Exponential Ratio Type Estimators In Stratified Random Sampling
EExponential Ratio–Type Estimators In Stratified Random Sampling †Rajesh Singh, Mukesh Kumar, R. D. Singh, M. K. Chaudhary Department of Statistics, B.H.U., Varanasi (U.P.)-India †Corresponding author
Abstract
Kadilar and Cingi (2003) have introduced a family of estimators using auxiliary information in stratified random sampling. In this paper, we propose the ratio estimator for the estimation of population mean in the stratified random sampling by using the estimators in Bahl and Tuteja (1991) and Kadilar and Cingi (2003). Obtaining the mean square error (MSE) equations of the proposed estimators, we find theoretical conditions that the proposed estimators are more efficient than the other estimators. These theoretical findings are supported by a numerical example.
Key words:
Stratified random sampling, exponential ratio-type estimator, bias, mean squared error. 1.
Introduction
Let a finite population having N distinct and identifiable units be divided into L strata. Let n h be the size of the sample drawn from h th stratum of size N h by using simple random sampling without replacement. Let = n and = N. ∑ = L1h h n ∑ = L1h h N Let y and x be the response and auxiliary variables respectively, assuming values y hi and x hi for the i th unit in the h th stratum. Let the stratum means be h Y = h N1 and ∑ = h N1i hi y h X = h N1 respectively. ∑ = h N 1i hi x A commonly used estimator for Y is the traditional combined ratio estimator defined as – stststCR X.xyy = (1.1) where, st y = ,yw hL1h h ∑ = st x = ,xw hL1h h ∑ = h y = h n1 and ∑ = h n 1i hi y h x = h n1 ,x h n 1i hi ∑ = w h = N h /N and .XwX L1h hh ∑ = = The MSE of CR y , to a first degree of approximation, is given by ]RS2SRS[w)y(MSE yxh2xh1h 22yhh2hCR −+γ≅ ∑ = L (1.2) where ),11( −=γ Nn hhh st XX st YYR == S xh S is the population ratio, is the population variance of a variate of interest in stratum h and is the population variance of auxiliary yh ariate in stratum h and is the population covariance between auxiliary variate and variate of interest in stratum h. yxh S Auxiliary variables are commonly used in survey sampling to improve the precision of estimates. Whenever there is auxiliary information available, the researchers want to utilize it in the method of estimation to obtain the most efficient estimator. In some cases, in addition to mean of auxiliary, various parameters related to auxiliary variable, such as standard deviation, coefficient of variation, skewness, kurtosis, etc. may also be known (Koyncu and Kadilar (2009). In recent years, a number of research papers on ratio type and regression type estimators have appeared, based on different types of transformation. Some of the contributions in this area are due to Sisodiya and Dwivedi (1981), Upadhyaya and Singh (1999), Singh and Tailor (2003), Kadilar and Cingi (2003, 2004, 2006), Singh et.al.(2004), Khoshnevisan et.al. (2007), Singh et.al. (2007) and Singh et.al. (2008). In this article, we study some of these transformations and propose an improved estimator. 2 . Kadilar and Cingi Estimator
Kadilar and Cingi (2003) have suggested following modified estimator ab,ststKC
Xyy = ab,st x (2.1) ( ) ,bw hL1h + ∑ = xa h where ab,st x = ab,st X ( = ) bXaw hLh h + = )x( ∑ . and a, b suitably chosen scalars, these are either functions of the auxiliary variable x such as coefficient of variation C x , coefficient of kurtosis β etc or some other constants. he MSE of the estimator KC y is given by MSE ( KC y ) = h1h 2h W γ ∑ = L [ ] yxhab2xh2ab2yh SR2SRS −+ (2.2) where )X.(X stab,stb,a a.Xw.YR hhhst ∑ = . Bahl and Tuteja (1991) suggested an exponential ratio type estimator ⎥⎦⎤⎢⎣⎡ +−= xX xXexpyy BT (2.3) BT y The estimator is more efficient than the usual ratio estimator under certain conditions. In recent years, many authors such as Singh et. al. (2007), Singh and Vishwakarma (2007) and Gupta and Shabbir (2007) have used Bahl and Tuteja (1991) estimator to propose improved estimators. Following Bahl and Tuteja (1991) and Kadilar and Cingi (2003), we have proposed some exponential ratio type estimators in stratified random sampling. 3.
Proposed estimators
The Bahl and Tuteja (1991) estimator in stratified sampling takes the following form ⎥⎦⎤⎢⎣⎡ +−= stst ststst xX xXexpyt (3.1) The bias and MSE of t, to a first degree of approximation, are given by )S21t(Bias yxh2xh − S8R3(wX1) hL1h 2hst γ= ∑ = (3.2) ]S4RRSS[w)t(MSE +−γ= ∑ = (3.3) .1 Sisodia- Dwivedi estimator
When the population coefficient of variation C x is known, Sisodia and Dwivedi (1981) suggested a modified ratio estimator for Y as- xxSD Cx CXyy ++= (3.4) In stratified random sampling, using this transformation the estimator t will take the form ⎥⎥⎥⎦⎢⎢⎢⎣ +++= ∑ ∑ = =
L1h L1h xhhhxhhhSD )Cx(w)CX(wexpyt ⎥⎤⎢⎡ +−+ ∑ ∑ = = L L
1h xhhhxhhh )Cx(w)CX(w ⎥⎦⎤⎢⎣⎡ +−
SDSD SDSDst xX xXexpy = (3.5) where = SD X )CX(w xhL1h hh + ∑ = and = SD x ).Cx(w xhL1h hh + ∑ = The bias and MSE of t
SD, are respectively given by – ) yxh S21S8R(X − = w1 SDhL 2h θγ ∑ Bias (t SD ) = (3.6) MSE (t SD ) = ]S4RSRS[w +−θ = ∑ (3.7) where ∑ = +
1h xhhhSD )CX(w ∑ = == L L 1h hhstSD
YwXYR
SDstSD XX = and θ . .2 Singh-Kakran estimator
Motivated by Sisodiya and Dwivedi (1981), Singh and Kakran (1993) suggested another ratio-type estimator for estimating Y as- )x(xyy β+= )x(X β+ (3.8) Using (3.8), the estimator t at (3.1) will take the following form in stratified random sampling- ⎥⎥⎦⎢⎢⎣ β++β+ ∑ ∑ = = L1h L1h h2hhh2hh ))x(x(w))x(X(w ⎥⎥⎤⎢⎢⎡ β+−β+= ∑ ∑ = =
L1h L1h h2hhh2hhSK ))x(x(w))x(X(wexpyt ⎥⎦⎤⎢⎣⎡ +−=
SKSK SKSKst xX xXexpy (3.9) where, = SK x ))x(x(w h2L hh β+ ∑ = and = SK X )).x(X(w h2L1h hh β+ ∑ = Bias and MSE of t SK , are respectively given by )S21 yxhxh − S8R(wX1 θγ ∑ = Bias (t SK ) = (3.10) MSE (t SK ) = ]S4RSRS[w +−γ ∑ = (3.11) .XX SKstSK =θ ∑ β+ = L h2hh stSK ))x(X(w Y where and = R .3 Upadhyaya-Singh estimator
Upadhyaya and Singh (1999) considered both coefficients of variation and kurtosis in their ratio type estimator as x2 x21US
C)x(x C)x(Xyy +β +β= (3.12) We adopt this modification in the estimator t proposed at (3.1) ⎟⎟⎟⎠⎜⎜⎜⎝ +β= ∑ = L1h xhh2hhst1US
C)x(x(wexpyt ⎟⎟⎞⎜⎜⎛ +β ∑ = xhh2L1h hh )C)x(X(w (3.13) t US1 at (3.13) can be re-written as ⎥⎦⎤⎢⎣⎡ +−= xX xXexpyt (3.14) ) xhL C)x(X(wX where
1h h2hh1US +β= ∑ = ).Cx(wx xhL 1h h2hh1US +β= ∑ = Bias and MSE of t
US1 , to first degree of approximation, are respectively given by ( ) ∑ ⎜⎛θγ= L 1US1USh2h1US
Rw1tBias = ⎟⎠⎞⎝ −
1h yxh2xhst
S21S8X (3.15) ( ) ⎥⎥⎦⎤⎢⎢⎣⎡ +−γ= ∑ = S4RSRSwtMSE (3.16) where ( ) ∑ = +β= L1h stxhh2hh1us
XC)x(XwR ∑ β h2hhst )x(.Xw.Y .X )x(.Xw β= ∑ θ and padhyaya and Singh (1999) proposed another estimator by changing the place of coefficient of kurtosis and coefficient of variation as )x(Cx )x(CXyy
2x 2x2US β+ β+= (3.17) Incorporating this modification in the proposed estimator t, we have- ⎥⎤⎢⎡ −= xXexpyt ⎦⎣ + xX (3.18) where ∑ = β+= L1h h2xhhh2us ))x(CX(wx and ∑ = β+= L1h h2xhhh2us ))x(CX(wX Bias and MSE of t
US2, are respectively given by – )S21S8R(wX1)t(Bias yxh2xh2US2UShL1h 2hst2US −θγ= ∑ = (3.19) ]S4RSRS[w)t(MSE +−γ= ∑ = (3.20) ( ) ∑ ∑ β+= L sth2xhhh xhhhst2US
X.)x(CXw C.Xw.YR where = .X CXw ∑ =θ and 3.4 . G.N. Singh Estimator ),x( β we propose following two estimators. Following Singh (2001), using values of and ⎥⎦⎤⎣= t ⎢⎡ +− GNS1GNSst xX xXexpy (3.21) here = X )X(w xhL1h hh σ+ ∑ = and = x )x(w xhL1h hh σ+ ∑ = The Bias and MSE of to a first degree of approximation, are respectively given by – t )S21S8R(wX1)t(Bias yxh2xh1GNS1GNSh θγ L1h 2hst1GNS − ∑ = (3.22) ]S4RSRS[ w)t(MSE +−γ= ∑ = (3.23) where ( ) ,Xw YR L1h xhhh st1GNS ∑ = σ+= .XX =θ Similarly, we propose another estimator ⎥⎦⎢⎣ + xX ⎤⎡ −= xXexpyt (3.24) where, ),)x(X(wX xhL1h h2hh2GNS σ+β= ∑ = )x(wx xhL 1h h2hh2GNS σ+β= ∑ = The Bias and MSE of to a first degree of approximation are respectively given by – t )S21S83R(X1)t(Bias yxh2xhGNS22GNSL 2st2GNS −θ= w h1h h γ ∑ = (3.25) ]S4RSRS[w)t(MSE +−γ= ∑ = (3.26) ( ) ,)x(Xw YR L xhh2hh st1GNS ∑ σ+β= X )x(.Xw β=θ ∑ = where, = . Improved Estimator
Motivated by Singh et. al. (2008), we propose a new family of estimators given by- α ⎤⎡ − ab,stab,st xX ⎥⎥⎦⎢⎢⎣ += ab,stab,ststMK xXexpyt (4.1) where a and b are suitably chosen scalars and α is a constant. The bias and MSE of MK t ( ) up to first order of approximation, are respectively given by ( ) ∑ = ⎟⎠⎞⎜⎝⎛ −+αα= L1h yxh2xhabab2stMK
S21S8 R2X1tBias θγ hh w (4.2) ( ) ⎥⎥⎦⎤⎢⎢⎣⎡ α+α−γ= ∑ = S4RSRSwtMSE )t(MSE (4.3) The is minimized for the optimal value of MK α given by- ∑ =
1i hab wR ∑ = γ=α L1i yxhh2h Sw γ L 2xhh2 S2 M Putting this value of α in equation (4.3), we get the minimum MSE of the estimator K t ρ−γ )1(S c as- ∑ = = L1i.minMK w)t(MSE (4.4) where, ρ is combined correlation coefficient in stratified sampling across all strata. It is calculated as .SSSw ∑ = ⎟⎟⎠⎞⎝ ργ=ρ M ⎜⎜⎛ wSw L11 L1i h2h2yhh2h ∑ ∑ = = γγ We note here that min MSE of K t is independent of a and b. therefore, we conclude that it same for any (all) values of a and b. . Efficiency comparisons
First we compare the efficiency of the estimator t at (3.1) with estimator t
SD.
We have MSE(t SD ) < MSE(t) ∑∑ ⎥⎥⎤⎢⎢⎡ +γ<⎥⎥⎤⎢⎢⎡ +−γ L 2xh2yxh2yhh2hL 2xh2SDyxhSD2yhh2h
S4RRSSwS4RSRSw == ⎦⎣⎦⎣ (5.1) ∑ ⎥⎤⎢⎡ +−γ L 22SD2
SRSRw < = ⎥⎦⎢⎣
1h xhyxhSDhh ∑ = ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡ +−γ L1h 2xh2yxhh2h
S4RRSw γ w Let and ∑ = γ= L1h yxhh2h
SwA ∑ = = L1h B SThen equation (5.1) can be re-written as-
B.R B.RA + < − +− -A (R SD – R) + B/4(R SD - R) (R SD + R) < 0 (5.2) From (5.2), we get two conditions ∑ ⎢⎡−θ L 2
SRw = ⎥⎦⎤⎣ +
1h 2xh2SDyxhSDhh
S4R < ∑ = ⎥⎥⎦⎤⎢⎢⎣⎡ +−θ L1h 2xh2yxhh2h
S4RRSw ∑ = θ=
1h yxhh2h
SwA ∑ = θ= L1h 2xhh2h
SwB (i) When (R SD -R) (R SD +R) > 0 B < 4A /(R SD +R) (5.3) (ii) When (R SD – R) (R SD + R) < 0 B > 4A / (R SD +R) (5.4) where and L When either of these conditions is satisfied, estimator t SD will be more efficient than the estimator t. he same conditions also holds true for the estimators t SK, t US1, t US2, t GNS1 and t
GNS2 if we replace R SD by R SK, R US1 , R
US2, R GNS1 and R
GNS2 respectively in conditions (i) and (ii). Next we compare the efficiencies of t opt with the other proposed estimators. ∑∑ == ⎥⎥⎦⎤⎢⎢⎣⎡ −+γ<ρ−γ < L1i yxab2x2ab2hh2hL1i 2c2yhh2h ab.minMK
SRS4RSw)1(Sw )t(MSE)t(MSE (5.5) On putting the value of and rearranging the terms we get c ρ >⎟⎟⎠⎞⎜⎜⎝⎛ γ−γ ∑∑ == (5.6) This is always true. Hence the estimator MK t under optimum condition will be more efficient than other proposed estimators in all conditions. 6. Data description and results
For empirical study we use the data set earlier used by Kadilar and Cingi (2003). Y is apple production amount in 854 villages of turkey in 1999, and x is the numbers of apple trees in 854 villages of turkey in 1999. The data are stratified by the region of turkey from each stratum, and villages are selected randomly using the Neyman allocation as hL 1h h hhh
SNn ∑ = = SN able 6.1: Data Statistics N =106 N =106 N =94 N =171 N =204 N =173 n =9 n =17 n =38 n =67 n =7 n =2 = = = = = = = = = = = = =β =β .26 =β =β x =β =β C x1 =2.02 C x2 =2.10 C x3 =2.22 C x4 =3.84 C x5 =1.72 C x6 =1.91 C y1 =4.18 C y2 =5.22 C y3 =3.19 C y4 =5.13 C y5 =2.47 C y6 =2.34 S x1 =49189 S x2 =57461 S x3 =160757 S x4 =285603 S x5 =45403 S x6 =18794 S y1 =6425 S y2 =11552 S y3 =29907 S y4 =28643 S y5 =2390 S y6 =946 =ρ = ρ =ρ =ρ =ρ =ρ =γ =γ =γ =γ =γ =γ = = = = = = N=854 n=140 x =β C x =3.85 C y =5.84 S x =144794 S y =17106 =ρ = = R =0.07793 R SD =0.07792 R SK =0.07784 R US1 =.07789 R
US2 =0.07786 R
GNS1 = 0.06632 R
GNS2 =0.07765 able 6.2 : Estimators with their MSE values
Estimators MSE values t t SD t SK t US1 t US2 t GNS1 t GNS2 t MK(opt)
From Table 6.2, we conclude that the estimator t MK has the minimum MSE and hence it is most efficient among the discussed estimators. 7. Conclusion
In the present paper we have examined the properties of exponential ratio type estimators in stratified random sampling. We have derived the MSE of the proposed estimators and also that of some modified estimators and compared their efficiencies theoretically and empirically. eferences
Bahl, S. and Tuteja, R.K. (1991): Ratio and product type exponential estimator. Infrm. and Optim.Sci.,XII,I,159-163. Kadilar, C. and Cingi, H. (2003): Ratio estimators in stratified random sampling. Biometrical Journal 45 (2003) 2, 218-225
Kadilar, C. and Cingi, H. (2004). Ratio estimators in simple random sampling. Applied Mathematics and Computation, 151, 893-902. Kadilar, C. and Cingi, H. (2006): New ratio estimators using correlation coefficient. InterStat, 1-11. Khoshnevisan, M., Singh, R., Chauhan, P., Sawan, N., Smarandache, F. (2007) : A general family of estimators for estimating population mean using known value of some population parameter(s). Far East J. Theor. Statist ....