VARIANCE ESTIMATION FOR COMBINED RATIO ESTIMATOR
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1 Sankyā : Te Indian Journal of Statistics 1995, Volume 57, Series B, Pt. 1, pp VARIANCE ESTIMATION FOR COMBINED RATIO ESTIMATOR By SANJAY KUMAR SAXENA Central Soil and Water Conservation Researc and Training Institute and A. K. NIGAM and N. D. SHUKLA Lucknow University SUMMARY. In tis article te properties of variance estimators of combined ratio estimator y RC = y st X/x st are studied. Te two estimators considered are te conventional and te balanced alf-samples estimators. It is sown tat te conventional variance estimator as well as te alf-samples variance estimator are unbiased to te first approximation. Te mean square errors (MSE) of te two estimators are evaluated. 1. Introduction Te balanced alf-samples metod (McCarty (1966, 1969)) as become popular for estimating te variance of non-linear statistics like ratios, regression and correlation coefficients from stratified sampling designs wit two primary units from eac stratum. Te metod as been extended to situations wit arbitrary number of units from different strata (Gurney and Jewett (1975), Gupta and Nigam (1987), Wu (1991)). Te bias of te balanced alf-samples estimator for combined ratio estimator as been worked out by Krewski and Rao (1981) and Rao and Wu (1985) in terms of first and second order analysis respectively. Wu (1985) derived te mean square error for te conventional variance estimator of a general combined ratio estimator. In te present investigation, we derive te mean square error of te balanced alf-samples estimator and compare it wit mean square error of te conventional variance estimator of te combined ratio estimator. Some empirical results are also reported. Paper received. April AMS (1991) subject classification. 62D05 Key words and prases. Combined ratio estimator, conventional variance estimator, balanced alf-samples variance estimator, mean square error.
2 86 sanjay kumar saxena, a.k. nigam and n.d. sukla 2. Conventional variance estimator Consider a population of L strata, te -t stratum consisting of N units. A simple random sample of size n units is drawn wit replacement from te -t stratum ( = 1, 2,..., L). Te total sample size is n =Σn and te population size is N =ΣN, (Te summation sign is taken from 1 to L, wenever te limits are not specified). Suppose now tat y and x are two caracteristics of interest, wic are correlated. Let y i an x i ( = 1, 2,..., L) be te values of y and x respectively for te i-t unit in te -t stratum, W = N /N be te stratum weigt, y, x te sample means and Y, X te population means respectively and X =ΣW X be known. One estimator of Y =ΣW Y is te combined ratio estimator. y RC = y st X/x st wit y st =ΣW y and x st =ΣW x Te mean square error (MSE) of y RC to te first approximation can be obtained as MSE(y RC )=ΣWn 2 1 (µ 02 + R 2 µ 20 2Rµ 11 )=V(y RC ),...(1) were R = Y/X and µ rs = N i=1 (X i X ) r (Y i Y ) s /N. Te conventional variance estimator for y RC, usually obtained by replacing te population values in (1) by teir sample values, is v c = Wn 2 1 (s 02 + rs20 2 2rs 11 ),...(2) were s 20,s 02 and s 11 are te sample variance of x, sample variance of y and sample covariance between x and y respectively, and r = y st /x st. To te first approximation, it follows from (1) and (A.1) in Appendix A tat E(v c )=MSE(Y RC )...(3) Hence, to te first approximation, variance estimator, v c, is unbiased and wen iger degree terms are retained, it is biased, a result also obtained by Wu (1985). In Appendix A, we prove tat te first approximation to te mean square error of v c is MSE(v c ) = Y 4 [4 W 2n 1 (c 02 + c 20 2c 11 ) W 4n 2.(c 20 c 11 ) 2 + W 4 n 3 {(c 40 + c 04 +6c 22 4c 31 4c 13 ) (n 3)(n 1) 1.(c c c c 20c 02 4c 11 c 20 4c 11 c 02 )} 4 W 5 n 3. (c 20 c 11 )(c 30 +3c 12 c 03 c 21 )],...(4)
3 variance estimation for combined ratio estimator 87 were c rs = µ rs /X r Y s. 3. Balanced alf-samples variance estimator (n =2) McCarty (1966, 1969) proposed te balanced alf-samples metod, wen n = 2 for all, based on a number of alf-samples formed by deleting one unit from te sample in eac stratum. Te set of k balanced alf-samples can always be constructed using te metod of Plackett and Burman (1946), were L k L + 3. Let y RC,α (α = 1, 2,..., k) denote te estimator of population mean Y computed from te α-t alf-sample. Tese estimators sould be of te same functional form as te parent sample estimator y RC. Te balanced alf-samples variance estimator based on k balanced alf-samples is given by k v b = (y RC,α y RC ) 2 /k... (5) α=1 wit y RC,α = y st,α X/x st,α and y RC = y st X/y st. To te first approximation, it follows from (1) and (A.7) in Appendix B tat E(v b )=MSE(y RC )...(6) Tus, to te first approximation, te balanced alf-samples variance estimator, v b, is also unbiased and wen iger degree terms are retained it is biased (see also, Rao and Wu, 1985). It is sown in Appendix B tat te first approximation to te mean squared error of v b is MSE(v b ) = Y 4 W 4 (c 40 + c 04 +6c 22 4c 31 4c 13 +c c c c 20c 02 4c 11 c 20 4c 11 c 02 )/8,...(7) were c rs is as defined in Section Stability of balanced alf-samples variance estimator For n =2, ( = 1, 2,..., L), it follows from (4) and (7) tat MES(v b ) = MSE(v c )+Y 4 [ W 5 (c 20 c 11 ). (c 30 +3c 12 c 03 c 21 ) W 2. (c 02 + c 20 2c 11 ) W 2 (c 20 c 11 ) 2 ]/2....(8) Using (4) and (8), after some algebra, we get MSE(v b )/M SE(v c )=1+[MSE(v b ) MSE(v c )]/M SE(v c )...(9)
4 88 sanjay kumar saxena, a.k. nigam and n.d. sukla As no clearcut comparison is possible between te mean square errors of v b and v c from (9), we studied it furter empirically. We consider a small artifical population taken from (Cocran, 1977; Table 6.1) by selecting 24 units from 49 units in te original population. Tese selected units were divided into 8 equal strata, eac wit 3 units (Table 1). TABLE 1. ARTIFICIAL POPULATION stratum x y x y x y x y x y x y x y x y units From (4), (7) and (8), we compute te following results troug computer for te population given in table 1. MSE(v c ) = MSE(v b ) = MSE(v b )/M SE(v c ) = Similar results were obtained for a few oter sets of data and are not reported ere to save space.
5 variance estimation for combined ratio estimator 89 Appendix A From (2), we ave v c = W 2 n 1 (s 02 + r 2 s 20 2rs 11 ) = W 2 n 1 [µ 02(1 + ε 4 )+R 2 µ 20 (1 + ε 3 ). (1 + ε 1 ) 2 (1 + ε 2 ) 2 2Rµ 11 (1 + ε 1 )(1 + ε 5 ).(1 + ε 2 ) 1 ] = V (y RC )+Y 2 W 2 n 1 [2(c 20 c 11 )ε 1 2(c 20...(A.1) c 11 )ε 2 + c 20 ε 3 + c 02 ε 4 2c 11 ε 5 + c 20 ε 2 1 +(3c 20 2c 11 )ε 2 2 2(2c 20 c 11 )ε 1 ε 2 +2c 20 ε 1 ε 3 2c 11 ε 1 ε 5 2c 20 ε 2 ε 3 +2c 11 ε 2 ε 5 + ], were ε 1 = (y st Y )/Y ; ε 2 =(x st X)/X ε 3 = (s 20 µ 20 )/µ 20 ; ε 4 =(s 02 µ 02 )/µ 02 ; and ε 5 = (s 11 µ 11 )/µ 11 Te expansion (A.1) can be used to obtain useful approximation to te bias and mean squared error weter te condition ε 2 < 1 is satisfied or not (Suktame, et al., 1984). From (A.1), we obtain te first approximation to te mean square error of v(y RC )as MSE(v c ) = Y 4 W 4 n 2 [4a2 E(ε 1 ) 2 +4a 2 E(ε 2 ) 2 +c 2 20.E(ε 3) 2 + c 2 02 E(ε 4) 2 +4c 2 11 E(ε 5) 2 8a 2.E(ε 1 ε 2 )+4ac 20 E(ε 1 ε 3 )+4ac 02 E(ε 1 ε 4 ) 8ac 11 E(ε 1 ε 5 ) 4ac 20 E(ε 2 ε 3 )...(A.2) 4ac 02 E(ε 2 ε 4 )+8ac 11 E(ε 2 ε 5 ) +2c 20 c 02 E(ε 3 ε 4 ) 4c 20 c 11 E(ε 3 ε 5 ) 4c 02 c 11 E(ε 4 ε 5 )], were a = c 20 c 11
6 90 sanjay kumar saxena, a.k. nigam and n.d. sukla Following Sukatme (1944) and Kendall and Stuart (1977), we can easily derive te moments and product moments as below : c 02 c 20 [c 40c 2 [c 04c 2 Eε 2 1 = W 2 n 1 Eε 2 2 = W 2 n 1 Eε 2 3 = n 1 Eε 2 4 = n 1 Eε 2 5 = n 1 E(ε 1 ε 2 ) = W 2 n 1 E(ε 1 ε 3 ) = W c 21 n 1 E(ε 1 ε 4 ) = W c 03 n 1 E(ε 1 ε 5 ) = W c 12 n 1 E(ε 2 ε 3 ) = W c 30 n 1 E(ε 2 ε 4 ) = W c 12 n 1 E(ε 2 ε 5 ) = W c 21 n 1 20 (n 3)(n 1) 1 ] 02 (n 3)(n 1) 1 ] [c 22c (n 1) 1 c 20 c 02 c 2 c 11 c 1 20 c 1 02 c 1 11 c 1 20 c 1 02 c (n 2)(n 1) 1 ] E(ε 3 ε 4 ) = (n c 20 c 02 ) 1 [c 22 c 20 c 02 +2c 2 11 (n 1) 1 ] E(ε 3 ε 5 ) = n 1 [c 31c 1 20 c 1 11 (n 3)(n 1) 1 ] E(ε 4 ε 5 ) = n 1 [c 13c 1 02 c 1 11 (n 3)(n 1) 1 ] Substituting te above expected values in (A.2), we get (4). We ave from (5) v b = k α=1 (y RC,α y RC ) 2 /k Appendix B = Y 2 k α=1 [(1 + ε 1α)(1 + ε 2α ) 1 (1 + ε 1 )(1 + ε 2 ) 1 ] 2 /k...(a.3) = Y 2 k α=1 [ε 1α ε 2α ε 1 + ε 2 + ε 2 2α ε 2 2 ε 1α ε 2α + ε 1 ε 2 + ] 2 /k,...(a.4) were ε 1α =(y st,α Y )/Y ; ε 2α =(x st,α X)/X; ε 1 and ε 2 are as defined earlier. As a first approximation to (A.4), consider v b = Y 2 k α=1 [(ε 1α ε 1 ) (ε 2α ε 2 )] 2 /k = Y 2 k α=1 [(ε 1α ε 1 ) 2 +(ε 2α ε 2 ) 2 2(ε 1α ε 1 )(ε 2α ε 2 )] 2 /k...(a.5)
7 variance estimation for combined ratio estimator 91 Using balanced alf-samples metod of variance estimation, we obtain te following results : k (ε 1α ε 1 ) 2 /k = Ws 2 02 /2Y 2 ; and α=1 k (ε 2α ε 2 ) 2 /k = Ws 2 20 /2X 2 ;...(A.6) α=1 k (ε 1α ε 1 )(ε 2α ε 2 )/k = Ws 2 11 /2XY α=1 From (A.5) and (A.6), we get v b = W 2 (s 02 + R 2 s 20 2Rs 11 )/2 = W 2 [µ 02(1 + ε 4 )+R 2 µ 20 (1 + ε 3 ) 2Rµ 11 (1 + ε 5 )]/2 = V (y RC )+ W 2(µ 02ε 4 + R 2 µ 20 ε 3 2Rµ 11 ε 5 )/2,...(A.7) were ε 3,ε 4 and ε 5 are as defined earlier. Terefore, from (A - 7), we get MSE(v b ) = W 4 [µ2 02 E(ε 4) 2 + R 4 µ 2 20 E(ε 3) 2 +4R 2 µ 2 11 E(ε 5) 2 +2R 2 µ 02 µ 20. E(ε 3 ε 4 ) 4Rµ 02 µ 11 E(ε 4 ε 5 )...(A.8) 4R 3 µ 20 µ 11 E(ε 3 ε 5 )]/4 Substituting te expected values from (A.3) in (A.8), we get (7). Acknowledgement. Te autors are grateful to Dr. R. Karan Sing for elpful discussions. References Cocran, W. G. (1977). Sampling Tecniques, 3rd ed. Wiley, New York. Gupta, V. K. and Nigam, A. K. (1987). Mixed ortogonal arrays for variance estimation wit unequal numbers of primary selection per stratum. Biometrika, 74, Gurney, M. and Jewett, R. S. (1975). Constructing ortogonal replications for variance estimation. J. Amer. Stat. Assoc., 70, Kendall, M. G. and Stuart, A. (1977). Te Advanced Teory of Statistics, 1, 4t ed. Krewski, D. and Rao, J. N. K. (1981). Inference from stratified samples: Properties of linearization, jackknife and balanced repeated replication metods. Ann. Statist., 9,
8 92 sanjay kumar saxena, a.k. nigam and n.d. sukla McCarty, P. J. (1966). Replication: An approac to te analysis of data from complex surveys. Vital and Healt Statistics, Ser. 2, No. 14, U. S. Deptt. of Healt, Education and Welfare, Wasington. (1969). Pseudoreplication: Half-samples. Review of te International Statist. Inst., 37, Plackett, R. L. and Burman, J. P. (1946). Te design of optimum multifactorial experiments. Biometrika, 33, Rao, J. N. K. and Wu, C. F. J. (1985). Inference from stratified samples: Second order analysis of tree metods for non-linear statistics. Jour. Amer. Stat. Assoc., 80, 391, Sukatme, P. V. (1944). Moments and product moments of moment statistics for samples of te finite and infinite population. Sankyā, 6, Sukatme, P. V.; Sukatme, B. V.; Sukatme, S. and Asok, C. (1984). Sampling Teory of Surveys wit Applications. 3rd ed. Iowa Statist. Univ. Press, Ames. Wu, C. F. J. (1985). Variance estimation for te combined ratio and combined regression estimators. J. Royal Statist. Soc. B, 47, No. 1, (1991). Balanced repeated replication based on mixed ortogonal arrays. Biometrika, 78, 1, pp Central Soil and Water Conservation Department of Statistics Researc and Training Institute Lucknow University 218, Kaulagar Road Lucknow Deradun India India
Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India
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