32017 mjis rajesh by Mathematical Journal of Interdisciplinary Sciences

DOI: 10.15415/mjis.2015.32017

A Generalised Family of Estimator for Estimating Unknown Variance Using Two Auxiliary Variables Viplav Kumar Singh and Rajesh Singh Department of Statistics, Banaras Hindu University, Varanasi -221005, India Email: rsinghstat@gmail.com Received: 25 March 2014| Revised: 19 February 2015| Accepted: 4 March 2015 Published online: March 30, 2015 The Author(s) 2015. This article is published with open access at www.chitkara.edu.in/publications

Abstract: The present study deals with the estimation of unknown variance using two auxiliary variates under simple random sampling without replacement (SRSWOR) scheme. We have also extended our work to the case of double sampling scheme. The asymptotic expressions for the mean squared errors of the proposed estimators are derived and their optimum values have been obtained. A comparison between the existing and the proposed class of estimators of S y2 has been made empirically with the help of two population data taken from [1] and [10]. Keywords and phrases: Proposed estimator, Double sampling technique, Bias(B), Mean Square Error (MSE), Optimum estimator. 1. Introduction In theory of sample surveys, we are estimating the population variance S y2 of study variable y. It is well known that use of auxiliary information improves the precision of proposed estimator. If information on an auxiliary variable is readily available then it is a well-known fact that the ratio-type and regression-type estimators can be used for estimation of parameters of interest, due to increase in efficiency of these estimators. The problem of estimating the population variance of S y2 of study variable y received a considerable attention of the statistician in survey sampling including [2], [3], [4], [5-7], [12], [13], [14], [15-16], [20] and [18] and have suggested improved estimators for estimation of S y2 Let ϕi (i=1,2,……..,N) be the population having N units such that y is highly correlated with the auxiliary variables x and z. We assume that a simple random sample without replacement (SRSWOR) of size n is drawn from the finite population of size N. Let (s ’2x , s ’2z )and( s x2 , sz2 , s y2 ) be the sample variances defined over n’ and n and S x2 , Sz2 and S y2 be the population variances of variables x, z and y respectively.

Mathematical Journal of Interdisciplinary Sciences Vol. 3, No. 2, March 2015 pp. 193–207

Singh, VK Singh, R

To estimate the population variance S y2 of study variable y, consider two cases: • The population variances S x2 and Sz2 of auxiliary variable x and z are known. • When both S x2 and Sz2 are unknown. Where, S x2 =

∑ (x − X) , S N −1 2

i =1

2 y

∑ (y −Y ) , S N −1 2

2 z

i =1

∑ (z − Z ) N −1

i =1

2. Large Sample Approximations let,s 2 = (1 + ε )S 2 , s 2 = (1 + ε )S 2 , s ′2 = (1 + ε ′ )S 2 , s 2 = (1 + ε )S 2and s ′2 y 0 y x 1 x x 1 x z 2 z z s′Z2 = (1 + ε ′ )S 2 , such that E (εi ) = E (ε ’i ) = 0,(i = 0,1, 2) . 2 z Also to the first order of approximation, we have E (ε02 ) = γ1∇*400 , E (ε12 ) = γ1∇*040 , E (ε22 ) = γ1∇*004 E (ε1ε2 ) = γ1∇*022 , E (ε0 ε1 ) = γ1∇*220 E (ε ’20 ) = γ 2 ∇*400 , E (ε ’12 ) = γ 2 ∇*040 , E (ε ’22 ) = γ 2 ∇*004 , E (ε ’1 ε ’2 ) = γ 2 ∇*022 , E (ε ’0 ε ’1 )

= γ 2 ∇*220 , E (ε ’0 ε ’2 ) = γ 2 ∇*202 and E (ε0 ε2 ) = γ1∇*202 . Where, γ1 =

1 n

, γ2 =

1 1  * p/2 q/2 r /2 , γ =  −  , ∇ pqr = (∇ pqr − 1) and ∇ pqrr = (µ pqr / µ200 µ020 µ002 )    n’ n n’ 1

p/2 q/2 r /2 and ∇ pqr = (µ pqr / µ200 µ020 µ002 ) and µpqr = 1 ∑ (y i − Y ) (X i − X ) (z i − Z ) ; N p,q,r, being non negative integers. p

2 3. Estimation of S y2 when both S x2 and Sz are Known

In case when S x2 and Sz2 are known, the conventional unbiased estimator is given as The conventional unbiased estimator, Sˆ y2 = s y2  S 2  S 2  Usual chain- ratio type estimator, respectively defined by, t R = s y2  2x  2z   s  s  x z 2 ˆ The expressions of the variances and mean square error (MSE) of S y and tR, up to first order of approximation, are given by

var(S y2 ) = γ1S y4 ∇*400

(1)

MSE (t R ) = γ1S y4 ∇*400 + ∇*040 + ∇*004 − 2∇*220 − 2∇*202 + 2∇*022 

(2)

194

A Generalised Motivated by [8], we proposed a generalised estimator for estimating the Family of Estimator population variance S y2 , as g   S 2 g  Sz2   c( sz2 − Sz2 )   a( s x2 − S x2 )  2  x    Tpr = s y  k1  2  exp  2 + k exp     2  (3) 2  sz2   S x + b( s x2 − S x2 )   Sz + d ( sz2 − Sz2 )    s x   2

where, k1,k2 are weights, g1 and g2 are constants a, b, c and d are either real numbers or the functions of known parameters. Note: Here we will use the notations Tp(1) and Tp(2), rather than Tpfor two different cases (ie: k1 + k2 = 1 and k1 + k2 ≠ 1 ). Bias and MSE of the suggested method are derived under two different conditions: • Case 1: When k1 + k2 = 1 , where k1 and k2 are weights. Theorem 2.1: Estimator Tpr (1) {equation (3)}in terms of εi ; i = 0,1 could be expressed as: Tpr (1) = S y2 1 + ε0 + k1 {ε1η1 + ε12 η2 + ε0 ε1η1 } + k2 {ε2 η3 + ε22 η 4 + ε0 ε2 η3 } (4)





 a 2 g ( g + 1)  η1 = (a − g1 ), η2 =  − ab − ag1 + 1 1 , η3 = (c − g2 )   2 2  c 2 g ( g + 1)  η4 =  − cd − cg2 + 2 2 .   2 2

Where,

and

Proof:

Tpr ( 1 ) = s y

 k  

 S   s

2 x 2 x

  

 exp   S

[

2 x

we − g1

(1 + ε1 )

assume −1

+ b( s x − S x

= (1 + ε0 ) S y k1 (1 + ε1 )

Here

a(sx − S x )

−g

 + k ) 

{

 S   s

exp aε1 (1 + bε1 )

−1

2 z 2 z

  

  S

exp 

2 z

ε1 , bε1 , ε2 and dε2 <

−g

+ d (sz − Sz

} + k (1 + ε ) 2

c( sz − S z )

  ) 

(5)

{

exp cε2 (1 + d ε2 )

that

the

−1

}] terms

− g2

,(1 + bε1 ) ,(1 + ε2 )

and (1 + dε2 )−1 are expandable. By expanding the right hand side of equation (5) and neglecting the terms of εi'shaving power greater than two, we have Tpr (1) = S y2  k1 (1 + ε0 ) {1 + ε1η1 + ε12 η2 } + k2 (1 + ε0 ) {1 + ε2 η3 + ε22 η 4 }

  (6) 2 2  Tpr (1) = S 1 + ε0 + k1 {ε1η1 + ε1 η2 + ε0 ε1η1 } + k2 {ε2 η3 + ε2 η 4 + ε0 ε2 η3 }   Theorem 2.2: Bias of Tp(1) is given as

2 y

195

for Estimating Unknown Variance Using Two Auxiliary Variables

Singh, VK Singh, R

B[Tpr (1) ] =

S y2  k1 {η2 ∇*040 + η1∇*220 } + k2 {η4 ∇*004 + η3∇*202 }  n 

(7)

Proof: B(Tpr (1) ) = E [Tpr (1) − S y2 ] = S y2 E  ε0 + k1 {ε1η1 + ε12 η2 + ε0 ε1η1 } + k2 {ε2 η3 + ε22 η4 + ε0 ε2 η3 }   S y2  k1 {η2 ∇*040 + η1∇*220 } + k2 {η3∇*004 + η3∇*202 }  n 

B(Tpr (1) ) =

Theorem 2.3: Mean square error of Tp(1), up to the first order of approximation : O  1  is  n 

MSE[Tpr (1) ] =

S y4

∇*400 + k12 A1 + k22 B1 + 2 k1C1 + 2 k2 D1 + 2 k1 k2 E1   n 

(8)

* * 2 * 2 * * Where, A1 = η1 ∇040 , B1 = η3 ∇004 , C1 = η1∇220 , D1 = η3∇202 and E1 = η1η3∇022

Proof: 2

MSE [Tpr ( 1 ) ] = E [Tpr ( 1 ) − S y ] 4

[

= S y E ε0 + k1 η1 ε1 + k 2 η 3 ε2 + 2 k1 η1 ε0 ε1 + 2 k 2 η 3 ε0 ε2 + 2 k1 k 2 η1 η 3 ε1 ε2

]

(9)

MSE [ t pr ( 1 ) ] =

Sy n

[∇

* 400

+ k1 A1 + (1 + k1 − 2 k1 ) B1 + 2 k1 C1 + 2 (1 − k1 ) D1 + 2 k1 (1 − k1 ) E1

]

Differentiating equation (9) w.r.t k1 and equating it to zero, we get the optimum value of k1 as k1* =

( B1 + D1 − C1 − E1 ) ( A1 + B1 − 2 E1 )

Putting the optimum value of k1in equation (9), we get the minimum MSE of the suggested method Tp(1). • Case 2: When k1 + k2 ≠ 1 , where k1and k2 are weights. Theorem 2.4: The estimator Tp(2) (equation.3) in terms of εi's, could be expressed as: Tpr ( 2 ) = S y2  k1 {1 + ε0 + ε1η1 + ε12 η2 + ε0 ε1η1 } + k2 {1 + ε0 + ε2 η3 + ε22 η 4 + ε0 ε2 η3 }





196

 a 2 g ( g + 1)  η1 = (a − g1 ), η2 =  − ab − ag1 + 1 1 ,η3 = (c − g2 ) annd   2 2 2  c g ( g + 1)  η4 =  − cd − cg2 + 2 2 .   2 2

Where,

Proof: T

pr ( 2 )

 = s  k  2

 S   s

2 x 2 x

  a( s − S )  exp    S + b( s − S g1

 + k ) 

 S   s

2 z 2

 c( s − S )   exp   S + d ( s − S  g2

  ) 

 −g −g = (1 + ε )S 2  k (1 + ε ) 1 exp aε (1 + bε )−1 + k (1 + ε ) 2 0 y1 1 1 1 2 2 (10)  −1  exp {cε2 (1 + d ε2 ) } 

{

Here

}

assume

ε1 , bε1 , ε2 and d ε2 < 1, so that the terms −1 (1 + ε1 ) ,(1 + bε1 ) ,(1 + ε2 )−g2 and (1 + dε2 ) are expandable. By expanding the right hand side of equation (10) and neglecting the terms of εi's having power greater than two, we have − g1

−1

Tpr (1) = S y2  k1 {1 + ε0 + ε1η1 + ε12 η2 + ε0 ε1η1 }  2  +k2 {1 + ε0 + ε2 η3 + ε2 η4 + ε0 ε2 η3 } 

(11)

Theorem 2.5: Bias of Tpr ( 2 ) is given as: S y2  k1 {1 + η2 ∇*040 + η1∇*220 } n  * * +k2 {1 + η4 ∇004 + η3∇202 } − 1 

B[Tpr ( 2 ) ] =

(12)

Proof: B[Tpr ( 2 ) ] = E [Tpr ( 2 ) − S y2 ] = S y2 E  k1 {1 + ε0 + ε1η1 + ε12 η2 + ε0 ε1η1 } + k2 {1 + ε0 + ε2 η3 + ε22 η 4 + ε0 ε2 η3 } − 1





B[Tpr ( 2 ) ] =

2 y

 k1 {1 + η2 ∇*040 + η1∇*220 } + k2 {1 + η 4 ∇*004 + η3∇*202 } − 1   n 

197

A Generalised Family of Estimator for Estimating Unknown Variance Using Two Auxiliary Variables

Singh, VK Singh, R

Theorem 2.6: Mean square error of Tpr ( 2 ) , up to the first order of approximation is: MSE [Tpr ( 2 ) ] = S y4 1 + k12 A2 + k22 B2 − 2 k1C2 − 2 k2 D2 + 2 k1k2 E2 

(13)

Where, A2 = 1 +

 ∇ + ( η + 2 η ) ∇ + 4 η ∇  , B = 1 + 1  ∇ + ( η + 2 η )∇ + 4 η ∇    n n

C2 = 1 +

1

E2 = 1 +

1

400

040

220

400

004

202

1 η2 ∇*040 + η1∇*220  , D2 = 1 +  η 4 ∇*004 + η3 ∇*202   n n ∇*400 + η2 ∇*040 + η 4 ∇*004 + 2η1∇*220 + 2η3 ∇*202 + η1η3 ∇*022   n

Proof: From equation (10), we have 2

[ {

}

{

} ]

[Tpr ( 2 ) − S y ] = S y k1 1 + ε0 + ε1 η1 + ε1 η 2 + ε0 ε1 η1 + k 2 1 + ε0 + ε2 η 3 + ε2 η 4 + ε0 ε2 η 3 − 1

After expanding the above equation up to the first order of approximation ie: 1 O   and then taking expectations of both sides, we get  n  MSE [Tpr ( 2 ) ] = S y4 1 + k12 A2 + k22 B2 − 2 k1C2 − 2 k2 D2 + 2 k1k2 E2  Minimising equation (13) with respect to k1 and k2, we get the optimum values k1 and k2, respectively given as k1* =

B2C2 − D2 E2 A D − C2 E 2 and k2* = 2 2 2 A2 B2 − E2 A2 B2 − E2 2

Putting this value of k1 and k2in equation (13), we get the minimum MSE of the suggested estimator Tpr ( 2 ) . 4.Estimation of S y2 when both S x2 and Sz2 are Unknown Consider a finite population with N (< ∞) identifiable units. Let y be the variable under study taking values yi (i = 1, 2,........... N ) for ith unit of the population. To estimate the population variance S y2 of y in the presence of two auxiliary variables, when the population variances of x and z are not known and SRSWOR scheme is used in selecting samples for both the phases. The following two phase sampling scheme may be recommended. 1) The first phase sample s' of size (n’< N) is drawn in order to observed x and z. 2) The second phase sample s of size n (n<n') is drawn in order to observe y, x and z.

198

A Generalised Let s x′ 2 and sz′ 2 be the unbiased estimators of S x2 and Sz2 respectively, based on 2 first phase sample s’; s y′ 2 , s x′ 2 and sz′ 2 be the unbiased estimators S y2 , S x2 and S x Family of Estimator for Estimating , respectively, based on second phase sample‘s’. When S x2 and Sz2 unknown, Unknown Variance then usual unbiased and ordinary ratio estimator are given as Using Two 2 2 ˆ Usual unbiased estimator S y = s y Auxiliary Variables Usual chain ratio-type estimator in two phase sampling, is defined as 2 2  s ’  s ’  t R’ = s y2  2x  2z   s x  sz  MSE’s expressions for the estimators Sˆ y2 and t R’ , up to the first order of approximation are ,respectively, given by

MSE (S y2 ) = S y4 γ1 ∇*400

(14)

MSE (t R’ ) = S y4 ∇*400 γ1 + γ ∇*040 + γ ∇*004 − 2 γ ∇*220 − 2 γ ∇*202 + 2 γ ∇*022  (15)

when S x2 and Sz2 are not known a priori, then under double sampling technique, suggested estimator Tp takes the following form:  T = s  k 

 s ’   a( s − s ’ )    exp  + k  s   s ’ + b( s − s ’ )  2

’

 s ’   c( s − s ’ )    exp    s   s ’ + d ( s − s ’ )  (16)  g2

’ 2

where, k1,k2are weights, g1 and g2 are constants and a, b, c and d are either real numbers or the functions of known parameters. Note: Here we have taken the notation T ’pr (1) and T ’pr ( 2 ) , rather than T ’pr for two different cases (ie: k1 + k2 = 1 and k1 + k2 ≠ 1 ). To obtain the bias and mean square error of the suggested estimator we have the following two conditions given as Case 1: When k1’ + k2’ = 1 , where k1’ and k2’ are defined weights. ’ Theorem 3.1: The suggested estimator Tpr’ (1) (say equation 16) in terms of εi s , could be expressed as: 1 + ε + k {( a − g )(ε − ε ) + ε Q + ε ’ Q + ε ε Q + ( a − g )(ε ε − ε ε )}  = S  + k ( c − g )( ε − ε ) + ε Q + ε ’ Q + ε ε Q + ( c − g )( ε ε − ε ε ) { }   ’

’

Tpr (1)

’

1 1

’

where, Q1 =

g ( g + 1) g ( g − 1) a2 a2 − ab − g1a + 1 1 , Q2 = + a − ab − ag1 + 1 1 2 2 2 2

199

Singh, VK Singh, R

Q3 = 2 ab − a − a 2 + 2 ag1 − g12 , Q4 =

g ( g + 1) c2 − cd − g2 c + 2 2 2 2

g ( g − 1) c2 − cd − g2 c + 2 2 , Q6 = 2cd − c − c 2 + 2 g2 c − g22 2 2

Q5 = c + Proof: T

’ pr (1)

g   2 g1  c(sz2 − s ’2z )   2  2  a(s x2 − s ’2x )   ’  s ’x  ’  s ’z     = s  k1  2  exp  2  + k   exp  2  s ’x + b(s x2 − s ’2x )  2  sz2   s ’z + d (sz2 − s ’2z )    s x       2 y

 k (1 + ε ’) (1 + ε ) exp {( aε − aε )(1 + ε + bε − bε ) } + k (1 + ε ’) (1 + ε )   = s   exp ( c ε − c ε )( 1 + ε + d ε − d ε ) { }   − g1

’

’ −1

’

− g2

’

−1

After expanding and arraigning the above equation up to the first order of approximation, we have: 1 + ε + k {( a − g )( ε − ε ) + ε Q + ε ’ Q + ε ε Q + ( a − g )( ε ε − ε ε )}    + k {( c − g )( ε − ε ) + ε Q + ε ’ Q + ε ε Q + ( c − g )( ε ε − ε ε )}  (17)   ’

’

Tpr ( 1 ) = S y

’

where, coefficient term of εi ’s are already defined. Theorem 3.2: Bias of Tpr’ (1) is given as B(Tpr’ (1) ) = S y2  k1’ {Q1 γ1∇*040 + γ 2 ∇*040 (Q2 + Q3 ) + ( a − g1 ) γ ∇*220 }



+k2’ {Q4 γ1∇*040 + γ 2 ∇*040 (Q5 + Q6 ) + (c − g2 )γ ∇*220 }



Proof: Subtracting S y2 from both sides and taking expectations of equation (17), we have  k ’ {Q γ ∇* + γ ∇* (Q + Q ) + (a − g ) γ ∇* }  2 040 2 3 1 220  1 1 1 040  B(T ) = S   (18) ’ * * * +k2 {Q4 γ1∇040 + γ 2 ∇040 (Q5 + Q6 ) + (c − g2 )γ ∇220 }   ’ Theorem 3.3: Mean square error of Tpr (1) , up to the first order of approximation is given as:  ∇*  MSE (Tpr’ (1) ) = S y4  400 + k1’2 A + k2 2 B + 2 k1’C + 2 k2’ D + 2 k1’k2’ E   n    ’ pr (1)

2 y

 ∇*400  + k1’2 A3 + (1 − k1’ )2 B3 + 2 k1’C3 + 2(1 − k1’ ) D3 + 2 k1’ (1 − k1’ ) E3   n   

MSE (Tpr’ (1) ) = S y4 

200

where, A3 = η12 γ ∇*040 , B3 = η32 γ ∇*004 , C3 = η1 γ ∇*220 , D3 = η3 γ ∇*202 , E3 = η1η3 γ ∇*022 such that η1 = (a − g1 ) and η3 = (c − g2 ) Proof: from equation (18), we have (Tpr’ (1) − S y2 )2 = S y4  εo + k1’η1 (ε1 − ε1’ ) + k2’ η3 (ε2 − ε2’ ) 2 After squaring and taking expectations of both sides of the above equation, up to the first order of approximation, we get ∇  + k A + (1 − k ) B + 2 k C + 2(1 − k ) D + 2 k (1 − k ) E  n *

’

MSE (Tpr (1) ) = S y

’2

400

’ 2

’

   (19)

Differentiating equation (19) partially w.r.t. to k1’ , we get the optimum value of k1’ for minimum MSE as:  B + D3 − C3 − E3   k1’* =  3  A3 + B3 − 2 E3  Putting the optimum value of k ’1* in (19), we get the minimum MSE of the ’ suggested estimator Tpr (1) . ’ ’ Case.2: When k1 + k2 ≠ 1 , where k1’ and k2’ are defined weights.   s ’   exp  a( s − s ’ )  + k  s ’ + b( s − s ’ )    s 

Tpr ( 2 ) = s y  k1  

’

 s ’   c( s − s ’ )    exp   (20)  s   s ’ + d ( s − s ’ )   2

’ 2

Theorem 3.4: The suggested estimator Tpr’ ( 2 ) in terms of εi ’s , is written as Tpr ( 2 ) = S y  k1 {1 + ε0 + η1 (ε1 − ε1 ) + ε1 Q1 + ε ’1 Q2 + ε1 ε1Q3 + η1 (ε0 ε1 − ε0 ε1 )} ’

’

+k2 {1 + ε0 + η3 (ε2 − ε2 ) + ε2 Q4 + ε ’2 Q5 + ε2 ε2 Q6 + η 3 (ε0 ε2 − ε0 ε2 )} ’

’

where, η1 , η3 , Q1 , Q2 , Q3 , Q4 , Q5 and Q6 are defined in theorem 3.1 and 3.2. Proof:

Tpr ( 2 ) = s y  k1 (1 + ε1 ’) (1 + ε1 ) ’

’

+k2 (1 + ε2 ’) (1 + ε2 )

− g2

− g1

exp {( aε1 − aε1 )(1 + ε1 + bε1 − bε1 ) ’

’ −1

’

exp {(cε2 − cε2 )(1 + ε2 + d ε2 − d ε2 ) ’

’

−1

}

(21)

After expanding and arraigning the above equation (21) up to the first order of approximation, we have

201

A Generalised Family of Estimator for Estimating Unknown Variance Using Two Auxiliary Variables

Singh, VK Singh, R

Tpr ( 2 ) = S y  k1 {1 + ε0 + η1 (ε1 − ε1 ) + ε1 Q1 + ε ’1 Q2 + ε1 ε1Q3 + η1 (ε0 ε1 − ε0 ε1 )} ’

’

+k2 {1 + ε0 + η3 (ε2 − ε2 ) + ε2 Q4 + ε ’2 Q5 + ε2 ε2 Q6 + η 3 (ε0 ε2 − ε0 ε2 )} ’

’

(22)

’

where, coefficient term of εi ’s are already defined. Theorem 3.5: Bias of suggested method Tpr’ (1) is given as b(T

’ pr ( 2 )

 k ’ {1 + Q γ ∇* + γ ∇* Q + γ ∇* Q + η γ ∇* }  1 1 040 2 040 2 2 040 3 1 220  1  )=S   ’ * * * *  + k2 {1 + Q4 γ1∇004 + γ 2 ∇004Q5 + γ 2 ∇004Q6 + η3 γ ∇202 }   2 y

Proof: Subtracting S y2 both side of equation (22), we have ’

Tpr ( 2 ) − S y

 k {1 + ε + η (ε − ε ) + ε Q + ε ’ Q + ε ε Q + η (ε ε − ε ε )}  (23)    + k {1 + ε + η (ε − ε ) + ε Q + ε ’ Q + ε ε Q + η ( ε ε − ε ε )} − 1   ’

= Sy

’

Taking expectations of both sides, we get the required bias of the suggested method as b(T

’ pr ( 2 )

 k ’ {1 + Q γ ∇* + γ ∇* Q + γ ∇* Q + η γ ∇* }  1 1 040 2 040 2 2 040 3 1 220  1  )=S    + k2’ {1 + Q4 γ1∇*004 + γ 2 ∇*004Q5 + γ 2 ∇*004Q6 + η3 γ ∇*202 }   2 y

Theorem 3.6: Mean square error of the suggested estimator Tpr’ ( 2 ) , is given by MSE (Tpr’ ( 2 ) ) = S y4 1 + k ’12 A + k ’22 B − 2 k1’C − 2 k2’ D + 2 k1’k2’ E  where A = 1 +

 

∇ 400

 

∇ 400

B = 1 +

 

C = 1 + ∇ 040 *

+ ∇ 004 η 3 γ +

{

Q1 n

 ∇ 1 + +∇ E= n    *

400

{ {

+ ∇ 040 η1 γ +

* 040

Q2 n’

{

2Q1 n 2 Q4 n

Q3 n’

η1 γ +

}

Q1 n

2Q2 n’ 2Q5 n’

2Q3 n’ 2Q6 n’

} }

 

 

Q2 n’

Q3 n’

}

 

+ 4 η 3 γ ∇ 202  ,

+ η1 γ ∇ 220  , D = 1 + ∇ 004 *

 

+ 4 η1 γ∇ 220  ,

{

Q4 n

Q5 n’

n’

}

 

+ η 3 γ ∇ 202  *

}

   n n’ n’  + 2 η γ ∇ + 2 η γ∇ + η η γ ∇  2

+ ∇ 004 η 3 γ +

202

Q6 *

220

022

Proof: we have  k {1 + ε + η (ε − ε ) + ε Q + ε ’ Q + ε ε Q + η (ε ε − ε ε )}   − S = S    + k {1 + ε + η (ε − ε ) + ε Q + ε ’ Q + ε ε Q + η (ε ε − ε ε )} − 1 ’

’

Tpr ( 2 )

’

1 1

’

After squaring and taking expectations 6f both sides of the above equation up to the first order of approximation, MSE of the estimator

MSE (Tpr’ ( 2 ) ) = S y4 1 + k ’12 A + k ’22 B − 2 k1’C − 2 k2’ D + 2 k1’k2’ E 

(24)

The optimum value of k1’ and k2’ is obtained by partially differentiating equation (24) w.r.t. k1’ and k2’ for minimum MSE, as  AD − CE   BC − DE  and k *’2 =  k *’1 =   2  AB − E 2   AB − E  Putting the optimum values of k ’1* and k ’*2 in (19), we get the minimum MSE (Tpr’ ( 2 ) ) . 6. Empirical Study Population 1: For empirical study we have taken the data of [10], which contain the data of 34 villages. The variables are: y- Area under wheat in 1964. x- Area under wheat in 1963. z- Cultivated area in 1961. Also, ∇400 = 3.726, ∇040 = 2.912, ∇004 = 2.808, ∇220 = 3.105, ∇202 = 2.979, ∇022 = 2.738

S y2 = 22564.56, S x2 = 197095.3, Sz2 = 2652.05, S yx = 60304.01, S yz = 22158.05 n = 7, n’= 15 N = 34 Population 2: The data for the empirical study are taken from [1]. The population consists of 340 villages. The variables are: y- Number of literate persons. x- Number of household. z- Total population in the village. To estimate the variance of y, we have used x and z as the prior information. From the data set, we have

203

A Generalised Family of Estimator for Estimating Unknown Variance Using Two Auxiliary Variables

Singh, VK Singh, R

Table 1: Percentage Relative Efficiency (PRE) for population-1. Estimator

PRE

S y2

100.00

135.2941

Tpr(1)

Tpr(2)

Tpr’ (1)

Tpr’ ( 2 )

-1

658.7851

-1

927.2479

897.380

-1

4417.39

-1

5256.36

-1

1365.2191

-1

182.6071

-1

190.078

-1

186.5093

-1

404.3805

-1

347.8099

-1

342.1697

Note: Here the notations Tpr (1) , Tpr ( 2 ) stands for suggested estimators in single phase and respectively Tpr’ (1) , Tpr’ ( 2 ) is for two phase sampling. N = 340, n ’= 120, n = 50, S y2 = 71379.47, S x2 = 11838.85, Sz2 = 691820.23 ∇*400 = 9.90334289, ∇*040 = 7.05448224, ∇*004 = 8.2552346, ∇*220 = 6.331398563

∇*202 = 8.12904924, ∇*022 = 6.13646859 The percentage relative efficiency (PRE) of estimator is defined as

PRE (*) =

VAR(S y2 ) MSE (*)

204

×100

Table 2: Percentage Relative Efficiency (PRE) for Population-2. Estimator

PRE

S y2

100.00

115.1561

Tpr (1)

Tpr ( 2 )

Tpr’ (1)

Tpr’ ( 2 )

-1

521.62661

-1

342.12

-1

409.4370

-1

868.2550

-1

2494.1349

-1

709.3792

-1

189.21

-1

169.3268

-1

315.53

-1

342.16

-1

404.38

170.3568

Note: Here the notations Tpr (1) , Tpr ( 2 ) stands for suggested estimators in single phase and Tpr’ (1) , Tpr’ ( 2 ) stands for suggested estimators in two phase sampling. Conclusion From the above Table 1 and Table 2, we observed that, the suggested estimators Tpr (1) , Tpr ( 2 ) (in single phase sampling) and Tpr’ (1) , Tpr’ ( 2 ) (in two phase sampling) perform much better than the usual unbiased and ratio estimator. References [1] Ahmed M.S.: Some Estimation Procedure Using Multivariate Auxiliary Information in Sample Surveys. Ph.D thesis, Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh-202002, India (1995).

205

A Generalised Family of Estimator for Estimating Unknown Variance Using Two Auxiliary Variables

Singh, VK Singh, R

[2] Grover, L. K .: A correction note on improvement in variance estimation using auxiliary information. Commun. Stat. Theo. Meth. 39:753–764 (2010). http://dx.doi.org/10.1080/03610920902785786 [3] Isaki, C.T.: Variance estimation using auxiliary information. J. Amer. Statist. Assoc.78:117–123 (1983). http://dx.doi.org/10.1080/01621459.1983.10477939 [4] Jhajj, H .S., Sharma, M. K. and Grover, L. K.: An efficient class of chain estimators of population variance under sub-sampling scheme. J. Japan Stat. Soc., 35(2), 273-286 (2005). http://dx.doi.org/10.14490/jjss.35.273 [5] Kadilar, C., Cingi, H.: Improvement in variance estimation using auxiliary information. Hacettepe J. Math. Statist. 35(1):111–115 (2006). [6] Kadilar, C., Cingi, H. : Improvement in variance estimation in simple random sampling. Commun. Statist. Theory. Meth. 36:2075–2081(2007). http://dx.doi.org/10.1080/03610920601144046 [7] Kadilar, C., Cingi, H.: A new estimator using two auxiliary variables. Applied mathematics and computation 162 901-908 (2005). http://dx.doi.org/10.1016/j.amc.2003.12.130 [8] Koyuncu, N.: Improved estimation of population mean in stratified random sampling using information on auxiliary attribute. Proceeding of 59-th ISI world Statistics Congress, Hong Kong, China, August, pp: 25-30 (2013). [9] Lu, J., Yan, Z. ,Ding, C., Hong, Z.: The chain ratio estimator using two auxiliary information. International Conference on Computer and Communication Technologies in Agriculture Engineering 586-589 (2010). [10] Murthy, M.N.: Sampling Theory and Methods (Statistical Publishing Society, India) (1967). [11] Shabbir, J. and Gupta, S. : On improvement in variance estimation using auxiliary information. Commun. Statist. Theor. Meth. 36(12):2177–2185 (2007). http://dx.doi.org/10.1080/03610920701215092 [12] Singh, H.P. and Singh, R.: Improved ratio-type estimator for variance using auxiliary information. J Indian Soc Agric Stat 54(3):276–287 (2001). [13] Singh, H. P. and Singh, R. : Estimation of variance through regression approach in two phase sampling. Aligarh Journal of Statistics, 23, 13-30 (2003). [14] Singh, H. P. and Solanki, R.S.: A new procedure for variance estimation in simple random sampling using auxiliary information. Stat. Paps. DOI 10.1007/s00362012-0445-2 (2012). http://dx.doi.org/10.1007/s00362-012-0445-2 [15] Singh, R., Chauhan, P., Sawan, N.: On linear combination of Ratio-product type exponential estimator for estimating finite population mean. Statistics in Transition 9(1) -115 (2008a). [16] Singh, R., Chauhan, P., Sawan, N. and Smarandache,F. : Almost unbiased ratio and product type estimator of finite population variance using the knowledge of kurtosis of an auxiliary variable in sample surveys. Octogon Mathematical Journal, Vol. 16, No. 1, 123-130 (2008b).

206

A Generalised [17] Singh, R., Kumar, M. , Smarandache, F. : Almost Unbiased Estimator for Estimating Population Mean Using Known Value of Some Population Family of Estimator for Estimating Parameter(s); Pak.j.stat.oper.res. 76 ( 2), 63-76 (2008). [18] Singh, R., Chauhan, P., Sawan, N. and Smarandache, F.: Improved Exponential Unknown Variance Using Two Estimator for Population Variance Using Two Auxiliary Variables. Italian Jour. Auxiliary Variables Of Pure and Appld. Math., 28, 103-110(2011). [19] Singh, V. K., Singh, R. and Florentin, S.: Some improved estimators for population variance using two auxiliary variables in double sampling. on improvement in estimating population parameter(s) using auxiliary information. Educational Publishing & Journal of Matter Regularity (Beijing) (2013). [20] Singh, V. K., Singh, R.: Predictive estimation of finite population mean using generalised family of estimators , accepted in JTSA(2014).

207