Int. J. Anal. Appl. (2023), 21:14

Estimation of Finite Population Mean by Utilizing the Auxiliary and Square of the

Auxiliary Information

Saddam Hussain1, Anum Iftikhar2, Kleem Ullah3, Gulnaz Atta4, Usman Ali5, Ulfat Parveen5,

Muhammad Yasir Arif5, Ather Qayyum5,∗

1Department of Statistics, University of Mianwali, Pakistan
2School of statistics, shanxi university of finance and economics Taiyuan, China

3Foundation University Medical College, Foundation University, Islamabad, Pakistan
4Department of Mathematics, University of Education Lahore, DGK campus, Pakistan

5Department of Mathematics, Institute of Southern Punjab Multan, Pakistan

∗Corresponding author: atherqayyum@isp.edu.pk

Abstract. This article fundamentally aims at the proposition of new family of estimators using auxiliary

information to assist the estimation of finite population mean of the study variable. The objectives

are achieved by devising dual use of supplementary information through straightforward manner. The

additional information is injected in mean estimating procedure by considering squared values of aux-

iliary variable. The utility of the proposed scheme is substantiated by providing rigorous comparative

account of the newly materialized structure with the well celebrated existing family of Grover and Kaur

(2014). The contemporary advents of the new family are documented throughout the article.

1. Introduction

The utility of auxiliary information to improve the effectiveness of estimation procedures estimating

the attributes of population under study is well cherished. The documented realizations of the advents

of employing auxiliary information while estimating study parameters can be traced back in eighteenth

century France. A keen review of literature substantiate that Pierre-Simon Laplace documented the

early advocacy of using supplementary information to assist the estimation of population of country.

Received: Oct. 8, 2022.

2020 Mathematics Subject Classification. 60K35.
Key words and phrases. auxiliary variable; bias; mean squared error; percentage relative efficiency; second raw moment

of auxiliary variable.

https://doi.org/10.28924/2291-8639-21-2023-14
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-14


2 Int. J. Anal. Appl. (2023), 21:14

He advised “The register of births, which are kept with care in order to assure the condition of the

citizens, can serve to determine the population of great empire without resorting a census of its

inhabitants. But for this it is necessary to know the ratio of population to annual the birth.” [16] Later

on, the applicability of the additional information to enhance the efficiency of underlying estimation

procedures was materialized by exploiting the correlation structure existent to govern both study

variable and auxiliary variable, [3]. Over the time, streams of propositions devising novel mechanism

expounding efficient estimation of study parameter can be in available literature. For a comprehensive

review of ongoing research efforts, one may consult to [4], [5], [8], [10], [11], [12], [13], [14], [21],

and [23]. It is noteworthy that these efforts comprehends the advances on two fronts, (i) – competent

use of auxiliary information and (ii) – introducing novel functional forms incorporating additional

information in estimation synergy. In adjacent past, [5] instigated the use of exponent-based formation

to inject supplementary information and thus proposed generalized family of estimators encapsulating

numerous exiting specifications. The optimality of the proposed scheme was delineated through

rigorous empirical evaluation. Motivated by the ongoing proceedings, this research targets the afore-

mentioned both fronts, simultaneously. We propose new family of estimators capable of entertaining

dual use of auxiliary information by the application of efficient exponent-based functional formation.

The comparative performance of the newly developed mechanism is documented with respect to

promising novel family of [5] estimators. The empirical evaluations are conducted by the consideration

of numerous data sets from statistics and allied research fields. This article is mainly divided into seven

major parts. Section 2 briefs about the preliminaries extensively used in this research whereas section

3 summarizes the contemporary methods. Section 4 is dedicated to expound the proposed scheme,

whereas section 5 documents the efficiency conditions. Section 6 explore the empirical performance

of the competing techniques. Lastly, section 7 comprehends the investigation by offering highlights of

the research along with few prospective research venues.

2. Notation and symbols

Let V be a finite population of N units, such as V = {V1,V2, . . . ,VN}. We draw a sample of size
n from the population through simple random sampling with out replacement scheme. Let Yi, Xi,

and Ui are study, auxiliary variable, and squared values of auxiliary variable, respectively, for the ith

(i = 1, 2, . . . ,N) unit of the population.

Let, ȳ = 1
n

n∑
i=1

yi, x̄ =
1
n

n∑
i=1

xi, and ū =
1
n

n∑
i=1

ui are sample means of the study, auxiliary variable,

and squared values of the auxiliary variable respectively.

Ȳ = 1
N

N∑
i=1

Yi, X̄ =
1
N

N∑
i=1

Xi, and Ū =
1
N

N∑
i=1

Ui, are population means of the study, auxiliary variable,

and squared values of the auxiliary variable respectively. On these grounds sample variances of study,



Int. J. Anal. Appl. (2023), 21:14 3

auxiliary variable, and squared values of auxiliary variable are defined as s2y =
1
n−1

n∑
i=1

(yi − ȳ)2, s2x =

1
n−1

n∑
i=1

(xi − x̄)2, and s2u =
1
n−1

n∑
i=1

(ui − ū)2.

Further more, let us define coefficients of variation of X, Y , and U as Cx, Cy, Cu, where

Cy = sy/Ȳ , Cx = sx/X̄, and Cu = su/Ū.

We now define error terms as e0 = (ȳ − Ȳ )/Ȳ , e1 = (x̄ − X̄)/X̄, e2 = (ū − Ū)/Ū,
such that E(ei ) = 0, i = 0, 1, 2. E(e20 ) = λC

2
y , E(e

2
1 ) = λC

2
x , E(e

2
2 ) = λC

2
u, where λ = (

1
n
− 1

N
)

commonly known as sample fraction.

In procession the error covariances are derived as E(e0e1) = λCyCxρyx, E(e0e2) = λCyCuρyu,

E(e1e2) = λCxCuρxu, where ρyx, ρyu, and ρxu, represents sample correlation coefficients defined

as ρyx =
Syx
SySx

, ρyu =
Syu
SySu

, and ρxu =
Sxu
SxSu

.

3. Some Existing Estimators

(i) The typically independent suggest estimator is Ȳ with variance

Var(ȳ) = λȲ 2C2y . (3.1)

(ii) [3] and [17] proposed traditional ratio type and product type estimators, ˆ̄YR and ˆ̄YP , respectively,

given by

ˆ̄YR = ȳ(
X̄

x̄
), (3.2)

ˆ̄YP = ȳ(
x̄

X̄
). (3.3)

It is a famous truth that, irrespective of the biasedness, the classical ratio and product estimator, ˆ̄YR
and ˆ̄YP , is more accurate than the mean per unit estimator Ȳ if there exist a high positive and negative

correlation between y and x, i.e ρyx > Cx/2Cy and ρyx < −Cx/2Cy.
The MSEs of ˆ̄YR and ˆ̄YP , respectively, given by

MSE( ˆ̄YR) ∼= λȲ 2(C2y + C
2
x − 2CyCxρyx ) (3.4)

MSE( ˆ̄YP ) ∼= λȲ 2(C2y + C
2
x + 2CyCxρyx ). (3.5)

Several authors have proposed a few converted ratio-kind estimators for estimating the finite population

mean with the aid of using the use of auxiliary information. Some suitable research on this path

include [2], [23], [24], [25], and many others authors. In a latest study, [15] Proposed a standard

elegance of estimators ˆ̄YK that consists of a few tailored ratio-kind estimators, given by

ˆ̄YK
∼= ȳ

{
aX̄ + b

α(aX̄ + b)) + (1 −α)(aX̄ + b)

}g
, . (3.6)



4 Int. J. Anal. Appl. (2023), 21:14

ˆ̄YK
∼= ȳ

{
aX̄ + b

α(aX̄ + b)) + (1 −α)(aX̄ + b)

}g
, (3.7)

In which a( 6= 0) and b are known recognized elements of any recognized population parameters, such
as Cx the coefficient of variation; ρyx the correlation between y and x; β1(x) the coefficient of skewness

and so on.

The minimal MSE of ˆ̄YK on the top of the line cost of (αϑg), where ϑ =
aX̄
aX̄+b

, is given by

MSEmin( ˆ̄YK) ∼= λȲ 2C2y (1 −ρ
2
yx ) (3.8)

(iii) The usual difference estimator ˆ̄YD is

ˆ̄YD = ȳ + t1(X̄ − x̄), (3.9)

In which t1 is an unknown elements. It is straightforward to expose that ˆ̄YD is unbiased. The minimal

variance of ˆ̄YD on the top of the line fee of t1, that is, t1(opt) = ρyx (Sy/Sx ), is given by

Varmin( ˆ̄YD) ∼= λȲ 2C2y (1 −ρ
2
yx ), (3.10)

which is same to the variance of the classical regression estimator ˆ̄Ylr = ȳ + b(X̄ − x̄), where b is
the slope estimator of the population regression coefficient β = t1(opt). The difference estimator

ˆ̄YD

is always better perform than the ratio type ˆ̄YR and product type ˆ̄YP estimators when estimating Ȳ .

(iv) [19] proposed an improved difference type estimatorof ˆ̄YD, given by

ˆ̄YR,D = t2ȳ + t3(X̄ − x̄), (3.11)

where t2 and t3 are selected quantities. The minimum MSE of ˆ̄YR,D at the optimum values,

t2(opt) =
1

1 + λC2y (1 −ρ2yx )
, (3.12)

and

t3(opt) =
Ȳ Cyρyx

X̄Cx [1 + λC
2
y (1 −ρ2yx )]

, (3.13)

is given by

MSEmin( ˆ̄YR,D) ∼=
λȲ 2C2y (1 −ρ2yx )

1 + λC2y (1 −ρ2yx )
. (3.14)

the above (11), it is shown that the ˆ̄YR,D is better perform than ˆ̄YD, i.e,

MSEmin( ˆ̄YR,D) ∼= Varmin( ˆ̄YD) −
λȲ 2C4y (1 −ρ2yx )2

1 + λC2y (1 −ρ2yx )
. (3.15)

(v) [1] proposed a ratio and product type exponential estimators, given by

ˆ̄YBT,R = ȳ exp
(X̄ − x̄
X̄ + x̄

)
, (3.16)

ˆ̄YBT,P = ȳ exp
(x̄ − X̄
x̄ + X̄

)
. (3.17)



Int. J. Anal. Appl. (2023), 21:14 5

The MSEs of ˆ̄YBT,R and ˆ̄YBT,P , respectively, are given by

MSEmin( ˆ̄YBT,R) ∼=
λȲ 2

4

(
4C2y + C

2
x − 4ρyxCyCx

)
, (3.18)

MSEmin( ˆ̄YBT,P ) ∼=
λȲ 2

4

(
4C2y + C

2
x + 4ρyxCyCx

)
. (3.19)

Following the work in [1], [22] proposed a generalized ratio type exponential estimator,

ˆ̄YS = ȳ exp(
a(X̄ − x̄)

a(X̄ + x̄) + 2b
). (3.20)

The minimum MSE of ˆ̄YS turns out to equivalent to Varmin( ˆ̄YD), i.e, MSEmin( ˆ̄YS) ∼= λȲ C2y (1 −ρ2yx ).
(vi) Based on the estimator [1], [19], [20], [22] proposed a estimator, given by

ˆ̄YSG = {t4ȳ + t5(X̄ − x̄)}exp
( X̄ − x̄
X̄ + x̄ + 2NX̄

)
, (3.21)

where t4 and t5 are suitably chosen constant.

Following these work, [4] proposed related estimator by combining [1] and [19]

ˆ̄YSG = {t6ȳ + t7(X̄ − x̄)}exp(
X̄ − x̄
X̄ + x̄

), (3.22)

in which t6 and t7 are elements.

In a recant study, proposed a estimators, given by

ˆ̄YGK,G = {t8ȳ + t9(X̄ − x̄)}exp(
a(X̄ − x̄)

a(X̄ + x̄) + 2b
), (3.23)

in which t8 and t9 are elements. Note that ˆ̄YGK,G contains the estimators given known in [4] and [9].

The minimum MSE of ˆ̄YGK,G at the optimum values,

t8(opt) =
8 −λϑ2C2x

8{1 + λC2y (1 −ρ2yx )}
,

and

t9(opt) =
Ȳ [λϑ3C3x + 8Cyρyx −λϑ2C2xC2yρyx − 4ϑCx

{
1 −λC2y (1 −ρ2yx )}]

8X̄Cx
{

1 + λC2y (1 −ρ2yx )
} .

is given by

MSEmin( ˆ̄YGK,G) ∼=
λȲ 2

{
64C2y (1 −ρ2yx ) −λϑ4C4x − 16λϑ2C2xC2y (1 −ρ2yx )

}
64
{

1 + λC2y (1 −ρ2yx )
} . (3.24)

The above equation can be written as, given by

MSEmin( ˆ̄YGK,G) ∼= Varmin( ˆ̄YD) −V1. (3.25)

Where V1 =
λ2Ȳ 2{ϑ2C2x +8C2y (1−ρ2yx )}2

64{1+λC2y (1−ρ2yx )}



6 Int. J. Anal. Appl. (2023), 21:14

4. Proposed Estimator

It is Well-recognized that the usage of auxiliary variables increase the precision of an estimator

each the estimation level and at the designing level. In many surveys, the auxiliary records is broadly

speaking to be had text-color the sampling design or frame. The concept is that if there exists the

ideal amount of correlation among the examine and auxiliary variables, the squared values of the

auxiliary variables also are correlated with the values of the examine variable. Thus, the squared

auxiliary variable (this is the squared values of auxiliary variable) may be taken into consideration a

brand new auxiliary variable, and this greater records may also assist us to growth the performance of

an estimator. By those notions, we endorse an advanced estimator of the finite populace mean. The

proposed estimator consists of the greater records withinside the shape of an auxiliary variable and

withinside the shape of the squared cost of the auxiliary variable.

Following [4], [5] and [20], we suggest a exponential type estimator ˆ̄Ypr, given by

ˆ̄YPr = {t10ȳ + t11(X̄ − x̄) + t12(Ū − ū)}exp
( a(X̄ − x̄)
a(X̄ + x̄) + 2b

)
, (4.1)

where t10, t11, and t12 are suitably constants, which will be determined later. Where a and b are

explained in Table1.
ˆ̄YPr also rewriting as

ˆ̄YPr = {t10Ȳ (1 + e0) − t11X̄e1 − t12Ūe2}
{

1 −
ϑe1

2
+

3ϑ2e21
8

+ . . .
}
. (4.2)

By expending (25) and upto two degree of approximation in ei, we can write

( ˆ̄YPr − Ȳ ) = −Ȳ + t10Ȳ + t10Ȳ e0 −
1

2
t10ϑȲ e1 − t11X̄e1 − t12Ūe2 −−

1

2
t10ϑȲ e0e1 +

3

8
t10ϑ

2Ȳ e21

+
1

2
t11ϑX̄e

2
1 +

1

2
t12ϑŪe2.

From (26), the bias and MSE of ˆ̄YPr up to first degree of approximation are, respectively, given by

Bias( ˆ̄YPr ) =
1

8

[
−8Ȳ + 4λϑCx (t11X̄Cx + t12ŪCuρxu) + t10Ȳ{8 + λϑCx (3ϑCx − 4Cyρyx )}

]
, (4.3)

MSEmin( ˆ̄YPr ) ∼= Ȳ 2 + t11λX̄C2x (−Ȳ ϑ + t11X̄) + t
2
12λŪ

2C2u + t12λŪCxCuρxu(−Ȳ ϑ + 2t11X̄)

+ Ȳ 2t210
[
1 + λ

{
C2y + ϑCx (ϑCx − 2Cyρyx )

}]
+

1

4
t10Ȳ

{
− 8Ȳ + λCx

{
ϑCx (−3ϑȲ

+ 8t11X̄) + 8t12ŪCuρxu + 4Cyρyx (Ȳ ϑ− 2t11X̄)
}
− 8t12λŪCyCuρyu

}
.

The values of t10, t11 and t12 obtained from (28) are, respectively, given by

t10(0pt) =
8 −λϑ2C2x

8{1 + λC2y (1 −ϕ2yxu)}
t11(opt) =

Ȳ


λϑC3x (−1 + ρ2xu) + (−8Cy + λϑ2C2xCy )(ρyx −ρyuρxu)

+4ϑCx (−1 + ρ2xu) − 1 + λC
2
y (1 −ϕ

2
yxu)




8X̄Cx (−1 + ρ2xu){1 + λC2y (1 −ϕ2yxu)}



Int. J. Anal. Appl. (2023), 21:14 7

and

t12(0pt) =
Ȳ (8 −λϑ2C2x )Cy (ρyxρxu −ρyu)

8ŪCu(−1 + ρ2xu){1 + λC2y (1 −ϕ2yxu)}
,

where ϕ2yxu =
ρ2yx +ρ

2
yu−2ρyxρyuρxu

1−ρ2xu
.

putting the above obtained values of t10, t11, and t12 in (28), and after a few simplifications, we get

the minimal MSE of ˆ̄YPr, given by

MSEmin( ˆ̄YPr ) ∼=
λȲ 2

{
64C2y (1 −ϕ2yxu) −λϑ4C4x − 16λϑC2yC2x (1 −ϕ2yxu)

}
64{1 + λC2y (1 −ϕ2yxu)}

. (4.4)

It can be shown that proposed estimator MSEmin( ˆ̄YPr ) is always better perform then the difference

estimator Varmin( ˆ̄YD), i.e.,

MSEmin( ˆ̄YPr ) ∼= Varmin( ˆ̄YD) −{V1 + V2}, (4.5)

where V1 is defined as before and V2, is given by

V2 =
λȲ 2C2y (ρyu −ρyxρxu)2(−8 + λϑ2C2x )2

64(1 −ρ2xu){1 + λC2y (1 −ρ2yx )}{1 + λC2y (1 −ϕ2yxu)}
.

Table 1. Some possible members of the suggested family of estimators

S.No a b ˆ̄YGK,G ˆ̄YPr
1 1 Cx ˆ̄YGK,G ˆ̄YPr
2 1 β2(x)

ˆ̄YGK,G
ˆ̄YPr

3 β2(x) Cx
ˆ̄YGK,G

ˆ̄YPr

4 Cx β2(x)
ˆ̄YGK,G

ˆ̄YPr

5 1 ρyx ˆ̄YGK,G ˆ̄YPr
6 Cx ρyx ˆ̄YGK,G ˆ̄YPr
7 ρyx Cx ˆ̄YGK,G ˆ̄YPr
8 β2(x) ρyx

ˆ̄YGK,G
ˆ̄YPr

9 ρyx β2(x)
ˆ̄YGK,G

ˆ̄YPr

10 1 NX̄ ˆ̄YGK,G ˆ̄YPr



8 Int. J. Anal. Appl. (2023), 21:14

Table 2. Some possible members of the suggested family of estimators

S.No a b ˆ̄YGK,G ˆ̄YPr
1 1 Cx ˆ̄YGK,G ˆ̄YPr
2 1 β2(x)

ˆ̄YGK,G
ˆ̄YPr

3 β2(x) Cx
ˆ̄YGK,G

ˆ̄YPr

4 Cx β2(x)
ˆ̄YGK,G

ˆ̄YPr

5 1 ρyx ˆ̄YGK,G ˆ̄YPr
6 Cx ρyx ˆ̄YGK,G ˆ̄YPr
7 ρyx Cx ˆ̄YGK,G ˆ̄YPr
8 β2(x) ρyx

ˆ̄YGK,G
ˆ̄YPr

9 ρyx β2(x)
ˆ̄YGK,G

ˆ̄YPr

10 1 NX̄ ˆ̄YGK,G ˆ̄YPr

5. Conclusion

This research persuaded the enhancement of efficiency of estimation procedures involving the es-

timation of finite population mean. The objectives are aimed by devising new class of estimators

facilitating the launch of dual use of auxiliary information through exponent-based formation. The du-

ality of the supplementary information is executed by considering the squares of information available

on supplementary variable. The comparative performance of the proposed approach is enumerated

by the application of well acknowledged data sets from multi-disciplinary research literature. Further-

more, rigorous evaluation analysis is conducted in comparison to the leading family of Grover and

Kaur (2014). The tedious comparative performance evaluation of contemporary methods reveals that

every instant of newly proposed family out performs every member of existing family. Moreover, this

distinction is witnessed with respect to all considered data sets. In future, it will be interesting to

extend the proposed scheme for more complicated study designs.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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https://doi.org/10.3934/math.2022547

	1. Introduction
	2. Notation and symbols
	3. Some Existing Estimators
	4. Proposed Estimator
	5. Conclusion
	References