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Bias and Mean Squared Error of Ratio Estimator

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Bias and Mean Squared Error of Ratio Estimator Chapter 5 Ratio and Product Methods of Estimation An important objective in any statistical estimation procedure is to obtain the estimators of parameters of interest with more precision. It is also well understood that incorporation of more information in the estimation procedure yields better estimators, provided the information is valid and proper. Use of such auxiliary information is made through the ratio method of estimation to obtain an improved estimator of the population mean. In ratio method of estimation, auxiliary information on a variable is available, which is linearly related to the variable under study and is utilized to estimate the population mean. Let Y be the variable under study and X be an auxiliary variable which is correlated with Y . The observations xi on X and yi on Y are obtained for each sampling unit. The population mean X of X (or equivalently the population total Xtot ) must be known. For example, xi 's may be the values of ysi ' from - some earlier completed census, - some earlier surveys, - some characteristic on which it is easy to obtain information etc. th th For example, if yi is the quantity of fruits produced in the i plot, then xi can be the area of i plot or the production of fruit in the same plot in the previous year. Let (x11 ,yxy ),( 2 , 2 ),...,( xynn , ) be the random sample of size n on the paired variable (X, Y) drawn, preferably by SRSWOR, from a population of size N. The ratio estimate of the population mean Y is y YXRXˆ ˆ R x N assuming the population mean X is known. The ratio estimator of population total YYtot i is i1 ˆ ytot YXRtot() tot xtot N n n where XXtot i is the population total of X which is assumed to be known, yytot i and xtot x i i1 i1 i1 ˆ are the sample totals of Y and X respectively. The YR()tot can be equivalently expressed as y YXˆ Rtot() x tot ˆ RX tot . Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 1 Y Looking at the structure of ratio estimators, note that the ratio method estimates the relative change tot Xtot yi that occurred after (,xiiy ) were observed. It is clear that if the variation among the values of and is xi y y nearly same for all i = 1,2,...,n then values of tot (or equivalently ) vary little from sample to sample xtot x and the ratio estimate will be of high precision. Bias and mean squared error of ratio estimator: Assume that the random sample (xii ,yi ), 1,2,..., n is drawn by SRSWOR and population mean X is known. Then N n ˆ 1 yi E()YXR N i1 xi n Y (in general). yy2 Moreover, it is difficult to find the exact expression for . So we approximate them and EEand 2 x x proceed as follows: Let yY 0 y (1o )Y Y xX x (1 )X . 11X Since SRSWOR is being followed, so E()0 0 E()1 0 1 E() 22EY (y ) 0 Y 2 1 Nn S 2 YNn2 Y f S 2 Y nY2 f C 2 n Y Nn 1 N S 22Y is the coefficient of variation related to Y. wherefS ,YiY ( YYC ) and NN1 i1 Y Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 2 Similarly, f EC() 22 1 n X 1 EExXyY() [()()] 01 XY 11Nn N ()()X iiXY Y XY Nn N 1 i1 1 f . S XY n XY 1 f SS XY n XY f SS XY nXY f CC n XY S where C X is the coefficient of variation related to X and is the population correlation coefficient X X between X and Y. ˆ Writing YR in terms of ',s we get y YXˆ R x (1 )Y 0 X (1 1 ) X 1 (101 )(1 ) Y 1 Assuming 1 1, the term (11 ) may be expanded as an infinite series and it would be convergent. xX Such an assumption means that 1, i.e., a possible estimate x of the population mean X lies X between 0 and 2 X . This is likely to hold if the variation in x is not large. In order to ensure that variation in x is small, assume that the sample size n is fairly large. With this assumption, ˆ 2 YYR (1011 )(1 ...) 2 Y (1011 10 ...). ˆ So the estimation error of YR is ˆ 2 YYYR (011 10 ...). Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 3 In case, when the sample size is large, then 01and are likely to be small quantities and so the terms involving second and higher powers of 01and would be negligibly small. In such a case ˆ YYYR () 01 and ˆ EY()0.R Y So the ratio estimator is an unbiased estimator of the population mean up to the first order of approximation. If we assume that only terms of 01and involving powers more than two are negligibly small (which is more realistic than assuming that powers more than one are negligibly small), then the estimation error of ˆ YR can be approximated as YYYˆ () 2 R 011 10 ˆ Then the bias of YR is given by ˆ ff2 EY()00RXXy Y Y C C C nn f Bias() Yˆˆ E ( Y Y ) YC ( C C ). RRn XXY upto the second order of approximation. The bias generally decreases as the sample size grows large. ˆ The bias of YR is zero, i.e., ˆ Bias( YR ) 0 2 if E(101 ) 0 Var() x Cov (,) x y or if 0 XXY2 1 X or if2 Var ( x ) Cov ( x , y ) 0 XY Cov(, x y ) or ifVar ( x ) 0 (assuming X 0) R YCovxy(, ) or if R XVarx() which is satisfied when the regression line of Y on X passes through the origin. Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 4 Now, to find the mean squared error, consider ˆˆ2 MSE() YRR E ( Y Y ) 22 2 EY(...)011 10 22 2 EY(2).01 01 Under the assumption 1 1 and the terms of 01and involving powers, more than two are negligibly small, ff2 f MSE() Yˆ Y22 C C 2 C C RXYXYnnn Yf2 CC222 CC n XY Xy up to the second-order of approximation. Efficiency of ratio estimator in comparison to SRSWOR Ratio estimator is a better estimate of Y than sample mean based on SRSWOR if ˆ MSE( YRSRS ) Var ( y ) ff or ifYCCCCYC222 ( 2 ) 22 nnXY XY Y 2 or ifCCCXXY 2 0 1 C or if X . 2 CY Thus ratio estimator is more efficient than the sample mean based on SRSWOR if 1 C X if R 0 2 C Y 1 C and X if R 0. 2 CY It is clear from this expression that the success of ratio estimator depends on how close is the auxiliary information to the variable under study. Upper limit of ratio estimator: Consider Cov(,) Rˆˆˆ x E ( R x ) E ()() R E x y ˆ E xEREx()() x YERX()ˆ . Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 5 Thus YCovRx(,)ˆ ER()ˆ XX Cov(,) Rˆ x R X Bias() Rˆˆ E () R R Cov(,) Rˆ x X ˆˆ Rx, R x X where is the correlation between Rxˆ and ; and are the standard errors of Rxˆ and Rxˆ , Rˆ x respectively. Thus ˆˆx Bias() Rˆ Rx, R X Rˆ x 1. X Rxˆ , assuming X 0. Thus Bias() Rˆ x ˆ X R Bias() Rˆ or C X Rˆ ˆ where CX is the coefficient of variation of X. If CX < 0.1, then the bias in R may be safely regarded as negligible in relation to the standard error of Rˆ. ˆ Alternative form of MSE ()YR Consider NN 2 2 ()()()YRXii YY i YRX i ii11 N 2 ()()(Using)YYii RXX Y RX i1 NN N 22 2 ()YYii R ( X X )2( R X ii XYY )() ii11 i 1 N 1 2222 ()YRXSiiY RS X 2. RS XY N 1 i1 Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 6 ˆ The MSE of YR has already been derived which is now expressed again as follows: f MSE() Yˆ Y22 ( C C 2 2 C C ) RYXXYn 22 f 2 SSYX S XY Y 222 nYXXY 22 fY22 Y Y 22SSYX2 S XY nY X X f 222 SRSRSYX2 XY n N f 2 (YRXii ) nN(1) i1 N Nn 2 (YRXii ) . nN(1) N i1 ˆ Estimate of MSE() YR ˆ Let UYRXiii i, 1,2,.., N then MSE of YR can be expressed as N ˆ f 1 22f MSE() YRiU ( U U )= S nN1 i1 n N 221 whereSUUUi ( ) . N 1 i1 ˆ Based on this, a natural estimator of MSE()YR is f MSE() Yˆ s2 Run n 221 wheresuuui ( ) n 1 i1 1 n 2 ()()yyRxxˆ ii n 1 i1 222ˆˆ sRsRsyx 2, xy y Rˆ . x Based on the expression N ˆ f 2 MSE() YRii ( Y RX ), nN(1) i1 ˆ an estimate of MSE() YR is f n MSE() Yˆ ( y Rxˆ )2 Riinn(1) i1 . f (2).sRsRs222ˆˆ n y xxy Sampling Theory| Chapter 5 | Ratio & Product Methods of Estimation | Shalabh, IIT Kanpur Page 7 Confidence interval of ratio estimator If the sample is large so that the normal approximation is applicable, then the 100(1-)% confidence intervals of Y and R are ˆˆˆˆ YRRRR Z Var(), Y Y Z Var () Y 22 and ˆˆˆˆ R Z Var(), R R Z Var () R 22 respectively where Z is the normal derivate to be chosen for a given value of confidence coefficient 2 (1 ).
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