳu )[(1+g) g(1+e zu )] = (1+eȳu )(1 ge zu ). Takig expectatio o both sides ad igorig higher orders, E(t u Ȳ) = Ȳg 1 ) E(eȳu e zu ) u B(t u ) = Ȳ u 1 ) gρ yz S y S z. (4.4) The bias of t m is derived i lemma 4.. Lemma 4.The bias) of t m is deoted by B(t m )give by B(t m )= Ȳ( 1 m 1 (Sx ρ yx S y S x )+ ) 1 [ gρyz S y S z ] Proof.Expressig (4.) i terms of e s, get t m = Ȳ(1+eȳm )(1+e x ) 1 (1 ge xm )(1 ge z ). Expadig the right had side ad eglectig the terms with power two or greater ad get t m = Ȳ[(1+eȳm e x + e x eȳm e x ) (1 geȳm ge z + g eȳm e z )]. Takig expectatio (4.5) o both sides, B(t m )= Ȳ 1 ) (Sx m ρ yxs y S x )+ 1 ) [ gρ yz S y S z ]. (4.5) Usig 4.1 ad 4., derive the bias of t dr. Theorem 4.1Bias of the estimator t dr to the first order approximatio is, B(t dr ) = ψb(t u )+(1 ψ)b(t m ), (4.6) where B(t u ) = Ȳ ad B(t m )= Ȳ u 1 ) gρ yz S y S z. 1 ) (S m x ρ yx S y S x )+ 1 ) ( gρ yz S y S z ). Proof.The bias of the estimator t dr is give by B(t dr ) = E(t dr Ȳ) B(t dr ) = ψe(t u Ȳ)+(1 ψ)e(t m Ȳ), (4.7) Usig lemma (4.1) ad (4.) ito equatio (4.7), the expressio for the bias of the estimator t dr as show i (4.6) We derive the MSE of t u i lemma 4.3. Lemma 4.3The ( mea ) square error of t u deoted by M(t u ) is give by M(t u )= Ȳ 1u [S 1 y + g Sz ] gρ yz S y S z
J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) / www.aturalspublishig.com/jourals.asp 53 Proof.Expressig t u i terms of e s, get t u = Ȳ(1+eȳu )(1 ge zu ). Expadig ad squarig(4.8), the right had side ad eglectig the terms with power two or greater, t u = Ȳ(1+eȳu ge zu ) (4.8) t u = Ȳ(1+e ȳ u + g e z u geȳu e zu ). (4.9) Takig expectatio (4.9) o both sides ad get M(t u ) M(t u )= Ȳ 1 ) [S u y+ g Sz ] gρ yz S y S z. (4.10) The MSE of t m is derived i lemma 4.4 Lemma 4.4The mea square error of t m deoted by M(t m ) is give by M(t m )= Ȳ [ m 1 1 ) (g S ) z gρ yz S y S z ]. ) Sy + 1 ) (g S ) x gρ yxs y S x + m Proof.Expressig t m i terms of e s, {[ (1+g) X g(1+eȳm ) X ] t m = Ȳ(1+eȳm ) (1+e x ) X } [(1+g) Z g(1+e z ) Z], Z = Ȳ(1+eȳm )(1 ge xm )(1 e x )(1 e z ). (4.11) Expadig (4.10), the right had side ad eglectig the terms with power two or greater, get t m = Ȳ(1+eȳm ge x ge xm ge z ). (4.1) Squarig o both sides (4.11) ad takig expectatio, MSE of the estimator t m upto first order of approximatio as, M(t m )= Ȳ [ 1 ) Sy m + 1 ) (g S ) x gρ yxs y S x + m 1 ) (g Sz gρ ) yzs y S z ]. (4.13) Usig lemma 4.3 ad 4.4, we derive the MSE of t dr Theorem 4.The mea square error of the estimator t dr to the first order approximatio is, M(t dr )=ψ M(t u )+(1 ψ) M(t m )+ψ(1 ψ)cov(t u,t m ) (4.14) where M(t u ) = Ȳ u 1 M(t m )= Ȳ [ m 1 1 ) (g Sz gρ ) yzs y S z ]. ad Cov(t u,t m )=0. ) [S y+ g S z + gρ yz S y S z ] ) Sy + 1 ) (g S ) x gρ yxs y S x + m
54. Beevi : Dual to ratio estimators for mea estimatio i... Proof.The mea square error of the estimator t dr is give by M(t dr )=E(t dr Ȳ) M(t dr )=E[ψ(t u Ȳ)+(1 ψ)(t m Ȳ)], (4.15) M(t dr )=ψ M(t u )+(1 ψ) M(t m )+ψ(1 ψ)cov(t u,t m ), usig lemma (4.3) ad (4.4) ito the equatio (4.15), the expressio for the MSE of the estimator t dr as show i (4.14) Remark 4.3The estimators, t u ad t m are based o two idepedet samples of sizes u ad m respectively, therefore the covariace term has bee vaished. 5 Miimum Mea Square Error of t dr To obtai the optimum value of ψ, partially differetiate the expressio (4.14) with respect to ψ, ad put it equal to zero, we get ψ opt = M(t m ) M(t u )+M(t m ) substitutig the values of M(t u ) ad M(t m ) from (4.10) ad (4.13) i (5.1), get ψ opt = (k 1+µk ) (k 1 + µ k ) = µ[(k 1+µk )] (k 1 + µ k ). Substitutio of ψ opt from (5.1) ito (4.14) gives optimum value of MSE of t dr as: (5.1) M(t dr ) opt = M(t m )M(t u ) M(t u )+M(t m ). Substitutig the values of M(t m ) ad M(t u ) from (4.9) ad (4.10) i (5.), get M(t dr ) opt = 1 [ k 1 + µk 1 k k 1 + µ k ],, where k 1 = 1+g gρ yz, k = g(ρ yx ρ yz ), here µ(= u ) is the fractio of fresh sample draw o the secod occasio. Agai M(t dr ) opt derived i equatio (5.3) is the fuctio of µ. To estimate the populatio mea o each occasio the better choice of µ is 1 (o matchig). However, to estimate the chage i mea from oe occasio to the other, µ should be 0 (complete matchig). (5.) (5.3) 6 Replacemet Policy I order to estimate t dr with maximum precisio a optimum value of µ should be determied so as to kow what fractio of the sample o the first occasio should be replaced ad miimize, M(t dr ) opt i (5.3) with respect to µ, the optimum value of µ is obtaied as, ˆµ = k 1± k1 + k 1k, (6.1) k where k 1 = 1+g gρ yz, k = g(ρ yx ρ yz ). From (6.1) it is obvious that for ρ yz ρ yx two values of ˆµ are possible, therefore to choose a value of ˆµ, it should be remembered that 0 ˆµ 1. All other values of ˆµ are iadmissible. If both the real values of ˆµ are admissible, the lowest oe will be the best choice as it reduces the total cost of the survey. Substitutig the value of ˆµ from (6.1) i (5.3), M(t dr ) opt = 1 [ k 1 + ˆµk 1k k 1 + ˆ µ k ]. (6.)
J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) / www.aturalspublishig.com/jourals.asp 55 7 Efficiecy Comparisos I this sectio, to compare t dr with respect to ȳ, (i) sample mea of y, whe a sample uits are selected at secod occasio without ay matched portio. (ii) differece estimator (Cochra 1977) whe o auxiliary iformatio is used at ay occasio, have bee obtaied for kow values of ρ yx ad ρ yz. Sice ȳ ad ˆȲ are ubiased estimators of Ȳ, their variaces for large are respectively give by V(ȳ)= S y, (7.1) V ˆ (Y)opt =[1+ (1 ρ yx )] S Y. (7.) For differet values ρ yx ad ρ yz, the below shows the optimum value of µ. That is ˆµ. The percet relative efficiecies, R 1 ad R of t opt with respect to ȳ ad ˆȲ respectively, where ad R 1 = V (y) M(t dr ) opt 100 R = V (Y) ˆ opt 100. M(t dr ) opt The estimator t pr (at optimal coditios) is also compared with respect to the estimators V(ȳ) ad V( ˆȲ), respectively. Where [ M(t pr ) opt = 1 k1 + ˆµk ] 1k k 1 + µ ˆ (7.3) k ad ˆµ = k 1± k 1 + k 1k k, where k 1 = 1+g gρ yz, k = g(ρ yx ρ yz ). (7.4) 7.1 Empirical Study The expressios of the optimum value of µ (i.e. ˆµ) ad the percet relative efficiecies R 1 ad R are i terms of populatio correlatio coefficiets ρ yx ad ρ yz. The Table 7.1. shows that the values of ˆµ, R 1 ad R for differet choices of ρ yx, ρ yz ad µ. Table 1: Table 7.1. optimum values of µ ad percet relative efficiecies of t dr with respect to ȳ ad ˆȲ ρ yz ρ yx 0. 0.4 0.6 0.8 ˆµ 0.4973 0.506 0.5081 0.5139 0.3 R 1 104.66 105.8 106.98 108.19 R 91.4 96.3 101.41 86.55 ˆµ 0.490 0.4973 0.508 0.5085 0.5 R 1 108.14 109.17 110.50 109.89 R 107.03 104.6 108.44 87.79 ˆµ 0.4864 0.4916 0.497 0.509 0.7 R 1 111.18 113.03 114.9 115.6 R 110.67 108.31 10.86 9.49 ˆµ 0.4805 0.4858 0.4913 0.4970 0.9 R 1 115.79 117.06 118.38 119.76 R 114.61 7.9 106.53 95.80
56. Beevi : Dual to ratio estimators for mea estimatio i... 8 umerical Illustratio The results obtaied i previous selectios are ow examied with the help of oe atural populatio set of data. Populatio Source: [Free access to the data by the Statistical Abstracts of the Uited States.] Let Y (study variable) be the level of cor productio (i per acre) ad Z (auxiliary variate) be the dosage of fertilizer usig i cor filed i 50 couties i the Uited states i 007 ad X be the cor productio i the year 006 i the States of Uited states. Based o the above descriptio, the values of the differet required parameters for populatio is, = 50, X = 139, Ȳ = 118, S Y = 3314., ρ yx = 0.987, ρ yz = 0.141, ˆµ = 0.4796, ψ opt = 0.4996. Table : Table 8.1. Percet Relative Efficiecies of t dr with Respect to ȳ ad ˆȲ f g Relative Efficiecies 0.5 1 R 1 = 4018.18 R =381.8 0.7.33 R 1 = 4063.95 R = 408.15 0.9 9 R 1 = 4068.33 R = 410.87 where R 1 = V(ȳ) M(t dr ) opt ad R = V ˆ (Y)opt M(t dr ) opt. ρ yz ρ yx -0. -0.4-0.6-0.8 ˆµ 0.4973 0.506 0.5081 0.5139-0.3 R 1 104.66 105.8 106.98 108.19 R 91.4 96.3 101.41 86.55 ˆµ 0.490 0.4973 0.508 0.5085-0.5 R 1 108.14 109.17 110.50 109.89 R 107.03 104.6 108.44 87.79 ˆµ 0.4864 0.4916 0.497 0.509-0.7 R 1 16.18 135.03 14.9 159.6 R 135.67 14.31 159.86 19.49 ˆµ 0.4805 0.4858 0.4913 0.4970-0.9 R 1 115.79 117.06 118.38 119.76 R 114.61 7.9 106.53 95.80
J. Stat. Appl. Pro. 7, o. 1, 49-58 (018) / www.aturalspublishig.com/jourals.asp 57 Table 3: Table 8.. MSE ad Bias of Differet Estimators. Bias MSE Bias Ȳ M(t dr ) opt = 3.585-734816.16 67.55 V(ȳ) = 13.5680 V ˆ (Y)opt = 78.5600 M(t dr ) opt =.375-484314.8 4104.36 V(ȳ) = 94.6914 V ˆ (Y)opt =56.1143 M(t dr ) opt = 1.8103-163808.63 1388.0 V(ȳ) = 73.6500 V ˆ (Y)opt = 43.6440 9 Iterpretatios of Empirical Results of t dr From Table 7.1., the relative efficiecy is observed that the suggested estimator is compared with mea per uit estimator ad Cochra (1977) estimator. So, the use of auxiliary iformatio at both occasios is justified. 1.For fixed values of ρ yx, the value of R 1 ad R are icreasig with icreasig values of µ ad the icreasig ρ yz..the values of R 1, R ad µ are icreasig with icreasig values of ρ yz. This is a agreemet with the results Sukhatme et.al (1984), which justifies that higher the value of ρ yx, higher the fractio of fresh sample required at the secod (curret) occasio. 3.For fixed values of ρ yx ad ρ yz, there is appreciable gai i the performace of the proposed estimator t dr over ȳ ad ˆȲ with the icreasig value of µ. 10 Coclusio From Table 7.1. clearly see that the value of ˆµ (at optimum coditio) also exist for both the cosidered populatios. Hece, it justifies that the suggested family of estimators t dr is feasible uder optimal coditios. Tables 8.1. ad 8.. idicates that the suggested estimators t dr at optimum coditios is preferable over sample mea per uit estimator ad also performs better tha the Cochra s estimator. Hece, it may be cocluded that the estimatio of mea at curret usig auxiliary iformatio o both occasios i successive samplig is highly i terms of precisio ad reducig the cost of survey.clearly idicates that the proposed estimators is more efficiet tha simple arithmetic mea estimator ad Cochra (1977) estimator. The followig coclusio ca be formed from Tables 7.1. For fixed ρ yx, ρ yxz ad µ, the values of R 1 ad R are icreasig. This pheomeo idicates that smaller fresh sample at curret occasio is required, if a highly positively correlated auxiliary characters is available. That is the performace of precisio of the estimates also reduces the cost of the survey. 11 Perspective Table 7.1. clearly idicates that the suggested estimators is more efficiet tha simple arithmetic mea ad Cochra (1977) estimators. The followig coclusio ca be made from Table 7.1. Fixed ρ yx, the values of R 1 ad R are icreasig while ˆµ is decreasig with the icreasig values of ρ yz. This pheomeo idicates that smaller fresh sample at curret occasio is required, if a highly positively correlated auxiliary characters is available. For Fixed ρ yz, the values of R 1 ad ˆµ are icreasig while R is decreasig for iitial values of the icreasig values of ρ yx. Thus behavior is i agreemet with Cochra (1977) results, which explais that more the value of ρ yx, more fractio of fresh sample is required at curret occasio. That is the performace of precisio of the estimates as well as reduces the cost of the survey. Uder the give framework (Tables 8.1.ad 8..) tables it is possible to reduce the bias ad mea square error of the estimator, the aalytical ad empirical results support the theoretical justificatio of the work. The estimatio of populatio mea o successive occasios should be ecouraged as there are umerous practical situatios that require the estimate of mea at differet poits of time as the characters are time depedet. Hece, the proposed estimators should be recommeded for their use i practice.
58. Beevi : Dual to ratio estimators for mea estimatio i... Refereces [1] Biradar, R.S. ad Sigh, H.P., Successive samplig usig auxiliary iformatio o both occasios. Cal. Stat. Assoc. Bull., 51, 43-51, (001). [] Cochra, W.G.: Samplig Techiques. 3 rd editio, (1977). [3] Chaturvedi, D.K. ad Tripathi, T.P. (1983). Estimatio of Populatio ratio o two occasios usig multivariate auxiliary iformatio. Jour. Id. Stat. Asso., 1, 113-10. [4] Das, K.: Estimatio of populatio ratio o two occasios. Jour. Id. Soc. Agri. Stat. 34(), 1-9, (198). [5] Feg, G.S, ad Zou, G.: Sample rotatio method with auxiliary variable. Commuicatio i Statistics-Theory ad Methods, 6, 6, 1497-1509, (1997). [6] Jesse, R.J : Statistical Ivestigatio of a sample survey for obtaiig farm facts. Iowa Agricultural Experimet Statistical Research Bulleti, 304, (194). [7] arai, R.D.: O the recurrece formula i samplig o successive occasios. Jour. Id. Soc. Agri. Stat., 5, 96-99, (1953). [8] Okafor, F.C. (199) The theory ad applicatio of samplig over two occasios for the estimatio of curret populatio ratio, Statistica1, 137-147. [9] Patterso, H.D.: Samplig o successive occasios with partial replacemet of uits. Jour. Roy. Stat. Soc., B 1, 41-55, (1950). [10] Rao, J..K. ad Mudholkar, G.S.: Geeralized multivariate estimators for the mea of fiite populatio parameters. Jour. Amer. Stat. Asso., 6, 1008-101, (1967). [11] Se, A.R.: Successive samplig with two auxiliary variables. Sakhya, Series B, 33, 371-378, (1971). [1] Se, A.R.: Theory ad applicatio of samplig o repeated occasios. Jour. Amer. Stat. Asso., 59, 49-509, (1973). [13] Se, A.R., Sellers, S. ad Smith, G.E.J.: The use of a ratio estimate i successive samplig.biometrics, 31, 673-683, (1975). [14] Srivekataramaa, T.: A dual to ratio estimator i sample surveys, Biometrika, 67(1), 199?04, (1980). [15] Sigh, G..: Estimatio of populatio mea usig auxiliary iformatio o recet occasio i h occasios successive samplig. Statistics i Trasitio, 6(4), 53-53, (003). [16] Sigh, G.. ad Sigh, V.K.: O the use of auxiliary iformatio i successive samplig. Jour. Id. Soc. Agri. Stat. 54 (1), 1-1, (001). [17] Yates, F.: Samplig Methods for Cesuses ad Surveys. Charles Griffl ad Co.,Lodo, (1949). azeema Beevi is a statistical ivestigator at Directorate of Ecoomics ad Statistics ad doe the Ph.D. i the Departmet of Statistics, Uiversity of Calicut. My research iterests are Samplig Theory (Two-Phase Samplig, Super Populatio Model, Predictive Model), Statistical Iferece ad Data Aalysis i particular, Small Area Estimatio ad Item Respose Theory. I am a lifelog member of Kerala Statistical Associatio (KSA) ad also Reviewer of Joural of Applied Probability ad Statistics, UK (016).