REVISTA INVESTIGACION OPERACIONAL VOL. 35, NO. 1, 49-57, 2014 ON AN IMPROVED RATIO TYPE ESTIMATOR OF FINITE POPULATION MEAN IN SAMPLE SURVEYS A K P C Swain Former Professor of Statistics, Utkal University, Bhubaneswar-751004, India ABSTRACT In this paper an alternative ratio type exponential estimator is suggested and is compared with Bahl and Tuteja’s ratio type exponential estimator and classical ratio estimator as regards bias and mean square error with large sample approximations both theoretically and with numerical illustration. KEYWORDS: Simple random sampling, ratio type estimators, Bias, Mean square error. MSC: 62D05 RESUMEN En este trabajo se sugiere un estimador del tipo razón- exponencial alternativo y se compara con el de Bahl y Tuteja el de razón clásico, considerando sesgo y error cuadrático medio, bajo aproximaciones basadas en muestras grandes y haciendo ilustraciones teóricas y numéricas. 1. INTRODUCTION: The use of auxiliary information dates back to year 1934 , when Neyman used it for stratification of the finite population. Cochran(1940) used auxiliary information in estimation procedure and proposed ratio method of estimation to provide more efficient estimator of the population mean or total compared to the simple mean per unit estimator under certain conditions when the auxiliary variable has positive correlation with the study variable in question. Let U = (U1, U2… UN) be the finite population of size N. To each unit Ui (i=1, 2…, N) in the population paired values (yi, xi) corresponding to study variable y and an auxiliary variable x, correlated with y are attached. Now, define the population means of the study variable y and auxiliary variable x as 1 N 1 N Y yi , X xi N i 1 N i 1 Thus, the population ratio is defined as R Y X Further, define the finite population variances of y and x and their covariance as S y2 1 N yi Y 2 , N 1 i 1 2 N 1 N xi X , and S yx 1 yi Y xi X , respectively. N 1 i 1 N 1 i 1 Thus, the squared coefficients of variation of y and x and their coefficient of covariation are respectively defined by C20, C02 and C11 ,where S x2 Crs 1 N ( yi Y )r ( xi X ) s , r, s 0,1,2,... N i 1 Define as the correlation coefficient between y and x in the population. A simple random sample ‘s’ of size n is selected from U without replacement and the values (yi, xi), i=1, 2 … n are observed on the sampled units. Assume that y and x are positively correlated. The simple mean y is an unbiased estimator of the population mean Y and variance of y is given by Var ( y ) Y 2C20 The classical ratio estimator of the population mean Y ,using auxiliary information on 49 x is given by yR where y X, x (1.1) y and x are sample means of y and x respectively, defined by yR The ratio estimator to O(1/ n) envisages advance knowledge of X y 1 n 1 n yi and x xi . n i 1 n i 1 . The bias( B ) and mean square error ( MSE )of yR are given by(see Sukhatme and Sukhatme,1970) B( yR ) Y (C02 C11) (1.2) MSE ( yR ) Y 2 (C20 C02 2C11) (1.3) where N n ( N 1)n The ratio estimator In large samples or if 1 2 Crs , yR yR 1 N ( yi Y )r ( xi X )s , r , s 0,1,2 N i 1 is a biased estimator and the bias decreases with increase in sample size. is more efficient than the simple mean per unit estimator when y if k C20 1 C02 2 C20 C02 . Bahl and Tuteja(1991) have suggested a ratio type exponential estimator given by yBTR y exp( To O(1/ n) X x ) X x ,the bias and mean square error of (1.4) yBTR are given by 3 1 B( yBTR ) Y ( C02 C11) 8 2 1 MSE ( yBTR ) Y 2 (C20 C02 C11) 4 1 3 yBTR is more efficient than yR and y if k 4 4 (1.5) (1.6) , (Bahl and Tuteja,1991): O(1 / n2 ) ,Bahl and Tuteja’s(1991) ratio type exponential estimator is more efficient than the classical ratio estimator y R when y and x follow bivariate normal distribution , coefficients of variation of y and x are equal and 3 0 . 4 Upadhyaya,Singh,Chatterjee and Yadav (2011) have shown that to In the following we suggest a ratio type estimator with square root transformation and compare it with the BahlTuteja’s (1991) ratio type exponential estimator and classical ratio estimator both theoretically and with the help of some natural and artificial populations. 2. A RATIO TYPE ESTIMATOR USING SQUARE ROOT TRANSFORMATION Define the ratio type estimator using square root transformation as 50 ySQR y ( 2.1. Bias and Mean square error of Write y Y (1 e0 ) and X 1/2 ) x (2.1) ySQR x X (1 e1) with E (e0 ) E (e1) 0 , V (e0 ) C20 , V (e1) C02 and Cov(eo , e1) C11 . Expanding ySQR in power series under the assumption that e1 1 for all possible samples and retaining terms up to fourth degree in e0 and e1 ,we have 1 3 15 105 4 1 3 15 ySQR Y Y (e0 e1 e12 e13 e1 e0e1 e0e12 e0e13 ) 2 8 48 384 2 8 48 1 ( ySQR Y )2 Y 2 (e02 e12 e0e1) e02e12 + 4 5 33 3 87 4 e0e12 e02e1 e0e13 e13 e1 4 24 8 192 (2.2) (2.3) Thus, 1 3 15 105 B( ySQR ) Y E (e0 ) E (e1) E (e12 ) E (e13 ) E (e14 ) 2 8 48 384 1 3 15 E (e0e1) E (e0e12 ) E (e0e13 ) 2 8 48 1 MSE ( ySQR ) E ( ySQR Y )2 Y 2 E (e02 ) E (e12 ) E (e0e1) 4 5 2 33 3 87 2 2 2 2 3 3 4 E ( e e ) e e E ( e e ) E ( e e ) E ( e ) E ( e ) +Y 0 1 0 1 0 1 0 1 1 1 4 24 8 192 (2.4) (2.5) Now, E (e14 ) N n N 2 N 6nN 6n2 3N ( N n 1)(n 1) 2 C C02 04 ( N 1)( N 2)( N 3) n3 ( N 1)( N 2)( N 3) E (e0e13 ) N n N 2 N 6nN 6n2 3N ( N n 1)(n 1) C C C 13 11 02 ( N 1)( N 2)( N 3) n3 ( N 1)( N 2)( N 3) E (e02e12 ) N n N 2 N 6nN 6n 2 N ( N n 1)(n 1) 2 C ( C C 2 C 22 20 02 11 ( N 1)( N 2)( N 3) n3 ( N 1)( N 2)( N 3) ( N n)( N 2n) C03 ( N 1)( N 2)n2 ( N n)( N 2n) E (e0e12 ) C12 ( N 1)( N 2)n2 E (e13 ) E (e02e1 ) ( N n)( N 2n) C21 ( N 1)( N 2)n2 where Crs 1 N ( yi Y )r ( xi X ) s / Y r X s N i 1 , (r , s 1,2,3,4) . 51 (See Sukhatme ,Sukhatme and Asok (1984)) Hence, N n 1 3 1 ( C C ) 02 11 N 1 n 8 2 N n 1A 3 15 B 105 15 Y ( C12 C03 ) 2 ( C04 C13 ) N 1 n n 8 48 48 n 384 3D 105 2 15 2( C02 C11C02 ) 48 n 384 N n 1 1 1 5 3 MSE ( ySQR ) Y 2 (C20 C02 C11) A( C12 C21 C03 ) N 1 n 4 n 4 8 B( ySQR ) Y Y 2 (2.6) (2.7) N n 1 1 87 33 1 33 87 2 2 B( C04 C22 C13 ) 2 D(C20C02 2C11 C11C02 C02 ) 2 N 1 n n 192 24 8 64 n where N 2 N 6nN 6n2 B ( N 2)( N 3) 2.2. Bias and Mean square error of yBTR N 2n A N 2 Expanding yBTR D N ( N n 1)(n 1) ( N 2)( N 3) in power series with the same assumptions used in expanding and including degree four in e0 ySQR and retaining terms up to and e1 ,we have e1 3 2 13 3 73 4 e0e1 3 2 13 3 e1 e1 e1 e0e1 e0e1 ) 2 8 48 384 2 8 48 5 3 79 4 2 2 31 3 ( yBTR Y )2 Y 2 (e02 1 e12 e0e1 e0e12 e02e1 e13 e1 e0 e1 e0e1 ) 4 4 8 192 24 N n 1 3 1 ( C C11 ) B( yBTR ) Y 02 N 1 n 8 2 yBTR Y Y (e0 Y N n 1A 3 13 B 73 13 ( C C ) ( C C13 ) 12 03 04 N 1 n n 8 48 48 n2 384 D 73 2 13 2( C02 C02C11) 16 n 128 MSE ( yBTR ) E ( yBTR Y ) Y 2 2 E (e02 1 2 e1 4 (2.8) e0e1) N n 1 1 1 5 3 5 3 79 4 2 2 31 3 Y 2 E ( e0e12 e02e1 e13 e1 e0 e1 e0e1 ) Y 2 (C20 C02 C11 ) A( C12 C21 C03 ) + 4 8 192 24 N 1 n 4 n 4 8 1 79 31 1 31 79 2 2 B( C04 C22 C13 ) 2 D(C20C02 2C11 C11C02 C02 ) 2 192 24 8 64 n n 2.3. Bias and Mean Square Error of yR 52 (2.9)) We find (see Sukhatme , Sukhatme and Asok,1984) to second order approximation B( yR ) Y N n 1 N n 1 1 1 1 2 (C02 C11) Y A (C12 C03 ) B 2 (C04 C13 ) 3D 2 (C02 C02C11) N 1 n N 1 n n n n MSE ( yR ) Y 2 Y 2 (2.10) N n 1 2 (C20 C02 2C11) A(2C12 C21 C03 ) N 1 n n N n 1 3 3 2 2 B(C04 C22 2C13 ) 2 D(C20C02 2C11 6C11C02 3C02 ) 2 N 1 n n n Under the assumption of bivariate normality of ( y, x) C20 C02 it may be shown that MSE ( yR ) Y 2 C02 C 2(1 ) 02 (12 18 6 2 ) n n (2.11) and equality of coefficients variation of y and x ,i.e. (2.12) C 5 C 143 31 MSE ( yBTR ) Y 02 ( ) 02 ( 2 2 ) n 4 n 64 8 (2.13) C 5 C 151 33 MSE ( ySQR ) Y 02 ( ) 02 ( 2 2 ) n 4 n 64 8 (2.14) 2 2 3. COMPARISON OF EFFICIENCY (i) Considering terms up to O(1 / n ) , MSE ( yR ) Y 2 (C20 C02 2C11) 1 MSE ( ySQR ) MSE ( yBTR ) Y 2 (C20 C02 C11) 4 B = B( yR ) Y (C02 C11) B1 = B( ySQR ) B( yBTR ) Y ( 3 C02 1 C11) 8 2 Now, MSE ( yR ) MSE ( ySQR ) MSE ( yR ) MSE ( yBTR ) 3 Y 2 ( C02 C11 ) 4 Thus, to O(1/ n) , yBTR and ySQR are equally efficient. They are more efficient than yR if C11 3 C02 4 or k 3 , 4 and more efficient than Hence , ySQR Further, to and y if yBTR O(1/ n) k 1 . 4 are more efficient than both the biases of ySQR and yBTR yR and y if . are equal and further they are less biased than is either greater than 5/4 or less than 11/12.. 2 (ii) To O(1 / n ) ,for large N, 53 yR if k 87 79 33 31 4 2 MSE ( ySQR ) MSE( yBTR ) ( ) E ( e ) ( ) E (e0e13 ) Y 2 1 ( 1 C02 C11C02 ) 1 192 192 24 24 4n 2 2 Hence, ySQR yBTR to O(1 / n2 ) will be more efficient than if 1 C 2 C C 0 02 11 02 2 C11 1 C02 2 or or 1 when C20 C02 2 Under bivariate normality of between y and x equal to y and x with the assumption that C20 C02 and the correlation coefficient we have 1 n 3 4 MSE ( yBTR ) MSE ( yR ) = Y 2C02 ( ) (see Upadhyaya et al,2011 and correct the coefficient of 1 n as -113/8 instead of -103/8 ) 3 4 MSE ( ySQR ) MSE ( yR ) = Y C02 ( ) 2 C02 625 113 ( 4 2 ) n 64 8 C02 617 111 ( 4 2 ) n 64 8 3 , both MSE ( yBTR ) MSE ( yR ) and MSE ( ySQR ) MSE ( yR ) are negative.Hence, 4 O(1 / n2 ) the sufficient condition that both yBTR and ySQR are more efficient than yR is that If 0 Further , ySQR will be more efficient than yBTR ySQR will be more efficient than yBTR 3 4 if Hence, to and 1 2 yR if y and 1 3 2 4 To sum up under the assumptions of bivariate normality of y and x ,the preferred estimators for ranges of x and the equality of coefficients of variation of re shown below in Table-1. Table-1 Comparison of Estimators Preferred estimator 0 1 4 y yBTR 1 1 4 2 1 3 2 4 ySQR 54 yR 3 1 4 4. NUMERICAL ILLUSTRATION: Illustration 1 Consider 12 natural populations described in Table-2 to compare the ratio type estimator with square root transformation estimator with classical ratio estimator as regards bias and mean square error.Bias and mean square error have been computed excepting the common constants(Table-3): Table-2 Natural Populations Pop no. Description N 1 Sampford (1962) 17 Acreage under oats in 1957 2 Singh and Chaudhary(1986) 16 3 4 Konijn(1973) Murthy(1967) 5 y yx Cx Cy Acreage of crops and grass in 1947 0.4 0.22 0.45 Area under wheat !979-80 Total cultivated area during 1978-79 0.96 0.74 0.69 16 16 Food expenditure Area under winter paddy(in acres) Total expenditure Geographical area (in acres) 0.95 0.29 0.08 0.09 0.11 0.47 Singh and Chaudhary(1986) 17 No. of milch animals in survey1977-78 No. of milch animals in census1976 0.72 0.02 0.01 6 Murthy(1967) 16 Output for factories(000Rs) Fixed capital(000Rs.) 0.84 0.15 0.09 7 Panse and Sukhatme(1967) 16 Progeny mean(mm) Parental plant value(mm) 0.68 0.07 0.06 8 Panse and Sukhatme(1967) Singh and Chaudhary((1986) Panse and Sukhatme(1967) 20 Parental plot mean(mm) 0.0.56 0.07 0.04 15 Area under wheat in 1973 0.23 0.82 0.67 25 Parental plot mean(mm) Parental plant value(mm) Area under wheat in 1971 Parental plant value(mm) 0.54 0.07 0.03 11 Panse and Sukhatme(1967) 20 Progeny mean(mm) Parental plant value(mm) 0.68 0.07 0.05 12 Swain(2003) 19 No. of milch cows in 1957 No. of milch cows census 1956 0.72 1.14 1.12 9 10 x Table 3 Comparison of Bias and Mean square Error Pop. no Bias( ySQR ) Y 1 2 3 4 5 6 7 0.34 0.07 0.28 0.38 0.20 0.12 0.09 Bias( yR Y 0.18 0.11 0.31 0.51 0.64 0.50 0.42 MSE ( ySQR ) Y 2 0.174912 0.125636 0.005316 0.210694 0.000056 0.002475 0.001983 55 MSE ( yR ) Y 2 MSE ( yR ) 100 MSE ( ySQR ) 0.171524 0.048972 0.001732 0.204538 0.000212 0.0081 0.002816 98.06 38.98 32.58 97.08 378.57 327.27 142.01 8 9 10 11 12 0.04 0.28 0.26 0.13 0.02 0.33 0.81 0.77 0.51 0.37 0.001257 0.489244 0.000998 0.001324 0.654131 0.003364 0.865788 0.003546 0.002598 0.703661 267.62 176.96 355.31 196.22 107.57 Comments : For the populations under consideration,the ratio type estimator with square root transformation is less biased than the classical ratio estimator.Further, For populations 5-12 , the ratio type estimator with square root transformation is more efficient than the classical ratio estimator in the sense of having lesser mean square error. Illustration 2 Consider a hypothetical population of size y: x N =5 as follows: 3 5 8 4 5 : 5 2 6 4 3 For a simple random sample (without replacement) of size ySQR and yBTR n =2,the exact biases and mean square errors of yR , are given in Table-4. Table 4: Comparison of Biases and Mean square errors Estimators Absolute Bias M. S .E 0 1.05 Relative Efficiency 100 y yR ySQR 0.1903 1.8667 56.25 0.0592 0.9996 105.04 yBTR 0.0569 0.9922 105.82 Comments: As per Illustration 2 (i)The ratio type estimator with square root transformation and ratio type exponential estimator have nearly equal bias and efficiency and further they are less biased and more efficient than the classical ratio estimator.. (ii) The classical ratio estimator is less efficient than the simple mean per unit estimator y ,where as the ratio type estimator with square root transformation and the ratio type exponential estimator are more efficient than y . 5. CONCLUSIONS: (i)To O(1/ n) ySQR and yBTR are equally efficient and more efficient than both 1 3 k . To same order of approximation both ySQR 4 4 11 5 greater than or less than . 12 4 (ii) Under the assumption of bivariate normality of and x ,to O(1 / n2 ) ySQR and yBTR ( y, x) is more efficient than both y R and 1 3 . However , for the values of 2 4 yBTR . in the range are less biased than yR if and k y if is either and equality of coefficients of variation of yBTR y for 3 1, yR 4 56 yR is to be preferred over y , ySQR and (iii) Numerical illustration with the help of twelve natural populations shows that there might arise situations when ySQR is less biased and more efficient than y R under large sample approximations. Also, for the artificial population under consideration ySQR happens to be less biased and more efficient than y R . However , the difference between ySQR and yBTR as regards bias and mean square error is only marginal. RECEIVED APRIL, 2013 REVISED NOVEMBER, 2014 REFERENCES 1 BAHL,S. and TUTEJA,R.K. (1991): Ratio and Product Type Exponential Estimator. ,Information and Optimization Sciences,12,159-163. 2 COCHRAN,W.G.(1940):The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce. Journ.Ag.Sc.,30,262-275. 3 KONIJN,H.S.(1973): Statistical Theory of Sample Survey Design and Analysis. North Holland Publishing Co. Amsterdam. 4 MURTHY,M.N.(1967): Sampling: Theory and Methods. Statistical Publ. Society,ISI, Calcutta. 5 NEYMAN, J. (1934):On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection. J. Royal Stat. Soc.,97,558-625. 6 PANSE,V.G. and SUKHATME,P.V.(1967): Statistical Methods for Agricultural Workers. Indian Council of Agril.Res, New Delhi 7 SAMPFORD,M.R.(1962):An Introduction to Sampling Theory. Oliver and Boyd. London. 8 SINGH,D and CHAUDHARY,F.S.(1986):Theory and Analysis of Sample Survey Designs, First Edition,Wiley Eastern Ltd. Calcutta. 9 SUKHATME, P.V. and SUKHATME , B.V.(1970): Sampling Theory of Surveys with Applications, Asia Publishing House, Calcutta. 10 SUKHATME, P.V., SUKHATME, B.V. and ASOK, C.(1984):Sampling Theory of Surveys with Applications,Indian Society of Agricultural Statistics, New Delhi. 11 SWAIN, A.K.P.C.(2003): Finite Population Sampling: Theory and Methods. South Asia Publishers, New Delhi. 12 UPADHYAYA,L.N.,SINGH,H.P.,CHATTERJEE,S. and YADAV,R.(2011): Improved Ratio and Product Exponential Type Estimator, Journ. Stat.Theo. and Pract. 5, 285-293. 57
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