Almost unbiased estimation procedures of population mean in two-occasion successive sampling

Transcription

1 Hacettepe Journal of Mathematics Statistics Volume 47 (5) (018), Almost unbiased estimation procedures of population mean in two-occasion successive sampling G. N. Singh, A. K. Singh Cem Kadilar Ÿ Abstract The objective of this paper is to construct some unbiased estimators of the current population mean in two-occasion successive sampling. Utilizing the readily available information on an auxiliary variable on both occasions, almost unbiased ratio regression cum exponential type estimators of current population mean have been proposed. Theoretical properties of the proposed estimation procedures have been examined their respective optimum replacement strategies are formulated. Performances of the proposed estimators are empirically compared with (i) the sample mean estimator, when no sample units were matched from the previous occasion (ii) natural successive sampling estimator when no auxiliary information was used on any occasion. mpirical results are critically interpreted suitable recommendations are made to the survey practitioners for their practical applications. Keywords: Successive sampling, Auxiliary information, Bias, Mean square error, Optimum replacement strategy. Mathematics Subject Classication (010): 6D05 Received : 09/01/016 Accepted : 08//016 Doi : HJMS Department of Applied Mathematics, Indian Institute of Technology(Indian School of Mines), Dhanbad, India, mail: gnsingh_ism@yahoo.com Corresponding Author Department of Applied Mathematics, Indian Institute of Technology(Indian School of Mines),Dhanbad, India, mail: aksingh.ism@gmail.com Ÿ Department of Statistics, Hacettepe University, Turkey, mail: kadilar@hacettepe.edu.tr

2 Introduction Change is the inherent phenomenon of the nature, if such change aects the human life, it is necessary to observe its behaviour pattern over the period of time. If the change is to be observed for a large group of individuals (population), one time survey does not provide the relevant information over a period of time. Successive (rotation) sampling happens to be more reasonable statistical tool to generate the estimates of unknown population parameters on dierent points of time (occasions) it is also capable of providing the information related to the patterns of variation in characteristics under study over the period of time. The problem of sampling on two successive occasions with a partial replacement of sampling units was rst considered by Jessen (194) in the analysis of a survey related to the farm data. The theory of successive sampling was further extended by Patterson (1950), Rao Graham (1964), Gupta (1979), Das (198) Chaturvedi Tripathi (1983) among others. Sen (1971, 197, 1973) used the information on auxiliary variables from previous occasion developed the estimators of current population mean in two occasions successive sampling. Later on Singh et al. (1991) Singh Singh (001) used auxiliary information on current occasion in two occasions successive sampling. In many situations, information on an auxiliary variable may be readily available on the rst as well as on the second occasion, for example, tonnage (or seat capacity) of each vehicle or ship is known in transportation survey, many other examples may be cited where the information on auxiliary variables are available on both the occasion in two occasions successive sampling. Utilizing the auxiliary information on both occasions, Feng Zou (1997), Birader Singh (001), Singh (005), Singh Priyanka (008, 010), Singh Karna (009), Singh Vishwakarma (009), Singh Prasad (013), Singh Homa (013), Singh Pal (015, 016) Srivastava Srivastava (016) among others have proposed varieties of estimators of population mean on current (second) occasions in two occasion successive sampling. It is to be mentioned that above works describe the biased estimation procedures of current population mean in two occasions successive sampling. Bias is an important factor in degrading the performance of estimators, keeping this point in mind motivated with the cited works we have suggested some unbiased estimation procedures of current population mean in two occasions successive sampling. Utilizing the information on readily available auxiliary information on both occasions, almost unbiased ratio regression cum exponential type estimators of current population mean have been proposed their properties are studied. The dominance of proposed estimation procedures have been shown over sample mean natural successive sampling estimators through empirical studies. The results of empirical studies are critically analyzed suitable recommendations are put forward to the survey practitioners.. Development of the stimators Consider a nite population U = (U 1, U,..., U N ) of N units which has been sampled over two occasions. The character under study is denoted by x(y) on the rst (second) occasion respectively. It is assumed that the information on an auxiliary variable z (stable over occasions) whose population mean is known, is readily available on both occasions has positive correlation with x y on the rst second occasions respectively. Let a simple rom sample (without replacement) of size n be drawn on the rst occasion. A rom sub-sample of size m = nλ is retained (matched) from the sample on rst occasion for its use on the second occasion, while a fresh simple rom sample (without replacement) of size u = (n m) = nµ is drawn on the second occasion from the entire population so that the total sample size on this occasion is also n. Here

3 170 λ µ (λ + µ = 1) are the fractions of the matched fresh samples, respectively, on the current (second) occasion. The values of λ or µ would be chosen optimally as they directly aect the cost of the survey. The following notations are considered for further use: X, Ȳ, Z: The population means of the variables x, y z respectively. x n, x m, ȳ u, ȳ m, z n, z m: The sample means of the respective variables based on the sample sizes shown in subscripts. ρ yx, ρ yz, ρ xz: The population correlation coecients between the variables shown in subscripts. S x=(n 1) 1 N (x i X) : The population variance of the variable x. Sy, Sz: The population variance of the variables y z respectively. s x(m): The sample variance of the variable x based on the matched sample of size m. C y, C x, C z: The coecients of variation of the variables shown in subscripts. b yz(u), b yx(m): The sample regression coecients between the variables shown in subscripts based on the sample size shown in braces. S yx, S xz, S yz: The population covariances between the variables shown in subscripts. s yz(u), s yx(m): The sample covariances between the variables shown in subscripts based on the sample sizes indicated in braces. To estimate the population mean Ȳ on the current (second) occasion, two sets of estimators are considered. The rst set of estimators S u = {T 1u, T u, T 3u} is based on sample of size u = nµ drawn afresh on the second occasion the second set of estimators S m = {T 1m, T m} is based on the sample of size m(= nλ) common with both the occasions, since, information on y is collected for the fresh sample of size u, therefore, we dene following exponential type estimators of set S u as { (.1) T 1u=ȳ u a ie (.) T u=ȳ u { b ie i[ Z zu Z+ zu i[ Z zu Z+ zu where ȳu = ȳ u + b (u) yz ( Z z u) (.3) T 3u=ȳ u { c ie i[ Z zu Z+ zu } } } where ȳu = ȳu z u Z, ai, b i c i (,,3) are suitably chosen scalars (weight) such that a i = 3 b i = 3 c i = 1 a i, b i, c i R(set of real numbers). Since, the information on study variable y is also available for the matched sample of size m retained on second occasion from the sample on rst occasion, therefore, again we suggest following ratio regression cum exponential type estimators of population mean Ȳ belong to the set Sm (.4) T 1m=ȳ m where ȳ m = ȳm x m x n (.5) T m=ȳ m { α ie { β ie i[ Z zm Z+ zm i[ Z zm Z+ zm } }

4 171 where ȳ m = ȳ m + b (m) yx ( x n x m) where α i β i (,,3) are suitably chosen scalars such that 3 α i = 3 β i = 1 α i, β i R Considering the convex linear combinations of the estimators of the sets S u S m, we have the nal estimators of the population mean Ȳ on the current (second) occasion as (.6) T ij = ϕ ijt iu + (1 ϕ ij)t jm(i = 1,, 3; j = 1, ) where ϕ ij(i = 1,, 3; j = 1, ) are the unknown constants to be determined under certain criterions..1. Remark. The estimators T iu(i = 1,, 3) T jm(j = 1, ) are proposed under the following conditions: (1) The sums of their respective weights are one. () The weights of the linear forms are chosen so that approximate biases are zero. (3) The approximate mean square errors attain minimum. 3. Properties of the Proposed stimators 3.1. Bias variance of the proposed estimators. Since the estimators T iu(i = 1,, 3) T jm(j = 1, ) dened in equations (.1)-(.5) are simple exponential, ratio regression cum exponential type estimators, they are biased estimators of Ȳ. Following the Remark.1, the proposed estimators may be converted in form of almost unbiased estimators of Ȳ. The variances V (.) up-to rst order of sample sizes of these estimators are derived under large sample approximations using the following transformations: ȳ u = (1 + e 1)Ȳ, ȳm = (1 + e)ȳ, xm = (1 + e3) X, x n = (1 + e 4) X, z u = (1 + e 5) Z, z m = (1 + e 6) Z, s (u) yz = (1 + e 7)S yz, s z(u) = (1 + e 8)Sz, s (m) yx = (1 + e 9)S yx, s x(m) = (1 + e 10)Sx such that (e k ) = 0 e k < 1, k = 1,,..., 10 Under the above transformations the estimators T iu(i = 1,, 3) T jm(j = 1, ) take the following forms: (3.1) T 1u = (3.) T u = (3.3) T 3u = (3.4) T 1m = (3.5) T m = a i Ȳ (1 + e 1)expi [ e5 ( e 5 ) b i[ȳ (1 + e1) Zβ yze 5(1 + e 7)(1 + e 8) 1 expi [ e5 c i Ȳ (1 + e 1)(1 + e 5) 1 expi [ e5 ( e 5 ) α i Ȳ (1 + e )(1 + e 4)(1 + e 3) 1 expi [ e6 ( e 6 ) β i[ȳ (1 + e) Xβ yx(e 4 e 3)(1 + e 9)(1 + e 10) 1 expi [ e6 ( e 5 ) ( e 6 ) where β yz β yx are population regression coecients between the variables shown in subscripts. To derived the bias variances/mean square errors of the proposed estimators, the following expectations ignoring the nite population corrections are used (e 1) = u 1 C y, (e ) = m 1 C y, (e 3) = m 1 C x, (e 4) = n 1 C x, (e 5) = u 1 C z, (e 6) = m 1 C z, (e 1e 5) = u 1 ρ yzc yc z, (e e 4) = n 1 ρ yxc yc x,

5 17 (e e 3) = m 1 ρ yxc yc x, (e 3e 4) = n 1 Cx, (e e 6) = m 1 ρ yzc yc z, (e 4e 6) = n 1 ρ xzc xc z, (e 3e 6) = m 1 ρ xzc xc z. To derive the expressions of bias mean square error of the estimator T 1u, we exp the right h side of equation (3.1) binomially exponentially neglecting terms of e s having power greater than two, we have (3.6) T 1u Ȳ = Ȳ {e1 A( e 5 + e1e5 where (3.7) A = ia i e 5 4 ( i )) 1 + } Taking expectation on both the sides of the equation (3.6) for large population size, ignoring nite population correction (f.p.c), we get the bias of the estimator T 1u to the rst order of approximations as (3.8) B(T 1u) = (T 1u Ȳ ) = Ȳ fua[ 1 ( i ) 1 + C z ρ yzc yc z 4 where f u = 1. u Squaring both sides of equation (3.6), neglecting the terms involving power of e s greater than two taking expectations, we get the mean square error (MS) of the estimator T 1u to the rst order of approximations as (3.9) M(T 1u) = (T 1u Ȳ ) = Ȳ [ f ucy + f ua Cz 4 fuaρyzcycz To minimize the MS of T 1u, we dierentiate M(T 1u) given in equation (3.9) with respect to A equating it to zero, we get the optimum value of A as, C y (3.10) A = ρ yz C z Substituting the optimum value of A in equation (3.9), we have the minimum MS of the estimator T 1u as (3.) Min.MS(T 1u) = Ȳ f ucy [ 1 ρ yz From equation (3.7) (3.10) we have C y (3.1) A = ia i = ρ yz C z From condition 3 equation (3.1), we have three unknowns to be determined from two equations only. It is therefore not possible to nd unique solutions of the constants a is(i = 1,, 3). Thus in order to get the unique solutions of the constants a is(i = 1,, 3), we shall impose a linear constraint as: (3.13) B(T 1u) = 0 which follows from equation (3.8) that (3.14) a 1 ( 1 ρyzcycz 3 8 C z Condition (3.15) 3 ) + a ( 1 ρyzcycz 1 C z ) ( 1 + 3a3 ρyzcycz 5 ) 8 C z = 0 = 1 equations (3.10) (3.1) can be written in matrix form as ρyzcycz 3 8 C z ( 1 ρyzcycz 1 C z ) 3( 1 ρyzcycz 5 8 C z ) a 1 1 a = C ρ y yz C z a 3 0

6 173 Solving equation (3.15) assuming C y = Cz we get the unique values of a is(i = 1,, 3) as: (3.16) a 1 = 4ρ yz 7ρ yz + 3, a = 1ρ yz 8ρ yz 3, a 3 = 1 + 4ρ yz 5ρ yz Substituting the values of a 1, a, a 3 from equation (3.16) into equation (.1), we have an almost unbiased optimum exponential type estimator say T1u written as T1u ={4ρ yz 7ρ yz + 3} y [ Z z u uexp + {1ρyz 8ρ Z yz 3} + z y [ Z z uexp u u Z + z (3.17) u + {1 + 4ρ yz 5ρ yz} y [ Z z u uexp3 Z + z u whose variance to the rst degree of approximation ignoring f. p. c. is given by (3.18) V (T1u) = Ȳ f ucy [ 1 ρ yz Proceeding as above, we have almost unbiased version of the estimator T u say Tu as Tu ={4δ + 3}ȳuexp [ Z z u {8δ + 3}ȳ u exp [ Z z u Z + z u Z + z u (3.19) + {4δ + 1}ȳ uexp3 [ Z z u Z + z u with variance (3.0) V (Tu) = Ȳ f ucy [ 1 ρ yz where ( Z δ = ρ ζ01 ) [ yz Sz S yz ζ 003 S z (xi X) r (y i Ȳ )s (z i Z) t ; (r,s,t are integer 0), Similarly almost unbiased version of the estimator T 3u say T3u is derived as (3.1) T3u = c 1ȳu exp [ Z z u + c ȳu Z exp [ Z z u + c 3 ȳu + z u Z exp3 [ Z z u + z u Z + z u with variance (3.) V (T3u) = Ȳ f ucy [ 1 ρ yz where c 1 = c 1 = (4ρ yz 9ρyz+5) 8ρ yz 1, c = c = (8ρ yz 4ρyz 45) 8ρ yz 1, c 3 = c 3 = (1ρ yz 5ρyz 1) 8ρ yz 1 Similarly, the unbiased version of the estimators of the set S m with their respective variances are derived as (3.3) T1m = α1ȳ mexp [ Z z m + α ȳmexp Z [ Z z m + α 3 ȳmexp3 + z m Z [ Z z m + z m Z + z m with variance (3.4) V (T1m) = Ȳ Cy [ fm(1 F1 ρ yz) + f (1 ρ yz) (3.5) Tm = β1 ȳm exp [ Z z m + β ȳm Z exp [ Z z m + β 3 ȳm + z m Z exp3 [ Z z m + z m Z + z m with variance (3.6) V (Tm) = Ȳ Cy [ fm(1 F ρ yz) + f ρyx(ρ yx ) ρ yzf f 1 where α 1 = α1 = 6RF 1ρ yz+8qf 1 ρ yz 6Q 3R+S β P 3R 4Q 1 = β1 = 3 7F ρ yz + 4 f m (P Q f )

7 174 α = α = F 1ρ yz(p +3R+4Q)+3P S P 3R 4Q β = β = 3(4F ρ yz 1) 8 f m (P Q f ) α 3 = α 3 = F 1P ρ yz+q P +S P 3R 4Q β 3 = β 3 = 1 5F ρ yz + 4 f m (P Q f ) P = 3 4 fm f1ρyz, Q = 1 fm f1ρyz, R = 5 fm f1ρyz, S = f(1 ρyx), 4 F 1 = f 1 f m, P = 1 (fmρyz fρyxρyz), Q = ρ yx f m = 1 m, f1 = 1 n, f = ( 1 m 1 n ). ( X ζ300 Sx Sx ζ 10 S yx ), F = 1 ρ yx f f m, 3.1. Theorem. The almost unbiased versions of the estimator T ij are given as T ij = ϕ ijt iu + (1 ϕ ij)t jm(i = 1..3; j = 1, ) Proof. Since the estimators Tiu(i = 1,, 3) Tjm(j = 1, ) derived in equations (3.17), (3.19), (3.1), (3.3) (3.5) are almost unbiased estimators of Ȳ. The nal estimator Tij(i = 1,, 3; j = 1, ) are the convex linear combinations of the estimators Tiu Tjm, therefore they are also unbiased estimators of Ȳ. 3.. Theorem. Variance of the estimator T ij(i = 1,, 3; j = 1, ) to the rst order of approximations are obtained as (3.7) V (T ij) = ϕ ijv (T iu) + (1 ϕ ij) V (T jm)(i = 1..3; j = 1, ) Proof. It is obvious that the variance of the proposed estimators Tij(i = 1,, 3; j = 1, ) are given by V (Tij) = (Tij Ȳ ) = [ϕ ijtiu + (1 ϕ ij)tjm Ȳ = [ϕ ij(tiu Ȳ )+(1 ϕij)(t jm Ȳ ) = ϕ ijv (Tiu) + (1 ϕ ij)v (Tjm) + ϕ ij(1 ϕ ij)[(tiu Ȳ )(T jm Ȳ ) where V (T iu) V (T jm) are obtained in equations (3.18), (3.0), (3.), (3.4) (3.6). It should be noted that the estimators T iu T jm are based on two nonoverlapping samples of sizes u m respectively, their covariance types of terms are of order N 1 ignored for large population size Remark. The above results are derived under the following assumptions: (i) Population size is suciently large(i.e., N ), therefore, nite population corrections (f.p.c.) are ignored. (ii) ρ xz = ρ yz, this is an intuitive assumptions, which has been also considered by Cochran (1977) Feng Zou (1997). (iii) since x y denote the same study variable over two occasions z is an auxiliary variable correlated to x y, therefore, looking on the stability nature of the coecient of variation (Reddy 1978) following Cochran (1977) Feng Zou (1997), the coecients of variation of variables x, y z are considered to be approximately equal i.e., C y = Cx = Cz Remark. It is to be noted that the suitable choices of the weights a i, b i, c i, α i β i(i = 1,, 3) derived earlier depend on unknown population parameters such asβ yz, β yx, C x, S yz, S yx, ρ yx, ρ yz, ζ 003, ζ 01 ζ 10. Thus to make such estimators practicable one has to use guessed or estimated values of these parameters. Guessed values of population parameters may be obtained either from past data or experience gathered over time; see for instance Reddy (1978). If such guessed values are not available it is advisable to use sample data to estimate these parameters as suggested by Singh et al. (007).

8 Minimum variances of the proposed estimators T ij(i = 1,, 3; j = 1, ). Since the variances of the estimators T ij(i = 1,, 3; j = 1, ) obtained in equation (3.7) are the functions of the unknown constants ϕ ij(i = 1,, 3; j = 1, ), therefore, to get the optimum values of ϕ ij, the variances of the estimators T ij are dierentiated with respect to ϕ ij equated to zero subsequently the optimum values of ϕ ij are obtained as V (Tjm) (3.8) ϕ ijopt = V (Tiu ) + V (T ; (i = 1,, 3; j = 1, ) jm ) Substituting the optimum values of ϕ ij(i = 1,, 3; j = 1, ) in equation (3.7), we get the optimum variances of the estimators Tij as (3.9) V (Tij) opt = V (T iu).v (Tjm) V (Tiu ) + V (T ; (i = 1,, 3; j = 1, ) jm ) Since V (T 1u) = V (T u) = V (T 3u), therefore, we have V (T ) opt = V (T 1) opt = V (T 31) opt V (T 1) opt = V (T ) opt = V (T 3) opt. Further, substituting the expressions of variances of the estimators T iu(i = 1,, 3) T jm(j = 1, ) from the equations (3.18), (3.0), (3.), (3.4) (3.6) in equations (3.8) (3.9), the simplied values of ϕ ijopt (i = 1,, 3; j = 1, ) V (T ij) opt are obtained as (3.30) ϕ opt = µ[a3 µ A + µ A 5 A 1 µ A 6 µ 3 A + µ A5 (3.31) V (T) opt = S y [ A 7 µ A 8 + µ A 9 n A 1 µ A 6 µ 3 A + µ A5 (3.3) ϕ 1opt = A1 µ 1A 15 µ 1A 16 A 1 µ 3 1 A15 µ 1 A16 (3.33) V (T 1) opt = S y n [ A 17 µ 1A 18 µ 1A 19 A 1 µ 3 1 A15 µ 1 A16 where A 1 = 1 ρ yz, A = ρ yz, A 3 = 1 A, A 4 = 1 ρ yx, A 5 = A +A 4, A 6 = A 1 A 3, A 7 = A 1A 3, A 8 = A 1A, A 9 = A 1A 5, A 10 = A A 8, A = A A 9, A 1 = A 6A 8 A 5A 9 +3A A 7, A 13 = A 1A 8 +A 5A 7, A 14 = A 1A 9 +A 7A 6, A 15 = ρ yzρ yx, A 16 = ρ yx ρ yzρ yx, A 17 = A 1, A 18 = A 1A 15, A 19 = A 1A 16, A 0 = A 15A 18, A 1 = A 15A 19, A = A 16A 19 3A 15A 17, A 3 = A 1A 18 A 16A 17, A 4 = A 1A 19 µ 1j is the fraction of fresh sample required for estimator T 1j(j = 1, ) Remark. Since the V (T 1u) = V (T u) = V (T 3u), therefore, we have V (T ) opt is similar to that of the V (T 1) opt, V (T 31) opt, V (T 1) opt is similar to that of the V (T ) opt V (T 3) opt. Hence onwards only the properties of the estimators T T 1 will be examined Optimum replacement strategies. To determine the optimum values of µ 1j(j = 1, ) so that the population mean Ȳ may be estimated with maximum precision, we minimize V (T1j) opt(j = 1, ) given in equations (3.31) (3.33) respectively with respect to µ 1j which result in 4th degree equations in µ 1 µ 1. The respective equations in µ 1j are obtained as (3.34) µ 4 A 10 µ 3 A µ A 1 + µ A 13 A 14 = 0 (3.35) µ 4 1A 0 + µ 3 1A 1 + µ 1A + µ 1A 3 + A 4 = 0

9 176 Solving equations (3.34) (3.35) we get the solutions of µ 1j(j = 1, ) say ˆµ 1j(j = 1, ). While choosing the values of ˆµ 1j, it should be remembered that 0 µ ˆ 1j 1 if more than one such admissible values of ˆµ 1j are obtained, the lowest one will be the best choice, because we have the same variance by replacing only the lowest fraction of total sample size on current occasion, which reduces the cost of the survey. All others values of ˆµ 1j(j = 1, ) are inadmissible. Substituting the admissible values of ˆµ 1j say µ 1j (j = 1, ) into the equations (3.31) (3.33) respectively, we have the optimum values of V (T1j) opt(j = 1, ), which are shown below (3.36) V (T ) opt = S y n (3.37) V (T 1) opt = S y n 4. Simulation Study [ A 7 µ A8 + µ A9 A 1 µ A6 µ3 A + µ A5 [ A 17 µ 1 A18 µ 1 A19 A 1 µ 3 1 A15 µ 1 A16 The percent relative eciencies of the estimators Tij(i = 1; j = 1, ) with respect to (i) sample mean estimator ȳ n when there is no matching from previous occasion (ii) natural successive sampling estimator ˆȲ = ϕ ȳ u + (1 ϕ )ȳm, where ȳm = ȳ m + b (m) yx ( x n x m), when no auxiliary information is used on any occasion have been computed for dierent choices of correlations. Since ȳ n is an unbiased estimator ˆȲ is a biased estimator of Ȳ, hence, following Sukhatme et.al (1984), the variance of ȳn the optimum mean square error of the estimator ˆȲ for large N are respectively given by (4.1) V (ȳ n) = S y n (4.) M( ˆȲ ) opt = [ 1 + (1 ρ yx) S y n 4.1. Simulation results. For dierent choices of ρ yz ρ yx, Tables 1- present the optimum values of µ 1j (j = 1, ) percent relatives eciencies 1 of the proposed estimators T1j(j = 1, ) with respect to ȳ n ˆȲ, where 1j = 1, ). V (ȳn) V (T 1j 100 ) j = M( ˆȲ ) opt opt V (T 1j 100(j = ) opt

10 * Table 1: Optimum values of PR s of the estimators T with respect to yn Ŷ yz yx

11 * Table : Optimum values of 1 PR s of the estimators T 1 with respect to yn Ŷ yz yx

12 Interpretation of simulation results. (1) From Table 1, it is clear that (a) For smaller values of ρ yx, the values of 1 are increasing µ are decreasing while for the higher values of ρ yx the values of 1 are increasing the values of µ do not follow any denite pattern when the values of ρ yz are increasing. This phenomenon indicates that smaller fraction of fresh sample on the current occasion is required, if information on a highly positively correlated auxiliary variable is used at the estimation stage. (b) For the xed values of ρ yz, the values of 1 are increasing while no denite trends are visible in the values of µ with the increasing values of ρ yx. (c) Minimum value of µ is obtained as 0.150, which indicates that only about 15 percent of the total sample size is to be replaced on the current (second) occasion for the corresponding choices of correlations. () From Table, it is observed that (a) For the xed values of ρ yx, the values of 1 are increasing while no denite patterns are seen in the values of µ 1 with the increasing values of ρ yz. (b) For the xed values of ρ yz, the values of 1 µ 1 are increasing while the values of are decreasing with the increasing values of ρ yx. (c) The minimum value of µ 1 is obtained as 0.086, which indicates that about 8 percent of the total sample size is to be replaced on the current (second) occasion for the corresponding choices of correlations. 5. Conclusions The work presented in this paper provides some unbiased ecient estimation procedures of current population mean in two-occasion successive sampling. The empirical results presented in Tables 1- their subsequent analyses interpretations vindicate that the use of auxiliary information on an auxiliary variable at estimation stage is highly rewarding in terms of the proposed estimators. It is also visible that if a highly correlated auxiliary variable is used at the estimation stage, relatively, only a small fraction of the sample on the current (second) occasion is required to be replaced by a fresh sample, which reduces the cost of the survey. Hence looking on the encouraging behaviours of the proposed estimators one may recommend them for their practical applications. Acknowledgements Authors are thankful to the reviewer for their valuable suggestions which helped in enhancing the quality of this paper. Authors are also thankful to the Indian Institute of Technology (Indian School of Mines), Dhanbad for providing nancial infrastructural support in completion of this work. References [1 Biradar, R. S. Singh, H. P. Successive sampling using auxiliary information on both occasions. Calcutta Statistical Association Bulletin, 51, 43-51, 001. [ Chaturvedi, D. K. Tripathi, T. P. stimation of population ratio on two occasions using multivariate auxiliary information. Journal of Indian Statistical Association, 1,3-10, [3 Cochran, W. G. Sampling Techniques, Wiley astern Limited, New Delhi, III dition, [4 Das, A. K. stimation of population ratio on two occasions.journal Indian Society Agricultures Statistics, 3 4, 1-9, 198

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