ISSN X A Modified Procedure for Estimating the Population Mean in Two-occasion Successive Samplings

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1 Afrika Statistika Vol. 1 3, 017, pages DOI: Afrika Statistika ISSN X A Modified Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Housila Prasad Singh and Suryal Kant Pal School of Studies in Statistics, Vikram University, Ujjain , India Received : August, 016; Accepted : October 0, 016 Copyright c 017, Afrika Statistika and Statistics and, the Probability African Society SPAS. All rights reserved Abstract.. This paper addresses the problem of estimating the current population mean in two occasion successive sampling. Utilizing the readily available information on two auxiliary variables on both occasions and the information on study variable from the previous occasion, some new estimation procedures have been developed. Properties of the proposed estimators have been studied and their respective optimum replacement policies are discussed. Relative comparison of efficiencies of the suggested estimators with the sample mean estimator when there is no matching from previous occasion and the optimum successive sampling estimator when no auxiliary information is used have been incorporated. Empirical study is carried out to judge the merits of the suggested estimators and suitable recommendations are given. We have also added a practical application in order to examine the performance of the proposed estimators. Key words: Successive sampling, Study variable, Auxiliary variable, Mean squared error, Optimum Replacement Policy. AMS 010 Mathematics Subject Classification : 6D05. Corresponding author : Surya Kant Pal: suryakantpal6676@gmail.com Housila Prasad Singh : hpsujn@gmail.com

2 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Abstract in French - Résumé Ce papier traite du problème de l estimation de la moyenne d un paramètre d une population dans le schéma d un échantillonnage en deux étapes successives. De nouvellesprocédures d estimation sont développées, basées sur l information fournies par les deux variables auxiliaires lors dans les deux éetapes et celle reçue de l estimation de la première étape.les propriétés de ces estimateurs ont été étudiées et la stratéegie optimale de leur remplacement sont discutées. L effcacitée relative des nouveaux estimateurs a été présentée, d abord par rapport à la moyenne empirique lorsqu il n y a pas de correspondances avec la première étape, ensuite et par rapport à l estimateur optimal de l échantillonnage successif lorsque les variables auxiliaires ne sont pas utilisées. Une étude de simulation a été entreprise pour montrer les mérites de cette méthode, ayant aboutit à des recommandations. Enfin, nous avons ajouté une application pratique pour examiner la performance de nos estimateurs. 1. Introduction Successive sampling is used widely in applied sciences, sociology, commerce, finance and economic researches.the sampling on two occasions is resorted to when the same variate is measured on two different occasions. In such situation, a subsample of units of the first occasion can be retained in the sample of the second occasion also which is known as matched portion and the estimators such ratio, regression, product and their ramification can be formed. These estimators are then combined with the estimators based on unmatched portion of the second occasion, for instance, see Cochran 1977, pp Sen 1971 considered the estimators for the population mean on the current occasion using information on two auxiliary variables available on previous occasion. Feng and 1997, and Biradar and Singh 001 and Singh et al. 011 used the auxiliary information on both the occasions for estimating the current population mean in two-occasion successive sampling. In some situations information on two or more than two auxiliary variables may be readily available or may be made available by diverting a small amount of fund available for the survey, for instance, see Singh and Vishwakarma 007, Singh and Vishwakarma 007 and Singh and Vishwakarma 009, Singh and Pal 015, Singh and Pal 015, Singh and Pal 015 and Singh and Pal 015, Singh and Pal 016, Singh and Pal 016 and Singh and Pal 016, and Singh et al The aim of the present work is to suggest a more efficient estimator of the population mean on current occasion using information on the two stable auxiliary variables on both the occasions. The behaviors of the suggested estimators are examined through empirical means and consequently suitable recommendations have been presented.. Formulation of Estimator We assume that the population consists of N units, which is supposed to be remaining unchanged in size over two occasions. The character under study is designated x, y on the first second occasion respectively. It is assumed that the information on two stable auxiliary variables z 1 and z whose population means are known and closely associated with x and y is available on first second occasion respectively. A simple random sample

3 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings of size n is drawn on the first occasion. A random subsample of size m = nλ is retained matched for its use on the second occasion; while a fresh unmatched simple random sample without replacement of size u = n m = nµ is drawn on the second occasion from the entire population so that the sample size on the current second occasion remains n; λ and µ λ + µ are the fractions of the matched and fresh samples, respectively, on the current second occasion. The optimum values of λ and µ should be chosen for the purpose. We have used the following notations throughout the paper. X, Y : Population means of the study variables x and y respectively. Z 1, Z : Population means of the auxiliary variables Z and Z 1 respectively. The sample means of the respective variables on the sample size shown in subscripts : x n x m, y u, y m, z jn, z ju, z jm j=1,. The population variance of the variable x: S x = 1 N 1 N x i X. The population variances S y, S Z 1, S Z of the variables y, Z 1 and Z respectively, The population coefficients of variation C y, C x, C z1, C z of the variables y, x, z 1 and z respectively, The sampling fraction f = n/n. The population partial regression coefficient of y on z 1 : β 01. = β 01 β 0 β 1 1 β 1 β 1 The population partial regression coefficient of y on z : β 0.1 = β 0 β 01 β 1 1 β 1 β 1. The sample estimate of β 01. based on the sample of the size u u 01. u 01 = 1 u u 0 1 u 1 u 1 The sample estimate of β 0.1 based on the sample of the size u, u 0.1 u 0 = 1 u 01 u 1 u 1 u 1.

4 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings The estimate of population regression coefficient β 01 of y on z 1 based on the sample of size u u 01 = s yz 1u, s z 1u The estimate of population regression coefficient β 0 of y on z based on the sample of size u, u 0 = s yz u, s z u The estimate of population regression coefficient β 1 of z 1 on z based on the sample of size u, u 1 = s z 1z u s z u The estimate of population regression coefficient β 1 of z on z 1 based on the sample of size u, u 1 = s z 1z u s z 1u where and m 01. s yz1u = 1 u 1 m n,, 0.1 We also have : u y i y u z 1i z 1u, s yzu = 1 u y i y u 1 u z i z u, s z 1u = 1 u u z 1i z 1u, s z = 1 u u z i z u u 1 s z1z u = 1 u 1 u z 1i z 1u z i z u, n and 0.1 are defined similarly. The sample estimate of population regression coefficient β 0x of y on x based on sample of size m m 0x = s 0xm s xm The sample estimate of population regression coefficient β 0x of y on x based on sample of size n, n 0x = s 0xn s xn

5 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings s 0xm = 1 m 1 m y i y m x i x m, s xm = 1 m x i x m m 1 The sample estimate of the partial regression coefficient β x1. of x on z 1 based on the sample of the size m m x1. m x1 = 1 m x m 1 m 1 The sample estimate of the partial regression coefficient β x.1 of x on z based on the sample of the size m s z 1m = 1 m 1 m x1 m x.1 β n x1. and βn x.1 are similarly defined. m x = 1 m x1 m 1 m 1 m 1 m 1 = s xz 1m m s, x = s xz m z 1m s z m,, m z 1i z, s z = 1 m m z i z, m 1 ρ 0x, ρ 01, ρ 0, ρ x1, ρ x, and ρ 1 are the correlation between y&x, y&z 1, y&z, x&z 1, x&z and z 1 &z respectively. Define also, ρ 01. = ρ 01 ρ 0 ρ 1 1 ρ 1, ρ 0.1 = ρ 0 ρ 01 ρ 1 1 ρ, ρ 0.1 = ρ 01 + ρ 0 ρ 01 ρ 0 ρ ρ 1 ρ x1. = ρ x1 ρ x ρ 1 1 ρ 1, ρ x.1 = ρ x ρ x1 ρ 1 1 ρ, ρ x.1 = ρ x1 + ρ x ρ x1 ρ x ρ ρ 1 Now to estimate the population mean Y on the current second occasion in two occasion some estimators are suggested. One is based on the fresh sample of size u = nµ drawn on the second occasion and structured as t u = y u + u 01. Z u 1 z 1u Z z u. 1 Three modified regression type estimators based on the sample of size m= nλ common to both the occasions are defined by t m1 = y m + t m = y m + t m3 = y m + m 0x x n x m m Z m 1 z 1m Z z m. m 0x x n x m m Z m 1 z 1m Z z m. 3 m 0x x n x m m Z m 1 z 1m Z z m. 4

6 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings. 135 where x n = x m = x n = x m = x n x m n x n + x1 Z 1 z 1n m x m + x1 Z 1 z 1m n x n + x Z z n m x m + x Z z m = [x n + = [ x m + β n Z z n, 5 Z Z1 z m, z 1n, 6 Z1 z 1m, β n x1. Z 1 z 1n + x.1 Z z n ], 7 m β x1. Z m 1 z 1m + β x.1 Z z m ]. 8 Combining the estimators t u and t mi i = 1,, 3, we have three estimators for the population mean Ȳ as t 1 = η 1 t u + 1 η 1 t m1, 9 t = η t u + 1 η t m, 10 and t 3 = η 3 t u + 1 η 3 t m3, 11 where the η i s, i = 1,, 3 are unknown scalars to be determined under certain criterions. 3. Mean Squared Errors of the Estimators t 1, t and t 3 In the following theorem we give the mean squared errors of the estimators t u, t m 1, t m and t m 3. Theorem 1. The mean squared errors MSEs of the estimators t u, t m 1, t m and t m 3, to the terms up to second order moments or alternatively up to the terms of order, are respectively given by 1 MSEt u = u 1 S N ya 3, 1 [ 1 MSEt m1 = Sy m A n A 1 ] N A 3, 13 [ 1 MSEt m = Sy m B n B 1 ] N A 3, 14

7 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings where MSEt m3 = S y [ 1 m D n D 1 ] N A 3, 15 A 1 = A 3 A, A { Cz Cx = ρ 0x ρ 1 ρ ρ 0x ρ C x C z { } Cx Cz ρ 0x ρ 0 + ρ 01 ρ x1 C z C x ρ 0x C z C x ρ x1 ρ 1 ρ x } Cz C x { C 1 x Cz 1 + ρ x1 + ρ x1 ρ 1 ρ x Cx C z }, A 3 = 1 ρ 01., B 1 = A 3 B, B { Cz1 Cx = ρ 0x ρ 1 ρ ρ 0x ρ C x C z1 { } Cx Cz1 ρ 0x ρ 01 + ρ 0 ρ x C z1 C x ρ 0x C z1 C x ρ x ρ 1 ρ x1 } Cz1 C x { C 1 x Cz 1 + ρ x + ρ x ρ 1 ρ x1 1 Cx C z1 }, D 1 = A 3 D, D = [ρ 0x1 + ρ x.1 ρ 0x ρ 01 ρ x1. + ρ 0 ρ x.1 ]. Proof. It is simple, and so, omitted. The covariance between the estimator t u and t mj, j = 1,, 3 to the first degree of approximation, is given by. Covt u, t mj = S y N 1 ρ 0.1 = S y N A 3 16 The mean squared error of the combined estimator t u and t mj, j = 1,, 3 to the first degree of approximation is obtained as MSEt j = Et j Ȳ = E [ η j t u + 1 η j t mj Ȳ ] = E [ η j t u Ȳ + 1 η jt mj Ȳ ] = [ η j Et u Ȳ + 1 η j Et mj Ȳ + η j 1 η j Et u Ȳ t mj Ȳ ] = [ ηj MSEt u + 1 η j MSEt mj + η j 1 η j Covt u, t mj ], j = 1,, 3. 17

8 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Putting the values of MSEt u, and MSEt mj j=1,, 3 from 1, 13, 14, 15 and 16, we get to the first degree of approximation in Corollary 1. Further we give the MSEs of the estimators t u and t mj, j = 1,, 3, and under the following assumption. Assumption 3.1: Since and denote the same study variable over two occasions and z 1, z are two auxiliary variables correlated to x and y, therefore, as mentioned in Murthy 1967, p.35, Reddy 1978, and Singh and Ruiz-Espejo 003, the coefficient of variation is stable over time and following Cochran 1977 and Feng and 1997, the coefficient of variation x, y, z 1 and z are approximately assumed to be the same i.e. C x C z1 C z C y. Corollary 1. : Under the Assumption 3.1, the MSEs of the estimators t u and t mj, j = 1,, 3, given by 1, 13, 14 and 15 respectively reduce to: where MSEt u = MSEt m1 = S y MSEt m = S y MSEt m3 = S y 1 u 1 S N ya 3, 18 [ 1 m A n A 1 ] N A 3, 19 [ 1 m B n B 1 ] N A 3, 0 [ 1 m D n D 1 ] N A 3, 1 A 1 = A 3 A, A = [ ρ 0x ρ 1 ρ ρ 0x ρ ρ x + ρ x1 ρ 1 ρ 0x ρ 0 + ρ 01 ρ x1 ρ 0xρ x1 ρ 1 ρ x ρ x1 ], B 1 = A 3 B, B = [ ρ 0x ρ ρ x1 + ρ x ρ 1 + ρ 0x ρ 1 ρ 0.1 ρ 0x ρ 01 + ρ x ρ 0 ρ 0xρ x ρ 1 ρ x1 ρ x ], and A 3, D 1 and D are same as defined earlier. The covariance between the estimators t u and t mj, j = 1,, 3, to the first degree of approximation under the Assumption 3.1 is same as given in 16. Putting the values of MSEt u, MSEt mj and Covt u, t mj from 16 and 18 to 1 one can easily get the MSEt j, j = 1,, 3 to the first degree of approximation under the Assumption 3.1.

9 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Minimum Mean Squared Errors of the Estimator t j, j = 1,, 3 Differentiating 17 with respect to η j and equating to zero i.e., MSEt j η j = 0, we get the optimum values of the η j s j=1,, 3 as η jopt = [MSEt mj Covt u, t mj ] [MSEt u + MSEt mj Covt u, t mj ]. Now inserting the value of η jopt from equation in 17, we get the minimum MSE of the estimator t j as min MSEt j = [MSEt umset mj {Covt u, t mj } ], j = 1,, 3. 3 [MSEt u + MSEt mj Covt u, t mj ] Substituting the values of MSEt u, MSEt mj and Covt u, t mj from equations 1, 13, 14, 15 and 16 in 3 we get the simplified values of η jopt and min.mset j opt, j = 1,, 3 as η 1opt = µa 3 µa A 3 µ A. 4 η opt = µ A 3 µ B A 3 µ B, 5 η 3opt = µ A 3 µ D A 3 µ D, 6 min.mset 1 = A 3[A 3 1 f µa + µ fa ]Sy na 3 µ, 7 A min.mset = A 3[A 3 1 f µ B + µ fb ]Sy na 3 µ, 8 B min.mset 3 = A 3[A 3 1 f µ C + µ fd ]Sy na 3 µ, 9 D where µ, µ, µ are the fractions of fresh samples drawn at the current second occasion. Under the Assumption 3.1, the expressions 4 to 9 respectively reduce to: η 1 1 opt = µ 1A 3 µ 1 A A 3 µ 1 A, 30 η 1 opt = µ 1 A 3 µ 1 B A 3 µ 1 B, 31 η 1 3 opt = µ 1 A 3 µ 1 D A 3 µ 1 D, 3

10 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings min.mset 1 1 = A 3[A 3 1 f µ 1 A + µ 1 fa ]S y na 3 µ 1 A, 33 min.mset 1 = A 3[A 3 1 f µ 1 B + µ 1 B f]s y na 3 µ 1 B, 34 min.mset 3 1 = A 3[A 3 1 f µ 1 C + µ na 3 µ 1 D 1 fd ]Sy, 35 where µ, µ 1, µ 1 are the fractions of fresh samples drawn at the current second occasion Optimum Replacement Strategies of the Estimators t j j = 1,, 3 To get the optimum values of the fractions µ, µ and µ of samples to be drawn afresh sample at the second occasion so that the population mean Y may estimated with maximum precision and minimum cost, it is desired to minimize min.mset 1, min.mset and min.mset 3 respectively with µ, µ and µ, which results in quadratic equations in µ, µ and µ, say µ, µ and µ, respectively µ A µa 3 + A 3 = 0, 36 µ = A 3 ± A 1 A 3 A, 37 µ B µ A 3 + A 3 = 0, 38 µ = A 3 ± B 1 A 3 B, 39 µ C µ A 3 + A 3 = 0, 40 µ = A 3 ± D 1 A 3 D. 41 It is obvious from equations 37, 39 and 41 that the real values of µ, µ and µ exist, iff the quantities under square root are greater than or equal to zero. For any combination of correlations, which satisfy the condition of real situations, two real values of µ, µ and µ, it should be noted that 0 µ 1, 0 µ 1, and 0 µ 1, and all the other values of µ, µ and µ are said to be inadmissible. If both the values of µ, µ and µ are admissible, lowest one will be the best choice as it reduces the cost of the surveys. Inserting the admissible values of µ, µ and µ say µ 0, µ 0 and µ 0 respectively in 7, 8 and 9, we get the optimum values of minimum mean squared errors of the estimators t 1, t and t 3, as:

11 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings min.mset 1 opt = A 3[A 3 1 f µ 0 A + µ 0fA ]Sy na 3 µ 0 A, 4 min.mset opt = A 3[A 3 1 f µ 0B + µ 0 fb ]S y na 3 µ 0 B, 43 min.mset 3 opt = A 3[A 3 1 f µ 0 D + µ 0 fd ]Sy na 3 µ 0 D, 44 Under Assumption 3.1, the expressions in 37, 39, 41, 4, 43 and 44 respectively reduce to: µ 1 = A 3 ± A 1 A 3 A, 45 µ 1 = A 3 ± B1 A 3 B, 46 ˆµ 1 = A 3 ± D 1 A 3 D, 47 A 3 [A 3 1 f µ 10 A + µ 10 fa Sy min.mset 1 1opt =, 48 n A 3 µ 10 A ] A 3 [A 3 1 f µ 10 B + µ 10 fb Sy minmset 1opt =, 49 n A 3 µ 10 B ] A 3 [A 3 1 f µ 10 D + µ 10 fd Sy min.mset 3 1opt =, 50 n A 3 µ 10 D where µ 10, µ 10 and µ 10 are optimum values of the fractions of fresh samples obtained from the equations 45, 46 and 47 respectively. ] 5. Efficiency Comparison The percent relative efficiencies PREs of the estimators t 1, t, and t 3 relative to i : sample mean estimator ȳ n, when there is no matching and ii : usual successive sampling estimator ˆȲ = η ȳ u + 1 η ȳ m, when there is no auxiliary information that is used at any occasion, where ȳ m = ȳ m + β 0x x n x m, have been computed for various choices of correlations and demonstrated in Table 1. The variance of ȳ n and optimum variance of ˆȲ are respectively given by V arȳ n = 1 f S n y, 51

12 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings and V ar ˆȲ opt = {1 + } 1 ρ 0x f n S y. Under the Assumption 3.1, we note that since, the optimum values of min.mset 1, min.mset and min.mset 3 respectively given by 48, 49 and 50 contain six correlations ρ 0x, ρ 0x, ρ 0, ρ x1, ρ x and ρ 1, therefore for simplifying the expressions and to show the empirical results in tabular form it is assumed that ρ x1 = ρ x = ρ 01 = ρ 0 = ρ 0, 5 which is an intuitive assumption and also considered by Cochran 1977 and Feng and 1997, see, Singh et al Under the above assumption 5, the values of the quantities A 1, A, A 3, B 1, B, D 1 and D respectively reduce to: [ ] A 1 = A 3 A, A ρ0x ρ 0 = 1 ρ 0 + ρ 0 ρ 1 + ρ 1 ρ 0x ρ ρ 0 + ρ 1 + ρ 0xρ 0 + ρ 0 ρ 1, 1 A 3 = and [ ] 1 ρ 0, B1 = A 1 + ρ 3 B, B = A, D1 = 1 ρ 0x 1 + ρ 0x ρ ρ 0x, ρ 1 Thus under the assumption 5, we have [ ] D = ρ 0x ρ 0x ρ 0 ρ 0x. 1 + ρ 1 min.mset 1 1opt = min.mset 1opt. 53 For the various choices of ρ 0x, ρ 0 and ρ 1, for a fixed value of the sampling fraction f = n/n, Table 1 shows the optimum values of µ 1, µ 1 and µ 1 and percent relative efficiencies E 1 1, E 1, E1, E and E1 3, E 3 of t 1, t and t 3 with respect to ȳ n and ˆȲ respectively, where E 1 V arȳ n 1 = 100, min MSEt 1 opt E V ar ˆȲ 1 = 100, min MSEt 1 opt E 1 V arȳ n = 100, min MSEt opt

13 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings E V ar ˆȲ opt = 100, min MSEt lopt E 1 V arȳ n 3 = 100, min MSEt 3 lopt V ar ˆȲ E opt 3 = 100 min MSE t 3 opt We note that µ 10 = µ 10, E1 1 = E 1 and E 1 = E. Findings are given in Table. Table 1 : The optimum values µ 10, µ 10 and µ 10 and P REs E 1 1, E 1 E 1, E and E 1 3, E 3 with respect to y n and ˆȲ for f = 0.1.

14 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Table 1. ρ 0 ρ 1 ρ 0x µ 10 = µ 10 E 1 1 = E 1 E 1 = E µ 10 E 1 3 E * * *

15 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings ρ 0 ρ 1 ρ 0x µ 10 = µ 10 E 1 1 = E 1 E 1 = E µ 10 E 1 3 E Table 1 Continued.

16 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings. 136 ρ 0 ρ 1 ρ 0x µ 10 = µ 10 E 1 1 = E 1 E 1 = E µ 10 E 1 3 E Table 1 Continued. Table 1 exhibits that the values of the percent relative efficiencies i.e E 1 j = E j ; j = 1,, 3; greater than 100%. It follows the proposed estimator t j s, j = 1,, 3 are better than the usual unbiased y n when there is no matching and the natural successive sampling estimator ˆȲ. It is interesting to note that the minimum value of µ 0 is 0.054, which reflects that the fraction to be replaced at the current occasion is as low as about = 1 percent of the total sample size, which leads substantial reduction in cost of the survey. Thus the use of auxiliary variable in the development of the estimators is very much beneficial. It is also evident from Table 1 that if highly correlated auxiliary variables are used, relatively only a fewer fraction of sample on the current second is to be replaced by a fresh sample which reduce the cost of the survey. Thus the proposal of the estimators are justified and recommended for their use in practice A Practical Application In this section we have examined the performance of the proposed estimator t j s j = 1,, 3 over i sample mean y n when there is no matching and, ii usual successive sampling ˆ estimator Y. defined in Section 6 through a natural population data earlier considered by Chaturvedi and Tripathi 1983, p.118. Population: Source :[Chaturvedi and Tripathi 1983].

17 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings The data under consideration were taken from Census 1961 and 1971, West Bengal, District Census Handbook, Malda District. The population consists of 78 villages under Gajole Police Stations with: y=number of agricultural labourers x= Number of agricultural labourers z 1 =Area of village in1961. z =Area of village in1961. N = 78, n = 30, f = , Y = , X = 5.111, Z 1 = , Z = , S y = , S x = , S Z 1 = , S Z = , S yx = , S yz1 = , S yz = , S xz1 = , S xz = , S Z1Z = , ρ x1 = , ρ x = , ρ 1 = , ρ 0x = , ρ 01 = , ρ 0 = We have computed the optimum values of µ 0, µ 0 and µ 0, and respectively by using the formulae 37, 39 and 41 and the percent relative efficiencies PREs the proposed estimators t j, j = 1,, 3 with respect to y n and ˆȲ using the formulae: P RE t 1, 1 f [ ] A 3 µ y n = 0A A 3 [A 3 1 f µ 0 A + µ 0 fa ] 100, P RE t, 1 f [ ] A 3 µ 0 B y n = A 3 [A 3 1 f µ 0 B + µ 0 fb ] 100, P RE t 3, y n = 1 f [ A 3 µ 0 D ] A 3 [A 3 1 f µ 0 D + µ { ρ P RE t 1, ˆȲ 0x f = 0 fd ] 100, } [A3 ] µ 0A A 3 [A 3 1 f µ 0 A + µ 0 fa ] 100, { 1 + [A3 ] 1 ρ P RE t, ˆȲ 0x f} µ 0 B = A 3 [A 3 1 f µ 0 B + µ 0 fb 100, ] { 1 + [A3 ] 1 ρ P RE t 3, ˆȲ 0x f} µ 0 D = A 3 [A 3 1 f µ 0 D + µ fd ] Findings are given in the Table.

18 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Table : Optimum values µ 0, µ 0, µ 0 and the PREs the proposed estimators t j, j = 1,, 3 under the optimum conditions with respect to y n and ˆȲ. ˆµ = µ 0 = P REt 1, ȳ n = P REt 1, ˆȲ = ˆµ = µ 0 = P REt, ȳ n = P REt 1, ˆȲ = ˆµ = µ 0 = P REt 3, ȳ n = P REt 3, ˆȲ = It is observed from Table that the proposed estimators t j, j = 1,, 3 are more efficient than the usual unbiased estimator y n when there is no matching and natural successive sampling estimator ˆȲ when there is no auxiliary information is used at any occasion with substantial gain in efficiency. Largest gain in efficiency is obtained by using the proposed estimator over both estimators y n, ˆȲ followed by the suggested estimator t. Thus all the three suggested estimators t j estimators y n, ˆȲ in practice., j = 1,, 3 are to be preferred over the Acknowledgements Authors are thankful to the Prof. Gane Samb LO, Founding Editor, and the learned referee for their valuable suggestions regarding improvement of the paper. References Biradar, R. S. and Singh, H.P Successive sampling using auxiliary information on both the occasions. Calcutta Statist. Assoc. Bull., 51, Chaturvedi, D. K. and Tripathi, T.P Estimation of population ratio on two occasions using multivariate auxiliary information. Jour. Ind. Statist. Assoc., 1, Cochran, W.G Sampling Techniques. Third Edition. Johan Wiley, New York. Feng, S. and Zou, G Sampling rotation method with auxiliary variable. Commun. Statist. Theo. Meth., 66, Murthy, M.N Sampling: Theory and Methods. Statistical Publishing Society, Calcutta India. Reddy, V. N A study on the use of prior knowledge on certain population parameters in estimation. Sankhya, C, 40, Sen A.R.1971 Successive sampling with two auxiliary variables. Sankhya, C, 33, Singh, G.N., Karna, J.P. and Prasad, S.011 On the Use of Multiple Auxiliary Variables in Estimation of Current Population Mean in Two-Occasion Successive Rotation Sampling. Sri Lankan Journ. Appli. Statist., 1, Singh, H. P. and Pal, S. K On the estimation of population mean in successive sampling. Int. Jour. Math. Sci. Applica., 51, Singh, H. P. and Pal, S. K On the estimation of population mean in rotation sampling. Jour. Statist. Appl. Pro. Lett.,,

19 Procedure for Estimating the Population Mean in Two-occasion Successive Samplings Singh, H. P. and Pal, S. K On the estimation of population mean in current occasions in two-occasion rotation patterns. Jour. Statist. Appl. Prob., 4, Singh, H. P. and Pal, S. K Improved estimation of current population mean over two-occasions. Sri Lankan Jour. Appl. Statist., 161, Singh, H. P. and Pal, S. K An efficient effective rotation patterns in successive sampling over two occasions. Commun. Statist. Theo. Meth., 4517, Singh, H. P. and Pal, S. K Search of good rotation patterns through exponential type regression estimator in successive sampling over two occasions. Italian Jour. Pure. Appl. Math., 36, Singh, H. P. and Pal, S. K Use of several auxiliary variables in estimating the population mean in a two occasion successive sampling. Commun. Statist. Theo. Meth. 453, Singh, H. P. and Vishwakarma, G.K Modified exponential ratio product estimators for finite population mean in double sampling. Austrian Jour. Statist., 36 3, Singh, H. P., Kim, J. M. and Tarray, T. A A family of estimators of population variance in two-occasion rotation patterns. Commun. Statist. Theo. Meth., 4514, Singh, H. P. and Ruiz-Espejo, M On linear regression and ratio-product estimation of a finite population mean. The Statistician 51, Singh, H. P. and Vishwakarma, G.K. 007, A general class of estimators in successive sampling. Metron, 65, Singh, H. P. and Vishwakarma, G. K A general procedure for estimating population mean in successive sampling. Commun. Statist. Theo. Meth., 38,