A note on estimation of population mean in sample survey using auxiliary information

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A note on estimation of population mean in sample survey using auxiliary information

  1. 1. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 9 A Note on Estimation of Population Mean in Sample Survey using Auxiliary Information R. K. Gangele and S.K. Choubey Department of Mathematics and Statistics Dr. H.S. Gour Central University, Sagar – 470003 (M.P.) INDIA E-mail- rkgangele@gmail.com Abstract In this paper we have suggested a class of estimators for population mean using auxiliary information in two- phase sampling. When the population mean X is not known, a class of estimators for finite population mean ∑= = N 1i iy N 1 Y of the study variable y has been suggested. Expressions of bias and mean squared error are obtained upto the first order of approximation. Asymptotically optimum estimators (AOE’s) are also identified with its mean squared error formula. We found that proposed class of estimator are better than usual ratio and other estimators. 1.1 Introduction Whenever there is auxiliary information available, the investigator wants to use it in the method of estimation which yields larger efficiency. Ratio, regression and product methods of estimation are good examples in this context. When the population mean of the auxiliary variable is known a large number of estimators for the population mean of the study variable is available in the literature for instance see Singh, H. P. (1986) and Singh, S. (2003) and the references cited therein. It is observed in the literature that the efficiency of the ratio and product estimators may be increased to the efficiency of regression estimator by making use of prior knowledge of = , where is the correlation coefficient between the study variable and auxiliary variable , and are the coefficients of variation of and respectively. Sacrificing the consistency of estimators researchers including Upadhyaya and Singh (1985), Srivastava (1974), Prasad (1989) and Lui (1990) have found the way by which the efficiency an estimator can be increased beyond the regression estimator of the mean in case of known population mean . When the population mean of the auxiliary variable is not known, it is often estimated from a preliminary large sample on which only the auxiliary variable is observed. The value of the population mean of the auxiliary variable is then replaced by this estimate. This procedure is known as the double sampling or two-phase sampling. Throughout, samples have been drawn by the method of simple random sampling without replacement (SRSWOR). The sample survey statisticians have presented several modifications of the classical two-phase sampling ratio and product estimators and studied their properties. They have shown that the efficiency of their estimators can be increased maximum upto usual two-phase sampling regression estimator by using of the well known optimum choice = , for instance see Srivastava (1970, 81), Gupta (1978) and Adhvaryu and Gupta (1983). In this paper we have made an effort to suggest a class of estimators using auxiliary information in two- phase sampling and studied its properties. When the population mean is not known, a class of estimators for finite population mean ∑= = N 1i iy N 1 Y of the study variable y has been suggested. A large number of estimators are identified as the member of the proposed class of estimators. Expressions of bias and mean squared error are obtained upto the first order of approximation. Asymptotically optimum estimators (AOE’s) are also identified with its mean squared error formula. 1.2 Two-Phase Sampling Procedure Consider the finite population = , , … , of identifiable units. Let and be the variable under study and the auxiliary variable respectively. Further let be the unknown real variable value of and be the known variable value of associated with , = 1,2, … , . The problem of estimating the population mean ∑= = N 1i iy N 1 Y of the study variable when the population mean ∑= = N 1i ix N 1 X of the auxiliary variable
  2. 2. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 10 is known has been dealt at a grater length. However, in many practical situations when the population mean is unknown a prior, the procedure of double sampling is used. Allowing, simple random sampling without replacement (SRSWOR) design in each phase, the two-phase (or double) sampling scheme will be as follows: (i) The first-phase sample ⊂ of fixed size ! is drawn to observe only in order to furnish a good estimate of the population mean . (ii) Given , the second phase sample ⊂ of fixed size ! is drawn to observe only. Let # = $% ∑$% ' , = $ ∑$ ' and # = $ ∑$ ' . Further we write = 1 + )* # = 1 + ) # = 1 + ) 1.2.1 Such that 2 1 21 2 1 20 2 10 2 1 2 2 22 1 22 0 210 11 )(, 11 )( 11 )(, 11 )( , 11 )(, 11 )( 0)()()( xx xx xy C Nn eeECk Nn eeE Ck Nn eeEC Nn eE C Nn eEC Nn eE and eEeEeE       −=      −=       −=      −=       −=      −= === where = + , , = + - , . = / ∑ −' and . = / ∑ −' . 1.3 Proposed Class of Estimators Motivated by Gangele (1995) and Gupta (1978), we define a class of estimators for population mean in two-phase sampling as 1 23 = ∑ 45 '* # #% 6 , 1.3.1 where 4′ = 0,1,2,3 are suitably chosen constants whose sum need not be unity, 9 is a suitably chosen scalar takes value +1 for product-type estimator and −1 for ratio-type estimator; # is a first phase sample mean based on ! observations and , # are second phase sample means of , respectively based on ! observations. Some members of the proposed class of estimators are shown in Table-1.3.1. 1.2.2
  3. 3. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 11 Table-1.3.1 Some members of the class of estimators 1 23 S. No. Estimator Choice of constants 4* 4 4 45 9 1 The mean per unit = $ ∑$ ' 1 0 0 0 - 2 The classical two-phase sampling ratio estimator :3 = #% # 0 1 0 0 -1 3 The classical two-phase sampling product estimator ;3 = # #% 0 1 0 0 1 4 Srivastava (1970) estimator 3 = #% # 6 0 1 0 0 −9 5 Chakraberty (1968), Vos (1980), Adhvaryu and Gupta (1983) type- estimator 3 = 4* + 1 − 4* < # # = 4* 1 − 4* 0 0 -1 6 Vos (1980), Adhvaryu and Gupta (1983) type estimator 53 = 4* + 1 − 4* < # # = 4* 1 − 4* 0 0 1 4, 9 being constants. 1.4 Bias and Mean Squared Error To obtain the bias and mean squared error of the proposed class of estimators 1 23, we express it in terms of )′ we have 1 23 = ∑ 45 '* 1 + )* 1 + ) 6 1 + )′ / 6 . or 1 23 = 1 + )* ∑ 45 '* 1 + ) 6 1 + )′ / 6 . 1.4.1 We assume that the sample size ! and ! ! > ! are large enough so that |) | < 1 and A)′ A < 1. i.e. B #/- - B < 1 and B #%/- - B < 1. and 1 + ) and 1 + )′ are expandable. Expanding the right hand side of (1.4.1) we have 1 23 = ∑ 45 '* 1 + )* C1 + 9) + 6 6/ ) + ⋯ E × C1 − 9)′ + 6 6G )′ + ⋯ E or 1 23 = H 4 5 '* I1 + )* + 9) + 9)*) + 9 9 − 1 2 ) − 9)′ − 9)*)′ − 9 ) )′ + 9 9 + 1 2 )′ + 9 9 − 1 2 )*) − 9 )*) )′ − 9 9 − 1 2 ) )′ + 9 9 + 1 2 )*)′ + 9 9 + 1 2 ) )′ + ⋯ J. Neglecting terms of )′ having power greater than two we have 1 23 = ∑ 45 '* K1 + )* + 9 ) − )′ + 9 )*) − )*)′ − 9 ) )′ + 6 6/ ) + 9 9+12)1′2.or 1 23 − = K∑ 45 '* K1 + )* + 9 ) − )′ + 9 )*) − )*)′ − 9 ) )′ + 6 6/ ) + 9 9+12)1′2−1.
  4. 4. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 12 1.4.2 Taking expectation of both sides of (1.4.2) and using the expected values given by (1.2.2) we get the bias of 1 23 to the first degree of approximation as L 1 23 = M∑ 45 '* M 1 + $ − $′ 9 + 6 6/ $ − − 6 6/ $′ − N − 1N = K∑ 45 '* K1 + $ − $′ 9 + $ − $′ 6 6G O − 1O i.e. L 1 23 = K∑ 45 '* K1 + $ − $′ 9 + 6/ O − 1O. 1.4.3 Squaring both sides of (1.4.2) and neglecting term of )′ having power greater than two we have PQR ST − PQ U = PQU V W W W W W W W W W W W W W W X Y + ∑ SZ U[ Z' ] ^ _ ^ ` Y + Ua + UZbcaY − aY ′ d +eZbcaaY − aaY ′ d + a U +ZU bU caY − aY ′ d U − UZU bU aYaY ′ +Zb Zb − Y aY U + Zb Zb + Y aY ′U f ^ g ^ h +U ∑ SZ [ Z Zij ' Sj ] ^ _ ^ ` Y + Ua + Z + j bcaY − aY ′ d +a U + U Z + j caaY − aaY ′ d − Z − j U bU aYaY ′ + b ZGj U b Z + j − Y aY U + b ZGj U b Z + j − Y aY ′U f ^ g ^ h −U ∑ k Y + a + ZbcaY − aY ′ d +ZbcaaY − aaY ′ d −ZU bU aYaY ′ + Zb Zb/Y U aY U + Zb ZbGY U aY ′U l[ Z' m n n n n n n n n n n n n n n o . 1.4.4 Taking expectation of both sides of (1.4.4) using the result in (1.2.2) we get the MSE of 1 23 to the first degree of approximation as p.q 1 23 = r1 + ∑ 45 '* s + 2 ∑ 45 it '* 4ts t − 2 ∑ 4 L5 '* u, 1.4.5 where s = K1 + $ − + $ − $′ 9 4 + 2 9 − 1 O = 0,1,2,3 , s t = K1 + $ − + $ − $′ 6 Gt 4 + 9 + w − 1 O < w = 0,1,2,3 , L = K1 + $ − $′ 9 + 6/ O = 0,1,2,3 . Expression (1.4.5) can be rewritten as p.q 1 23 = M 1 + 4*s* + 4 s + 4 s + 45s5 +2 4*4 s* + 4*4 s* + 4*45s*5 + 4 4 s + 4 45s 5 + 4 45s 5 −2 4*L* + 4 L + 4 L + 45L5 N, 1.4.6 where L* = 1, L = K1 + $ − $′ 4 C + 6/ E O, L = K1 + $ − $′ 24 C + 6/ E O, L5 = K1 + $ − $′ 34 C + 56/ E O, s* = K1 + $ − O, s = K1 + $ − + $ − $′ 9 4 + 29 − 1 O, s = K1 + $ − + $ − $′ 29 4 + 49 − 1 O, (
  5. 5. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 13 s5 = K1 + $ − + $ − $′ 39 4 + 69 − 1 O, s* = K1 + $ − + $ − $′ 6 4 + 9 − 1 O, s* = K1 + $ − + $ − $′ 9 4 + 29 − 1 O, s*5 = K1 + $ − + $ − $′ 56 4 + 39 − 1 O, s = K1 + $ − + $ − $′ 56 4 + 39 − 1 O, s 5 = K1 + $ − + $ − $′ 29 4 + 49 − 1 O, and s 5 = K1 + $ − + $ − $′ y6 4 + 59 − 1 O. Differentiating the p.q 1 23 given in (1.4.6) with respect to 4′ , = 0,1,2,3 partially and equating them to zero we have             =                         3 2 1 3 2 1 0 3231303 2321202 .1312101 0302010 1 B B B AAAA AAAA AAAA AAAA α α α α 1.4.7 Solving (1.4.6) we get the optimum values of constants 4′ , = 0,1,2,3 as ∆ ∆ = ∆ ∆ = ∆ ∆ = ∆ ∆ = ∗∗ ∗∗ 3 3 2 2 1 1 0 0 , αα αα 1.4.8 where             =∆ 3231303 2321202 .1312101 0302010 AAAA AAAA AAAA AAAA             =∆ 323133 232122 .131211 030201 0 1 AAAB AAAB AAAB AAA             =∆ 323303 232202 .1312101 03020 1 1 AABA AABA AABA AAA             =∆ 331303 2321202 .131101 03010 2 1 ABAA ABAA ABAA AAA . 1 3231303 221202 .112101 02010 3             =∆ BAAA BAAA BAAA AAA Putting the optimum values 4∗ of 4 = 0,1,2,3 in (1.6) we get the minimum MSE of proposed estimator 1 23 as p.q 1 23 = |1 − 4* ∗ − 4∗ L − 4∗ L − 45 ∗ L5}. 1.4.9 or p.q 1 23 = K1 − ∆•G∆%€%G∆•€•G∆‚€‚ ∆ O. 1.4.10 Thus we established the following theorem. Theorem 1.4.1. To the first degree of approximation, p.q 1 23 ≥ K1 − ∆•G∆%€%G∆•€•G∆‚€‚ ∆ O,
  6. 6. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 14 with equality holding if 4* ∗ = ∆• ∆ , 4∗ = ∆% ∆ , 4∗ = ∆• ∆ , 45 ∗ = ∆‚ ∆ . It is to be noted that the biases and mean squared errors of the estimators belonging to the class 1 23 can be obtained easily from (1.4.3) and (1.4.6) respectively just by putting the suitable values of the scalars 4 = 0,1,2,3 and 9. It is to be noted that any member of the class will not have MSE/minimum MSE smaller than that of the proposed class of estimators 1 23. It is to be noted that some empirical studies need to be conducted in order to find out situations where some members of the proposed class 1 23 are found to be better than usual unbiased estimator , double sampling ratio :3, product ;3, regression estimator ( „…3 = + †‡ # − # , †‡ being the sample estimate of the population regression coefficient †), Srivastava (1970) estimator „3 and other existing estimators. Probably a well designed Monte Carlo study will be quite useful and may throw some light on it. Remark 1.4.1 In similar fashion the properties of the proposed class of estimator 1 23 can be discussed in the case “when the second phase sample of size ! is drawn independently of the first phase sample of size ! ”. References Adhvaryu, D. and Gupta, P.C. (1983): On some alternative sampling strategies using auxiliary information. Metrika, 30, 217-226. Gangele, R. (1995): Study of some estimators using a priori/ auxiliary information. Ph. D. thesis submitted to Vikram University, Ujjain, India. Gupta, P.C. (1978): On some quadratic and higher degree ratio and product estimators. Jour. Ind. Soc. Agril. Statist. , 30, 71-80. Lui, K.J. (1990): Modified product estimators of finite population mean in finite sampling. Commun. Statist. Theory Meth., 19(10), 3799-3807. Prasad, B. (1989): Some improved ratio-type estimators of population mean and ratio in finite population sample surveys. Commun. Statist. -Theory Meth. , 18,379-392. Singh, H.P. (1986): Estimation of ratio, product and mean using auxiliary information in sample surveys. Alig. Jour. Statist., 6, 32-44. Singh, S. (2003): Advanced sampling theory with application. How Michael selected Amy. Vol. I and II, Kluwer Academic Publishers, Dordrecht, The Netherlands. Srivastava, S.K. (1970): Two-phase sampling estimator in sample surveys. Amt. Jour. Statist., 12, 23-27. Srivastava, S.K. (1981). A general ized two-phase sampling estimator. Jour. Ind. Sac. Agri. Statist., 33, (1) , 38- 46. Srivastava, V. K. (1974): On ratio method of estimation. Trab. Estadist. Invest. Operat. , 25, 113-117. Upadhyaya, L.N., Singh, H.P. and Vos, J.W.E. (1985): On the estimation of population means and ratios using supplementary information. Statistica Neerlandica, 39(3), 309-318.
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