EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6019
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Use of Predictive Technique for Estimation of
Population Mean Using Auxiliary Information:

Application on Real Data Sets

Sajid Khan1, Muhammad Farooq1, Sohaib Ahmad2,∗, Manahil SidAhmedMustafa3,
Elsiddig Idriss Mohamed3, Elfarazdag M. M.Hussein3,
Fatima Ibrahim Abdallah Albadwi4

1 Department of Statistics, University of Peshawar, Peshawar, Pakistan
2 Department of Statistics Abdul Wali Khan Univerity, Mardan, Pakistan
3 Department of Statistics, Faculty of Science, University of Tabuk, Kingdom of Saudi Arabia
4 Department of Management Information Systems, College of Business and Econometrics,
Qassim University, Buraydah, Saudi Arabia

Abstract. In this article, we used model base method (predictive approach) using auxiliary vari-
ables under simple random sampling. Based on the predictive approach the unobserved population
along with the observed ones is considered. We found the results for bias, mean squared error both
theoretically and numerically. The research analyzes sophisticated through predictive models be-
cause it successfully models complex non-linear research variable interactions combined with their
auxiliary variables. The current work is evaluated through numerous applications which produce
better improved estimation performances. The corresponding optimal strategies of the suggested
estimators are explored together with their empirical and graphical analogues with various contem-
porary estimators of population mean. Some actual data sets are used in an empirical research.
By means of a comparison between the mean square error of the proposed estimators with the
mean square error of existing estimators under which the suggested class of estimator dominates
the existing estimators. The favorable empirical results clearly demonstrate the superiority of the
proposed estimators over the existing estimators.

2020 Mathematics Subject Classifications: 62D05

Key Words and Phrases: Predictive approach, estimation of mean, bias, means squared error,
efficiency

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6019

Email addresses: sajidkhan2022@uop.edu.pk (S. Khan), m.farooq@uop.edu.pk (M. Farooq),
sohaib ahmad@awkum.edu.pk (S. Ahmad), msida@ut.edu.sa (M. S. Mustafa),
eidriss@ut.edu.sa (E. I. Mohamed), e.hussain@ut.edu.sa (E. M. M. Hussein),
F.albadwi@qu.edu.sa (F. I. A. Albadwi)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 2 of 18

1. Introduction

To achieve more efficient estimators, survey sampling is the technique that contains
design based and model based method using auxiliary information. Sample survey can
be obtain through prediction theory that is a general frame work for statistical inference
referring to a character of a finite population. In many cases, the assessment of the
study variable is established for every unit of the population before the sample is selected.
After selecting and observing the sample units, we get information about the sampled
values only; while non-sampled values remain unknown. This ignorance of y-values (study
variable) leads to a need of predicting some function of those values mathematically in
order to have an estimator or predictor for the full population. For example, it is typical
in many scientific disciplines, such as economics and medicine, for an outcome or research
variable to be costly to detect while its covariates are relatively cheap. For example,
it is very costly to measure consumption poverty since it involves administering a long
questionnaire that people must fill out over a long period of time. However, its variate,
such as asset holdings, water and lightning resources are less expensive to observe. When
destructive testing is too costly and inaccurate, non-destructive testing might be utilized
instead in some industries. This is where prediction estimators as a replacement. It must
be noted that we use the term “prediction” in the sense of making statistical guess about
the non-sampled (un-observed) y-values, not in the sense of forecasting future values. The
reactivity of the customary estimators of the population mean of sampling is really better
understood with the help of this approach. The predictive approach has been applied
by researchers for examining the existing estimators, or developing new estimators. The
authors [1–4] used the auxilary variables for estimation of population parameters under
differenct sampling schemes. The readers can find some related findings in predictive
approach under simple random sampling include [5] proposed for stratified two phase
random sampling a generalised exponential chain ratio estimator. [6] proposed a two-
phase sample predictive estimate of finite population mean in case of incomplete data. [7]
proposed ranked set sampling-based new predictive estimators. [8] proposed both internal
and external validation of predictive models by means of a small sample bias and precision
simulation analysis. [9] addressed a developing enhanced predictive estimator for finite
population mean with auxiliary information. [10] addressed population mean predictive
estimate in ranked set sampling. [11] effective finite population mean estimators derived
from predictive estimating in simple random sampling. [12–18] discussed an enhanced
estimators based on auxiliary information for predictive approach under different sampling
scheme. Authors in [19] proposed a better estimator based on predictive technique under
PPS sampling for population mean estimate.

In survey sampling, the method which is designed based rely survey scheme like strati-
fied, cluster and randomization to achieve inferences about the unknown population. These
techniques provide unbiased estimators. These methods ensure unbiased estimators and
avoid constructing assumptions about the distribution of the population by trusting sim-
ply on the sample design for inferential processes. Whenever the designed prepared well
for the collection of sample, it is more suitable and having more application. Alternatively



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 3 of 18

the model based utilizes the helping information and having statistical models for the sur-
vey scheme. To boost the efficiency of estimate, we utilize predictive approach whenever
the sample size is small. By integrating information from numerous bases, model-based
approach provides more precise estimates. In many circumstances when consistent data is
accessible, then model-based technique may outclass design-based approach; though, it is
important to carefully deliberate model assumptions. For good estimation of population
mean, predictive approaches are important for making learned predictions about the tar-
get variable using more information. By integrating information from causes other than
the sample, predictive models are capable to diminish estimation errors. With predictive
approaches, properties are assigned competently and interferences are focused where they
are most desirable, leading to better-informed policies and actions. For the assessment
of population mean the predictive approach based on simple random sampling is indis-
pensable in many domains. Mostly in situation when the sample numbers are limited,
the predictive approach boosts the efficiency of estimators. Enhancements in efficiency
and a reduction in sample inconsistency lead to more consistent estimations of population
parameters. Using this prediction method, researchers are capable to assign possessions
appropriately and make knowledgeable conclusions based on estimated demographic influ-
ences. Eventually, the social sciences, public health, and economics stand to help from a
well sympathetic of population dynamics, better decision-making, and well-organized use
of resources when simple random sampling is supplemented by predictive techniques used
to assignment population means. The research gap that this work fills is produced by the
comprehensive integration of finite population mean estimate into an integrated structure
incorporating auxiliary information considerations and a predictive technique under simple
random sampling. While some studies have looked at these components alone or in pairs,
very few have integrated them all. The new method that our study introduces combines
predictive approach estimate with basic random sampling techniques is a significant con-
tribution to the field. In sophisticated sampling circumstances, this complete background
provides a more precise technique of measuring population mean. By examining all these
fundamentals collectively, our study not only fills extant holes in the literature but also
gives a more thorough and practical solution for real-world survey circumstances where
these aspects often co-exist. Several fields, such as healthcare, weather prediction, and
finance, make use of this research in their estimation of improved population mean. As a
result, healthcare planning, weather forecasting, and financial decision-making are all en-
hanced. Improving estimating approaches and ensuring higher precision, the study takes
into account variables like predictive approach, simple random sampling, and population
mean in sample surveys.

Most current survey estimation methods including ratio, regression and difference esti-
mators require strong assumptions of linear relationships and normal data distribution to
function. The assumptions fail to maintain validity when used to analyze actual datasets
which show characteristics of complexity and non-linear behavior and contain substan-
tial noise levels. Research on predictive approaches for parameter estimation shows few
applications regarding population mean estimation with auxiliary variables. Many aca-
demic works neglect to perform comparative research on traditional estimators and modern



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 4 of 18

predictive approaches particularly when they are used with genuine datasets. The avail-
able literature lacks adequate guides for actual implementation as well as performance
standards when sampling different scenarios. Studies fail to demonstrate how machine
learning methods can be properly adapted to survey sampling applications that retain
interpretability features together with minimal bias effects. A thorough research of pop-
ulation mean estimation through predictive techniques needs immediate attention using
diverse empirical datasets for verification. The development of such research leads to
improved estimation techniques that better match modern data collection scenarios.

The continuing of the article is structured as: Section 2, includes the methods and ma-
terials regarding the considered study. Section 3, includes some relevant existing adopted
estimator based on a predictive approach under simple random sampling. The suggested
generalized estimator are given in Section 4. Section 5 include numerical study. Applica-
tions of the study are given in section 6. Argumentation of the article is given in Section
7. Lastly, Section 8, contain the concluding remarks of the article.

2. 2. Methodology

LetK = K1,K1, · · · ,KN contains of N diverse units. Suppose y be the study variable
and x and z be the auxiliary variables. Thus (yj , xj , zj), j = 1, 2, . . . N denote the jth

observations for the main and auxiliary variables. We are concerned in approximating the
population mean: Ȳ = 1

N

∑N
j=1 Yj in the occurrence of the auxiliary variables x and z.

Further, let S signify the fixed size from the population K. Let s be a member of S and
let ns signify the effective sample size in s and s̄ denote the gathering of units of K which
are not included in s.

Ȳ =
1

n

∑
jϵs

yj (sample mean)

Ȳs̄ =
1

N − n

∑
jϵs̄

yj (mean of non− population values of the study variable)

For any sϵS, we have : Ȳ = n
N Ȳs +

n
(N−n) Ȳs , ȳ = 1

n

∑
jϵs yj

That is,ȳ = Ȳs
Thus the Ȳs can be replaced byȳ as:

t =
n

N
ȳ +

n

N − n
T

where T is the predictor of the population mean Ȳs̄ of unobserved units of the popu-
lation. We introduce the following symbols and notations:

X̄S̄ = NX̄−nx̄
N−n , Non-sampled population mean of x.

Z̄S̄ = NZ̄−nz̄
N−n , Non-sampled population mean of z.

Ȳ = 1
N

∑N
j=1 yj , X̄ = 1

N

∑N
j=1 xj , and Z̄ = 1

N

∑N
j=1 zj , donates the population means.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 5 of 18

ȳ = 1
N

∑N
j=1 yj , Sample mean of y, X̄ = 1

N

∑N
j=1 xj , Sample mean of x, and z̄ =

1
N

∑N
j=1 zj , Sample mean of z.

Cy =
Sy

Ȳ
, Cx = Sx

X̄
, Cz =

Sz

Z̄
, the coefficient of variation of variation of y, x and z.

S2
x =

∑n
j=1

xj−X̄
N−1 , S

2
y =

∑n
j=1

yj−Ȳ
N−1 , S

2
z =

∑n
j=1

zj−Z̄
N−1 , denote the population variance

of the study variable y.

Sxy =
∑N

j=1
(xj−X̄)(yj−Ȳ )

N−1 , Szy =
∑N

j=1
(zj−z̄)(yj−Ȳ )

N−1 , Sxz =
∑N

j=1
(xj−X̄)(zj−Z̄)

N−1 , denote
the population covariance ,

where

C = ρxy
Cy

Cx
, λ =

1− f

n
, f =

n

N

Let

e0 =
ȳ − Ȳ

Ȳ
, e1 =

x̄− X̄

X̄
, e2 =

z̄ − Z̄

Z̄

Such that E(e0) = 0 = E(e1) = E(e2)
E(e20) = λC2

y , E(e21) = λC2
x, E(e22) = λC2

z , E(e0e1) = λρxyCxCy, E(e0e2) = λρzyCzCy,
E(e1e2) = λρxzCxCz,

3. Adopted existing estimators

In this section, we have deliberated some existing adopted estimators for estimation
of mean using predictive approach under simple random sampling.

(i) [20] proposed the predictive estimators of usual mean, ratio and product estimators
for predicting the mean Ȳs̄ is given by:

t1 =
nȳ

N
+ (

N − n

N
)Ȳs̄, where Ȳs = ȳ (1)

The bias and variance of t1 are given by:

Bias(t1) = 0 (2)

V ar(t1) ∼= λȲ 2C2
y (3)

t2 =
nȳ

N
+ (

N − n

N
)(ȳ

x̄s̄
x̄
) (4)

t3 =
nȳ

N
+ (

N − n

N
)(ȳ

x̄s̄
ȳ
) (5)

The bias and MSE of t2 and t3 are given by:

Bias(t2) ∼= λȲ C2
x[1− C] (6)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 6 of 18

Bias(t3) ∼= λȲ C2
x[C +

f

1− f
] (7)

and

MSE(t2) ∼= λȲ 2[C2
y + C2

x(1− 2C)] (8)

MSE(t3) ∼= λȲ 2[C2
y + C2

x(1 + 2C)] (9)

(ii) [21] recommended the following enhanced estimators, which are given by:

t4 =
nȳ

N
+

N − n

N
ȳexp(

X̄s̄ − x̄

X̄s̄ + x̄
) (10)

t5 =
nȳ

N
+

N − n

N
ȳexp(

x̄− X̄s̄

X̄s̄ + x̄
) (11)

The bias and MSEs of t3 and t4 are given by:

Bias(t4) ∼=
λ

8
Ȳ C2

x[3− 4(C + f)] (12)

Bias(t4) ∼=
λ

8
Ȳ C2

x[3− 4(C + f)] (13)

and

MSE(t4) ∼= λȲ 2[C2
y +

C2
x

4
(1− 4C)] (14)

MSE(t5) ∼= λȲ 2[C2
y +

C2
x

4
(1 + 4C)] (15)

(iii) [7]recommended efficient exponential estimators, which are given by:

t6 = k1[
nȳ

N
+ (

N − n

N
)ȳexp

{
X̄s̄ − x̄

X̄s̄ + x̄

}
] (16)

t7 = k2[
nȳ

N
+ (

N − n

N
)ȳexp

{
x̄− X̄s̄

X̄s̄ + x̄

}
] (17)

The bias and mean square errors of t6 and t7 are given by:

Bias(t6) ∼= Ȳ [(k1 − 1) + k1

{
3

8
− 1

2
f

}
λC2

x − 1

2
λCC2

x] (18)

Bias(t7) ∼= Ȳ [(k2 − 1) + k2

{
1

8(1− f)

}
λC2

x − 1

2
λCC2

x] (19)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 7 of 18

and

MSE(t6) ∼= Ȳ 2(1− A2
1

B2
1

) (20)

MSE(t7) ∼= Ȳ 2(1− A2
2

B2
2

) (21)

where

A1 = 1 +
1

8
(3− 4f)λC2

x − 1

2
λCC2

x

A2 = 1− 1

8(1− f)
λC2

x +
1

2
λCC2

x

B1 = 1 + λC2
x − 2λCC2

x +
1

8
(5− 4f)λC2

x

B2 = 1 + λC2
x − 2λCC2

x +
f

4(1− f)
λC2

x

(iv) The author in [22] recommended improved exponential estimators, which are given
by:

t8 =
nȳ

N
+ (

N − n

N
)ȳexp

{√
X̄ s̄ −

√
x̄√

X̄s̄ +
√
x̄

}
(22)

t9 =
nȳ

N
+ (

N − n

N
)ȳexp

{√
x̄−

√
X̄ s̄√

X̄s̄ +
√
x̄

}
(23)

The bias and MSE are given by:

Bias(t8) ∼= λȲ 2C2
x(

(3− 4f)

(32(1− f))
− 1

4
C) (24)

Bias(t9) ∼= λȲ 2C2
x(

(1)

(4C))
− 1− 4f

32(1− f)
) (25)

and

MSE(t8) ∼= λȲ 2(C2
y +

C2
x

16
− CC2

x

2
) (26)

MSE(t8) ∼= λȲ 2(C2
y +

C2
x

16
+

CC2
x

2
) (27)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 8 of 18

4. The suggested estimator

Research surveys and population investigations require the evaluation of population
mean as their core objective. The ratio and regression methods serve as traditional tools
to enhance the estimate efficiency when working with auxiliary information. Increasing
amounts of complex datasets demand new predictive techniques to detect intricate vari-
ables interactions due to rising needs in robust forecasting. Research teams now seek to
find contemporary predictive modeling techniques which boost the accuracy and reliabil-
ity of population mean estimation. We suggest a new enhanced generalized estimator for
population mean using two auxiliary variables under simple random sampling using pre-
dictive approach. This work is considered unique as, to the best of our knowledge, no one
has before explored the predictive method of estimating the finite population mean. Such
techniques provide improved accuracy and enhanced robustness for prediction through
their ability to handle complicated or dimensionally complex auxiliary information. The
suggested estimator is given in equation (28):

tp =
nȳ

N
+

(
N − n

N

)[
ȳ + k1

(
X̄s̄ − x̄

)
+ k2

(
Z̄z̄ − z̄

)]
exp

{
γ(X̄s̄ − x̄)

X̄s̄+x̄

}
(28)

where k1, and k2 are unknown constants.
MSE of the above estimator when γ may take the values 1 and -1. We get (29):

tp =
nȳ

N
+ (

N − n

N
)ȳexp

{
γ(X̄s̄ − x̄)

X̄s̄+x̄

}
+ (

N − n

N
)k1

(
X̄s̄ − x̄

)
exp

{
γ(X̄s̄ − x̄)

X̄s̄+x̄

}

+(
N − n

N
)k2

(
Z̄z̄ − z̄

)
exp

{
γ(X̄s̄ − x̄)

X̄s̄+x̄

}
(29)

where

X̄S̄ =
NX̄ − nx̄

N − n

Simplifying the right-hand side of Equation (29), multiplying and expressing the above
equation in terms of e’s as follows:

tp = Ȳ (1 + e0)

[
(f + (1− f)exp

{
− γe1
2(1− f)

(1 +
1− 2f

2(1− f)
)−1

}]
−k1(1− f)X̄e1

1− f
exp

{
− γe1
2(1− f)

(1 +
1− 2f

2(1− f)
)−1

}
−k2(1− f)Z̄e2

1− f
exp

{
− γe1
2(1− f)

(1 +
1− 2f

2(1− f)
)−1

}
After simplifying the above equation, we got equation (30):

tp = Ȳ (1 + e0)(f + 1− f) + (1− f)

{
1− γe1

2(1− f)
+

γ2 + 2γ − 4γf

8(1− f)2

}
e21



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 9 of 18

−k1(1− f)X̄e1
1− f

{
1− γe1

2(1− f)

}
− k2(1− f)Z̄e2

1− f

{
1− γe1

2(1− f)

}
(30)

Expanding and approximating terms of (30) up to the first degree, we got equation
(31):

tp − Ȳ ∼= Ȳ e0 −
1

2
Ȳ γe1 − X̄k1e1 − Z̄k2e2 −

1

2
Ȳ γe0e1 +

Ȳ (γ2 + 2γ − 4γf)e21
8(1− f)

+
X̄k1γe

2
1

2(1− f)
+

Z̄k1γe1e2
2(1− f)

(31)

Taking expectations on both sides equation (31), we develop the bias which is given in
equation (32):

Bias(tp) ∼= Ȳ E(e0)−
1

2
Ȳ γE(e1)− X̄k1E(e1)−

1

2
Ȳ γE(e0e1)

+
Ȳ (γ2 + 2γ − 4γf)

8(1− f)
E(e21) +

X̄k1γ

2(1− f)
E(e21) +

Z̄k1γ

2(1− f)
E(e1e2)

Bias(tp) ∼= −1

2
Ȳ γρxyCxCy+

Ȳ (γ2 + 2γ − 4γf)

8(1− f)
λC2

x+
X̄k1γ

2(1− f)
λC2

x+
Z̄k1γ

2(1− f)
γλρxzCxCz

(32)
Expanding square and taking expectations of equation (31), we got equation (33):

MSE(t) ∼= Ȳ 2E(e20)−
2Ȳ 2γE(e0e1)

2
− 2X̄Ȳ k1E(e0e1)− 2Z̄Ȳ k2E(e0e1) +

Ȳ 2γ2E(e21)

4

+
2X̄Ȳ k1γE(e21)

2
+ X̄2k21E(e21) + Z̄k22E(e22) +

2Z̄Ȳ k2γE(e1e2)

2
+ 2X̄Z̄k1k2E(e1e2) (33)

Substituting the values of expectations in equation (33), we got equation (34):

MSE(t) ∼= Ȳ 2λC2
y − Ȳ 2γλρxyCxCy − 2X̄Ȳ k1λρxyCxCy − 2Z̄Ȳ k2λρzyCzCy +

Ȳ 2γ2λC2
x

4

+X̄Ȳ k1γλC
2
x + X̄2k21λC

2
x + Z̄k22λC

2
z + Z̄Ȳ k2γλρyzCyCz + 2X̄Z̄k1k2λρxzCxCz (34)

To find values of k1 and k2 , we partially differentiate equation (34) with respect to
sk1 and k2and set to zero i.e

∂MSE(tP )

∂k1
= 0

∂MSE(tp)

∂k2
= 0



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 10 of 18

We get following two normal equations.

−2X̄Ȳ λρxyCxCy + X̄Ȳ γλC2
x + 2X̄2k1λC

2
x + 2X̄Z̄k2λρxzCxCz = 0

−2Z̄Ȳ λρzyCzCy + 2Z̄k2λC
2
z + Z̄Ȳ γλρyzCyCz + 2X̄Z̄k2λρxzCxCz = 0

Solving these equations, we get the values of k1 and k2, i.e.

k1opt = − Ȳ (γCxρ
2
xz − 2Cyρxzρyz − γCx + 2Cyρxy)

2 ¯XCx(ρ2xz − 1)

and

k2opt =
Ȳ Cy(ρxyρxz − ρyz)

Z̄Cz(ρ2xz − 1)

Putting values in equation (34), we get bias and MSE of the proposed estimator, which
are given by:

Bias(tp)opt ∼=
γλCxȲ (4f2Cx − fγCx + 4fCyρxy − 6fCx + 2Cx)

8(1− f)
(35)

and

MSE(tp)opt ∼= Ȳ 2λC2
y − Ȳ 2γλρxyCxCy +

1

4
Ȳ 2γ2λC2

x

+
Ȳ 2(γCxρ

2
xz − 2Cyρxzρyz − γCx + 2Cyρxy)λρxyCy

(ρ2xz − 1)

−
2Ȳ 2λC2

yρyz(ρxyρxz − ρyz)

(ρ2xz − 1)
+

Ȳ 2λC2
y (ρxyρxz − ρyz)

(ρ2xz − 1)

− Ȳ 2γλCx(γCxρ
2
xz − 2Cyρxzρyz − γCx + 2Cyρxy)

2(ρ2xz − 1)

+
Ȳ λ(γCxρ

2
xz − 2Cyρxzρyz − γCx + 2Cyρxy)

2

4(ρ2xz − 1)

+
Ȳ 2γλCxCyρxz(ρxyρxz − ρyz)

(ρ2xz − 1)

+
Ȳ 2λ(γCxρ

2
xz − 2Cyρxzρyz − γCx + 2Cyρxy)

4(ρ2xz − 1)

− Ȳ 2λCyρxz(γCxρ
2
xz − 2Cyρxzρyz − γCx + 2Cyρxy)(ρxyρxz − ρyz)

(ρ2xz − 1)

After simplification the above equation we got the minimum MSE of tp, which is given
in equation (36):



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 11 of 18

MSE(tp)opt ∼=
Ȳ 2λC2

y (1− ρ2xy − ρ2xz − ρ2yz + 2ρxyρxzρyz)

(1− ρ2xy)
(36)

For different values of k1, k2, and γ in equation (28), some members of the family of
the proposed estimator can be obtained, which is given in Table 1.

Table 1: Some members of the generalized family of regression cum exponential type estimators.

Estimators γ k1 k2
tp1 =

nȳ
N + N−n

N Ȳs̄ = ȳ , sample mean 0 0 0

tp2 =
nȳ
N + N−n

N

{
ȳ + b(X̄s̄ − x̄)

}
, Srivastava [8] 0 B 0

tp3 =
nȳ
N + N−n

N ȳexp( X̄s̄−x̄
X̄s̄+x̄

) , Singh et al. [6] 1 0 0

tp4 =
nȳ
N + N−n

N ȳexp( x̄−X̄s̄

X̄s̄+x̄
) , Singh et al. [6] -1 0 0

tp5 =
nȳ
N + N−n

N

{
ȳ + b(X̄s̄ − x̄)

}
exp( X̄s̄−x̄

X̄s̄+x̄
) 1 B 0

tp6 =
nȳ
N + N−n

N

{
ȳ + b(X̄s̄ − x̄)

}
exp( x̄−X̄s̄

X̄s̄+x̄
) -1 b 0

tp7 =
nȳ
N + N−n

N

{
ȳ + k1(X̄s̄ − x̄)

}
+ k2(Z̄s̄ − z̄) 0 k1 k2

tp8 =
nȳ
N + N−n

N

{
ȳ + b(Z̄s̄ − z̄)

}
exp( X̄s̄−x̄

X̄s̄+x̄
) 1 0 b

tp9 =
nȳ
N + N−n

N

{
ȳ + b(Z̄s̄ − z̄)

}
exp( x̄−X̄s̄

X̄s̄+x̄
) -1 0 0

Case 1: When γ = 1 in equation (28) the proposed estimator becomes the predictive
regression-cum-exponential ratio type estimator, which is given in equation (37):

tPR =
nȳ

N
+

N − n

N

{
ȳ + k1(X̄s̄ − x̄) + k2(Z̄s̄ − z̄)exp(

X̄s̄ − x̄

X̄s̄ + x̄
)

}
(37)

Optimum bias and MSE of equation (37), after simplifying we got:

Bias(tPR)opt ∼=
θC2

xȲ (4f2 + 4fC − 7f + 1)

8(1− f)
(38)

and

MSE(tPR)opt ∼=
Ȳ 2C2

y (1− ρ2xy − ρ2xz − ρ2yz + 2ρxyρxzρyz)

1− ρ2xz
(39)

Case 2: For γ = −1, the proposed estimator becomes the predictive regression-cum-
exponential product type estimator, which is given in equation (40):

tPR =
nȳ

N
+

N − n

N

{
ȳ + k1(X̄s̄ − x̄) + k2(Z̄s̄ − z̄)exp(

x̄− X̄s̄

X̄s̄ + x̄
)

}
(40)

The bias and MSE of the proposed predictive regression-cum-exponential product type
estimator are given by:

Bias(tPP )opt ∼=
θC2

xȲ (4f2 + 4fC − 5f + 3)

8(1− f)
(41)



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 12 of 18

and

MSE(tPP )opt ∼=
Ȳ 2C2

y (1− ρ2xy − ρ2xz − ρ2yz + 2ρxyρxzρyz)

1− ρ2xz
(42)

5. 1. Numerical study

This section contains comparisons of estimators using some actual data sets. To check
the relative performances of estimators we compute PRE. The PRE is expressed as:

PRE(tl) = V ar(ȳ)/MSE(tu)100, u = [1.2, · · · , p]

To check the bias of different estimators, we calculated the following quantities:

Q(t2) = |Bias(t2)

λȲ C2
x

|, Q(t3) = |Bias(t3)

λȲ C2
x

|, Q(t4) = |Bias(t4)

λȲ C2
x

|, Q(t5) = |Bias(t5)

λȲ C2
x

|

Q(t6) = |Bias(t6)

λȲ C2
x

|, Q(t7) = |Bias(t7)

λȲ C2
x

|, Q(t8) = |Bias(t8)

λȲ C2
x

|, Q(t9) = |Bias(t9)

λȲ C2
x

|

Q(t10) = |Bias(t10)

λȲ C2
x

|

As the term (λȲ C2
x) is common in bias terms of all estimators, so we divide the each

bias with this common quantity to simplify the calculations. Values of quantities are found
numerically and findings are shown in the following Table 2.

6. Application of the study

Applications of the suggested estimator are validated using some real data sets, which
is given below:

Data 1: [Source:[2]]
Y = Number of kids, X = number of polio cases, Z = Number of polio cases.
N = 34, n = 15, Ȳ = 4.92, X̄ = 2.59, Z̄ = 2.91, ρxy = 0.7326, ρyz = 0.643, ρxz =

0.6837, Cx = 1.23187, Cy = 1.01232, Cz = 1.053516
Data 2: [Source: [3]]
Y = Number of fish caught in 1995, X = Number of fish caught in 1994, Z = Number

of fish caught in 1993.
N = 69, n = 15, Ȳ = 4514.899, X̄ = 4954.435, Z̄ = 4591.072, ρxy=0.9601, ρyz = 0.9564, ρxz =

0.9729, Cx = 1.4247, Cy = 1.3508, Cz = 1.3755.
Data 3: [Source: [4]]
Y = Tomato supply in tons of Pakistan for in 2003,X = Tomato supply in tons of

Pakistan in 2002, Z = Tomato supply in tons of Pakistan in 2001.
N = 97, n = 30, Ȳ = 3135.6168, X̄ = 3050.2784, Z̄ = 2743.9587, ρxy = 0.9872, ρyz =

0.8501, ρxz = 0.6122, C2
x = 5.4812, C2

y = 4.8674, C2
z = 6.2422



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 13 of 18

Table 2: Biases of all considered estimators based on real data sets

Quantities Data1 Data2 Data3

Qt2 0.3979 0.0034 0.2393

Qt3 0.1466 0.2683 0.1496

Qt4 1.6031 3.6445 2.7324

Qt5 2.7046 0.0211 1.2113

Qt6 0.2471 0.1743 0.56

Qt7 0.1583 0.2795 0.2133

Qt8 0.2319 0.4173 0.2993

Qt9 0.1553 0.8142 0.1335

Qtp 0.5894 0.5356 0.4864

QtPR 0.0553 0.0814 0.0335

QtPP 0.5894 0.5356 0.4864

Figure 1: Biases of all considered estimators based on real data sets

7. Discussion

Table 1, included some members of our generalized class of estimators. Table 2, con-
sists the results of bias for all the considered estimators in the article. The results of
mean squared error for all the adopted and recommended estimators are given in Table
3. Table 4, include the result of percentage relative efficiency. We observed from Table 2
the following points: The proposed regression-cum exponential ratio-type estimator tPR

is less biased than all considered estimators. For population 2, suggested predictive esti-
mator is less biased than other estimator. Also for population 3, the suggested estimator
is less biased than all estimators. For all three populations the proposed regression-cum
exponential product-type estimator tPP is less biased. The result provide by t3, t5andt9
for all the three population is maximum mean squared error. Also we observed from Table
4, that estimators t3, t5andt9 provide less efficient result as compared to all considered es-



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 14 of 18

Table 3: MSEs of all considered estimators using real data sets

Estimators Data 1 Data 2 Data 3

t1 44.39061 47526246 34042153

t2 39.34197 3993781 14057273

t3 219.2408 193161470 130697896

t4 20.6411 12997169 14465855

t5 110.5905 107581013 72786167

t6 22.34624 190873849 7112580

t7 23.48469 17226857 18578610

t8 27.20956 27070996 21858040

t9 72.18426 74362919 51018195

tp 18.88242 3274143 4963220

Figure 2: MSEs of all considered estimators using real data sets

timators. The MSEs for both tPR and tPP are the same, so we consider of tp comparing its
MSE with all other considered estimator in Table 3. The result of bias for all estimators
are visualize, and are given in Figure 1. The mean squared error and percentage relative
efficiency of estimators are also shown in Figures 2 and Figure 3. It is detected from Table
3 that the mean squared error of the proposed estimator is minimum than all considered
adopted estimators in all populations. It is detected from Table 4 that the suggested
predictive regression-cum exponential estimator performs improved in terms of PRE than
all considered estimators. The research value of this study rests in both its programmable
applications together with advancements in methodology. Real datasets benefit from pre-
dictive techniques as the research creates an alignment between theory development and
real-world implementation. Modern predictive tools allow the proper use of auxiliary data
to enhance population mean estimation processes. The technique delivers essential con-
sequences for disciplines including public health and agriculture and economics together
with the social sciences since their policy implementation usually depends on accurate pop-
ulation estimations. The research adds important knowledge to current academic support



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 15 of 18

Table 4: PREs of all considered estimators using real data sets

Estimators Data 1 Data 2 Data 3

t1 100 100 100

t2 112.8327 1190.006 242.1676

t3 20.24743 24.60441 26.0464

t4 215.0593 365.6661 235.3276

t5 40.13962 44.17717 46.7701

t6 198.6491 24.8993 478.6189

t7 189.0194 275.8846 183.2334

t8 163.1434 175.5615 155.742

t9 61.49625 63.91122 66.7251

tp 235.0896 1451.563 685.8881

Figure 3: PREs of all considered estimators using real data sets

which promotes combining data science technologies with traditional statistical practices.
The research promotes adoption of data-oriented and adaptable procedures for survey
analysis. Practitioners and policymakers can learn from these findings to improve their
use of existing auxiliary information during situations that require either expensive or
time-consuming or difficult data collection processes. This research contributes evidence
to the creation of better yet efficient and precise and extendable estimation methods for
current data setups.

8. Conclusion

This articles aims to estimate population mean using predictive approach under simple
random sampling. The terminologies of the bias and MSE are computed to first order. We
found the bias and MSE of the suggested and existing counterparts both theoretically and
numerically. To check the efficiency of estimators we utilized three actual data sets. Based



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 16 of 18

on the numerically result, it is shown that the suggested estimators has minimum MSE
and higher PRE. By applying the suggested estimator to future studies that attempt to es-
timate the means of finite populations will yield drastically better results. Using auxiliary
variables and predictive modeling approaches, this estimator improves the efficiency and
accuracy of estimates. Reducing sampling variability and improving precision is especially
achieved when there is a high correlation among the study variable and auxiliary variables.
The current work can be easily extended to estimation population mean using predictive
approach under stratified random sampling. Predictive methods should add the com-
bination of linear regression and ratio estimators through non-parametric and ensemble
approaches to include random forests along with gradient boosting and neural networks.
The models flourish at detecting complicated patterns between study variables and auxil-
iary variables that deviate from linear structures. The evaluation of performance metrics
needs to be established under various sampling approaches (including stratified and sys-
tematic together with adaptive methods) to widen practical use for complicated survey
methodologies. The technique needs to be analyzed for real-time estimation applying
streaming data with dynamic populations to properly serve environments that experience
rapid changes such as health surveillance and economic monitoring systems. Testing pre-
dictive estimators across different population structures through simulation studies and
selecting targets in agriculture and demographic and health sectors to conduct authentic
validations will promote their practical implementation between sectors.

Data availability The datasets generated and/or analyzed during the current study
are available in the current study are available from the corresponding author on reasonable
request.

Conflict of interest
The authors declare no conflict of interest

References

[1] MFA. Crops area production. The Journal of Agricultural Science, 2004. Islamabad,
Pakistan: Ministry of Food and Agriculture.

[2] Shashi Bahl and R. K. Tuteja. Ratio and product type exponential estimators. Journal
of Information and Optimization Sciences, 12(1):159–164, 1991.

[3] Sarjinder Singh. Advanced sampling theory with applications: how Michael “selected”
Amy. Springer, 2003. 2 Vols.

[4] W. G. Cochran. The estimation of the yields of cereal experiments by sampling for the
ratio of grain to total produce. The Journal of Agricultural Science, 30(2):262–275,
1940.

[5] Aamir Sanaullah, Humera Ameer Ali, Muhammad Noor ul Amin, and Muhammad
Hanif. Generalized exponential chain ratio estimators under stratified two-phase ran-
dom sampling. Applied Mathematics and Computation, 226:541–547, 2014.

[6] Lovleen Grover and Anchal Sharma. Predictive estimation of finite population mean
in case of missing data under two-phase sampling. Journal of Statistical Theory and
Applications, 22, 2023.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 17 of 18

[7] Shashi Bhushan and Anoop Kumar. Novel predictive estimators using ranked set sam-
pling. Concurrency and Computation: Practice and Experience, 35(3):e7435, 2023.

[8] Ewout W. Steyerberg, Sacha E. Bleeker, Henriëtte A. Moll, Diederick E. Grobbee,
and Karel G. M. Moons. Internal and external validation of predictive models: a sim-
ulation study of bias and precision in small samples. Journal of Clinical Epidemiology,
56(5):441–447, 2003.

[9] Subhash Kumar Yadav and S. S. Mishra. Developing improved predictive estimator
for finite population mean using auxiliary information. Statistika: Statistics and
Economy Journal, 95:76–84, 2015.

[10] Shakeel Ahmed, Javid Shabbir, and Sat Gupta. Predictive estimation of population
mean in ranked set sampling. REVSTAT-Statistical Journal, 17(4):551–562, 2019.

[11] Rajesh Singh, Hemant Verma, and Prayas Sharma. Efficient estimators of finite
population mean using predictive estimation in simple random sampling. Journal of
Statistics, 23:1–11, 2016.

[12] Shashi Bhushan, Anoop Kumar, Md Tanwir Akhtar, and Showkat Ahmad Lone.
Logarithmic type predictive estimators under simple random sampling. AIMS Math-
ematics, 7(7):11992–12010, 2022.

[13] Sohaib Ahmad, Javid Shabbir, Erum Zahid, and Muhammad Aamir. Improved fam-
ily of estimators for the population mean using supplementary variables under PPS
sampling. Science Progress, 106(2):00368504231180085, 2023. PMID: 37341780.

[14] Sohaib Ahmad, Sardar Hussain, Aned Al Mutairi, Mustafa Kamal, Masood Ur
Rehman, and Manahil SidAhmed Mustafa. Improved estimation of population distri-
bution function using twofold auxiliary information under simple random sampling.
Heliyon, 10(2):e24115, 2024.

[15] M. K. Pandey, G. N. Singh, Tolga Zaman, Aned Al Mutairi, and Manahil SidAhmed
Mustafa. Improved estimation of population variance in stratified successive sampling
using calibrated weights under non-response. Heliyon, 10(6):e27738, 2024.

[16] Ashok K. Jaiswal, M. K. Pandey, and G. N. Singh. Optimizing population mean
estimation using regression and factor type estimators in the presence of non-response.
Franklin Open, 7:100096, 2024.

[17] Anoop Kumar and Asra Sayyed Siddiqui. Enhanced estimation of population mean
using simple random sampling. Research in Statistics, 2(1):2335949, 2024.

[18] Abdullah Ali H. Ahmadini, Rajesh Singh, Yashpal Singh Raghav, and Anamika Ku-
mari. Estimation of population mean using ranked set sampling in the presence of
measurement errors. Kuwait Journal of Science, 51(3):100236, 2024.

[19] Sajid Khan, Muhammad Farooq, Sohaib Ahmad, and Sardar Hussain. Improved
estimator for estimation of population mean using predictive approach under PPS
sampling. VFAST Transactions on Mathematics, 12, 2024.

[20] S. K. Srivastava. Predictive estimation of finite population mean using product esti-
mator. Metrika, 30(1):93–99, 1983.

[21] Housila P. Singh, Ramkrishna Solanki, and Alok Singh. Predictive estimation of finite
population mean using exponential estimators. Statistika: Statistics and Economy
Journal, 94:41–53, 2014.



S. Khan et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6019 18 of 18

[22] Viplav Kumar Singh and Rajesh Singh. Predictive estimation of finite population
mean using generalised family of estimators. Istatistik: Journal of the Turkish Sta-
tistical Association, 7(2):43–54, 2014.