IMPROVED ROTATION PATTERNS USING TWO AUXILIARY VARIABLES IN SUCCESSIVE SAMPLING

Karna, Jaishree Prabha; Nath, Dilip Chandra Article IMPROVED ROTATION PATTERNS USING TWO AUXILIARY VARIABLES IN SUCCESSIVE SAMPLING Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Karna, Jaishree Prabha; Nath, Dilip Chandra (2018) : IMPROVED ROTATION PATTERNS USING TWO AUXILIARY VARIABLES IN SUCCESSIVE SAMPLING, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, NY, Vol. 19, Iss. 1, pp. 25-44, https://doi.org/10.21307/stattrans-2018-002 This Version is available at: https://hdl.handle.net/10419/207885 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ STATISTICS IN TRANSITION new series, March 2018 25 STATISTICS IN TRANSITION new series, March 2018 Vol. 19, No. 1, pp. 25–44, DOI 10.21307/stattrans-2018-002 IMPROVED ROTATION PATTERNS USING TWO AUXILIARY VARIABLES IN SUCCESSIVE SAMPLING Jaishree Prabha Karna 1 , Dilip Chandra Nath ABSTRACT The present paper emphasizes the role of two auxiliary variables on both the occasions to improve the precision of estimates at the current (second) occasion in two-occasion successive sampling. Information on two auxiliary variables, which are positively correlated with the study variable, has been used with the aid of exponential type structures and an efficient estimation procedure of population mean on the current (second) occasion has been suggested. The behaviour of the proposed estimator has been studied and compared with the sample mean estimator, when there is no matching from the previous occasion and natural successive sampling estimator, which is a linear combination of the means of the matched and unmatched portions of the sample at the current (second) occasion. Optimal replacement strategy is also discussed. The concluding remarks are discussed justifying utility of the proposed sampling scheme. The results have been well supported analytically as well as empirically by using real life data. Key words: exponential type estimators, bias, mean squared error, optimum replacement strategy. Mathematics subject classification: 62D05 1. Introduction Repeated surveys over years or seasons or months are commonly used on many occasions for estimating same characteristics at different points of time. The information collected on previous occasion can be used to study the change or the total value over occasion for the character and also in addition to study the average value for the most recent occasion. There are several possibilities: (i) the same sample may be used on each occasion (ii) a new sample may be drawn on each occasion or (iii) a part of the sample may be retained while the remainder of the sample may be drawn afresh. Intuition suggests that for estimating changes from one occasion to the next, it may be best to retain the same sample on each occasion, while for estimating the mean on each occasion it may be advised to draw a fresh sample on each occasion. If it is desired to estimate the population mean on each occasion and also the change from one occasion to the next, it is 1 Department of Statistics, Gauhati University, Guwahati – 781014, India. E-mail: [email protected] 26 J. P. Karna, D. C. Nath: Improved rotation patterns… always better to retain a part of the sample and draw the remainder of the sample afresh. In successive (rotation) sampling, it is more advantageous to utilize the entire information collected in the previous investigations (occasions), to cite one may refer the papers by Jessen (1942), Patterson (1950), Rao and Graham (1964), Gupta (1979), Das (1982) and Chaturvedi and Tripathi (1983). Sen (1971) has also used this technique successfully with the utilization of information on two auxiliary variables, which was readily available on previous occasion, and proposed the estimators of population mean on the current occasion in twooccasion successive sampling. Sen (1972, 1973) further extended his work for several auxiliary variables. 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 case of public health and welfare of a state or country, several factors are available that can be treated as auxiliary variables, such as the number of beds in different hospitals may be known, number of doctors and supporting staffs may be available, the amount of funds available for medicine etc. may be known. Likewise, there is a wealth of information available, which if used efficiently can improve the precision of estimates. Utilizing the auxiliary information on both the occasions Feng and Zou (1997), Biradar and Singh (2001), Singh (2005) and Singh and Karna (2009 a, b) proposed ratio and regression type estimators for estimating the population mean on the current (second) occasion in two-occasion successive sampling. More recently, the contributions of Ralte and Das (2015), Singh and Pal (2015), Karna and Nath (2016) and Beevi and Chandran (2017) established beneficial results by using the auxiliary variables on both the occasions. Exponential type estimators support increasing the precision of the estimates of population parameters such as mean, median, total, etc. The exponential ratio and product type estimators in the estimation of finite population mean was introduced by Bahl and Tuteja (1991), when variable of interest and auxiliary variable is negatively or positively correlated. Further, the works of Upadhyaya et al. (2011), Yadav and Cadilar (2013), Singh and Pal (2017) examine the advantageous property of the exponential type estimators. In line with the preceding works, we propose more an effective and relevant estimator using exponential type estimators for population mean at the current occasion in two-occasion successive sampling. Properties of the proposed estimator and optimum replacement policy have been discussed. Empirical support have been given to validate the theoretical results. 2. Formulation of the estimator Δ Let U = (U1, U2, …, UN) be the finite population of N units, which has been sampled over two occasions. The character under study is denoted by x (y) on the first (second) occasion respectively. Assume that the information on two auxiliary variables z1 and z2, whose population means 1 Z and 2 Z are known, is available on the current (second) occasion and is closely related (positively correlated) to y on the second occasion. Let a simple random sample (without replacement) of size n be selected on the first occasion. A random sub-sample of STATISTICS IN TRANSITION new series, March 2018 27 m = n units is retained (matched) for its use on the second occasion, while a fresh simple random sample (without replacement) of u = (n-m) = nμ units is drawn on the second occasion from the entire population so that the sample size on the second occasion is also n.  and μ ( + μ = 1) are the fractions of matched and fresh samples at the second (current) occasion. Further, we consider the following notations throughout this work: 12 X, Y, Z , Z : population means of the variables x, y, z1 and z2 respectively; n m u m 1u 2u 1n 2n x , x , y , y , z , z , z , z : sample means of the respective variables based on the sample sizes shown in suffices; byx: sample regression coefficient of the variable y on x; βyx : population regression coefficient of the variable y on x; yx, yz1 , yz2, xz1, xz2, z1z2 : correlation coefficients between the variables shown in suffices; N 2 -1 2 Xi i=1 S = (N-1) (x -X)  : population mean square of the variable x; 2 2 2 2 X Y Z1 Z2 S , S , S , S : population mean squares of the variables x, y, z1 and z2 respectively. Utilizing information on two auxiliary variables, two different estimators of the population mean Y on the current (second) occasion may be considered. Motivated with the work of Bahl and Tuteja (1991), the estimator based on the fresh sample of size u drawn on the second occasion is defined by 1 1u 2 2u uu 1 1u 2 2u Z -z Z -z Δ = y exp exp Z +z Z +z             . (1) The second estimator Δm is also a modified chain ratio type estimator based on the sample of size m, common with both the occasions, and is defined as    1 1n 2 2n m m yx n m 1 1n 2 2n Z -z Z -z Δ = y +b m x -x exp exp Z +z Z +z               . (2) Now, considering the convex linear combination of Δu and Δm , we define the final estimator of population mean Y on the current occasion as Δ = φΔu + (1-φ)Δm (3) where φ is an unknown constant to be determined so as to minimize mean square error of the estimator Δ. 28 J. P. Karna, D. C. Nath: Improved rotation patterns… For estimating the mean on each occasion the estimator Δu is suitable, which implies that more belief on Δu could be shown by choosing φ as 1 (or close to 1), while for estimating the change from one occasion to the next, the estimator Δm could be more useful so φ might be chosen as 0 (or close to 0). For asserting both the problems simultaneously, a suitable (optimum) choice of φ is required. 3. Properties of the proposed estimator Δ 3.1 Bias and mean square error of estimator Δ Since, um and  are ratio and chain-type regression in ratio estimators respectively, they are biased for population mean Y .Therefore, the resulting estimators Δ defined in (3) is also biased estimator of Y . To obtain bias B (.) and mean square errors M (.) of Δ up to the first order of approximations we consider the following transformations: Y)e(1y1u  , Y)e(1y2m  , m3 x (1 e )X , n4 x (1 e )X , 1u 5 1 z =(1+e )Z , 1n 6 1 z =(1+e )Z , 2u 7 2 z =(1+e )Z , 2n 8 2 z =(1+e )Z , yx 9 yx s (m) = (1+e )S , 22 x 10 x s (m) = (1+e )S ; such that k E(e ) = 0 and k e <1 k = 1, 2, 3, ..., 10. Relative variances and covariances are derived as 22 1y 11 E(e ) = - C , uN    22 2y 11 E(e ) = - C , mN    22 3x 11 E(e ) = - C , mN    22 4x 11 E(e ) = - C , nN    22 5 z1 11 E(e ) = - C , uN    22 6 z1 11 E(e ) = - C , nN    22 7 z2 11 E(e ) = - C , uN    22 8 z2 11 E(e ) = - C , nN    2 3 4 x 11 E(e e ) = - C , nN    1 5 yz1 y z1 11 E(e e ) = - ρ C C , uN    1 7 yz2 y z2 11 E(e e ) = - ρ C C , uN    5 7 z1z2 1 z2 11 E(e e ) = - ρ C C , uN z    2 6 yz1 y z1 11 E(e e ) = - ρ C C , nN    2 8 yz2 y z2 11 E(e e ) = - ρ C C , nN    STATISTICS IN TRANSITION new series, March 2018 29 6 8 z1z2 z1 z2 11 E(e e ) = - ρ C C , nN    2 3 yx y x 11 E(e e ) = - ρ C C , mN    2 4 yx y x 11 E(e e ) = - ρ C C , nN    3 6 xz1 x z1 11 E(e e ) = - ρ C C , nN    4 6 xz1 x z1 11 E(e e ) = - ρ C C , nN    3 8 xz2 x z2 11 E(e e ) = - ρ C C , nN    4 8 xz2 x z2 11 E(e e ) = - ρ C C , nN    2100 39 yx α 11 E(e e ) = - , m N XS    2100 49 yx α 11 E(e e ) = - , n N XS    3000 3 10 2 x α 11 E(e e ) = - , m N XS    3000 4 10 2 x α 11 E(e e ) = - . n N XS    where         q r s t qrst 1 1 2 2 α =E x-X y-Y z -Z z -Z ;   ((q, r, s, t)  0 are integers) Under the above transformations um and  take the following forms: 57 u1 57 -e -e Δ = (1+e )Yexp exp 2+e 2+e             (4)     -1 68 m 2 9 4 3 10 yx 68 -e -e Δ = (1+e )Y+ 1+e e -e 1+e β X exp exp 2+e 2+e               . (5) Subsequently, we have the following theorems: Theorem 3.1: Bias of the estimator Δ to the first order of approximations is obtained as um B(T) = φ B( ) + (1-φ) B( ) (6) where   yz2 yz1 2 z1z2 uy ρρ ρ 1 1 1 3 BΔ = - - - + S Y u N 4 2 2 4       (7) 30 J. P. Karna, D. C. Nath: Improved rotation patterns…   yz2 yz1 3000 yx 2100 z1z2 m24 xx ρ ρ α S α ρ 1 1 1 3 1 1 B = - - - + + - + Y n N 4 2 2 4 m n S S                   xz2 yx ρS 1 1 1 -- 2 n N X    (8) Proof: The bias of the estimator Δ is given by um B( ) = E -Y = φE( -Y) + (1-φ) E( -Y)         um = φB( )+ 1-φ B( ) (9) where uu B( ) = E -Y    and mm B( ) = E -Y    . The bias of Δu and Δm is derived as follows: uu B( ) = E -Y    . Substituting the expression of Δu from from equation (4), we get 57 u1 57 -e -e B( ) = E (1+e )Yexp exp -Y 2+e 2+e                . Now, expanding the right hand side of the above expression, taking expectations and retaining the terms up to the first order of approximations, we have   yz2 yz1 z1z2 u ρρ ρ 1 1 1 3 BΔ = - - - + Y u N 4 2 2 4       . (10) Similarly   mm B = E Y            -1 68 2 9 4 3 10 yx 68 -e -e = E (1+e )Y+ 1+e e -e 1+e β X exp exp Y 2+e 2+e                yz2 yz1 3000 yx 2100 z1z2 24 xx ρ ρ α S α ρ 1 1 1 3 1 1 = - - - + + - + Y n N 4 2 2 4 m n S S                  xz2 yx ρS 1 1 1 -- 2 n N X    . (11) STATISTICS IN TRANSITION new series, March 2018 31 Substituting the values of B(Δu) and B(Δm) from equations (10) and (11) in the equation (9) we get the bias of estimator Δ as shown in equation (6). Theorem 3.2: Mean square error of the estimator Δ to the first order of approximations is obtained as           2 2u m u m M( ) = φ M + 1-φ M +2φ 1-φ Cov ,     (12) where   22 u u 1 y 11 M( ) = E -Y = - A S uN     (13)   22 m m 2 3 4 y 1 1 1 M(Δ ) = E Δ -Y = A + A - A S m n N                      (14) and      2 1y u m u m AS Cov , = E -Y -Y = - N       (15) where     1 z1z2 yz2 yz1 31 A = + ρ -2 ρ +ρ 22 , 2 2 yx A =1-ρ,     2 3 z1z2 yz2 yz1 yx 1 A = ρ -2 ρ +ρ +ρ 2 and     4 z1z2 yz2 yz1 1 A = ρ -2 ρ +ρ +1. 2 Proof: It is obvious that mean square error of the estimator Δ is given by       2 2 um M( ) = E -Y = E φ -Y + 1-φ -Y               2 2u m u m = φ M( )+ 1-φ M +2φ 1-φ Cov( , )    (16) where 2 uu M( ) = E -Y    , 2 mm M( ) = E -Y    and    u m u m Cov( , ) = E -Y -Y       . The mean square errors of Δu and Δm are derived as follows: 2 uu M( ) = E -Y    . Substituting the expression of Δu from equation (4), we get 2 57 u1 57 -e -e M( ) = E (1+e )Yexp exp -Y 2+e 2+e                . 32 J. P. Karna, D. C. Nath: Improved rotation patterns… Now, expanding the right-hand side of above expression, taking expectations and retaining the terms up to the first order of approximations, we have       22 u u z1z2 yz2 yz1 y 1 1 3 1 M( ) = E -Y = - + ρ -2 ρ +ρ S u N 2 2          . (17) Similarly   2 mm M = E -Y          2 -1 68 2 9 4 3 10 yx 68 -e -e = E (1+e )Y+ 1+e e -e 1+e β X exp exp Y 2+e 2+e                          2 2 2 yx z1z2 yz2 yz1 yx z1z2 yz2 yz1 y 1 1 1 1 1 = 1-ρ + ρ -2 ρ +ρ +ρ - ρ -2 ρ +ρ +1 S m n 2 N 2                              (18) and          2 y u m u m z1z2 yz2 yz1 S 31 Cov , = E -Y -Y = - + ρ -2 ρ +ρ 2 2 N          . (19) Substituting the values of M(Δu), M(Δm) and Cov(Δu, Δm) from equations (17), (18) and (19) in the equation (16) we get mean square error of estimator Δ as shown in equation (12). Remark 1: Since x and y are the same study variable over two occasions and z1 and z2 are auxiliary variables positively correlated with x and y, therefore, considering the stable behaviour of coefficient of variations [Reddy (1978)] the expressions of bias mean square errors in equations (6) and (12) are derived under the assumption that the coefficients of variation of x, y and z1 and z2 are approximately equal, i.e. Cx = Cy = Cz1 = Cz2. 3.2. Minimum mean square error of Δ The mean square error of the estimator Δ in equation (12) is the function of unknown constant φ, therefore, it is minimized with respect to φ and subsequently the optimum value of φ is obtained as     m u m u m u m opt M( )-Cov , φ = M( )+M( )-2Cov ,        . (20) STATISTICS IN TRANSITION new series, March 2018 39 Remark 3:”*” in the Tables 1-3 indicates that μ* do not exist for the corresponding combinations of correlations. The following results may be extracted from Tables 1-3: (a) For the fixed value of ρyx and ρz1z2, the value of μ decreases but E1 and E2 increases with the increasing values of ρyz1 and ρyz2. This phenomenon is expected as availability of highly correlated auxiliary variables results in reducing the cost of survey and enhancing the precision of estimates. (b) For the fixed value of ρz1z2, ρyz1 and ρyz2, the value μ, E1 and E2 increases with the increasing values of ρyx. This behaviour indicates that when the study characters over the occasions are highly correlated, a larger fresh sample at the current occasion is required but the efficiency of estimates increases. (c) For fixed values of ρyz1, ρyz2 and ρyx, the values of μ are increasing whereas decreasing trends are observed in the efficiencies E1 and E2. (d) The lowest value of μ is observed as 0.4682, which indicates that the fraction of fresh sample to be drawn at the current occasion is as low as 46% of the total sample size. (e) Percent relative gain in efficiencies E1 and E2 seems to be appreciably high under optimal conditions. Highest values of E1 and E2 are observed as 1831.7 and 1257.7 respectively. (f) A close perusal of Figure 1 suggests that for the fixed value of ρyx and ρz1z2, the value of μ decreases with the increasing values of ρyz1 and ρyz2. This behaviour supports the fact that if highly correlated auxiliary variables are available, the less fresh sample is required to be drawn at the current occasion, which subsequently reduces the cost of survey. (h) It can be clearly seen from Figure 2 that for the fixed value of ρyx and ρz1z2, E1 increases with the increasing values of ρyz1 and ρyz2. This phenomenon justifies the use of two auxiliary variables, which results in increasing the precision of estimates. 6. Illustrations using real life data To illustrate the comparison of relative efficiency of the proposed estimator with respect to nˆ y and Y , using real life approach, the data from the Census of India (2001) and (2011) were taken into account. We define the variables as x (y): the total number of workers in villages in the district of Ranchi, India in 2001 (2011) z1: the total number of literate people in villages in the district of Ranchi, India. z2: the total number of females in the district of Ranchi, India. Using successive sampling strategy defined in Section 2 for the above data set we have taken n (sample drawn at first occasion) = 70, m (matched portion of the sample over two occasions) = 35 and u (fresh sample drawn at second ρyz2 40 J. P. Karna, D. C. Nath: Improved rotation patterns… occasion) = 35. The values of the different estimators computed from the sample along with their corresponding mean square errors and efficiency of the proposed estimator ∆ with respect to nˆ y and Y have been shown in the Table 4. Table 4. Relative Efficiency (%) of estimator ∆ with respect to nˆ y and Y using real life data Estimators Estimates MSE % Efficiency ∆ 481 341.86 100 n y 465 1926.13 563.42 ˆ Y 422 1360.48 397.96 The above table validates the conclusions observed from the Tables 1, 2 and 3 that the proposed estimator ∆ is more efficient than nˆ y and Y with maximum gain in efficiency occurring while comparing it with mean per unit estimator, which is the expected phenomenon. 7. Conclusion In the context of the preceding interpretations, it may be concluded that the use of two auxiliary variables for the estimation of population mean at the current occasion in two-occasion successive sampling is highly appreciable, as demonstrated through empirical results. It is also observed and justified through this study that the use of exponential type estimator is highly rewarding in terms of precision. Hence, the proposed estimator Δ may be recommended for its practical use by survey practitioners. Acknowledgement Authors are thankful to the honourable referees and University Grants Commission, New Delhi (PDFWM-2014-15-GE-ASS-28217) for providing necessary infrastructure to carry out the present work. STATISTICS IN TRANSITION new series, March 2018 41 REFERENCES BEEVI, N. T., CHANDRAN, C., (2017). Efficient family of ratio-type estimators for mean estimation in successive sampling using auxiliary information on both occasions. Statistics In Transition – New Series, 18 (2), pp. 227–246. BIRADAR, R. S., SINGH, H. P., (2001). Successive sampling using auxiliary information on both occasions. Cal. Statist. Assoc. Bull. 51, pp. 243–251. CENSUS OF INDIA, (2001). www.cesusindiagov.in. CENSUS OF INDIA, (2011). www.cesusindiagov.in. CHATURVEDI, D. K., TRIPATHI, T. P., (1983). Estimation of population ratio on two occasions using multivariate auxiliary information. Journal of Indian Statistical Association 21, pp. 113–120. COCHRAN, W. G., (1977). Sampling Techniques, Wiley Eastern Limited, New Delhi, III Edition. DAS, A. K., (1982). Estimation of population ratio on two occasions, Jour Ind. Soc. Agr. Statist. 34, pp. 1–9. FENG, S., ZOU, G., (1997). Sample rotation method with auxiliary variable. Communications in Statistics-Theory and Methods, 26 (6), pp. 1497–1509. GUPTA, P. C., (1979). Sampling on two successive occasions. Jour. Statist. Res. 13, pp. 7–16. JESSEN, R. J., (1942). Statistical Investigation of a Sample Survey for obtaining farm facts, Iowa Agricultural Experiment Station Research Bulletin No. 304, Ames, Iowa, USA, pp. 1–104. KARNA, J. P., NATH, D. C., (2016). Rotation sampling scheme using transformed auxiliary variable. Journal of Statistics & Management Systems, 19 (6), pp. 739–754. PATTERSON, H. D., (1950). Sampling on successive occasions with partial replacement of units, Journal of the Royal Statistical Society, 12, pp. 241– 255. RALTE, Z., DAS, G., (2015). Ratio-to-regression estimator in successive sampling using one auxiliary variable. Statistics in Transition – new series, 16 (2), pp. 183–202. RAO, J. N. K., GRAHAM, J. E., (1964). Rotation designs for sampling on repeated occasions, Journal of the American Statistical Association, 59, pp. 492–509. REDDY, V. N., (1978). A study on the use of prior knowledge on certain population parameters. Sankhya, 40, C, pp. 29–37. SEN, A. R., (1971). Successive sampling with two auxiliary variables, Sankhya, 33, Series B, pp. 371–378. 42 J. P. Karna, D. C. Nath: Improved rotation patterns… SEN, A. R., (1972). Successive sampling with p   p1 auxiliary variables, Ann. Math. Statist., 43, pp. 2031–2034. SEN, A. R., (1973). Some theory of sampling on successive occasions, Australian Journal of Statistics, 15, pp. 105–110. SINGH, G. N., (2005). On the use of chain-type ratio estimator in successive sampling, Statistics in Transition, 7, pp. 21–26. SINGH, G. N., KARNA, J. P., (2009a). Estimation of Population mean on the current in two-occasion successive sampling. Metron, 67(1), 2009, pp. 87– 103. SINGH, G. N., KARNA, J. P., (2009b). Search of effective rotation patterns in presence of auxiliary information in successive sampling over two occasions. Statistics in Transition-new series, 10, pp. 209, 59–73. SINGH, H. P., PAL, S. K., (2015). An efficient effective rotation pattern in successive sampling over two occasions. Communication in Statistics: Theory and Methods, 45 (17), pp. 5017–5027. SINGH, H. P., PAL, S. K., (2017). A class of exponential type estimators of a general parameter. Communication in Statistics – Theory and Methods, 46 (8), pp. 3957–3984. SUKHATME, P. V., SUKHATME, B. V., SUKHATME, S., ASOK, C., (1984). Sampling theory of surveys with applications. Iowa State University Press, Ames, Iowa (USA) and Indian Society of Agricultural Statistics, New Delhi (India),III Revised Edition. UPADHYAYA, L. N., SINGH, H. P., CHATTERJEE, S., YADAV, R., (2011). Improved ratio and product exponential type estimators. J. Stat. Theo. Pract. 5 (2), pp. 285–302. YADAV, S. K., KADILAR, C., (2013). Improved exponential type ratio estimator of population variance. Revis. Colum. de Estadist. 36 (1), pp. 145–152. STATISTICS IN TRANSITION new series, March 2018 43 APPENDIX Bias of estimator Δu uu B( ) = E -Y    1 1u 2 2u u 1 1u 2 2u Z -z Z -z = E y exp exp -Y Z +z Z +z                57 1 57 -e -e = E (1+e )Yexp exp -Y , from eq. (4) 2+e 2+e                57 1 57 -e -e = YE e exp exp 2+e 2+e                11 5 5 7 7 1 e e e e = YE e exp - 1 exp - 1 2 2 2 2                                       Retaining the terms up to the first order of approximations, we get u B( ) 22 5 5 7 7 1 e 3e e 3e = YE e 1- + 1- + 2 8 2 8             22 5 7 5 7 1 7 1 5 5 7 1 e e 3e 3e e e e e e e = YE e - - + + - - + 2 2 8 8 2 2 4    22 5 7 5 7 1 7 1 5 5 7 1 e e 3e 3e e e e e e e = YE e - - + + - - + 2 2 8 8 2 2 4    Taking expectations as discussed in Section 3.1 and using Note 1, we have   yz2 yz1 2 z1z2 uy ρρ ρ 1 1 1 3 BΔ = - - - + S Y u N 4 2 2 4       1 2 1 2 22 5 5 5 5 7 7 7 7 1 e e e e e e e e 11 = YE e 11 . 1 ... 11 . 1 ... 2 2 2 4 2 2 2 2 4 2                                                        44 J. P. Karna, D. C. Nath: Improved rotation patterns… Bias of estimator Δm mm B( ) = E -Y        -1 68 2 9 4 3 10 yx 68 -e -e = E (1+e )Y+ 1+e e -e 1+e β X exp exp -Y , from eq (5) 2+e 2+e                  Expanding the exponentials as in the case of bias of estimator Δu and retaining the terms up to first order of approximations, we get       22 8 6 6 8 m 2 4 3 4 9 3 9 4 10 3 10 yx 8 6 e e e e 33 B(Δ ) = E 1+e Y+ e -e +e e -e e -e e +e e β X 1- + e - + + e Y 2 8 2 4 8        22 2 8 2 6 6 8 4 6 3 8 3 6 8 6 4 9 3 9 4 10 3 10 4 8 yx e e e e e e e e e e e e 33 = E e e Y+ e e -e e -e e +e e -e e β X 8 8 2 2 4 2 2 2                       Now, taking expectations as discussed in Section 3.1 and using Note 1, we have   yz2 yz1 3000 yx 2100 z1z2 m24 xx ρ ρ α S α ρ 1 1 1 3 1 1 B = - - - + + - + Y n N 4 2 2 4 m n S S                   xz2 yx ρS 1 1 1 -- 2 n N X     The mean square errors of the estimators Δu and Δm have been obtained in the similar manner as the bias of the estimators.  The variance of the estimator natural successive sampling estimator ˆ Y has been discussed in detail in Sukhatme et al. (1984), pages 256-260.