BEST LINEAR UNBIASED ESTIMATORS OF POPULATION MEAN ON CURRENT OCCASION IN TWO-OCCASION ROTATION PATTERNS

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1 STATISTICS IN TRANSITIONnew series, Spring 57 STATISTICS IN TRANSITIONnew series, Spring Vol., No., pp BST LINAR UNBIASD STIMATORS OF POPULATION MAN ON CURRNT OCCASION IN TWOOCCASION ROTATION PATTRNS G. N. Singh, S. Prasad ABSTRACT Best linear unbiased estimators have been proposed to estimate the population mean on current occasion in twooccasion successive (rotation) sampling. Behavior of the proposed estimators have been studied and their respective imum replacement policies are discussed. mpirical studies are carried out to examine the performance of the proposed estimators and consequently the suitable recommendations are made. Key words: successive sampling, auxiliary information, unbiased, variance, imum replacement policy.. Introduction Often in sample surveys on successive occasions for the same population, the current or most recent estimates are of the primary interest if the characteristics of the population are likely to change rapidly over time. For example, monthly surveys are carried out to collect data on prices of goods to determine the consumer price index, labor force surveys are conducted on monthly basis to estimate the numbers of people in employment and industries, collect information at regular intervals to know popularity of their products, etc. In such studies, successive (rotation) sampling may be an impressive statistical tool to generate reliable and cost effective estimates of different population parameters on successive points of time (occasions) in chronological order. It also provides effective estimates of changing patterns over a period of time. The problem of successive (rotation) sampling with a partial replacement of sampling units was initiated by Jessen (9) in the analysis of agricultural survey data. He pioneered using the entire information collected during the previous investigations. The theory of successive (rotation) sampling was further Department of Applied Mathematics, Indian School of Mines, Dhanbad86, India. mail: gnsingh_ism@yahoo.com.

2 58 G. N. Singh, S. Prasad: Best linear extended by Patterson (95), Rao and Graham (96), Gupta (979), Das (98) and Chaturvedi and Tripathi (98), among others. Sen (97) applied this theory with success in designing the strategies for estimating the population mean on the current occasion using information on two auxiliary variables readily available on the previous occasion. Sen (97, 97) extended his work for several auxiliary variables. Singh et al. (99) and Singh and Singh () used the auxiliary information available on the current occasion and proposed estimators for the current population mean in twooccasion successive (rotation) sampling. Singh () generalized his work for hoccasion successive sampling. In many situations, information on an auxiliary variable may be readily available on the first as well as on the second occasion; for example, tonnage (or seat capacity) of each vehicle or ship is known in survey sampling of transportation, number of beds in different hospitals may be known in hospital surveys, number of polluting industries and vehicles is known in environmental surveys, nature of employment status, educational status, food availability and medical aids of a locality is well known in advance for estimating various demographic parameters in demographic surveys. Utilizing auxiliary information on both the occasions, Feng and Zou (997), Biradar and Singh (), Singh (5), Singh and Priyanka (6, 7, 8), Singh and Karna (9a, b) have proposed several estimators for estimating the population mean on current (second) occasion in twooccasion successive (rotation) sampling. Recently Singh and Vishwakarma (9) have suggested a general estimation procedure for population mean in successive (rotation) sampling. Motivated with the above works and utilizing the information on an auxiliary variable, readily available on both the occasions, we have proposed best linear unbiased estimators for estimating the current population mean in twooccasion successive (rotation) sampling. Behaviors of the proposed estimators are examined through empirical means of comparison and subsequently the suitable recommendations are made.. Sample structures and notations on two occasions Let U = (U, U,, U N ) be the finite population of N (large) units which is assumed to remain unchanged over two occasions. Let x (y) be the character under study on the first (second) occasion respectively. It is assumed that the information on an auxiliary variable z (stable over occasion), is readily available for both the occasions, whose population mean is known and it is highly positively correlated to x and y on the first and second occasions respectively. A simple random sample (without replacement) of size n units is drawn on the first occasion and a random subsample of size m = nλ units from the sample on the first occasion is retained (matched) for its use on the current (second) occasion. A fresh (unmatched) sample of size u = (nm) = nμ units is drawn on the current occasion from the entire population by simple random sampling (without replacement) method so that the sample size on the current occasion is

3 STATISTICS IN TRANSITIONnew series, Spring 59 also n. λ and μ (λ+μ =) are the fractions of the matched and fresh samples, respectively, on the current occasion. We consider the following notations for further use: X, Y, Z : Population means of the variables x, y and z respectively. x, x, y, y, z, z, m n u m u m z n : Sample means of the respective variables based on the sample sizes shown in suffices. ρ yx, ρ yz, ρ xz : Correlation coefficients between the variables shown in suffices. N S x = (N) (xix) : Population mean square of x. i = S y, S Z : Population mean squares of y and z respectively.. Formulation of the estimator To estimate the population mean Y on the current (second) occasion, we consider the following minimum variance linear unbiased estimator of Y, which is as follows: T = { ay u+ a y m} + { ax m+ a x n} + { a5z u+ a 6z m+ a 7z n+ a8z } () where a, a, a, a, a 5, a 6, a7 and a 8 are constants to be determined so that (i) T becomes an unbiased estimator of Y and (ii) the variance of T attains a minimum value. For unbiasedness condition, we must have a +a =, ( a +a ) = and( a +a +a +a ) =. ( ) Substituting a = φ, a 8= a 5+ a 6+ a 7, the estimator T defined in equation () reduces to the following form a = β and ( ) { } { ( ) } { } ( ) ( ) ( ) T = φ y + φ y +β x x + a z Z +a z Z +a z Z u m m n 5 u 6 m 7 n ( ) ( ) { } ( ) ( ) = φ y u+k zuz + φ y m+k x mx n +k zmz +k znz = φt u + ( φ ) T () m where T u = y u + k ( u ) z Z ; an estimator based on the fresh sample of size u

4 6 G. N. Singh, S. Prasad: Best linear and T m = y m +k ( xmx n ) +k ( zmz ) +k ( zn Z) ; an estimator based on the a5 β a6 a7 matched sample of size m, k =, k =, k =, k = φ φ φ φ φ are the unknown constants to be determined under certain criterions. Remark.. For estimating the population mean on each occasion the estimator T u is suitable, which implies that more belief on T u could be shown by choosing φ as (or close to ), while for estimating the change over the occasions, the estimator T m could be more useful and hence φ might be chosen as (or close to ). For asserting both the problems simultaneously, the suitable (imum) choice of φ is desired. and. Properties of the estimator T T is an unbiased estimator of Y whose variance, ignoring finite population corrections, is derived in the following theorem. Theorem.. Variance of the estimator T is obtained as V( T ) = φ V( T u ) + ( φ ) V( T m ) () where V(T u ) = ηs y u () V(T m ) = η+ η+ η Sy m m n n (5) ( η = +k +kρ yz ), η = ( +k +kρ yz ), η = ( k +k ρ yx +k kρ xz ) and ( yz ) η = k +k ρ +k k. Proof: It is obvious that the variance of the estimator T is given by V(T ( ) ( )( ) ) = T Y = φ Tu Y + φ TmY = φ V(T u ) + ( φ ) V( T m ) +φ ( φ ) C (6) where V ( T u ) = T u Y, V(T m) = T my and C = ( Tu Y)( TmY) Substituting the expressions of T u and T m in equation (6), taking expectations and ignoring finite population corrections, we have the expression of the variance of the estimator T as given in equation ().

5 STATISTICS IN TRANSITIONnew series, Spring 6 Remark.. Results in equation () are derived under the assumption that the population mean squares of the variables x, y and z are almost equal. Remark.. T u and T m are based on two independent samples of sizes u and m respectively and they are unbiased estimators of Y, hence the covariance term C between T u and T m vanishes. 5. Minimum variance of the estimator T Since the variance of the estimator T in equation () is the function of the unknown constants k, k, k, k and φ, therefore it is minimized with respect to these constants, and subsequently the imum values of k, k, k, k and φ are obtained as k = ρ yz (7) ρyzρ xz ρ yx k = (8) ρxz ρyxρ xz ρ yz k = (9) ρ xz ( ) ρ ρ ρ ρ xz yz xz yx ρxz k = V(T ) () m φ = () V(T u )+V(T m ) Substituting the values of k, k, k and the imum variances of T u and T m as V(T u ) = ASy u where A = ρ yz, and A = k. k in equations () and (5), we get V(T ) = A + A + A S m m n n m y ( ) ( ρxz ) +ρ ρ +ρ +ρ ρ ρ ρ ρ xz xz yx yz yz yx xz yz A =, () () A = k

6 6 G. N. Singh, S. Prasad: Best linear Further, substituting the values of V(T u ) and V(T m) from equations () and () in equation (), we get the imum value φ with respect to k, k, k and k as V(T ) m φ = () V(T u ) +V(T m ) Again from equation () substituting the value of φ in equation (), we get the imum variance of T as V( T m ).V( Tu ) = (5) + ( ) ( ) ( ) u Further, substituting the values from equations () and () in equations () and (5), the simplified values of φ and V( T ) are obtained as φ = μ ( A +μ A ) 5 6 A+μ A 7+μA6 ( A+μA ) m V(T ) A = S 5 6 y n A+μ A 7+μA6 (6) (7) where A 5 = A +A, A 6 = AA, A 7 = A5A and μ is the fraction of fresh sample for the estimator T. 6. Optimum replacement policy To determine the imum value of μ (fraction of a sample to be drawn afresh on the current occasion) so that the population mean Y may be estimated given in equation (7) with the maximum precision, we minimize the ( ) with respect to μ, which result in a quadratic equation in μ and respective solutions of μ say μ is given below: Qμ +Qμ +Q = (8) where Q ± Q Q Q μ= Q Q = A 6, Q = A5A 6 and Q = A7A5AA 6. (9)

7 STATISTICS IN TRANSITIONnew series, Spring 6 From equation (9), it is obvious that the real values of μ exist if the quantity under square root is greater than or equal to zero. Two real values of are possible. Hence, while choosing the value ofμ, it should be remembered that μ. All other values of μ are inadmissible. Substituting the admissible value of μ say from equation (9) into equation (7), we have the imum value of V( T ) as 7. fficiency comparison ( A+μA ˆ ) V(T ) A = S 5 6 y ˆ ˆ n A+μ A 7+μA6 To study the performance of the estimator T the percent relative efficiencies of the estimator T with respect to (i) y n, when there is no matching, and (ii) the estimator T, when no auxiliary information is used at any occasion, have been computed for different choices of correlations. The estimator T is defined under the same circumstances as the estimator T, but in the absence of the auxiliary variable z on both the occasions and proposed as μ () T = { by u+ by m} + { bx m+ bx n} () where b, b, b and b are constants to be determined so that (i) T becomes an unbiased estimator of Y and (ii) The variance of T attains a minimum value. For unbiasedness condition, we must have ( b +b ) = and ( ) b +b =. Substituting b = φ and b = β, the estimator T defined in equation () reduces to the following form T = φ y + φ y +β x x { ( ) } { } u m m n ( ) { } = φ y + φ y +k x x u m 5 m n = φt u + ( φ ) T () m where T u = y u ; an estimator based on the fresh sample of size u

8 6 G. N. Singh, S. Prasad: Best linear and T m = y m +k5 ( xmx n ) ; an estimator based on the matched sample of size m, β k 5 = and φ φ are the unknown constants to be determined in such a way that they minimize the variance of the estimator T. Following the methods discussed in Sections, 5 and 6, the imum values of k 5, μ (fraction of fresh sample for the estimator T ), variance of y n and imum variance of T for large N are given by k 5 = ρ yx () = ± ρ ρyx yx () Sy V y = (5) n ( ) n μˆ ρ V(T ) = S yx y n μˆ ρyx (6) For different choices of ρ yx, ρ xz and ρ yz, Table shows the imum values of μ and percent relative efficiencies and of the estimator T with respect to the estimators ( ) V y = ( n ) y n and T respectively, where and ( ) ( ) =. 8. Analysis of results for estimator T The following conclusions can be read out from Table. (a) For fixed values of ρ xz and ρ yz, the values of μ and are increasing with the increasing values of ρ yx. The values of are decreasing for the lower values of ρ yx while increasing pattern may be seen for the higher values of ρ yx. (b) For fixed values of ρ xz and ρ yx, the values of μ are decreasing with the increasing values of ρ yz.. Values of and are increasing with the increasing values of ρ yz.. This behavior is highly desirable, since it concludes that if highly

9 STATISTICS IN TRANSITIONnew series, Spring 65 correlated auxiliary variable is available, it pays in terms of enhance precision of the estimates as well as it reduces the cost of the survey. (c) For fixed values of ρ yz and ρ yx, the values of μ are decreasing with the increasing values of ρ xz. Similar patterns are visible for the efficiencies and. (d) Minimum value of μ is.9, which indicates that only percent of the total sample size is to be replaced on the current occasion for the corresponding choices of the correlations. 9. Use of auxiliary variable only at the current occasion In section we have formulated the estimator T on the assumption that information on a stable auxiliary variable z was readily available on both the occasions. If the duration between two successive occasions is small then one may expect the stability of the auxiliary variable but the stability character of the auxiliary variable may not sustain if the duration between two successive occasions is appreciably large. In such situation it may not be wise to use the auxiliary information from the previous occasion. Motivated with the above argument, we formulate the estimator T when the information on an auxiliary variable z is available only on the current (second) occasion. The estimator T is formulated as T = { cy u+ cy m} + { cx m+ cx n} + { c5z u+ c6z m+ c7z } (7) where c, c, c, c, c 5, c6 and c 7 are constants to be determined so that (i) T becomes an unbiased estimator of Y and (ii) The variance of T attains a minimum value. For unbiasedness condition, we must have ( c +c ) =, ( c +c ) = and ( 5 6 7) c = β and c = ( c + c ) c +c +c =. Substituting c = φ, 7 5 6, the estimator T defined in equation (7) reduces to the following form T = φ y + φ y +β x x + c z Z +c z Z { } { ( ) } { } ( ) ( ) u m m n 5 u 6 m ( ) ( ) { } ( ) = φ y u+l zuz + φ y m+l x mx n +l zmz = φt u + ( φ ) T (8) m

10 66 G. N. Singh, S. Prasad: Best linear where T u = y u + l ( zu Z) ; an estimator based on the fresh sample of size u and T m= y m+l( xmx n) +l( zmz ) ; an estimator based on the matched sample c5 β c6 of size m, l =, l =, l = and φ are the unknown constants to φ φ φ be determined under certain criterions. 9.. Properties of the estimator T T is an unbiased estimator of Y whose variance is given in the following theorem. Theorem 9.. Variance of the estimator T is obtained as ( ) ( ) ( ) ( ) = φ + φ (9) u m u = +l +lρyz Sy u V( T ) = ( +l +l ρ ) + ( l +l ρ +l l ρ ) S m m n where ( ) ( ) m yz yx xz y Proof: It is obvious that the variance of the estimator T is given by where ( ) ( ) ( ) ( )( ) = T Y = φ Tu Y + φ TmY u ( ) ( ) ( ) ( ) u m () () = φ + φ +φ φ R () = T Y ( )( ) R = Tu Y TmY u, ( ) m = TmY and Substituting the expressions of T u and T m in equation (), taking expectations and ignoring finite population corrections, we have the expression of the variance of T as given in equation (9). Remark 9.. Results in theorem 9. is derived similar to the results obtained in theorem..

11 STATISTICS IN TRANSITIONnew series, Spring Minimum variance of the estimator T Since the variance of the estimator T in equation (9) is the function of the unknown constants l, l, l and φ, therefore it is minimized with respect to these constants and subsequently the imum values of l, l, l and φ are obtained as l = ρ yz () ρ yz ρ xz ρyx l = () μ ρ l = xz μ ρ ρ ρ yx xz yz μρxz V( Tm ) ( ) ( ) u m (5) φ = + (6) Now, substituting the values of l, l and l in equations () and (), we get the imum variances of T u, T m as where = B S u ( ) u y B+μ B 5+μB V( T m ) = S m ( μρxz ) B = ρ yz, xz ( xz yx yz yx xz ) B = ρ ρ +ρ ρ ρ ρ, B ( ) = ρyzρ xz ρ yx and B 5 = B +B. y B = ρxzb, (7) (8) Further, substituting the values of V(T u ) and V(T m) from equations (7) and (8) in equation (6), we get the imum value φ with respect to l, l and l as V( T ) m ( ) ( ) φ = + (9) Again, from equation (9) substituting the value of get the imum variance of T as u m φ in equation (9), we

12 68 G. N. Singh, S. Prasad: Best linear ( ) ( m ) ( ) u ( ) ( ) = () m + u Further, substituting the values from equations (7) and (8) in equations (9) and (), the simplified values of φ and V( T ) are obtained as where μ ( B +μ B +μ B ) 5 μb 6+μB 7+μB +B φ = B+μ B +μ B V( T ) = S B 6 = BBρ xz, B = BB and 8 9 y n μb 6+μB 7+μB +B B 7 = Bρ xz +Bρ xz +B 5, B 8= B, B 9= BB 5, μ is the fraction of fresh sample for the estimator T. () () 9.. Optimum replacement policy To determine the imum value of μ (fraction of a sample to be drawn afresh on the current occasion) so that population mean Y may be estimated given in equation () with the maximum precision, we minimize the ( ) with respect to μ, which result in fourth degree equation in μ and respective solutions of μ is discussed below: Pμ +Pμ +Pμ +Pμ +P 5 = () where P = B6B, P = B6B 9, P = B B B B, P = B B B B B B, ( ) P 5 = BB9B B8 7 8 From equations () it is obvious that the four real values of μ are possible. Hence, while choosing the values of μ, it should be remembered that μ. All the other values of μ are inadmissible. If more than one admissible values are obtained, the lowest admissible value is the best choice as it reduces the cost of the survey. From equation (), substituting the admissible value of μ say into equation (), we have the imum value of V( T ) as

13 STATISTICS IN TRANSITIONnew series, Spring 69 ( Bμˆ B +μˆ B ) B V( T ) = S 5 y n μˆb ˆ ˆ 6+μB 7+μB 8+B () 9.. fficiency comparison To study the performance of the estimator T, the percent relative efficiencies of the estimator T with respect to (i) y n, when there is no matching, and (ii) the estimator T, when no auxiliary information is used at any occasion, have been obtained for different choices of correlations. For different choices of ρ yx, ρxz and ρ yz, Table shows the imum values of μ and percent relative efficiencies and of the estimator T with respect to the estimators and T respectively, where ( ) V y n = and ( ) ( ) ( ) =. y n 9.5. Analysis of results for estimator T The following conclusions can be read out from Table : (a) For fixed values of ρ xz and ρ yz, the values of μ and are increasing with the increasing values of ρ yx. fficiencies are decreasing for the increasing values of ρ yx. (b) For fixed values of ρ xz and ρ yx, the values of μ increase for the lower values of ρ yz and decrease for the higher values of ρ yz. fficiencies and are increasing with the increasing values of ρ yz. (c) For fixed values of ρ yz and ρ yx, the values of μ are increasing with the increasing values of ρ xz. fficiencies and increase for the lower values of ρ xz while decreasing patteren may also be seen for the higher values of ρ xz. (d) Minimum value of μ is.565, which indicates that only 5 percent of the total sample size is to be replaced at the current occasion for the corresponding choices of the correlations.

14 7 G. N. Singh, S. Prasad: Best linear Table. Optimum values of μ and percent relative efficiencies of T with respect to yn and T ρ xz ρ yz ρ yx Note: indicates do not exist

15 STATISTICS IN TRANSITIONnew series, Spring 7 Table. Optimum values of μ and percent relative efficiencies of T with respect to yn and T ρ xz ρ yz ρ yx Note: indicates do not exist

16 7 G. N. Singh, S. Prasad: Best linear. General conclusions The estimators T and T proposed in this work are proved to be the best linear unbiased estimators of population mean Y with their respective minimum variance. These estimators may be seen as new innovative ideas in survey literature as they nicely utilized the information on an auxiliary variable in order to improve the precision of the estimates. From the analysis of the results shown in Tables, the propositions of the estimators T and T are vindicated because it enhances the precision of estimates as well as reduces the cost of the survey. Therefore, the proposed estimators may be recommended to survey practitioners for use in real life problems. Acknowledgements Authors are thankful to the referee for his valuable suggestions. Authors are also thankful to the UGC, New Delhi and Indian School of Mines, Dhanbad for providing financial assistance and necessary infrastructures to carry out the present research work.

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