A new improved difference-cum-exponential ratio type estimator in systematic sampling using two auxiliary variables

(1)

RESEARCH ARTICLE

J.Natn.Sci.Foundation Sri Lanka 2020 48 (1): 27 - 36 DOI: http://dx.doi.org/10.4038/jnsfsr.v48i1.9931

A new improved difference-cum-exponential ratio type estimator in systematic sampling using two auxiliary variables

J Shabbir^1*, S Masood² and S Gupta³

1 Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan.

2 Department of Mathematics and Statistics, PMAS-Arid Agriculture University Rawalpindi, Pakistan.

3 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, USA.

Submitted: 22 May 2018; Revised: 23 January 2019; Accepted: 27 September 2019

* Corresponding author ([email protected]; https://orcid.org/0000-0002-0035-7072) Abstract: A difference-cum-exponential ratio type estimator

was proposed for estimating the finite population mean using two auxiliary variables under systematic sampling. Expressions for the biases and mean square errors (MSEs) were derived up to first order of approximation. It was observed that the proposed estimator is more efficient than the usual sample mean estimator, traditional ratio estimator, exponential-ratio estimator and many other recently proposed difference type estimators in terms of MSEs. Four datasets were used for efficiency comparisons.

Keywords: Bias, efficiency, MSE, systematic sampling.

INTRODUCTION

The use of auxiliary information in sample survey in an appropriate way may increase the precision of estimators by taking advantages of correlation between the study variable and the auxiliary variable. The ratio, product, exponential and regression estimators have frequently been used by many researchers in different forms either at the estimation stage or at designing stage or at both stages. In daily life information on single as well as two auxiliary variables are commonly used to enhance the precision of estimators. For example: (i) let y be the yield of a particular crop based on the area of a crop (x) and the amount of water utilisation (z); (ii) let y be the electricity consumed in the households (HHDs) based on income of the HHDs (x) and size of the HHDs (z) and (iii) let y be the average grades of the undergraduate students based

on the number of study hours used by students (x) and intelligence coefficient (IQ) level of the students (z). In all these examples, we are interested to use the auxiliary information in parallel to the study variable. Madow and Madow (1944), Cochran (1946) and Gautschi (1957) were among the earlier contributors in the area of systematic sampling. Other authors who have contributed in this area include: Swain (1964), Kushwaha and Singh (1989), Banarasi et al. (1993), Singh and Singh (1998), Singh et al. (2011), Singh and Jatwa (2012), Tailor et al.

(2013), Khan and Singh (2015), Khan (2016), Pal and Singh (2017), Kocyigit and Cingi (2017), Riaz et al.

(2017), Tailor and Mishra (2018), Qureshi et al. (2018) and Mishra et al. (2018). The main objective of this study was to construct a new estimator by taking advantage of two auxiliary variables to improve the efficiency of the estimator in systematic sampling.

METHODOLOGY

Consider a finite population U={U U1, ,... ,...,2 U_i N} of Nordered identifiable units. A sample of size n units is selected by first selecting an observation from the first k units and then selecting every k^th unit thereafter. Thus, we can obtain k possible samples each of size n such that nk N= . Let y_ij and ^{( , )}x zij ij

( 1,2,3,..., ;i= k j=1,2,3,..., )n be the observed values of the study variable ( )y and the auxiliary variables ( , )x z , respectively for the j^th unit in the i^th possible

(2)

sample. Let ⁽ ⁾

1 1

1 ^k ⁿ

sys

i j ij

y y

n = =

=

∑∑

^, ⁽ ⁾

1 1

1 ^k ⁿ

sys

i j ij

x x

n = =

=

∑∑

^and

( )

1 1

1 ^k ⁿ

sys ij

i j

z z

n = =

=

∑∑

be the systematic sample means corresponding to population means

1 1

1 ^k ⁿ ij,

i j

Y y

N = =

=

∑∑

1 1

1 ^k ⁿ

i j ij

X x

N = =

=

∑∑

^and

1 1

1 ^k ⁿ

i j ij

Z z

N = =

=

∑∑

, respectively.

Let ρ_y^∗= + −1 ( 1) ,n ρ_yy ρ_x^∗= +1 (n−1)ρ_xx and 1 ( 1) ,

z n zz

ρ^∗= + − ρ where ⁽ ⁾⁽ ₂ ⁾,

( )

ij ij

yy

ij

E y Y y Y E y Y

ρ = ⁻ ^′⁻

−

2

( )( )

( )

ij ij

xx

ij

E x X x X E x X

ρ = ⁻ ^′⁻

− and ⁽ ⁾⁽ ₂ ⁾

( )

ij ij

zz

ij

E z Z z Z E z Z

ρ = ⁻ ^′⁻

− are

the intra-class correlation coefficients between pairs of units within the same systematic sample for y, x and ( , )y z, respectively. Let ρyx^, ρ_yz and ρxz be the usual population correlation coefficients between ( , ),y x

( , )y z and ( , )x z , respectively. Let C_y ^S^y

= Y , _C_x ^S^x

=X and C_z ^S^z

= Z be the coefficients of variation of y, x and ( , )y z, respectively, where

2

1 1

1 ( ) ,

1

k n

y ij

i j

S y Y

N = =

= −

−

∑∑

²

1 1

1 ( )

1

k n

x ij

i j

S x X

N = =

= −

−

∑∑

and ²

1 1

1 ( )

1

k n

z ij

i j

S z Z

N = =

= −

−

∑∑

be the population

standard deviations for y, x and ( , )y z, respectively.

We also define the following error terms.

Let ₀ y⁽^sys⁾ 1

ζ = Y − , 1 x⁽^sys⁾ 1

ζ = X − , 2 z⁽^sys⁾ 1

ζ = Z − , such that E^{( ) 0}ζi = for ( 0,1,2)i = . Also E( )ζ0² = Φρ_y^∗C_y²,

2 2

( )1 x x

Eζ = Φρ^∗C , E( )ζ2² = Φρ_z^∗C_z², E(ζ ζ0 1)= ΦC_yx ρ ρ_{y x}^{∗ ∗}, ( 0 2) yz y z

Eζ ζ = ΦC ρ ρ^{∗ ∗} and E(ζ ζ_{1 2})= ΦC_xz ρ ρ_{x z}^{∗ ∗}, where

yx yx y x

C =ρ C C , Cyz=ρyzC Cy z, C_xz=ρ_{xz x z}C C , and

1 N

nN Φ = − .

Now we discuss some of the existing estimators.

The usual sample mean estimator is given by

( )

ˆ sys

Y = y ^...(1)

The variance of ˆY₀ is given by 2 2

ˆ0

( ) _{y y}

Var Y = ΦY ρ^∗C ...(2) Single auxiliary variable

(i) The traditional ratio estimator is given by

( ) ( )

ˆ ^sys

R sys

Y y X x

 

=  

  ...(3)

The bias and MSE of ˆ

YR to first order of approximation are respectively given by

{

²

}

( )ˆ_R _x _x _yx _{y x}

B Y ≅ ΦY ρ^∗C −C ρ ρ^∗ ^∗ ...(4) and

{ }

2 2 2

( )ˆ_R _y _y _{x x} 2 _yx _{y x}

MSE Y ≅ ΦY ρ^∗C +ρ^∗C − C ρ ρ^∗ ^∗ ...(5) (ii) The usual exponential ratio type estimator is given by

( ) ( )

( )

ˆ_RE ^sys exp X x^sys_sys Y y

X x

 − 

=  +  ...(6)

The bias and MSE of Y^ˆ_RE to first order of approximation are respectively given by

3 2 1

(ˆ )

8 2

RE x x yx y x

B Y ≅ ΦY^ ρ^∗C − C ρ ρ^∗ ^∗^

  ...(7)

and

2 2 1 2

(ˆ )

RE y y 4 x x yx y x

MSE Y ≅ ΦY ^ρ^∗C + ρ^∗C −C ρ ρ^∗ ^∗^

  ...(8)

(iii) The usual difference estimator is given by

( )

( ) ( )

ˆ ^sys ^sys

YD =y +d X x− , ...(9) where d is the constant.

The minimum variance of Y^ˆ_D at optimum value of d

i.e.

yx y * *y x opt

x x

Y C

d XC

ρ ρ ρ

ρ^∗

= , is given by

Var Y( )ˆD min ≅ ΦY²ρy y^∗C²

(

1−ρ²yx

)

=MSE Y( )ˆD min

...(10)

(3)

unbiased linear estimator which is given by

( ) ( )

( )

1 2 3

ˆ ^sys ^sys ^sys

KC

ax b ax b

Y y q q q

aX b aX b

α δ

  +   +  

 

=  +  +  +  +  , ...(11) where ( , ) Rα δ ∈ and q i =_i( 1,2,3) are constants and a and b are the functions of known population parameters, which may be the population mean, population coefficient of variation, population coefficient of skewness and population coefficient kurtosis of the auxiliary variable x.

The minimum MSE of ˆ

YKC at optimum value of

2 3

( )

AG= qα+qδ i.e.

*

( ) *

yx y y

G opt

x x

A C

gC

ρ ρ

= − ρ , where

g aX

=aX b

+ , is given by

( )

2 2 2

ˆ min

( _KC) _{y y} 1 _yx

MSE Y ≅ ΦY ρ^∗C −ρ , ...(12) which is equal to the variance of the linear regression estimator.

(v) Riaz et al. (2017) suggested the following class of estimators in systematic sampling:

Yˆ_RDS

{

t y¹ ⁽^sys⁾ t X x²

(

⁽^sys⁾

) }

exp X x⁽(^sys_sys⁾)

γ X x

 − 

= + −  + ,

where t i =_i( 1,2) are constants and ^γ is the scalar quantity.

For γ=1, the above estimator becomes:

( )

{

¹ ⁽ ⁾ ² ⁽ ⁾

}

⁽⁽ ⁾⁾

ˆRDS ^sys ^sys exp ^syssys

Y t y t X x X x

X x

 − 

= + −  + ^,

...(13)

The bias of ˆ

YRDS to first order of approximation is given by

2 2

1 1 3 1 1 2

(ˆ ) ( 1)

8 2 2

RDS x x yx y x x x

B Y ≅ t − Y t Y+ Φ ^ ρ^∗C − C ρ ρ^∗ ^∗^+ t X CΦ ρ^∗

 

2 2

1 1 3 1 1 2

(ˆ ) ( 1)

8 2 2

RDS x x yx y x x x

B Y ≅ t − Y t Y+ Φ ^ ρ^∗C − C ρ ρ^∗ ^∗^+ t X CΦ ρ^∗

  ...(14)

The MSE of ˆ

YRDS is given by

2 2 2 2 2 2 2

1 1 1 1 2 1 2

(ˆRDS) ( 1) 2

MSE Y ≅ t − Y +t Y A t X B t Y C t YXB+ − − + t t D

2 2 2 2 2 2 2

1 1 1 1 2 1 2

(ˆ_RDS) ( 1) 2

MSE Y ≅ t − Y +t Y A t X B t Y C t YXB+ − − + t t D,

where ^A^{= Φ}

(

^ρ^y^∗^C^y²⁺^ρ^x^∗^C^x²⁻²^C^yx ^{ρ ρ}^{y x}^∗ ^∗

)

^,

x x2

B= Φρ^∗C , ³ ²

4 ^x ^x ^yx ^{y x} C= Φ^ ρ^∗C −C ρ ρ^∗ ^∗^,

(

^{x x}² ^yx ^{y x}

)

D= Φ ρ^∗C −C ρ ρ^∗ ^∗ .

The minimum MSE of ˆ

YRDS at optimum values of ( 1,2)

t i =i i.e. ₁₍ ₎ ⁽ ₂ ²⁾

2( )

opt

B C D

t AB D B

= − +

− + and

2( ) 2

( 2 )

2 ( )

opt

Y AB CD B D

t X AB D B

− + −

= − + is given by

(

² ² ²

)

min 2 2

2 4 4 4

ˆ 1

( ) 1

RDS 4

AB BC BCD B BC BD B

MSE Y Y

AB D B

 + − + + − + 

 

≅  − − + 

(

² ² ²

)

min 2 2

2 4 4 4

ˆ 1

( ) 1

RDS 4

AB BC BCD B BC BD B

MSE Y Y

AB D B

 + − + + − + 

 

≅  − − + 

...(15)

Two auxiliary variables

(i) The traditional ratio-ratio estimator using two auxiliary variables is given by

( )

( ) ( )

ˆ ^sys

RR sys sys

X Z

Y y

x z

  

=   

   ...(16)

The bias and MSE of ˆ

YRR to first order of approximation are respectively given by

{

² ²

}

( )ˆ_RR _{x x} _z _z _xz _{x z} _yx _{y x} _yz _{y z} B Y ≅ ΦY ρ^∗C +ρ^∗C +C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗

...(17)

and^{MSE Y}^{( )}^ˆ^RR ^{≅ Φ}^Y²

{

^ρ^y^∗^C^y²⁺^ρ^{x x}^∗^C²⁺^ρ^z^∗^C^z²⁺²^C^xz ^{ρ ρ}^{x z}^∗ ^∗⁻²^C^yx ^{ρ ρ}^{y x}^∗ ^∗⁻²^C^yz ^{ρ ρ}^{y z}^∗ ^∗

}

{ }

2 2 2 2

( )ˆRR y y x x z z 2 xz x z 2 yx y x 2 yz y z

MSE Y ≅ ΦY ρ^∗C +ρ^∗C +ρ^∗C + C ρ ρ^∗ ^∗− C ρ ρ^∗ ^∗− C ρ ρ^∗ ^∗

...(18) (ii) Tailor et al. (2013) suggested the ratio-product estimator which is given by

(4)

( ) ( )

( )

ˆ ^sys ^sys

RP sys

X z Y y

x Z

 

=   

   ...(19)

(iii) The bias and MSE of Y^ˆRP to first order of approximation are given by

{

²

}

( )ˆ_RP _{x x} _xz _{x z} _yx _{y x} _yz _{y z} B Y ≅ ΦY ρ^∗C −C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗+C ρ ρ^∗ ^∗

...(20)

and^{MSE Y}^{( )}^ˆ^RP ^{≅ Φ}^Y²

{

^ρ^y^∗^C²^y⁺^ρ^{x x}^∗^C²⁺^ρ^z^∗^C^z²⁻²^C^xz ^{ρ ρ}^{x z}^∗ ^∗⁻²^C^yx ^{ρ ρ}^{y x}^∗ ^∗⁺²^C^yz ^{ρ ρ}^{y z}^∗ ^∗

}

{ }

2 2 2 2

( )ˆ_RP _y _y _{x x} _z _z 2 _xz _{x z} 2 _yx _{y x} 2 _yz _{y z} MSE Y ≅ ΦY ρ^∗C +ρ^∗C +ρ^∗C − C ρ ρ^∗ ^∗− C ρ ρ^∗ ^∗+ C ρ ρ^∗ ^∗

...(21) (iv) The exponential ratio-ratio type estimator is given by

( ) ( )

( )

( ) ( )

ˆ_RRE ^sys exp X x ^sys_sys exp Z z ^sys_sys

Y y

X x Z z

 −   − 

=  +   +  ...(22)

The bias and MSE of Y^ˆRRE to first order of approximation are given by

(

² ²

)

3 1 1 1

(ˆ )

8 4 2 2

RRE x x z z xz x z yx y x yz y z

B Y ≅ ΦY^ ρ^∗C +ρ^∗C + C ρ ρ^∗ ^∗ − C ρ ρ^∗ ^∗− C ρ ρ^∗ ^∗^

 

(

² ²

)

3 1 1 1

(ˆ )

8 4 2 2

RRE x x z z xz x z yx y x yz y z

B Y ≅ ΦY^ ρ^∗C +ρ^∗C + C ρ ρ^∗ ^∗− C ρ ρ^∗ ^∗ − C ρ ρ^∗ ^∗^

  ...(23)

and

2 2 1 2 1 2 1

(ˆ )

4 4 2

RRE y y x x z z xz x z yx y x yz y z

MSE Y ≅ ΦY ^ρ^∗C + ρ^∗C + ρ^∗C + C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗^

 

2 2 1 2 1 2 1

(ˆ )

4 4 2

RRE y y x x z z xz x z yx y x yz y z

MSE Y ≅ ΦY ^ρ^∗C + ρ^∗C + ρ^∗C + C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗^

  ...(24)

(v) Tailor and Mishra (2018) suggested the exponential ratio-product type estimator, which is given by

( ) ( )

( )

( ) ( )

ˆ_RPE ^sys exp X x ^sys_sys exp z ^sys_sys Z

Y y

X x z Z

 −   − 

=  +   +  ...(25)

The bias and MSE of Y^ˆ_RPE to first order of approximation are given by

(^ˆ ) ³ ² ¹ ² ¹ ¹ ¹

8 8 4 2 2

RPE x x z z xz x z yx y x yz y z

B Y ≅ ΦY^ ρ^∗C − ρ^∗C − C ρ ρ^∗ ^∗− C ρ ρ^∗ ^∗+ C ρ ρ^∗ ^∗^

 

2 2

3 1 1 1 1

(ˆ )

8 8 4 2 2

RPE x x z z xz x z yx y x yz y z

B Y ≅ ΦY^ ρ^∗C − ρ^∗C − C ρ ρ^∗ ^∗ − C ρ ρ^∗ ^∗ + C ρ ρ^∗ ^∗^

  ...(26)

and

2 2 1 2 1 2 1

(ˆ )

4 4 2

RPE y y x x z z xz x z yx y x yz y z

MSE Y ≅ ΦY ^ρ^∗C + ρ^∗C + ρ^∗C − C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗+C ρ ρ^∗ ^∗^

 

2 2 1 2 1 2 1

(ˆ )

4 4 2

RPE y y x x z z xz x z yx y x yz y z

MSE Y ≅ ΦY ^ρ^∗C + ρ^∗C + ρ^∗C − C ρ ρ^∗ ^∗−C ρ ρ^∗ ^∗+C ρ ρ^∗ ^∗^

  ...(27)

(vi) The traditional difference-difference type estimator is given by

( ) ( )

( ) ( ) ( )

1 2

ˆ ^sys ^sys ^sys

YDD=y +d X x− +d Z z−

, ...(28) where d₁ and d₂ are constants.

The minimum variance of Y^ˆ_DD, at optimum values

of d₁ and d₂ i.e.

( )

*

1( ) * 1 2

y y yx yz xz

opt

x x xz

d YC

XC

ρ ρ ρ ρ

ρ ρ

= −

− and

( )

*

2( ) * 1 2

y y yz yx xz

opt

z z xz

d YC

ZC

ρ ρ ρ ρ

ρ ρ

= −

− , is given by

( )

2 2 2

min . min

ˆ ˆ

( _DD) _{y y} 1 _{y xz} ( _DD) Var Y ≅ ΦY ρ^∗C −R =MSE Y ,

...(29)

where

2 2

2. 2

2 1

yx yz yx yz xz

y xz

xz

R ρ ρ ρ ρ ρ

ρ

+ −

= − is the multiple

correlation coefficient of y on x and z.

(vii) Khan and Singh (2015) and Khan (2016) suggested the following estimators respectively:

1 ( ) 2

( )

ˆ ( )

yz sys sys

KS sys

yx

Z b z Z Y y X

X b x X Z

δ δ

   + − 

   

=  + −      , where δ =_i( 1,2)i are constants and b_yx and b_yz are the sample regression coefficients and

( )

( ) ( )

( )

( ) (

( )

( ) ( )

ˆ exp exp

1 1

sys sys

sys

K sys sys

f X x h Z z

Y y

X g x Z η z

 −   − 

   

=  + −   + − ,

where −∞ < < ∞f , −∞ < < ∞h , g >0 and η >0. Note that minimum MSEs of Y^ˆ_KS and Y^ˆ_K are exactly equal to the minimum variance of Y^ˆ_DD to first order of approximation but the estimator _Y^ˆ_DD is preferable because of unbiasedness.

(viii) Qureshi et al. (2018) introduced the following estimator for population mean

(5)

( )

1 (

( )

( ) ( )

2

ˆ exp

1

sys

QKH sys sys sys

X Z z

Y y

x Z z

ν

α ν

 − 

   

=    + − , ...(30)

where v₁ and v₂ are constants and α is the scalar quantity, which takes values (0, 1, 1)− + .

The bias of Y^ˆ_QKH reported by Qureshi et al. (2018) is given by

( ) { }

2 2

1 2 1

(ˆ ) 1 2( 1) ,

QKH x yx yx 2 z yx yx zx zx

B Y Y C H C a H H

a

ν ρ^∗ α α ρ^∗ ν ρ^∗

 

≅  − + − + − − 

( ) {

}

2 2

1 2 1

(ˆ ) 1 2( 1) ,

QKH x yx yx 2 z yx yx zx zx

B Y Y C H C a H H

a

ν ρ^∗ α α ρ^∗ ν ρ^∗

 

≅  − + − + − − 

where Hij=ρijC Ci/ j, 1 ( 1) 1 ( 1) ⁱ

ij

j

n n ρ ρ

ρ

∗= + −

+ − , a=α α/ ,^*

* (1 2)

yx yz zx

yx

xz

H H H

α ρ

ρ

∗ −

= − .

Note: It is observed that the bias expression of estimator ˆQKH

Y by Qureshi et al. (2018) is not correct. Using equation (30), the correct bias expression is:

( ) 1 1

2 2

(ˆ_{QKH c}) _yx _{y x} _yz _{y z} _xz _{x z}

B Y Y νC ρ ρ α C ρ ρ αν C ρ ρ

ν ν

∗ ∗ ∗ ∗ ∗ ∗

  

≅ Φ − − + 

  



( ) 1 1

2 2

(ˆ_{QKH c} ) _yx _{y x} _yz _{y z} _xz _{x z}

B Y Y νC ρ ρ α C ρ ρ αν C ρ ρ

ν ν

∗ ∗ ∗ ∗ ∗ ∗

  

≅ Φ − − + 

  



2 2 2

1 1 2 2

2 2 2

1 ( 1)

2ν ν ρ^{x x}C α α 2α ρ^zC^z

ν ν ν

∗   ∗ 

+ + + − +  

  

2 2 2

1 1 2 2

2 2 2

1 ( 1)2ν ν ρ^{x x}C α α 2α ρ^zC^z

ν ν ν

∗   ∗ 

+ + + − +  

   ...(31)

The minimum MSE of ˆ

YQKH at optimum values of ν₁ and _{( / )}α ν₂ i.e.

( ) ( )

( )

*

1 * 1 2

y y yx yz xz

opt

x x xz

v C

C

ρ ρ ρ ρ

ρ ρ

= −

− and

( )

*

2 * 2

( / )

1

y y yz yx xz

opt

z z xz

C C

ρ ρ ρ ρ

α ν ρ ρ

= −

− is given by

( )

2 2 2

min .

(ˆQKH) y y 1 y xz

MSE Y ≅ ΦY ρ^∗C −R

...(32) (ix) Mishra et al. (2018) suggested the following class of estimators:

0 0 1 1

ˆ ˆ ˆ ˆ

MSS RP RPE

Y =δ Y +δY +δY _{, ...(33)}

where ²

0 i 1

i

δ

=

∑

= are constants; Y iˆ ( 0, ,_i = RP RPE) are defined earlier.

Mishra et al. (2018) derived the bias and minimum MSE of Y^ˆMSSup to first order of approximation as:

( ) (

²

)

2 2

1 1

(ˆ )

2 4

MSS yz y z yx y x x x yz y z

B Y ≅ ΦY^^δ +δ ^ C ρ ρ^{∗ ∗}−C ρ ρ^{∗ ∗} +^δ +δ ^ C ρ^∗−C ρ ρ^{∗ ∗} ^

( ) (

²

)

2 2

1 1

(ˆ )

2 4

MSS yz y z yx y x x x yz y z

B Y ≅ ΦY^^δ +δ ^ C ρ ρ^{∗ ∗}−C ρ ρ^{∗ ∗} +^δ +δ ^ C ρ^∗−C ρ ρ^{∗ ∗} ^ ...(34)

and

( )

²

2 2

min 2 2

(ˆ ) 3

2

yz y z yx y x

MSS y y

x x z z xz x z

C C

MSE Y Y C

C C C

ρ ρ ρ ρ

ρ ρ ρ ρ ρ

∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗

 − 

 

≅ Φ  + 

+ −

 

 

...(35)

The optimum value of 1 2² δ δ

 + 

 

  is:

( )

1 22 2 2 2

yz y z yx y x

opt x x z z xz x z

C C

C C C

ρ ρ ρ ρ

δ δ

ρ ρ ρ ρ

∗ ∗ ∗ ∗

 +  = −

 

  + −

...(36)

Expressions given in equations (34), (35) and (36) are not correct.

Solving equation (33), the correct expressions of bias and MSE of estimator Y^ˆ_MSS to first order of approximation are:

( )

2 2

( ) 1 1

(ˆ )

2 4

MSS c yz y z yx y x xz x z

B Y ≅ ΦY^δ +δ ^ C ρ ρ^{∗ ∗}−C ρ ρ^{∗ ∗} −^δ +δ ^C ρ ρ^{∗ ∗}

( )

2 2

( ) 1 1

(ˆ )

2 4

MSS c yz y z yx y x xz x z

B Y ≅ ΦY δ +δ  C ρ ρ^{∗ ∗}−C ρ ρ^{∗ ∗} −δ +δ C ρ ρ^{∗ ∗}

2 2

1 3

8 C^{x x} 8 C^z ^z

δ δ

δ ρ^∗ ρ^∗^

 

+ +  − 

...(37)

and

2 2

( )

²

( ) min 2 2

(ˆ )

2

yz y z yx y x

MSS c y y

x x z z xz x z

C C

MSE Y Y C

C C C

ρ ρ ρ ρ

ρ ρ ρ ρ ρ

∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗

 − 

 

≅ Φ  − + − 

 

...(38)

The correct optimum value of ₁ ² 2 δ δ

 + 

 

  is:

2 2

1 1

2 _{opt c}( ) 2 _opt

δ δ

 +  = − + 

   

    ...(39)

Proposed estimator

Motivated by Khan (2016) and Riaz et al. (2017), the following difference-cum-exponential ratio type estimator is proposed for the population mean under systematic sampling: