Exponential 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.

th Let nh be the size of the sample drawn from h stratum of size Nh by using simple random sampling without replacement. Let

L L ∑ n h = n and ∑ N h = N. =1h =1h Let y and x be the response and auxiliary variables respectively, assuming values yhi

th th and xhi for the i unit in the h stratum.

Let the stratum means be

N 1 h 1 N h Y y X h = ∑ hi and h = ∑ x hi respectively. N h =1i N h =1i

A commonly used estimator for Y is the traditional combined ratio estimator defined as –

yst yCR = X. st xst (1.1) L L where, yst = ∑ h h ,yw xst = ∑ h h ,xw =1h =1h

nh n h 1 y 1 ,x y h = ∑ hi and x h = ∑ hi n h =1i n h =1i

L wh = Nh/N and = ∑ hh .XwX =1h

The MSE of yCR , to a first degree of approximation, is given by

L 2 22 2 CR ∑ h yhh xh −+γ≅ yxh ]RS2SRS[w)y(MSE (1.2) =1h

11 Y Yst 2 where h ( −=γ ), R == is the population ratio, S yh is the population Nn hh XX st

2 variance of a variate of interest in stratum h and S xh is the population variance of auxiliary variate in stratum h and S yxh is the population covariance between auxiliary variate and variate of interest in stratum h.

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

yst yKC = X ab,st (2.1) x ab,st

L L x w xa + ,b X where ab,st = ∑ h ( h ) ab,st = ∑ h ( h + bXaw ) =1h h=1 . and a, b suitably chosen scalars, these are either functions of the auxiliary variable x such as

coefficient of variation Cx, coefficient of kurtosis β2 )x( etc or some other constants. The MSE of the estimator is given by yKC

L MSE ( y ) = W 2 γ 2 + 2 2 − SR2SRS KC ∑ h h [ yh ab xh yxhab ] (2.2) =1h

∑ a.Xw.Y hhhst where R b,a = . stab,st )X.(X

Bahl and Tuteja (1991) suggested an exponential ratio type estimator

⎡ − xX ⎤ BT = expyy ⎢ ⎥ ⎣ + xX ⎦ (2.3)

The estimator yBT 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

⎡ − xX stst ⎤ = st expyt ⎢ ⎥ + xX ⎣ stst ⎦ (3.1)

The bias and MSE of t, to a first degree of approximation, are given by

L 1 2 R3 2 1 t(Bias ) = ∑ h γh (w Sxh − yxh )S (3.2) Xst =1h 8 2 L 2 2 2 R 2 ∑ h h yh RSS[w)t(MSE yxh +−γ= xh ]S (3.3) =1h 4 3.1 Sisodia- Dwivedi estimator

When the population coefficient of variation Cx is known, Sisodia and Dwivedi (1981) suggested a modified ratio estimator for Y as-

+ CX = yy x SD + Cx x (3.4)

In stratified random sampling, using this transformation the estimator t will take the form

⎡ L L ⎤ ⎢∑∑xhhh −+ + xhhh )Cx(w)CX(w ⎥ ==1h 1h SD = expyt ⎢ L L ⎥ ⎢ ++ + )Cx(w)CX(w ⎥ ⎢∑∑xhhh xhhh ⎥ ⎣ ==1h 1h ⎦

⎡ SD − xX SD ⎤ = st expy ⎢ ⎥ (3.5) ⎣ SD + xX SD ⎦

L L where X = and x = + ).Cx(w SD ∑ hh + xh )CX(w SD ∑ hh xh =1h =1h

The bias and MSE of tSD, are respectively given by –

L 1 2 RSD 2 1 Bias (tSD) = ∑ w h θγ SDh ( Sxh − Syxh ) (3.6) Xst =1h 8 2

L 2 2 2 RSD 2 MSE (tSD) = ∑ h yhh SRS[w yxhSD +−θ xh ]S (3.7) =1h 4

L ∑ Yw hh Y Xst where R st == =1h and θ = . SD L SD X SD XSD ∑ + xhhh )CX(w =1h

3.2 Singh-Kakran estimator

Motivated by Sisodiya and Dwivedi (1981), Singh and Kakran (1993) suggested another ratio- type estimator for estimating Y as-

β+ )x(X = yy 2 SK β+ )x(x 2 (3.8)

Using (3.8), the estimator t at (3.1) will take the following form in stratified random sampling-

⎡ L L ⎤ ⎢ ∑∑h2hh −β+ β+ h2hh ))x(x(w))x(X(w ⎥ = expyt ⎢ ==1h 1h ⎥ SK ⎢ L L ⎥ ⎢ ∑∑h2hh +β+ β+ h2hh ))x(x(w))x(X(w ⎥ ⎣ ==1h 1h ⎦

⎡ − xX SKSK ⎤ = st expy ⎢ ⎥ (3.9) + xX ⎣ SKSK ⎦ where,

L L xSK = ∑ hh β+ h2 ))x(x(w and XSK = ∑ hh β+ h2 )).x(X(w =1h =1h

Bias and MSE of tSK, are respectively given by

L 1 2 RSK 2 1 Bias (tSK) = ∑ h θγ SKh (w Sxh − yxh )S (3.10) Xst =1h 8 2

L 2 2 2 RSK 2 MSE (tSK) = ∑ h h yh −γ SRS[w yxhSK + xh ]S (3.11) =1h 4

Xst where Yst and =θ . RSK = SK L XSK ∑ β+ h2hh ))x(X(w =1h

3.3 Upadhyaya-Singh estimator

Upadhyaya and Singh (1999) considered both coefficients of variation and kurtosis in their ratio type estimator as

2 +β C)x(X x 1US = yy (3.12) 2 +β C)x(x x

We adopt this modification in the estimator t proposed at (3.1)

⎛ L ⎞ ⎜ ∑ hh h2 +β xh )C)x(X(w ⎟ ⎜ ⎟ = expyt =1h (3.13) st1US ⎜ L ⎟ ⎜ ⎟ ⎜ ∑ h2hh +β C)x(x(w xh ⎟ ⎝ =1h ⎠ tUS1 at (3.13) can be re-written as

⎡ − xX 1US1US ⎤ = st1US expyt ⎢ ⎥ ⎣ + xX 1US1US ⎦ (3.14) L L X where X = ∑ h2hh1US +β C)x(X(w xh ) x 1US = ∑ h2hh +β xh ).Cx(w =1h =1h

Bias and MSE of tUS1, to first degree of approximation, are respectively given by

L 1 2 ⎛ R 1US 2 1 ⎞ ()tBias 1US = ∑wh θγ 1USh ⎜ Sxh − Syxh ⎟ (3.15) Xst =1h ⎝ 8 2 ⎠

L ⎡ 2 ⎤ 2 2 R 1US 2 ()tMSE 1US ∑ h h ⎢ yh −γ= SRSw yxh1US + Sxh ⎥ (3.16) =1h ⎣⎢ 4 ⎦⎥

∑ β h2hhst )x(.Xw.Y ∑ β h2hh )x(.Xw where R = and θ = . 1us L 1US X 1US ∑ ()+β XC)x(Xw stxhh2hh =1h Upadhyaya and Singh (1999) proposed another estimator by changing the place of coefficient of kurtosis and coefficient of variation as

β+ 2x )x(CX 2US = yy (3.17) β+ 2x )x(Cx

Incorporating this modification in the proposed estimator t, we have-

⎡ − xX 2US2US ⎤ = st2US expyt ⎢ ⎥ ⎣ + xX 2US2US ⎦ (3.18)

L L where x = ∑ β+ h2xhhh2us ))x(CX(w and X= ∑ β+ h2xhhh2us ))x(CX(w =1h =1h

Bias and MSE of tUS2, are respectively given by –

L 1 2 R 2US 2 1 2US )t(Bias = ∑ w h θγ 2USh ( Sxh − yxh )S Xst =1h 8 2 (3.19)

L 2 2 2 R 2US 2 2US ∑ h h yh −γ= SRS[w)t(MSE yxh2US + xh ]S =1h 4 (3.20)

∑ C.Xw.Y xhhhst where R = 2US L ∑ ()β+ X.)x(CXw sth2xhhh =1h

∑ CXw xhhh

and 2US =θ . X 2US

3.4. G.N. Singh Estimator

Following Singh (2001), using values of and β2 ),x( we propose following two estimators.

⎡ − xX GNS1GNS 1 ⎤ t 1GNS = st expy ⎢ ⎥ + xX ⎣ 1GNS1GNS ⎦ (3.21) L L where X 1GNS = ∑ hh σ+ xh )X(w and x 1GNS = ∑ hh σ+ xh )x(w =1h =1h

The Bias and MSE of t 1GNS to a first degree of approximation, are respectively given by –

L 1 2 R 1GNS 2 1 1GNS )t(Bias ∑ w h θγ GNSh 1( Sxh − yxh )S (3.22) Xst =1h 8 2

L 2 2 2 R 1GNS 2 1GNS ∑ w)t(MSE h h S[ yh −γ= SR yxh1GNS + xh ]S (3.23) =1h 4

Y X where R = st , =θ st . 1GNS L 1GNS X 1GNS ∑ ()Xw σ+ xhhh =1h

Similarly, we propose another estimator

⎡ − xX 2GNS2GNS ⎤ = st2GNS expyt ⎢ ⎥ + xX ⎣ 2GNS2GNS ⎦ (3.24)

L L where, X = ∑ h2hh2GNS σ+β xh ),)x(X(w x 2GNS = ∑ x(w h2hh σ+β xh ) =1h =1h

The Bias and MSE of t 2GNS to a first degree of approximation are respectively given by –

L 1 2 3RGNS2 2 1 2GNS )t(Bias = ∑ wh γhθ 2GNS ( Sxh − yxh )S Xst =1h 8 2 (3.25)

L 2 2 2 R 2GNS 2 2GNS ∑ h h yh −γ= SRS[w)t(MSE yxh2GNS + xh ]S =1h 4 (3.26)

L ∑ hh β h2 )x(.Xw Y where, R = st , =θ =1i 1GNS L 2GNS X 2GNS ∑ ())x(Xw σ+β xhh2hh =1h 4. Improved Estimator

Motivated by Singh et. al. (2008), we propose a new family of estimators given by-

α ⎡ − xX ab,stab,st ⎤ = stMK expyt ⎢ ⎥ ⎢ + xX ab,stab,st ⎥ ⎣ ⎦ (4.1) where a and b are suitably chosen scalars and α is a constant.

The bias and MSE of tMK up to first order of approximation, are respectively given by

L 1 2 ⎛ ()+αα R2 ab 2 1 ⎞ ()tBias MK = ∑wh hθγ ab⎜ Sxh − Syxh ⎟ (4.2) Xst =1h ⎝ 8 2 ⎠

L ⎡ 22 ⎤ 2 2 α R ab 2 ()tMSE MK ∑ h ⎢ SRSw yxhab +α−γ= S ⎥ h ⎢ yh 4 xh ⎥ (4.3) =1h ⎣ ⎦

The MK )t(MSE is minimized for the optimal value of α given by-

L 2 ∑ h γ Sw yxhh =α 2=1i L 2 2 ab ∑ wR h γ hSxh =1i

Putting this value of α in equation (4.3), we get the minimum MSE of the estimator tMK as-

L 2 2 2 )t(MSE .minMK = ∑ w h h yh ρ−γ c )1(S =1i (4.4) where,ρc is combined correlation coefficient in stratified sampling across all strata. It is 2 ⎛ L ⎞ ⎜ 2 ⎟ ⎜ ∑ h ργ SSw xhyhhh ⎟ calculated as 2 =ρ ⎝ =1i ⎠ . c L L 2 2 2 2 ∑∑h γ h yh wSw h γ hSxh ==11 1i

We note here that min MSE of tMK is independent of a and b. therefore, we conclude that it same for any (all) values of a and b.

5. Efficiency comparisons

First we compare the efficiency of the estimator t at (3.1) with estimator tSD. We have

MSE(tSD) < MSE(t)

L ⎡ 2 ⎤ L ⎡ 2 ⎤ 2 2 R SD 2 2 2 R 2 ∑ w h ⎢ yhh −γ SRS yxhSD + Sxh ⎥ ∑ h γ< ⎢ yhh RSSw yxh + Sxh ⎥ (5.1) =1h ⎣⎢ 4 ⎦⎥ =1h ⎣⎢ 4 ⎦⎥

2 L ⎡ 2 ⎤ L ⎡ R ⎤ 2 RSD 2 2 ⎢ 2 ⎥ ∑ h ⎢−γ SRw yxhSDh + Sxh ⎥ < ∑ h h RSw yxh +−γ Sxh =1h ⎢ 4 ⎥ =1h ⎢ 4 ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥

L L 2 2 2 Let A ∑ h γ= Sw yxhh and B = ∑ w h γ hSxh =1h =1h

Then equation (5.1) can be re-written as-

R 2 R 2 − R A + SD B. < RA +− B. SD 4 4

-A (RSD – R) + B/4(RSD - R) (RSD + R) < 0 (5.2)

From (5.2), we get two conditions

L 2 L ⎡ 2 ⎤ 2 ⎡ R SD 2 ⎤ 2 R 2 ∑ hh ⎢−θ SRw yxhSD + Sxh ⎥ < ∑ wh h ⎢ RSyxh +−θ Sxh ⎥ =1h ⎣ 4 ⎦ =1h ⎣⎢ 4 ⎦⎥

(i) When (RSD-R) (RSD+R) > 0

B < 4A /(RSD+R) (5.3)

(ii) When (RSD – R) (RSD + R) < 0

B > 4A / (RSD+R) (5.4)

L L 2 2 2 where ∑ h θ= SwA yxhh and ∑ h θ= SwB xhh =1h =1h

When either of these conditions is satisfied, estimator tSD will be more efficient than the estimator t. The same conditions also holds true for the estimators tSK, tUS1, tUS2, tGNS1 and tGNS2 if we replace RSD by RSK, RUS1, RUS2, RGNS1 and RGNS2 respectively in conditions (i) and (ii).

Next we compare the efficiencies of topt with the other proposed estimators.

)t(MSE .minMK < ab )t(MSE L L ⎡ R 2 ⎤ 2 2 2 2 ⎢Sw)1(Sw 2 ab 2 −+γ<ρ−γ SRS ⎥ ∑ h h yh c ∑ h h h x yxab =1i =1i ⎢ 4 ⎥ ⎣ ⎦ (5.5)

On putting the value of ρc and rearranging the terms we get

2 ⎛ L R L ⎞ ⎜ ∑ 2 Sw −γ ab ∑ 2 γ 2 ⎟ > 0Sw ⎜ h yxhh 2 h h xh ⎟ ⎝ =1i =1i ⎠ (5.6)

This is always true. Hence the estimator tMK 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

SN hh n h = L ∑ h SN h =1h Table 6.1: Data Statistics

N1=106 N2=106 N3=94 N4=171 N5=204 N6=173 n1=9 n2=17 n3=38 n4=67 n5=7 n6=2

1 = 24375X 2 = 27421X 3 = 72409X

4 = 74365X 5 = 26441X 6 = 9844X

1 = 536iY 2 = 2212Y 3 = 9384Y

4 = 5588Y 5 = 967Y 6 = 404Y

x1 =β 71.25 βx 2 = .34 57 β x 3 = .26 14

x 4 =β 60.97 β x 5 = 47.27 βx 6 = 10.28 Cx1=2.02 Cx2=2.10 Cx3=2.22 Cx4=3.84 Cx5=1.72 Cx6=1.91 Cy1=4.18 Cy2=5.22 Cy3=3.19 Cy4=5.13 Cy5=2.47 Cy6=2.34 Sx1=49189 Sx2=57461 Sx3=160757 Sx4=285603 Sx5=45403 Sx6=18794 Sy1=6425 Sy2=11552 Sy3=29907 Sy4=28643 Sy5=2390 Sy6=946

1 =ρ 82.0 ρ2 = 86.0 ρ3 = .0 90 =ρ 99.0 ρ = .0 71 ρ = .0 89 4 5 6 γ1 = 102.0 γ2 = 049.0 γ3 = .0 016 γ4 = 009.0 γ5 = .0 138 γ6 = .0 006 2 2 2 1 = 015.0w 2 = .0w 015 3 = 012.0w 2 = 04.0w 2 = 057.0w 2 = 041.0w 4 5 6 N=854 n=140 β = 07.312 x Cx=3.85 Cy=5.84 Sx=144794 Sy=17106 ρ = 92.0 = 37600X = 2930Y R =0.07793 RSD=0.07792 RSK=0.07784 RUS1=.07789 RUS2=0.07786 RGNS1= 0.06632 RGNS2=0.07765

Table 6.2 : Estimators with their MSE values

Estimators MSE values

t 359619.594

tSD 359649.688

tSK 359890.313

tUS1 359739.875

tUS2 359830.125

tGNS1 360007.985

tGNS2 360479.192

tMK(opt) 218374.8898

From Table 6.2, we conclude that the estimator tMK 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.

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