An Improved Generalized Estimation Procedure of Current Population Mean in Two-Occasion Successive Sampling

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1 Journal of Modern Alied Statistical Methods Volume 15 Issue Article An Imroved Generalized Estimation Procedure of Current Poulation Mean in Two-Occasion Successive Samling G. N. Singh Indian School of Mines, Dhanbad, India, Alok Kumar Singh Indian School of Mines, Dhanbad, India, Anu Kumar Sharma Indian School of Mines, Dhanbad, India, Follow this and additional works at: htt://digitalcommons.wayne.edu/masm Part of the Alied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Singh, G. N.; Singh, Alok Kumar; and Sharma, Anu Kumar (016) "An Imroved Generalized Estimation Procedure of Current Poulation Mean in Two-Occasion Successive Samling," Journal of Modern Alied Statistical Methods: Vol. 15 : Iss., Article 14. DOI: 10.37/masm/ Available at: htt://digitalcommons.wayne.edu/masm/vol15/iss/14 This Regular Article is brought to you for free and oen access by the Oen Access Journals at DigitalCommons@WayneState. It has been acceted for inclusion in Journal of Modern Alied Statistical Methods by an authorized editor of DigitalCommons@WayneState.

2 An Imroved Generalized Estimation Procedure of Current Poulation Mean in Two-Occasion Successive Samling Cover Page Footnote Authors are thankful to the Indian School of Mines, Dhanbad for roviding financial and infrastructural suort in comletion of this research work. This regular article is available in Journal of Modern Alied Statistical Methods: htt://digitalcommons.wayne.edu/masm/vol15/ iss/14

3 Journal of Modern Alied Statistical Methods November 016, Vol. 15, No., doi: 10.37/masm/ Coyright 016 JMASM, Inc. ISSN An Imroved Generalized Estimation Procedure of Current Poulation Mean in Two-Occasion Successive Samling G. N. Singh Indian School of Mines Dhanbad, India Alok Kumar Singh Indian School of Mines Dhanbad, India Anu Kumar Sharma Indian School of Mines Dhanbad, India The resent work is an attemt to make use of several auxiliary variables on both occasions for imroving the recision of estimates for the current oulation mean in two-occasion successive samling. A generalized exonential-cum-regression tye estimator of the current oulation mean is roosed and its otimum relacement strategy has been discussed. Emirical studies are carried out to show the dominance of the roosed estimation rocedure over the samle mean estimator and natural successive samling estimator. Emirical results have been interreted and suitable recommendations are ut forward to survey ractitioners. Keywords: Successive samling, auxiliary information, bias, mean square error, otimum relacement strategy Introduction There are many roblems of ractical interest in different fields of the alied and environmental sciences where the various characters of interest have tendencies to change over time. It is often required to monitor the behaviors of such characters at different oints of time (occasions) and the atterns of variations occurring over the eriod of time. For examle, an investigator or owner involved in the cold drinks industry may be interested (a) to know the average or total sale of cold drinks in the different seasons, (b) to know the attern of change in average or total sale of cold drinks in two different seasons, or (c) they may be simultaneously interested to know both (a) and (b). These kinds of roblems are well answered by the tools of successive (rotation) samling. A. K. Sharma is in the Deartment of Alied Mathematics. at: aksharma.ism@gmail.com. Alok Kumar Singh is a Junior Research Fellow in the Deartment of Alied Mathematics. at: aksingh.ism@gmail.com. G. N. Singh is in the Deartment of Alied Mathematics. at: gnsingh_ism@yahoo.com. 187

4 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE The theory of successive (rotation) samling was initiated by Jessen (194), where the idea of using the available information gathered on revious occasions during the ast surveys was suggested. Jessen (194) used ast information in order to make current estimates more recise in agronomical surveys. This idea was further exlored by Patterson (1950), Rao and Graham (1964), Guta (1979), Das (198), and Chaturvedi and Triathi (1983), among others. Sen (1971) extended this theory by utilizing the information on two auxiliary variables, which was available on revious occasions, and suggested estimators of current the oulation mean in two-occasion successive samling. Sen (197; 1973) generalized his idea for several auxiliary variables. V. K. Singh, Singh, and Shukla (1991) and G. N. Singh and Singh (001) used the auxiliary information from the current occasion for estimating the current oulation mean in two-occasion successive samling. G. N. Singh (003) extended this work for h-occasion successive samling. In many situations, information on an auxiliary variable may be readily available on the first as well as on the second occasion. For instance, to study the roblems related to the ublic health and welfare of a state or a country, several factors that can be treated as auxiliary variables, such as the number of beds, doctors, and suorting staff in different hositals, the amount of funds available for medicine, etc. may be known well in advance. Likewise, in other cases, there may be information available on several auxiliary variables and, if efficiently utilized, the estimates could be made more recise. Utilizing the auxiliary information on both occasions, Feng and Zou (1997), Biradar and Singh (001), G. N. Singh (005), G. N. Singh and Priyanka (006; 007; 008; 010), G. N. Singh and Karna (009), H. P. Singh and Vishwakarma (009), G. N. Singh and Prasad (010), G. N. Singh, Karna, and Prasad (011), H. P. Singh, Tailor, Singh, and Kim (011), G. N. Singh and Prasad (013), and G. N. Singh and Homa (013) roosed varieties of estimators of the oulation mean on the current (second) occasions in two-occasion successive samling. Motivated with these arguments, the obective of the resent work is to roose a more recise estimator of the oulation mean on the current occasion using the information on ( ) stable auxiliary variables which are readily available on both occasions. Utilizing the information on auxiliary variables, a generalized exonential-cum-regression tye estimator of the current oulation mean in two-occasion successive samling has been roosed. The dominance of the roosed estimator has been shown over the samle mean and natural successive samling estimators. Emirical studies have been carried out to ustify the roosition of estimator. Results are interreted, and suitable recommendations have been made. 188

5 SINGH ET AL. Formulation of Estimator Let U = (U 1, U,, U N ) be a finite oulation of N units which has been samled over two occasions, and let the character under study be denoted by x (y) on the first (second) occasion. It is assumed that the information on stable (non-negative integer constant) auxiliary variables z ( = 1,,, ), whose oulation means are known and closely related to x and y, are available on the first (second) occasion. Let a simle random samle (without relacement) of size n be drawn on the first occasion. A random subsamle of size m = nλ is retained (matched) for its use on the second occasion, while a fresh simle random samle (without relacement) of size u = (n m) = nμ is drawn on the second occasion from the entire oulation so that the samle size on the second occasion is also n. Here λ and μ (λ + μ = 1) are the fractions of the matched and fresh samles, resectively, on the current (second) occasion. The values of λ or μ would be chosen otimally. The following notations have been considered for use below: X (Y ): The oulation mean of the study variable x (y) on the first (second) occasion. Z : Poulation mean of the th ( = 1,,, ) auxiliary variable. x n, x m, y u, y m, z n, z u, z m, ( = 1,,, ): The samle means of the resective variables based on the samle sizes shown in the subscrit.,,, : Poulation correlation coefficients between the variables xz z zk shown in the subscrit. S 1 N 1 x N x X i1 i : Poulation variance of the variable x. S, S : Poulation variances of the variables y and z ( = 1,,, ), y z resectively. To estimate the oulation mean Y on the current (second) occasion, two indeendent estimators are suggested. One is a generalized exonential tye estimator based on a samle of size u (= nµ) drawn afresh on the second occasion and given by T u Z z u yuex (1) 1 Z zu 189

6 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE The second estimator is a generalized exonential-cum-regression tye estimator based on the samle of size m (= nλ) common to both the occasions and is defined as where * m m * m m 1 T y b Z z () Z z m xmex 1 Z zm m * m * * * Z z n ym ym b xn xm, xn xn ex 1 Z zn x Combining the estimators T u and T m, we have the final estimator T of Y given as T T 1 T (3) u where φ (0 φ 1) is an unknown constant (scalar) to be determined under certain criterion. Remark 1: The estimator T u is suitable for estimating the oulation mean on the current occasion, while the estimator T m is more aroriate for estimating change over two occasions. These two estimators may be derived from the estimator T by choosing φ as 1 or 0, resectively. To handle both roblems simultaneously, an otimum choice of φ is required. Proerties of the Proosed Estimator Bias and Mean Square Error Because the estimators T u and T m are generalized exonential and generalized exonential-cum-regression tye estimators, they are biased estimators of the oulation mean Y. Therefore, the resulting estimator T is also a biased estimator of Y. The bias B(.) and mean square error M(.) of the estimator T is derived under m 190

7 SINGH ET AL. large samle assumtion and u to the first order of aroximations using the following transformations: y 1 e Y, y 1 e Y, x 1 e X, x 1 e X, u 1 m m 3 n , 1, 1, z 1 e Z, z 1 e Z, z 1 e Z, s 1 e S, u m n s m e S s m e S s m e S z 9 z xz 10 xz 11 1 s m e S x 1 x such that E(e i ) = 0 and E(e h ) = 0, e i 1 for i = 1,, 3, 4, 11, 1 and e h 1 for h = 5, 6,, 10, = 1,, 3,,. Under the above transformations, the estimators T u and T m take the following forms: Tu Y 1 e1 ex 1 1 e5 e5 (4) 1 1 e7 e7 1e4 ex 1 1 e e1 e6 e6 1 e3 ex 1 1 Tm Y 1 e X (5) Thus, there are the following theorems: Theorem 1: obtained as The bias of the estimator T to the first order of aroximations is T T T B B 1 B (6) u m where B a u N 8 1 z 8 1 z zk 1 Yz T Y u (7) 191

8 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE and B T m X X X z 8 z 8 z zk z S m n z Sx S S z S S x (8) where α qr = E[(x i X ) (y i Y ) q (z Z ) r ] for integers, q, r 0 and = 1,,,. Proof: The bias of the estimator T is given by T T Y Tu Y Tm Y B E E 1 E B 1 B T T u m (9) where B(T u ) = E(T u Y ) and B(T m ) = E(T m Y ). To derive the B(T u ), roceed as follows: 1 e5 e5 ETu Y EY 1 e1 ex 1 Y (10) 1 Now, exanding the right hand side of (10) binomially and exonentially and taking exectations and retaining the terms u to the first order of aroximations, we have the exression of the bias of the estimator T u as given in (7). Similarly, the bias of the estimator T m is written as E T m Y E Ye 1 e7 e7 1e4 ex 1 1 e 1 11 (11) X 1 1 e 1 e6 e6 1 e3 ex

9 SINGH ET AL. Exanding (11) binomially and exonentially, taking exectations both sides, and retaining the terms u to the first order of aroximations yields the exression of the bias of the estimator T m as shown in (8). Theorem : The mean square error of the estimator T to the first degree of aroximation is obtained as where T T T T T M M 1 M 1 C, (1) u m u m 1 1 MTu Sy 1 4 z zk u N 1 k1 (13) M T 1 k z z k m S y m N 1 4 z z k 1 k 1 (14) 1 C, 1 T T S u m y N 1 Proof: The mean square error of the estimator T is given by (15) T T Y Tu Y Tm Y M E E 1 T T T T M u 1 M m 1 C u, m (16) where C(T u, T m ) = E[(T u Y )(T m Y )], M(T u ) = E(T u Y ), M(T m ) = E(T m Y ). To derive the M(T u ), roceed as follows: 1 e5 e5 ETu Y EY 1 e1 ex 1 Y (17) 1 193

10 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE Now, exanding the right hand side of (17) binomially and exonentially and taking exectations and retaining the terms u to the first order of aroximations, we have the exression of the mean square error of the estimator T u as given in (13). The mean square error of the estimator T m is written as E e 7 1 e4 ex 1 e 1 11 e7 Tm Y EYe X 1 e1 e 6 1e3 ex 1 e 6 (18) Exanding (18) binomially and exonential, taking exectations both sides, and retaining the terms u to the first order of aroximations, the exression is derived for the mean square error of the estimator T m as shown in (14). Similarly, the exectation of C(T u, T m ) may be derived in the form shown in (15). Remark : The above results are derived under the assumtion that the coefficients of variation of variables x, y, z, and z k are aroximately equal. We have also considered the intuitive assumtions ( = 1,, 3,, ), as suggested by Cochran (1984) and Feng and Zou (1997). In the light of these assumtions, the exression of M(T m ) takes the form as shown in (14). Minimum Mean Square Errors of the Estimator T Because the mean square error of the estimator T in (1) is a function of the unknown constant (scalar) φ, it can be minimized with resect to φ and, subsequently, the otimum value of φ is obtained as xz ot MTm C Tu, Tm T T T T M M C, u m u m (19) From (19), substituting the value of φ ot in (1), we get the otimum mean square error of the estimator T as M T ot Tu Tm Tu Tm T T T T M M C, M M C, u m u m (0) 194

11 SINGH ET AL. Further substituting the values from (13)-(15) into (19) and (0), the simlified values of φ ot and M(T) ot are obtained as A A 9 8 ot A9 A1 A13 (1) M T ot S y A18 A0 A 1 A9 A13 A1 n () where 1 1, 1, 1 k z z A k z zk k 1 A A3, 4 1, 5 1 3, 6 3, A A A A A A A A 1 1,,, 1 1 A7 A1 A4 A8 A6 fa7 A9 A5 A8 A k 1 A A A, A A A fa, A A fa, A A A A A, A A A, A A A, A A A, A A A fa, A A A A fa, A fa f A, A fa A z zk, where f = n/n. Otimum Relacement Strategy of the Estimator T The otimum mean square error M(T) ot in () is a function on µ, the fraction of the samle to be drawn afresh at the second occasion. It is an imortant factor in reducing the cost of the survey, therefore, to determine the otimum value of µ so that Y may be estimated with maximum recision and minimum cost. We thus minimize M(T) ot with resect to µ which results in a quadratic equation in µ, which is shown as D D D 0 (3)

12 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE where D 1 = A 1 A 0 + A 13 A 1, D = A 10 A 0 + A 13 A 18, D 3 = A 10 A 1 A 1 A 18. Solving (3) for μ, the solutions of μ (say ˆ ) are given as D D D1 D3 ˆ (4) D 1 From (4), it is clear that the real values of ˆ exist IFF the quantity under the square root is greater than or equal to zero. For any combinations of correlations which satisfy this condition for real solutions, two real values of ˆ are ossible. Hence, while choosing the values of ˆ, it should be remembered that 0 ˆ 1, and that all other values of ˆ are said to be inadmissible. If both the values of ˆ are admissible, the lowest one is the best choice as it reduces the cost of the survey. From (4), substituting the admissible value of ˆ (say μ 0 ) in (), we have the otimum value of mean square error of the estimator T, which is shown below: M T * ot S y A A A A A A n (5) Secial Case When the auxiliary variates are mutually uncorrelated, i.e., 0 for k = 1,, 3,,, then the exression of the otimum values of μ and M(T) ot reduce to zz k D D D D ˆ (6) * * * * 1 3 * D1 and M T * * * * ot * * * A A A S A A A n y (7) where 196

13 SINGH ET AL. D A A A A, D A A A A, D A A A A, * * * * * * * * * * * * * * * * * * * * * * A1 1, A 1, 5 1 3, 6 3, A A A A A A A * * * * * * * * * A7 A1 A4, A8 A6 fa7, A9 A5 A8, A10 1, 4 A * 11 A A, A A A fa, A A fa, A A A A A, * * * * * * * * * * * * * A A A, A A A, A A A, A A A fa, * * * * * * * * * * * * A A A A fa, A fa f A, A fa A * * * * * * * * * * * Efficiency Comarison The ercent relative efficiencies of the estimator T with resect to (i) the samle mean estimator y n when there is no matching and (ii) Y ˆ * 1 * u yu y m when no additional auxiliary information is used at any occasion, where y m ym xn xm, have been obtained for different choices of the correlations involved. Since y n and Y ˆ are unbiased estimators of Y following Sukhatme, Sukhatme, Sukhatme, and Asok (1984), the variance of y n and otimum variance of Y ˆ are given by 1 1 n N y S V n y ˆ S S V Y ot 1 1 n N y y (8) (9) The ercent relative efficiencies E 1 and E of T (under otimal condition) with resect to y n and ˆ Y, resectively, are given by V y V Y ˆ E 100, 100 n ot 1 E * * M T M T ot ot 197

14 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE Emirical Study The exression of the otimum μ (i.e., μ 0 ) and the ercent relative efficiencies E 1 and E are in terms of oulation correlation coefficients. Therefore, the values of μ 0, E 1, and E have been comuted for different choices of ositive correlations, while the value of f (= n/n) (samling fraction) is chosen to be 0.1. For emirical studies, cases of = and 3 have been considered. Case 1 For = and assuming that the two auxiliary variables are correlated, i.e., the values of A 1, A, A 3, A 4, A 9, and A 10 take the form 0, zz 1 3 A1 1 1, 1 z1z A z, 1z 1 A 1 1 A, 1, A z1z 1 Substituting these values in (4) and (5) yields the values of otimum MT *, ot E 1, and E. For different choices of correlations, Tables 1- show the otimum values of μ (i.e., μ 0 ) and ercent relative efficiencies E 1 and E of the estimator T (under otimal condition) with resect to y n and ˆ Y, resectively. Case For = and assuming that the two auxiliary variables are uncorrelated, i.e., 0, the values of A, A, and A take the form zz 1 * 1 * * A 1, A, A * * * Using these values in (6) and (7), the otimum values of μ, E 1, and E are shown in Table

15 SINGH ET AL. Table 1. Otimum values of μ and ercent relative efficiencies of T with resect to y n and ˆ Y for ρ = 0.3 ρ ρ 1 ρ zz μ 0 E 1 E μ 0 E 1 E μ 0 E 1 E Table. Otimum values of Otimum values of μ and ercent relative efficiencies of T with resect to y n and ˆ Y for ρ = 0.5 ρ ρ 1 ρ zz μ 0 E 1 E μ 0 E 1 E μ 0 E 1 E * Note: * indicates μ 0 does not exists and -- indicates no gain. 199

16 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE Table 3. Otimum values of Otimum values of μ and ercent relative efficiencies of T with resect to y n and Y ˆ for 0.0 zz 1 ρ 1 ρ ρ μ 0 E 1 E μ 0 E 1 E μ 0 E 1 E * * * * * * * * * * * * Note: * indicates μ 0 does not exists and -- indicates no gain. Case 3 For = 3 and assuming that the two auxiliary variables are correlated, i.e., 0 for k = 1,, 3, the values of A 1, A, A 3, A 4, and A 10 take the form z1z 1 3 z1z 3 3 zz , 4, A 1, z z z z z z A 1, A A A z z z z z z In this case there are seven different correlations. For a few sets of these seven correlations, the otimum value of μ (i.e., μ 0 ) and ercent relative efficiencies E 1 and E of the estimator T (under otimal condition) with resect to y n and Y ˆ have been comuted and shown below: zz k 00

17 SINGH ET AL. Set 1: 0.3, 0.5, 0.6, 0.5, 0.3, 0.4, 1 3 z1z z1z3 zz 0.6, , E , E Set : 0.3, 0.6, 0.6, 0.5, 0.3, 0.4, 1 3 z1z z1z3 zz 0.6, 0.900, E , E Set 3: 0.3, 0.7, 0.6, 0.5, 0.3, 0.4, 1 3 z1z z1z3 zz 0.6, 0.393, E , E Set 4: 0.3, 0.8, 0.6, 0.5, 0.3, 0.4, Case z1z z1z3 zz 0.6, 0.105, E , E For = 3 and assuming that the two auxiliary variables are uncorrelated, i.e., 0 for k = 1,, 3, the values of A 1, A, and A 10 take the form zz 1 7 A1 1 1, A 3, A For a few sets of the above four correlations, the values of the otimum value of μ (i.e., μ 0 ) and ercent relative efficiencies E 1 and E are shown below: Set 1: 0.3, 0.5, 0.6, 0.4, , 38.4, E1 E

18 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE Set : 0.4, 0.5, 0.6, 0.4, , , E E Set 3: 0.5, 0.5, 0.6, 0.4, , 449.1, E E Set 4: 0.6, 0.5, 0.6, 0.4, , , E E Conclusion 1. From Tables 1- it is vindicated that: a. For the fixed values of, zz 1, and 1, the values of μ 0 decrease. Similarly, and E 1 and E increase with the increasing values of for fixed values of, zz, and 1, the otimum value of μ 0 decrease and E 1 and E increase with the increasing values of. 1 These atterns indicate that a smaller fresh samle on the current occasion is required if highly correlated auxiliary variables are available. b. For the fixed values of,, and 1, the values of μ 0, E 1, and E decrease with the increasing values of zz ; this means that the 1 auxiliary variables are quite sensitive with resect to the relation between them.. From Table 3, i.e., when the auxiliary variables are uncorrelated, it has been observed that a. For fixed values of and 1, the values of E 1 and E increase with increasing value of, while no definite atterns are observed in μ 0. 0

19 SINGH ET AL. b. For fixed values of and with increasing value of 1, the values of E 1 and E increase, while no definite atterns are observed in μ 0. Similar atterns are visible for the case when the values of and are fixed and increasing values of are observed. 3. For = 3 and when the three auxiliary variables are uncorrelated, for fixed values of,,,,, and, the values of μ 0 decrease zz 1 zz 3 zz 1 3 while E 1 and E increase with the increasing values of 1. Similar atterns are observed if the case for the increasing values of or 3 is taken into account. 4. For = 3 and when the three auxiliary variables are mutually correlated, we observed that no secific attern is seen as for so many combinations of correlations the otimum values of μ 0 do not exist. This behavior suggests that the correlation between the auxiliary variable do not lay a significant role in terms of the roosed estimator. 5. It could be seen that the results are more areciable for one and two auxiliary variables, while when the number of auxiliary variables increases, the exressions become comlex due to the increase in the number of correlations. Hence, ractically, it is more realistic to use two auxiliary variables out of several available auxiliary variables. Thus, it is clear that the use of the auxiliary variables is highly rewarding in terms of the roosed estimator. It is also clear that, if the information on highly correlated auxiliary variables is used, only a relatively small fraction of the samle on the current (second) occasion is desired to be relaced by a fresh samle, which reduces the cost of the survey. Hence, it can be recommended for future use. 3 1 References Biradar, R. S., & Singh, H. P. (001). Successive samling using auxiliary information on both occasions. Bulletin of the Calcutta Statistical Association, 51(03-04), Chaturvedi, D. K., & Triathi, T. P. (1983). Estimation of oulation ratio on two occasions using multivariate auxiliary information. Journal of the Indian Statistical Association, 1,

20 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE Cochran, W. G. (1984). Samling techniques (3rd ed.). New Delhi, India: Wiley Eastern. Das, A. K. (198). Estimation of oulation ratio on two occasions. Journal of the Indian Society of Agricultural Statistics, 34(), 1-9. Feng, S., & Zou, G. (1997). Samle rotation method with auxiliary variable. Communications in Statistics Theory and Methods, 6(6), doi: / Guta, P. C. (1979). Samling on two successive occasions. Journal of Statistical Research, 13(13), Jessen, R. J. (194). Statistical Investigation of a Samle Survey for obtaining farm facts. Agricultural Research Bulletins, 304, Retrieved from htt://lib.dr.iastate.edu/ag_researchbulletins/7/ Patterson, H. D. (1950). Samling on successive occasions with artial relacement of units. Journal of the Royal Statistical Society. Series B (Methodological), 1(), Available from htt:// Rao, J. N. K., & Graham, J. E. (1964). Rotation design for samling on reeated occasions. Journal of the American Statistical Association, 59(306), doi: / Sen, A. R. (1971). Successive samling with two auxiliary variables. Sankhyā: The Indian Journal of Statistics, Series B, 33(3/4), Available from htt:// Sen, A. R. (197). Successive samling with ( 1) auxiliary variables. The Annals of Mathematical Statistics, 43(6), Available from htt:// Sen, A. R. (1973). Note: Theory and alication of samling on reeated occasions with several auxiliary variables. Biometrics, 9(), doi: /59401 Singh, G. N. (003). Estimation of oulation mean using auxiliary information on recent occasion in h-occasion successive samling. Statistics in Transition, 6(4), Singh, G. N. (005). On the use of chain-tye ratio estimator in successive samling. Statistics in Transition, 7(1), 1-6. Singh, G. N., & Homa, F. (013). Effective rotation atterns in successive samling over two occasions. Journal of Statistical Theory and Practice, 7(1), doi: /

21 SINGH ET AL. Singh, G. N., & Karna, J. P. (009). Estimation of oulation mean on current occasions in two-occasion successive samling. Metron, 67(1), Singh, G. N., Karna, J. P., & Prasad, S. (011). On the use of multile auxiliary variables in estimation of current oulation mean in two-occasion successive samling. Sri Lankan Journal of Alied Statistics, 1, doi: /slastats.v1i Singh, G. N., & Prasad, S. (010). Some estimator of oulation mean in two-occasion rotation atterns. Association for the Advancement of Modelling & Simulation Techniques in Enterrises, 47(), Singh, G. N., & Prasad, S. (013). Best linear unbiased estimators of oulation mean on current occasion in two-occasion successive samling. Statistics in Transition New Series, 14(1), Singh, G. N., & Priyanka, K. (006). On the use of chain-tye ratio to difference estimator in successive samling. International Journal of Alied Mathematics and Statistics, 5(S06), Singh, G. N., & Priyanka, K. (007). On the use of auxiliary information in search of good rotation atterns on successive occasions. International Journal of Statistics and Economics, 1(A07), Singh, G. N., & Priyanka, K. (008). On the use of several auxiliary variates to imrove the recision of estimates at current occasion. Journal of the Indian Society of Agricultural Statistics, 6(3), Retrieved from htt://isas.org.in/isas/s/volume/vol6/7-g.n.singh.df Singh, G. N., & Priyanka, K. (010). Estimation of oulation mean at current occasion in resence of several varying auxiliary variates in two-occasion successive samling. Statistics in Transition New Series, 11(1), Singh, G. N., & Singh, V. K. (001). On the use of auxiliary information in successive samling. Journal of the Indian Society of Agricultural Statistics, 54(1), 1-1. Retrieved from htt:// Singh, H. P., Tailor, R., Singh, S., & Kim, J.-M., (011). Estimation of oulation variance in successive samling. Quality & Quantity, 45(3), doi: /s Singh, H. P., & Vishwakarma, G. K. (009). A general rocedure for estimating oulation mean in successive samling. Communications in Statistics Theory and Methods, 38(), doi: /

22 AN IMPROVED GENERALIZED ESTIMATION PROCEDURE Singh, V. K., Singh, G. N., & Shukla, D. (1991). An efficient family of ratio-cum-difference tye estimators in successive samling over two occasions. Journal of Scientific Research, 41(C), Sukhatme, P. V., Sukhatme, B. V., Sukhatme, S., & Asok, C. (1984). Samling theory of surveys with alications. Ames, IA: Iowa State University Press. 06