Estimation of finite population mean in simple random ...

Communications in Statistics - Theory and Methods

ISSN: 0361-0926 (Print) 1532-415X (Online) Journal homepage: http://www.tandfonline.com/loi/lsta20

Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables Siraj Muneer, Javid Shabbir & Alamgir Khalil To cite this article: Siraj Muneer, Javid Shabbir & Alamgir Khalil (2017) Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables, Communications in Statistics - Theory and Methods, 46:5, 2181-2192, DOI: 10.1080/03610926.2015.1035394 To link to this article: http://dx.doi.org/10.1080/03610926.2015.1035394

Accepted author version posted online: 22 Mar 2016. Published online: 22 Mar 2016. Submit your article to this journal

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Date: 22 February 2017, At: 03:00

COMMUNICATIONS IN STATISTICS—THEORY AND METHODS , VOL. , NO. , – http://dx.doi.org/./..

Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables Siraj Muneera , Javid Shabbirb , and Alamgir Khalila a Department of Statistics, University of Peshawar, Peshawar, Pakistan; b Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan

ABSTRACT

ARTICLE HISTORY

In this article, we propose a new class of estimators to estimate the finite population mean by using two auxiliary variables under two different sampling schemes such as simple random sampling and stratified random sampling. The proposed class of estimators gives minimum mean squared error as compared to all other considered estimators. Some real data sets are used to observe the performances of the estimators. We show numerically that the proposed class of estimators performs better as compared to all other competitor estimators.

Received  December  Accepted  March  KEYWORDS

Auxiliary variables; stratification; bias; MSE; efficiency. MATHEMATICS SUBJECT CLASSIFICATION

D

1. Introduction The use of auxiliary information either at estimation stage or at designing stage or at both stages increases precision of the estimators, when there exists a relationship between the study variable and the auxiliary variable. The use of suitable auxiliary information results in considerable reduction in mean squared error (MSE) of the ratio and regression estimators. For example, height is highly correlated with age and the yield of wheat crop is correlated with fertilizer and water. When auxiliary information is available, statisticians often used it at various stages to obtain most efficient estimates. For detail, see the following references: Olkin (1958), Singh (1965, 1967), Mohanty (1967), Chand (1975), Srivastava (1980), Agarwal (1980), Agarwal and Kumar (1985), Srivastava and Jhajj (1981), Estevao and Sarndal (2002), Singh and Espejo (2003), Kadilar and Cingi (2003b), Abu-Dayyeh et al. (2003), Unnikrishan and Kunte (1995), Gupta and Shabbir (2008a, 2008b), Shabbir and Gupta (2011), Diana and Perri (2007), Singh and Singh (2014), etc. These authors proposed a large number of ratio and regression type estimators for estimating the finite population mean under different sampling schemes using one or more auxiliary variables. Consider a finite population U = (U1 , U2 , . . . , UN ) of size N units. We draw a sample of size n units from U , by using simple random sample without replacement (SRSWOR) sampling scheme. Let yi and (xi , zi ) be the values of the study and the auxiliary variables, respec¯ ¯ Z), ¯ z) ¯ be the sample means corresponding to population mean Y¯ and (X, tively. Let y¯ and (x, respectively.

CONTACT S. Muneer Pakistan.

sirajmuneer@yahoo.com

©  Taylor & Francis Group, LLC

Department of Statistics, University of Peshawar, Peshawar ,

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¯

¯

¯ X ¯ Z Let e0 = ( y−Y¯ Y ), e1 = ( x− ), and e2 = ( z− ) such that E(ei ) = 0, (i = 0, 1, 2.), E(e20 ) = X¯ Z¯ 2 2 2 2 2 λCy , E(e1 ) = λCx , E(e2 ) = λCz , E(e0 e1 ) = λCyx , E(e0 e2 ) = λCyz , E(e1 e2 ) = λCxz , where Cy2 S2y , C2 Y¯ 2  x N

S2x , Cz2 X¯ 2 (x −X¯ )2 S2x = i=1N−1i , N ¯ (xi −X¯ )(zi −Z)

=

i=1

N−1

=

= S2z

S2z ,C Z¯ 2 yx

=

, and λ =

= ρyxCyCx ,

N (z −Z) ¯ 2 i=1 i

N−1 1 ( n − N1 ).

Cyz = ρyzCyCz , Cxz = ρxzCxCz , S2y =

, Syx =

N

¯

¯

i=1 (yi −Y )(xi −X )

N−1

,

Syz =

N

¯

N

¯ 2 i=1 (yi −Y )

¯

i=1 (yi −Y )(zi −Z)

N−1

N−1

,

,

Sxz =

2. Selected estimators in simple random sampling We discuss the usual mean per unit estimator and some existing estimators using two auxiliary variables (x, z). (i) The usual mean per unit estimator is 1 yi n i=1 n

y¯0 =

(1)

The variance of y¯0 is given by V (y¯0 ) = λY¯ 2Cy2

(2)

(ii) Olkin (1958) proposed the following multivariate ratio type estimator:   Z¯ X¯ y¯MR = y¯ w1 + w2 x¯ z¯

(3)

where w1 and w2 are constants that satisfy the condition w1 + w2 = 1. The minimum MSE of y¯MR , to first order of approximation, is given by   (Cz2 + Cyx − Cyz − Cxz )2 2 2 2 ¯ λ Y MSE(y¯MR )min ∼ + C − 2C − C (4) = yz y z Cx2 + Cz2 − 2Cxz C2 +C −C −C

yx yz xz and w2(opt ) = The optimum values of w1 and w2 are given by w1(opt ) = z C2 +C 2 x z −2Cxz (1 − w1(opt ) ) (iii) Singh (1965) proposed the following chain ratio-ratio type estimator:   ¯ Z X¯ (5) y¯S1 = y¯ x¯ z¯

The MSE of y¯S1 , to first order of approximation, is given by MSE(y¯S1 ) ∼ = λY¯ 2 (Cy2 + Cx2 + Cz2 − 2Cyx + 2Cxz − 2Cyz ) (iv) Singh (1967) proposed the following chain ratio-product type estimator:   X¯ z¯ y¯S2 = y¯ x¯ Z¯

(6)

(7)

The MSE of y¯S2 , to first order of approximation, is given by MSE(y¯S2 ) ∼ = λY¯ 2 (Cy2 + Cx2 + Cz2 − 2Cyx − 2Cxz + 2Cyz )

(8)

(v) The usual classical regression estimator is given by ¯ y¯Reg = y¯ − byx (x¯ − X¯ ) − byz (z¯ − Z)

(9)

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where byx =

syx s2x

and byz =

syz s2z

are the sample regression coefficients whose population

regression coefficients are βyx = The MSE of y¯Reg is given by

Syx S2x

and βyz =

Syz . S2z

 2 2 − ρyz + 2ρyx ρyz ρxz MSE(y¯Reg ) ∼ = λY¯ 2Cy2 1 − ρyx

S

2183

(10)

S

where ρyx = SyyxSx , ρyz = SyyzSz , and ρxz = SSxxzSz are the population correlation coefficients between their respective subscripts. (vi) The usual difference estimator is given by ¯ y¯D = y¯ − d1 (x¯ − X¯ ) − d2 (z¯ − Z)

(11)

where d1 and d2 are unknown constants, whose values are to be determined. The minimum MSE of y¯D is given by

2 2 ρyx + ρyz − 2ρyx ρyz ρxz 2 2 (12) MSE(y¯D )min ∼ = λY¯ Cy 1 − 2 1 − ρxz The optimum values of d1 and d2 are d1(opt ) =

¯ y (ρyx − ρyz ρxz ) YC 2 ) ¯ x (1 − ρxz XC

d2(opt ) =

¯ y (ρyz − ρyx ρxz ) YC ¯ z (1 − ρ 2 ) ZC

and

xz

(vii) Recently, Singh and Singh (2014) proposed the following new estimator:

α  ¯ − z) ¯ aX¯ +k 1 a( Z ¯ yx (X¯ − x)] ¯ ¯ exp α2 + w4 [y¯ + byz (Z¯ − z)] y¯SS = w3 [y+b ¯ ax+k ¯ + 2b a(Z¯ + z) (13) where byx and byz are the sample regression coefficients; a, b, k, α1 , and α2 are suitably chosen constants; and w3 and w4 are the weights such that w3 + w4 = 1. The minimum MSE of y¯SS , to first order of approximation, is given by

2 2 MSE(y¯SS )min ∼ = λ Y¯ 2Cy2 + w3(opt ) η1 + w4(opt ) η2 − 2w3(opt ) η3  − 2w4(opt ) η4 + 2w3(opt ) w4(opt ) η5 (14) The optimum values of w3 and w4 are w3(opt ) =

η3 η2 −η4 η5 η1 η2 −η52

η1 = A24Cx2 , η2 = A25Cz2 , η3 = Y¯ A4Cyx , η4 = Y¯ A5Cyz , η5 = aZ¯ , ¯ 2(aZ+b)

A2 α 2

η1 η4 −η3 η5 where η1 η2 −η52 ¯ aX A4 A5Cxz , A1 = aX+k , A2 = ¯

and w4(opt ) =

¯ where βyx ¯ and A5 = α2 A2Y¯ + βyz Z, A3 = 22 2 + A22 α2 , A4 = α1 A1Y¯ + βyx X, and βyz are defined earlier.

3. Proposed class of estimators in simple random sampling Motivated by Gupta and Shabbir (2008a) and Singh and Singh (2014), we propose the following general class of estimators. Also we can generate many estimators by substituting different

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values of k1 , k2 , and α. y¯PR





z¯ − Z¯ = [k1 y¯ − k2 (x¯ − X¯ )] α 2 − exp z¯ + Z¯





Z¯ − z¯ + (1 − α) exp Z¯ + z¯

 (15)

where (ki , i = 1, 2) are unknown constants whose values are to be determined. Putting α = 1 and α = 0 in above proposed class of estimators, we get the following two new estimators. (i) For α = 1, the proposed class of estimators reduces to   z¯ − Z¯ (16) y¯PR(α=1 ) = [k1 y¯ − k2 (x¯ − X¯ )] 2 − exp z¯ + Z¯ (ii) For α = 0, the proposed class of estimators reduces to y¯PR(α=0)



Z¯ − z¯ = [k1 y¯ − k2 (x¯ − X¯ )] exp Z¯ + z¯

 (17)

Rewriting y¯PR in terms of e s, we have

   e2 3 α 2 ¯ ¯ ¯ ¯yPR ∼ − e − ··· = [k1Y + k1Y e0 − k2 Xe1 ] 1 − + 2 8 4 2

Keeping terms up to power 2 in e s, we have     e e 3 α e e 2 0 e2 1 2 2 ∼ − e2 − + k2 X¯ − e1 y¯PR − Y¯ = (k1 − 1)Y¯ + k1Y¯ e0 − + 2 8 4 2 2 The bias and MSE of y¯PR are given by B(y¯PR ) ∼ = (k1 − 1) Y¯ + k1Y¯



 λCyz 3 α 1 ¯ 2 − λCz − + k2 XλC xz 8 4 2 2

(18)

(19)

(20)

and



  α 2 Cz − 2Cyz + k22 X¯ 2 λCx2 MSE(y¯PR ) ∼ = Y¯ 2 + Y¯ 2 k21 − 2k1Y¯ 2 + k21Y¯ 2 λ Cy2 + 1 − 2   α 3 1 2 2 ¯ ¯ ¯ − Cz − Cyz + 2k1 k2Y¯ Xλ(C − 2k1Y¯ λ xz − Cyx ) − k2Y XλCxz 8 4 2 (21)

Minimizing Equation (21) with respect to k1 and k2 , we get the optimum values of k1 and k2 , i.e., U1 k1(opt ) = U2 and k2(opt ) where

 U1 = 1 +

and

Y¯ = X¯



U1 Cxz − 2 2Cx U2



Cxz − Cyx Cx2



λCxz (Cxz − Cyx ) 3 α 1 − λCz2 − λCyz − 8 4 2 2Cx2

 λ(Cxz − Cyx )2 α 2 λCz − 2λCyz − U2 = 1 + λCy2 + 1 − 2 Cx2

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Substituting the optimum values of k1 and k2 in Equation (21), we get the minimum MSE of y¯PR , which is given by MSE(y¯PR )min

 2 U12 λCxz 2 ∼ ¯ − =Y 1− 4Cx2 U2

(22)

3.1. Numerical comparison Now we obtain the MSE values of all considered estimators by using the following four data sets. 1. Data set 1 [Source: Hair (2002)]y: Perceived level of price charged by product suppliers. x: Overall level of service necessary for maintaining a satisfactory relationship between supplier and purchaser. z: Overall image of the manufacturer/supplier. N = 100, n = 29, Y¯ = 2.3640, X¯ = 2.9250, Z¯ = 5.2390, ρyx = 0.1602, ρyz = 0.0829, ρxz = 0.0846, Cy2 = 2.5582, Cx2 = 0.0661, Cz2 = 0.0461. 2. Data set 2 [Source: Singh (2003), pages 1119–1121] y: Tobacco yield (metric tons) in specified countries during 1998. x: Tobacco area (hectares) in specified countries during 1998. z: Tobacco production (metric tons) in specified countries during 1998. N = 106, n = 31, Y¯ = 1.5507, X¯ = 34438.61, Z¯ = 52444.56, ρyx = −0.0077, ρyz = 0.0304, ρxz = 0.9912, Cy2 = 0.2629, Cx2 = 18.8364, Cz2 = 23.3405. 3. Data set 3 [Source: MFA (2004)] y: District wise tomato production in tonnes of Pakistan for 2003. x: District wise tomato production in tonnes of Pakistan for 2002. z: District wise tomato production in tonnes of Pakistan for 2001. N = 97, n = 30, Y¯ = 3135.6186, X¯ = 3050.2784, Z¯ = 2743.9587, ρyx = 0.8072, ρyz = 0.8501, ρxz = 0.6122, Cy2 = 4.8674, Cx2 = 5.4812, Cz2 = 6.2422. 4. Data set 4 [Source: Jhonston (1963)] y: Percentage of lives affected by diseases. x: Mean January temperature. z: Date of flowering of particular summer flowering species (number of days from January 1). N = 10, n = 2, Y¯ = 52, X¯ = 42, Z¯ = 200, ρyx = 0.7966, ρyz = −0.9364, ρxz = −0.7333, Cy2 = 0.4997, Cx2 = 0.2440, Cz2 = 0.0021. We use the following expression to obtain the percent relative efficiency (PRE): PRE =

V (y¯0 ) × 100, i = MR, S1, S2, Reg, D, SS, PR MSE(y¯i )min

The results are given in Table 1. Note: The minimum MSE values of y¯SS are given in Table 1 for (α1 , α2 , a, b, k) = 1.

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Table . PRE of different estimators with respect to y¯0 . Estimator

Data 

Data 

Data 

Data 

y¯0 y¯MR y¯S1 y¯S2 y¯Reg y¯D y¯SS y¯PR

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

y¯PR

.

.

.

.

(α=1) (α=0)

3.2. Conclusion We have developed a class of estimators which is found to be more efficient than all other considered estimators. In Table 1, we see that the proposed estimators y¯PR(α=1) and y¯PR(α=0) are more efficient than the usual multivariate ratio type estimator (y¯MR ), ratio-ratio estimator (y¯S1 ), ratio-product estimator (y¯S2 ), regression estimator (y¯Reg ), difference estimator (y¯D ), and Singh and Singh (2014) estimator (y¯SS ). The difference estimator (y¯D ) and Singh and Singh (2014) estimator (y¯SS ) are equally efficient. It is noted that y¯D and y¯SS are more efficient than all other considered estimators in Data set 4.

4. Selected estimators in stratified random sampling Several estimators exist in literature to estimate the finite population mean Y¯ = Y¯st =  L ¯ h=1 WhYh in stratified random sampling. Diana (1993), Kadilar and Cingi (2003a), Sahoo and Bala (2000), Gupta and Shabbir (2006), Shabbir and Gupta (2005, 2011), Koyuncu and Kadilar (2009a, b), Singh and Vishwakarma (2008), Singh et al. (2008), and Singh and Singh (2014) proposed different estimators in stratified random sampling. In order to estimate Y¯ , we propose a new class of estimators using two auxiliary variables in stratified random sampling. Generally the proposed class of estimators performs better than all other considered estimators. Let a population  of size N be divided into L strata with the hth stratum consisting of 2, . . . , N ). Let a sample of size nh be drawn by Nh units, such that Lh=1 Nh = N, (h = 1,  SRSWOR from the hth stratum, such that Lh=1 nh = n. Let yhi and (xhi , zhi ) be the values of the study and the auxiliary variables, respectively, in the hth stratum of the ith unit. We ¯ assume that X Z¯ are knownin advance. and L L ¯ ¯ means corLet yst = h=1 Wh y¯h , x¯st = Lh=1 W h x¯h , and z¯st = h=1 LWh zh be the sample L ¯ ¯ ¯ responding to population means Y = h=1 WhYh , X = h=1 Wh X¯ h , and Z¯ = Lh=1 Wh Z¯ h ,  yhi nh xhi nh zhi Nh yhi ¯ ¯ respectively, where y¯h = nh i=1 nh , x¯h = i=1 nh , z¯h = i=1 nh , Yh = i=1 Nh , Xh = Nh zhi Nh yhi ¯ i=1 Nh , Zh = i=1 Nh . ¯

¯

¯

¯

Y x¯st −X z¯st −Z Let e0st = ( ystY− ¯ ), e1st = ( X¯ ), and e2st = ( Z¯ ), such that E(eist ) = 0, (i = 0, 1, 2),

L E(e20st )

=

E(e22st ) =

2 2 h=1 Wh λh Syh

L =

2 2 h=1 Wh λh Sxh X¯ 2

= V020 , Y¯ 2  L 2 2 W 2 λh Syxh h=1 Wh λh Szh = V002 , E(e0st e1st ) = h=1 h = V110 , Y¯ X¯ Z¯ 2

L

= V200 ,

E(e21st )

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L E(e0st e2st ) =

2 h=1 Wh λh Syzh

2187

L = V101 , E(e1st e2st ) =

2 h=1 Wh λh Sxzh

= V011 , ¯ X¯ ¯ X¯ Y Y  1 1 − λh = nh Nh  N E[(y¯ −Y¯ )r (x¯ −X¯ )s (z¯ −Z¯ )t ] where Vrst = Lh=1 Wh(r+s+t ) h h Y¯ rhX¯ s Z¯ht h h , and Wh = Nh is the known stratum weight in the hth stratum. Let ρyxh , ρyzh , and ρxzh be the population correlation coefficients with their respective subscripts in the hth stratum. Now we discuss some selected estimators in stratified sampling. (i) The variance of the mean per unit estimator is given by V (y¯st ) = Y¯ 2V200 (ii) The multivariate ratio estimator in stratified sampling is given by    Z¯ X¯ + w2st y¯MR(c) = y¯st w1st x¯ z¯st

(23)

(24)

where w1st and w2st are the weights that satisfy the condition w1st + w2st = 1. +V110 −V101 −V011 ) and The minimum MSE y¯MR(c) at optimum values w1st (opt ) = (V002 (V020 +V002 −2V011 ) w2st (opt ) = (1 − w1st (opt ) ) is given by   (V002 +V110 − V101 −V011 )2 2 ¯ MSE(y¯MR(c) )min = Y λ (V200 + V002 −2V101 )− (25) (V020 + V002 − 2V011 ) (iii) Singh (1965) suggested the following chain ratio-ratio estimator in stratified sampling:   ¯  X¯ Z (26) y¯S1(c) = y¯st x¯st z¯st The MSE of y¯S1(c) , to first order of approximation, is given by MSE(y¯S1(c) ) ∼ = Y¯ 2 (V200 + V020 + V002 − 2V110 + 2V011 − 2V101 ) (iv) Singh (1967) ratio-product estimator using stratified sampling is given by   X¯ z¯st y¯S2(c) = y¯st x¯st Z¯

(27)

(28)

The MSE of y¯S2(c) , to first order of approximation, is given by MSE(y¯S2(c) ) ∼ = Y¯ 2 (V200 + V020 + V002 − 2V110 − 2V011 + 2V101 )

(29)

(v) The classical regression estimator in stratified random sampling is given by y¯Reg(c) = y¯st + byx(st ) (X¯ − x¯st ) + byz(st ) (Z¯ − z¯st )

(30)

where byx(st ) and byz(st ) are the sample regression coefficients across strata whose pop¯ 110 ¯ 101 YV ulation regression coefficients are βyx(st ) = YV ¯ 002 . ¯ 020 and βyz(st ) = ZV XV The MSE of y¯Reg(c) , to first order of approximation, is given by   2 2 V110 V101 V110V101V011 2 ∼ ¯ MSE(y¯Reg(c) ) = Y V200 1 − (31) − +2 V200V020 V200V002 V200V020V002

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(vi) The usual difference estimator in stratified random sampling is given by ¯ y¯D(c) = y¯st − d1(st ) (x¯st − X¯ ) − d2(st ) (z¯st − Z) where d1(st ) and d2(st ) are unknown constants. The minimum MSE of y¯D(c) is given by    2 2 V110V002 + V101 V020 − 2V101V110V011 2 ∼ ¯ MSE(y¯D(c) )min = Y V200 − 2 V000V020 − V011

(32)

(33)

The optimum values of d1(st ) and d2(st ) are d1(st )opt =

Y¯ (V002V011 − V001V110 ) 2 ) X¯ (V020V002 − V011

and d2(st )opt =

2 ) Y¯ (V020V110 − V001 2 ¯ Z(V020V002 − V011 )

(vii) Singh and Singh (2014) proposed a new estimator in stratified random sampling as α  aX¯ + k 1 y¯SS(c) = w3st [y¯st + byx(st ) (X¯ − x¯st )] ax¯st + k

a(Z¯ − z¯st ) ¯ + w4st [y¯st + byz(st ) (Z − z¯st )] exp α2 (34) a(Z¯ + z¯st ) + 2b where byx(st ) and byz(st ) are sample regression coefficients across strata whose popu¯ 110 ¯ 101 YV lation regression coefficients are βyx(st ) = YV ¯ 002 ; a, b, k, α1 , and α2 ¯ 020 and βyz(st ) = ZV XV are suitably chosen constants, and w3st and w4st are weights that satisfy the condition: w3st + w4st = 1. The minimum MSE of y¯SS(c) , to first order of approximation, is given by 2 2 MSE(y¯SS(c) )min ∼ = λ[Y¯ 2V200 + w3st (opt ) γ1 + w4st (opt ) γ2 − 2w3st (opt ) γ3 − 2w4st (opt ) γ4 + 2w3st (opt ) w4st (opt ) γ5 ]

(35)

γ1 γ4 −γ3 γ5 4 γ5 The optimum values of w3st and w4st are w3st (opt ) = γγ3 γγ2 −γ 2 and w4st (opt ) = γ γ −γ 2 1 2 −γ5 1 2 5 where γ1 = B24V020 , γ2 = B25V002 , γ3 = Y¯ B4V110 , γ4 = Y¯ B5V101 , γ5 = B4 B5V011 , B1 = ¯ B2 α 2 aX¯ ¯ and B5 = α2 B2Y¯ + , B2 = a¯Z , B3 = 2 2 + B22 α2 , B4 = α1 B1Y¯ + βyx(st ) X, ¯ aX+k

¯ βyz(st ) Z.

2(aZ+b)

2

4.1. Proposed class of estimators in stratified random sampling Motivated by Singh and Singh (2014), we propose a class of combined estimators in stratified sampling. We can use both combined and separate estimates. Cochran (1977) suggested that with only a small sample in each stratum, the combined estimate is to be recommended. Also for ease of computation, we use the combined estimator in our proposed setup. The proposed estimator is given by       ¯ − z¯st ¯st − Z¯ z Z y¯PR(c) = [k3st y¯st − k4st (x¯st − X¯ )] α 2 − exp + (1 − α) exp z¯st + Z¯ Z¯ + z¯st

COMMUNICATIONS IN STATISTICS—THEORY AND METHODS

2189

where α is a scalar, whose values are (0, 1), and (kist , i = 3, 4) are unknown constants whose values are to be determined. Putting α = 1 and α = 0 in above proposed class of estimators, we get the following two new estimators: i. For α = 1, the proposed class of estimators reduces to   z¯st − Z¯ (36) y¯PR(c)(α=1) = [k3st y¯st − k4st (x¯st − X¯ )] 2 − exp z¯st + Z¯ ii. For α = 0, the proposed class of estimators reduces to y¯PR(c)(α=0)



Z¯ − z¯st = [k3st y¯st − k4st (x¯st − X¯ )] exp Z¯ + z¯st



The bias and MSE of y¯PR(c) , to first order of approximation, are given by  1 1 3 α ∼ ¯ ¯ 011 ¯ − V002 − V101 + k4st XV B(y¯PR(c ) = (k3st − 1) Y + k3st Y 8 4 2 2 and



 α MSE(y¯PR(c) ) ∼ = Y¯ 2 + Y¯ 2 k23st − 2k3st Y¯ 2 + k23st Y¯ 2 (V200 + 1 − V002 − 2V101 2  3 α 1 + k24st X¯ 2V020 − 2k3st Y¯ 2 − V002 − V101 8 4 2  ¯ ¯ ¯ ¯ + 2k3 k4st Y Xλ(V011 − V110 ) − k4Y XV011

(37)

(38)

(39)

The optimum values of k3st and k4st are k3st (opt ) = and k4st (opt ) = where

 Ast = 1 +

and

Y¯ X¯



Ast Bst

Ast V011 − 2V020 Bst



V011 − V110 V020



3 α 1 V011 (V011 − V110 ) − V002 − V101 − 8 4 2 2V020

 α (V011 − V110 )2 V002 − 2V101 − Bst = 1 + V200 + 1 − 2 V020

By substituting the optimum values of k3st and k4st in (39), we get minimum MSE of y¯PR(c) :  2 A2st V011 2 ¯ Y − 1 − (40) MSE(y¯PR(c) )min ∼ = 4V020 Bst 4.2. Numerical comparisons We use the following data sets for comparison. (1) Data set 1 [Source: Sarndal et al. (1992), p. 529] y : Population in thousands during 1985.

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x : Population in thousands during 1975. z : Total number of seats in municipal council. N = 199,

n = 47,

N1 = 25,

N2 = 48,

N3 = 32,

N4 = 38,

N5 = 56, n1 = 6, n2 = 11, n3 = 8, n4 = 9, n5 = 13, Y¯1 = 62.4400, Y¯2 = 29.6042, Y¯3 = 24.0625, Y¯4 = 31.0000, Y¯5 = 29.41071, X¯ 1 = 59.5200, X¯ 2 = 29.1667, X¯ 3 = 23.9375, X¯ 4 = 30.6316, X¯ 5 = 28.7143, Z¯ 1 = 51.1600, Z¯ 2 = 47.6667, Z¯ 3 = 50.2500,

Z¯ 4 = 48.4737,

S22y = 1320.2442,

S23y = 445.3508,

S21x = 16564.7600, S25x = 3565.1169, S24z = 82.0939,

Z¯ 5 = 46.3572,

S24y = 1536.3784,

S22x = 1228.2270,

S25y = 3219.8464,

S23x = 437.1573,

S21z = 197.9734, S25z = 97.9065,

S21y = 15521.5900,

S22z = 106.3546,

S24x = 1721.1579, S23z = 106.7742,

S1yx = 10125.1783,

S2yx = 1270.2376,

S3yx = 440.4234, S4yx = 1622.6216, S5yx = 3385.5013, S1xz = 383.6633, S2xz = 409.8440,

S3xz = 189.1129,

S4xz = 232.3954,

S5xz = 417.5584,

S1yz = 150.0933,

S2yz = 422.9929,

S3yz = 189.1129,

S4yz = 229.8378,

S5yz = 399.9052. (2) Data set 2 [Source: National Horticulture Board (2010)] y : Productivity (MT/hectare). x : Production (000 tons). z : Area (000 hectare). N = 10, n = 7, N1 = 5, N2 = 5, n1 = 3, n2 = 4, Y¯1 = 1.70, Y¯2 = 3.67, X¯ 1 = 10.41, X¯ 2 = 309.14, Z¯ 1 = 6.20, Z¯ 2 = 80.67, S21y = 0.2916, S22y = 1.9881, S21x = 1.4116, S22x = 3486.6916, S2yx = 83.47,

S21z = 1.4116,

S1xz = 1.7500,

S22z = 116.8561,

S2xz = 64.9700,

S1yx = 1.6000,

S1yz = −0.2000,

S2yz = 5.5800. Table . PREs of different estimators with respect to y¯st . Estimator

Data 

Data 

y¯c y¯MR(c) y¯S1(c) y¯S2(c) y¯Reg(c) y¯D(c) y¯SS(c) y¯PR(c)

. . . . . . . .

. . . . . . . .

y¯PR(c)

.

.

(α=1) (α=0)

COMMUNICATIONS IN STATISTICS—THEORY AND METHODS

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We use the following expression to obtain the PRE: PRE =

V (y¯st ) × 100, MSE(y¯i (c))min

i = MR, S1, S2, Reg, D, SS, PR

The results are given in Table 2. Note: The minimum MSE values of y¯SS(c) are given in Table 2 for (α1 , α2 , a, b, k) = 1. 4.3. Conclusion We have proposed a class of estimators for estimating the finite population mean using information on two auxiliary variables in stratified sampling. Expressions for bias and MSE of the proposed class of estimators are derived up to first degree of approximation. The proposed estimators y¯PR(c)(α=1) and y¯PR(c)(α=0) are compared with usual mean estimator and other considered estimators. A numerical study is carried out to support the theoretical results. In Table 2, the proposed class of estimators performs better than all competitor estimators in stratified sampling. Both difference estimator y¯D(c) and Singh and Singh (2014) estimator y¯SS(c) are equally efficient in both data sets.

Acknowledgments Authors are thankful to referees for their valuable suggestions and comments which helped to improve the research paper.

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