EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6477
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Developing a Generalized Class of Estimators for
Estimation of Population Mean Using Neutrosophic

Approach

Sanaa Al-Marzouki1, Sohaib Ahmad2,∗

1Department of Statistics, Faculty of Science, King Abdul Aziz University, Jeddah,
Kingdom of Saudi Arabia
2Department of Statistics, Abdul Wali Khan University, Mardan, Pakistan

Abstract. Preceding studies have only been conducted with clear, determinate data, as the clas-
sical statistics approach is unable to accommodate ambiguity and uncertainty. As a generalization
and alternative to classical statistics for such indeterminate or uncertain data, neutrosophic statis-
tics is concerned with dealing such ambiguity in data. Making use of neutrosophic data, we propose
a generalized class of estimators for the population mean under simple random sampling. Based
on the numerical outcome it is presented that the suggested estimators achieved well in terms of
minimum MSE. In order to more accurately represent the range of values throughout which our
population parameter occurs, the results of these estimators are provided as intervals rather than a
single value. We use simulation and interval data from the Islamabad Stock Exchange, focused on
the UBL, to further investigate the efficiency of the suggested neutrosophic estimator. The numer-
ical results confirm the suggested generalized neutrosophic estimators are superior to the existing
methods. The mean square error (MSE) and percentage relative efficiency (PRE) performance
measures demonstrate that the developed neutrosophic regression type estimator is always bet-
ter than the conventional neutrosophic ratio estimator, neutrosophic product estimators, and the
neutrosophic exponential ratio estimator. This paper addresses concerning how the neutrosophic
regression estimator can make estimates more accurate when working with data that is unclear or
uncertain and has a wide range of correlation between the study and the auxiliary variables that
are being examined.

2020 Mathematics Subject Classifications: 62D05, 62f10

Key Words and Phrases: Neutrosophic statistics, generalized estimator, mean estimation, bias,
MSE, PRE

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6477

Email addresses: salmarzouki@kau.edu.sa (S. Al-Marzouki),
sohaib ahmad@awkum.edu.pk (S. Ahmad)

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



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 2 of 22

1. Introduction

In conventional statistics, the information being analyzed is known and measurable. Many
authors in the field of classical statistics spent considerable time investigating and devel-
oping estimators of the mean of a population in the presence of auxiliary information.
Research reveals that the sampling error for ratio is greatly decreased when the auxiliary
variable is incorporated in addition to the research variable when there is a high degree
of correlation between the study variable and the auxiliary variable. This has significant
implications for the minimum sample size required by the ratio estimation method, or the
degree to which the sample size can be reduced by the approach without compromising
accuracy. However, there are situations in which the available data is not random but
rather indeterminate, hazy, confusing, or imprecise, and for which classical statistics and
its methodologies are inadequate. Estimation based on classical statistical approaches
performs poorly under these conditions. One approach to such a problem is to use fuzzy
logic [31, 32], however this still doesn’t account for uncertainty. Neutrosophic approaches
are significantly more trustworthy in such circumstances. They have to contend with un-
certainty as well as randomness. Unlike estimate problems in classical random sampling
schemes, where the data is predetermined, the subject of neutrosophic estimating is new
and consequently unexplored [34], and [35]. Its broad applicability, however, has given it
a higher profile than classical statistical methods, leading to its use in areas like decision
making [36]. In classical statistics, the emphasis is on determined data, which is obtained
after all possible measurement errors have been ruled out. This holds true when there
is zero margin for error in the measurements being made. Therefore, we require cutting-
edge approaches to make sense of the unknown information. When precise bounds on the
variable of interest are unavailable, fuzzy logic can be used as a means of disseminating
such knowledge.
However, neutrosophic logic can be viewed of as a generalization of fuzzy logic; it allows for
the evaluation of inconclusive alongside a definite part of the observations, and it can be
applied to analysis performed under imprecise or uncertain circumstances. Recently, there
has been a meteoric rise in the sophistication of the approach surrounding the application
of uncertain logic in decision building [3, 9]. Following on from fuzzy collections is the
next stage of development known as a complicated neutrosophic set [11]. In [10], we
find a thorough presentation of fuzzy collections and its generalizations, including an
analysis of their attributes and operations and a look into interval valued neutrosophic
sets. When the fuzzy set is inadequate to resolve the ambiguity of a given decision-making
situation, the neutrosophic set should be considered. A great deal of differentiation exists
between neutrosophic cliques. In a study [14], researchers devised a trapezoidal bipolar
neutrosophic quantity and a classification scheme for it to be used in decision-making
contexts. In a single paper [15], the theory behind generalized spherical fuzzy numbers
was laid forth alongside a thorough analytic scheme and other methods. To execute
mathematical and geometric operations on pentagonal neutrosophic numbers, another
study suggests using a mobile communication application [13]. Neutrosophic numbers are
gaining more and more attention from scientists in recent years; for instance, a method



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 3 of 22

was devised for application within the cylindrical neutrosophic domain [12]. When the
data are ambiguous, statisticians turn to the neutral approach known as neutrosophic
statistics. In contrast to traditional statistics, neutrosophic statistics are appropriate when
working with data or a sample that includes neutrosophic. Statisticians use neutrosophic
approaches [16] when there is a lack of clarity in either the population or the sample
observations. The term ”neutrosophic data” is used to describe a set of information where
the exact values are unknown; neutrosophic numerical processes are used to analyze this
type of information. The sample size may not be presented as an expected value [16] in
neutrosophic statistics. Incorporating neutrosophic statistics into the current uncertainty
framework was found to be highly beneficial by the group [23, 24]. By using neutrosophic
numbers, rock engineers can investigate the gauge consequence and anisotropy of the
combined roughness coefficient. Because of this, there is less data loss and more fitted
functions are generated than would otherwise be possible [25]. A novel method of analyzing
neutrosophic data via analysis of variance was introduced [30]. These authors [26-29] were
the pioneers in their fields when it comes to neutrosophic interval statistics. Researchers
interested in neutrosophic data might learn more by consulting the sources [17-22].

1.1. Research gap

Previous research on survey sampling has solely focused on gathering facts and figures
that may be defined as accurate, definite, and unambiguous. A single correct answer is
obtained by employing such methods; nonetheless, the result may be incorrect, overstated,
or understated. However, in many cases and under particular conditions, the data are of
a neutrosophic kind; on this occasion, neutrosophic statistics is applied, but more typical
statistical processes were unsuccessful. Data with a neutrosophic flavor feature murky
interval values, confusing logic, and ambiguous findings. In this way, data from experi-
ments or populations might represent interval values in neutrosophic numbers. Although
its precise value was unknown at the time of data collection, it was assumed that the true
observation would lie within that range. There is more uncertain data than definite data
available in the real world. Therefore, more statistical approaches specific to neutrosophic
study are required. Due to the complete amount of potential study variables in the actual
world, data collection can become prohibitively expensive, particularly in cases when the
confirmation is vague. As a result, attempting to keep the population under tight control
using techniques not optimized for ambiguous data will be both risky and costly. It’s
impossible to provide an explanation when the study design and any relevant covariates
are neutrosophic. In order to estimate the anonymous population mean in the occurrence
of auxiliary variables via survey sampling, enhanced generalized class estimators have not
been investigated previously, according to a literature review. This finding is a product
of academics poring over past studies that have already been published. There are not
enough promising articles out there to satisfy the demand in the field of statistics just yet.
Beginning with this research, we can now go further in this area.



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 4 of 22

1.2. Scope of the suggested idea

When working with imperfect or unclear data sets, neutrosophic analysis might be helpful.
Furthermore, this method allows for competing viewpoints to coexist. Certain observa-
tions may be given in a range of unknown values if certain instruments are used to collect
the data. Because of statistical uncertainty, traditional methods of investigation are now
worthless. Survey sampling based on neutrosophic has come a long way in recent years,
but the generalized class of estimating is still novel and must be approached with great
caution due to uncertainty in the underlying data source. Products with insignificant mea-
surement errors or manufacturing mistakes will be accepted if they fall inside a particular
measurement range, such as when a machine produces nuts or bolts and we measure those
things. Using outdated statistical approaches that provide only a single value increases
the likelihood of discarding perfectly serviceable things. Neutrosophic statistics delivers
precise assessment of interval outcomes.
Neutrosophic statistics is a new way of looking at things that can work with datasets
that aren’t completely clear or have missing information. This strategy lets people have
different beliefs and works with a range of uncertain figures that could stand for some
observations, even a precise measurement. On the other hand, standard statistics don’t
work well when there is ambiguity. This is where neutrosophic statistics come in, giving us
a new way to look at data analysis. In actual life, we often lack knowledge of the population
parameters. In these situations, statistical inference methods may need to be more useful.
Instead, reasonable estimates are utilized to solve the problem of an unknown parameter
value by guessing its values. This practical technique gives the statistician confidence that
the resulting data are not clear but still valuable. Neutrosophic statistics are a solid way
to solve these problems since they can find the optimal interval value with the least mean
square error. Interested authors may read [37-44] to learn about neutrosophic approach
based on different sampling design.

2. Terminlogy

Consider a neutrosophic sample of size nN∈ (nL, nU ), which is chosen from a population
of NN units (Ω1, Ω2, . . . , ΩN ). Let yNi is the ith sample observation of neutrosophic
data, which is of the form yN ∈ (yL, yU ) and similarly for auxiliary variable xN∈ (xL, xU ),
and zN∈ (zL, zU ). As Y N∈ (YL, YU ) and XN∈ (XL, XU ), ZN∈ (ZL, ZU ) be the popu-
lation mean of neutrosophic study and auxiliary variables. The neutrosophic coefficient
of variation for yN , xN zN are denoted by CyN∈ (CyNL, CyNU ), CxN∈ (CxNL, CxNU )

and CzN∈ (CzNL, CzNU ). Let s2yN =
∑n
i=1 (yiN−yN )2

nN−1 , s2xN =
∑n
i=1 (xiN−xN )2

nN−1 and s2zN =∑n
i=1 (ziN−zN )2

nN−1 , be the unbiased sample variances conforming to population variances S2
yN

=
∑N
i=1 (yiN−Y N )

2

NN−1 , S2
xN =

∑N
i=1 (xiN−XN)

2

NN−1 , and S2
zN =

∑N
i=1 (xiN−XN)

2

NN−1 of YN , XN , and
ZN correspondingly.
Let CyN , CxN and CzN , denotes the population coefficient of variation of yN , xN and zN ,



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 5 of 22

where CyN=
SyN
Y

, CXN=SxN
X

and CZN=SzN
Z

.

ρyxN =
SyxN

SyNSxN
, ρyzN =

SyzN
SyNSzN

, ρxzN =
SxzN

SxNSzN

Where SyN =

√∑N
i=1 (yiN−Y N )

2

NN−1 , SxN=

√∑N
i=1 (xiN−XN)

2

NN−1 and SzN=

√∑N
i=1 (ziN−ZN)

2

NN−1

Syx =

∑N
i=1

(
yiN − Y N

) (
xiN −XN

)
NN − 1

, Syx =

∑N
i=1

(
yiN − Y N

) (
ziN − ZN

)
NN − 1

, Sxz =

∑N
i=1

(
xiN −XN

) (
ziN − ZN

)
NN − 1

λN =

(
1

nN
− 1

NN

)
.

3. Existing estimators

In this section, we have presented some existing counterparts.

(i) The conventional estimator for population mean based on neutrosophic statistics, is
given by:

Ŷ UN = yN (1)

The variance of Ŷ UN , is given by:

V ar(Ŷ UN ) = λNY
2
NC

2
yN (2)

Ŷ RN = yN

(
XN

xN

)
(3)

Ŷ RN = Y N (1 + eo)

(
XN

XN (1 + e1)

)
Ŷ RN = Y N (1 + eo) (1 + e1)

−1

Ŷ RN = Y N (1 + eo)
(
1− e1 + e21

)
Ŷ RN = Y N

(
1− e1 + e21 + eo − eoe1 − eoe

2
1

)
(4)

Ŷ RN − Y N = Y N

(
1− e1 + e21 + eo − eoe1 − eoe

2
1

)
(5)

Apply expectation both sides of the equation(5), we have:

E
(
Ŷ RN − Y N

)
= Y NE

(
1− e1 + e21 + eo − eoe1 − eoe

2
1

)
(6)

E
(
Ŷ RN − Y N

)
= Y NE

(
−e1 + e21 + eo − eoe1 − eoe

2
1

)
(7)



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 6 of 22

Bias
(
Ŷ RN

)
=Y N

(
E (e1) + E

(
e21
)
+ E (eo)− E (eoe1)− E

(
eoe

2
1

))
Where

E (ei) = 0

Ignore higher order approximation, we have:

Bias
(
Ŷ RN

)
=Y N

(
E
(
e21
)
+ E (eoe1)

)
Apply expectation, we have:

The bias of Ŷ RN , are given by:

Bias(Ŷ RN ) ∼= ?NY N

(
C2
yN − ρyxNCyNCxN

)
,

Now

Squaring and apply expectation of equation (6), we have

E
(
Ŷ RN − Y N

)2
=Y NE

(
−e1 + e21 + eo − eoe1 − eoe

2
1

)
Ignore higher order, we have:

MSE
(
Ŷ RN

)
= Y

2
NE(eo−e1)2

MSE
(
Ŷ RN

)
= Y

2
NE
(
e20 + e21 − 2eoe1

)
Apply expectation, we have:

MSE
(
Ŷ RN

)
= Y

2
NE
(
E
(
e20
)
+ E

(
e21
)
− 2E (eoe1)

)
Finally after applying the expected values, we got the mean squared error of Ŷ RN :

MSE(Ŷ RN ) ∼= λNY
2
N

(
C2
yN + C2

xN − 2ρyxNCyNCxN

)
(8)

(ii) The product estimator, given by:

Ŷ PN = yN

(
xN

XN

)
(9)

The properties of Ŷ PN , are given by:

Bias(Ŷ PN ) ∼= λNY N ρyxNCyNCxN ,

and

MSE(Ŷ PN ) ∼= λNY
2
N

(
C2
yN + C2

xN + 2ρyxNCyNCxN

)
(10)



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 7 of 22

(iii) The adopted difference estimator is given by:

Ŷ DN = yN +Q
(
XN − xN

)
, (11)

where Q is constant.

Qoptimum = ρyxN

(
SyN
SxN

)
The variance of Ŷ DN , at the optimum value of Qoptimum is given by:

Var(Ŷ DN ) min
∼= λNY

2
NC

2
yN (1− ρ2yxN ) (12)

(iv) The [6] recommended as:

Ŷ BTRN = yNexp

(
XN − xN

XN + xN

)
, (13)

Ŷ BTPN = yNexp

(
xN −XN

XN + xN

)
(14)

The properties of Ŷ BTN , are given by:

Bias
(
Ŷ BTRN

)
∼= λN Y N

(
3
8 C

2
xN − 1

2ρyxNCyNCxN

)
,

MSE
(
Ŷ BTRN

)
∼= λNY

2
N

(
C2
yN +

1

4
C2
xN − ρyxNCyNCxN

)
(15)

Bias
(
Ŷ BTPN

)
∼= λN Y N

(
3
8 C

2
xN − 1

2ρyxNCyNCxN

)
,

MSE
(
Ŷ BTPN

)
∼= λNY

2
N

(
C2
yN +

1

4
C2
xN + ρyxNCyNCxN

)
(16)

(v) The exponential estimator developed by [33] given by:

Ŷ SN = yNexp

(
a
(
XN − xN

)
a
(
XN + xN

)
+ 2b

)
(17)

Bias
(
Ŷ SN

)
∼= λN Y N

(
3
8 θ

2
NC

2
xN − 1

2θNρyxNCyNCxN

)
,

MSE
(
Ŷ SN

)
= λNY

2
N

(
C2
yN +

1

4
θ2NC

2
xN + θNρyxNCyNCxN

)
, (18)

where θN= aXN

aXN+b
.



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 8 of 22

4. Suggested estimator:

Classical statistics can only be utilized on investigations with clear, unambiguous data.
It fails to apply with data that is hazy or uncertain. Neurosophic statistics is a way to
handle data that is ambiguous or not certain. It is both an alternative to and a gen-
eralization of classical statistics for this specific type of data. Taking motivation from
[4], we recommended an improved generalized neutrosophic estimators under simple ran-
dom sampling using twofold auxiliary information. The neutrosophic estimators may be
superior to classical estimators in situations where the study variable’s observations are
nondeterministic; they may perform inadequately in situations where the observations are
deterministic. The suggested neutrosophic estimators have been shown to be more trust-
worthy than the traditional method of estimating. Simulation studies and medical uses in
neutrosophic settings have shown that the suggested estimators are even more accurate.
The recommended class of estimators is given by:

Ŷ
(∗)a,b
Prop, N =

[
ψ15yN + ψ16

(
XN − xN

)
+ ψ17

(
ZN − zN

)]
exp

(
a
(
XN − xN

)
a
(
XN + xN

)
+ (a− 1) aXN (a+ 1)b

)
(19)

where ψ15, ψ16, and ψ17 are constants. Some members of the suggested class of estimators
are given in Table 1.

Table 1: Some estimators of Ŷ
(∗)a,b
Prop, N

a b Ŷ
(∗)
Prop, N

1 CxN Ŷ
1

Prop, N

1 β2(xN) Ŷ
2

Prop, N

β2(xN) CxN Ŷ
3

Prop, N

CxN β2(xN) Ŷ
4

Prop, N

1 ρyxN Ŷ
5

Prop, N

CxN ρyxN Ŷ
6

Prop, N

ρyxN CxN Ŷ
7

Prop, N

β2(xN) ρyxN Ŷ
8

Prop, N

ρyxN β2(xN) Ŷ
9

Prop, N

1 NNXN Ŷ
10

Prop, N

Solving equation (19), we have:

Ŷ
(∗)a,b
Prop, N =

[
ψ15Y N (1 + ξ0)− ψ16XNξ2 − ψ17ZNξ3

] [
1− θξ1 +

3

8
θ2ξ21 + . . .

]
(20)



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 9 of 22

where θ= αXN

[(α+1)(aXN+β)]
, is a known quantity.

Ŷ
(∗)a,b
Prop, N − Y N =

[
− Y N+ Y Nψ15 + Y Nψ15ξ0 +

5

8
Y Nψ15ξ

2
1 − ψ16ZNξ2+ XNψ17θ

2ξ21

]
(21)

Taking expectations of equation (21), we have:

Bias(Ŷ
(∗)a,b
Prop, N ) = Y N

[
ψ15 − 1 +

5

8
λψ15C

2
xN + λψ17ZNθC

2
xN

]
(22)

Squaring equation (21), we have:(
Ŷ

(∗)a,b
Prop, N − Y N

)2

= Y
2
N+Y

2
Nψ

2
15+Y

2
Nψ

2
15ξ

2
0+

5

4
Y

2
Nψ

2
15ξ

2
1+Z

2
Nψ

2
16ξ

2
2+X

2
Nψ

2
17ξ

2
1−2Y NZNψ15ψ16ξ0ξ2

+2 Y NXNψ15ψ17θξ
2
1 − 2 Y NXNψ15ψ17ξ0ξ1 + 2 Y NZNψ16ψ17ξ1ξ2 − 2 Y

2
Nψ15 −

5

4
Y

2
Nξ

2
1

−2 Y NXNψ17θξ
2
1 (23)

MSE(Ŷ
(∗)a,b
Prop, N ) = Y

2
N


1 + ψ2

15

{
1 + λ

(
C2
yN +

5

4
C2
xN

)
+ λZ

2
Nψ

2
16C

2
zN − λR1ψ17C

2
xN (2θ+R1ψ17)

}
− 2R2ψ15λ16ρCyNCzNCyNCzN − 2λR1ψ15ψ17

(
ρCyNCxNCyNCxN − θC2

xN

)
+ 2λR1R2ψ16ψ17ρCyNCzNCyNCzN

(24)

Where R1=
XN

Y N
, R2=

ZN
Y N

.

Differentiate equation (24) w.r.t ψ15, we have:

∆MSE

∆λ15
= 2Y

2
N

[
λ15 λC

2
yN − R2λ16ρCyNCzNCyNCzN − λR1λ17ρCyNCxNCyNCxN

]
= 0

Differentiate equation (24) w.r.t ψ16, we have:

∆MSE

∆λ16
= 2Y

2
N

[
λ15 λ Z

2
Nψ

2
16C

2
zN − R2λ15ρCyNCzNCyNCzN − λR1R2λ17ρCyNCzNCyNCzN

]
= 0

Differentiate equation (24) w.r.t ψ17, we have:

∆MSE

∆λ17
= 2Y

2
N

[
−λR1C

2
xNλ+R1 − λR1λ15ρCyNCxNCyNCxN + λR1R2λ16ρCyNCzNCyNCzN

]
= 0

This is done by taking partial derivatives of equation (24) with respect to ψ15, ψ16 and ψ17

and equate to zero, and solving the resulting system of equation to find their optimal values
The optimal values of ψ15, ψ16 and ψ17, are given by:

ψ15 =

(
1− 1

2θ
2C2

xN

1 + λC2
yN (1−Q2

yN.xNzN )

)
,



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 10 of 22

ψ16 =

YN

[
λθ3

(
−1 + ρ2xNzN − CyN

(
1 − 1

2
λθ2C2

xN

) (
ρyNxN − ρxNzNρyNzN

)
+ θCxN

{
−1 + ρ2xNzN

})(
−1 + λC2

xN

{
1 −Q2

yN.xNzN

})]
XNCxN

{
−1 + ρ2

xNzN

}(
1 + λC2

xN

{
1 −Q2

yN.xNzN

})
 ,

ψ17 =

 Y N

(
1− 1

2λθ
2C2

xN

)
CyN (ρyNzNρyNxN − ρxNzN )

ZNCzN

{
−1 + ρ2xNzN

} [
1 + λC2

xN

{
1−Q2

yN.xNzN

}]


Substituting these optimal values into equation (24), which gives us minimized MSE as
shown in equation (25).

MSE

(
Ŷ

(∗)a,b
Prop, N

)
=
λY

2
N

[
C2
yN

{
1−Q2

yN.xNzN

}
− 1

2λθ
4C4

xN − λθ2C2
xN

{
1−Q2

yN.xNzN

}]
1 + λC2

yN

{
1−Q2

yN.xNzN

} ,

(25)
where

Q2
yN.xNzN =

ρ2yNxN+ρ
2
yNzN − 2ρyNxNρyNzNρxNzN

1− ρ2xNzN

.

5. Numerical study

In this section, we consider neutrosophic data for numercial comparisions of the suggested
and exisitng estimators. In Table 2, we see the data sets and their description.

PRE= V ar(Ŷ UN )

MSE
(
Ŷ i,NN

) ×100,

where i = Ŷ RN , Ŷ PN , Ŷ DN , , Ŷ BTRN , Ŷ BTPN , Ŷ SN , Ŷ
(∗)
Prop, N (∗ = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).

We performed a numerical analysis of UBL data from the Islamabad Stock Exchange,
which included interval data with uncertain values. The data were taken from [1] and [2].
The summary statistics of the data are presented in Table 2.

Table 2: Summary statistics using neutrosophic data

Parameter Values Parameter Values

NN [239,239] S2
yN [2968.468,3131.635]

nN [35,35] S2
xN [3110.931,3156.542]

λN [0.02438733,
0.02438733]

S2
zN [4779.904, 4783.409]

Y N [131.5651,135.6154] ρyxN [0.8680465,0.5235659]

XN [149.6231,153.5848] ρyzN [0.1637699, 0.1761911]

ZN [119.6402,
119.4603]

ρxzN [0.3753044, 0.3712835]



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 11 of 22

Table 3: MSE using neutrosophic data

EstimatorMSE Values of ex-
isting work

Ŷ SN (Singh esti-
mator)

Ŷ propN(Proposed
work)

Ŷ UN [0.004330113,
0.004330113]

[0.003172306,
0.001862044]

[ 0.003087266,
0.0009190622]

Ŷ RN [0.003702557,
0.001067901]

[0.00317687,
0.001885676]

[0.003088775,
0.0009195191]

Ŷ PN [0.01170995,
0.01428829]

[0.003172302,
0.001862026]

[0.003087265,
0.0009190619]

Ŷ DN [0.003143137,
0.001067352]

[0.004017153,
0.003797651]

[0.003139938,
0.0009346095]

Ŷ BTRN [0.0031723,
0.001862012]

[0.003173376,
0.001871448]

[0.003087632,
0.0009192456]

Ŷ BTPN [0.008472206,
0.007175997]

[0.003590381,
0.003309825]

[0.003130105,
0.0009332394]

[0.003172311,
0.001862049]

[0.003087268,
0.0009190623]

[0.00317279,
0.001866334]

[0.003087433,
0.0009191462]

[0.003181196,
0.001889243]

[0.003090096,
0.000919587]

[0.00433003,
0.004329979]

[0.003141473,
0.0009350626]



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 12 of 22

Figure 1: Showing mean squared error of all considered estimators based on real data set



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 13 of 22

Table 4: PRE using neutrosophic data

Estimator PRE Values of
existing work

Ŷ SN (Singh esti-
mator)

Ŷ propN(Proposed
work)

Ŷ UN [100, 100] [136.4973, 232.5462] [140.2572, 471.1447]

Ŷ RN [116.9493,
405.4787, ]

[136.3012, 229.6319] [140.1887, 470.9106]

Ŷ PN [30.30533,
36.97806]

[136.4975, 232.5484] [140.2573, 471.1449]

Ŷ DN [137.764,
405.6874]

[107.7906, 114.0208] [137.9044¸ 463.3072]

Ŷ BTRN [136.4976,
232.5503]

[136.4513, 231.3777] [140.2406, 471.0507]

Ŷ BTPN [51.10963,
60.34162]

[120.6031, 130.8261] [138.3376, 463.9874]

[136.4971, 232.5456] [140.2571, 471.1447]

[136.4765¸232.0116] [140.2496, 471.1017]

[136.1159, 229.1983] [140.1288, 470.8759]

[100.0019, 100.0031] [137.837, 463.0827]



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 14 of 22

Figure 2: Showing percentage relative efficiency of all considered estimators based on real data set

6. Simulation study

In this section, a simulation study is conducted for the validation of numerical results.
We used simulated neutrosophic data and comparison is done on the basis of MSE and
PRE. YN , XN and ZN are neutrosophic random variables, following the neutrosophic
normal distribution. The YN has mean µY N and variance σ2Y N . The XN has neutrosophic
normal distribution with mean µXN and variance σ2XN . The ZN has neutrosophic normal
distribution with mean µZN and variance σ2ZN . Where YN ∈ (YL, YU ), XN ∈ (XL, XU )
and ZN ∈ (zL, zU ). We have generated 5000 random normal variate by using neutrosophic
normal distribution, i.e. YN ∼ NN ((5, 8),((0.9)2 , (1.2)2) and XN ∼ NN ((15, 18),( (0.5)2

, (0.7)2 )).
A neutrosophic normal distribution is typically denoted as:
XN ∼N

[
(µL, µU ) ,

(
σ2L, σ2U

)]
where
The mean of the distribution lies in the interval (µL, µU )
The variance lies in the interval (σ2L, σ2U )
The random variable XN ∈ (XL, XU ), derived from the above parameters.
For simulation purpose we generate 5000 random samples from neutrosophic normal dis-
tributions defined as:

YN∼NN ((5, 8), ((0.9)2, (1.2)2)

XN∼NN ((15, 18), ( (0.5)2, (0.7)2)).

For each simulated value, first randomly sample a mean and variance within the specified
intervals.



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 15 of 22

For Y N

YN ∼ U (5, 8)
σ2Y ∼ U(0.81, 1.44)
For XN

XN ∼ U (15, 18)
σ2X∼ U(0.25, 0.49)
By utilizing µ and σ2 to generate values for standard normal distribution.
xi ∼ NN

(
µX , σ2X

)
and yi ∼ NN

(
µY , σ2Y

)
Repeat the above steps for 5000 iteration to generate 5000 simulated values for each of
XN and YN .

Table 5: MSE using simulated neutrosophic data

EstimatorMSE Values of
existing work

Ŷ SN (Singh estima-
tor)

Ŷ propN(Proposed
work)

Ŷ UN [0.2619458,
0.2619458]

[0.2022682, 0.1893483] [0.02465215,
0.02063134]

Ŷ RN [0.1290996,
0.1497684]

[0.2082852, 0.1952096] [0.02465263,
0.02063211]

Ŷ PN [0.448552,
0.4089126]

[0.2017871, 0.1890021] [0.02465212,
0.02063127]

Ŷ DN [0.02065194,
0.02466429]

[0.2204352¸ 0.2113867] [0.02465366,
0.02063334]

Ŷ BTRN [0.1888027,
0.2015085]

[0.2040597, 0.1911544] [0.0246523,
0.02063158]

Ŷ BTPN [0.3310805,
0.3485289]

[0.2098011, 0.1985855] [0.02465288,
0.02063229]

[0.2022997, 0.1893758] [0.02465215,
0.02063134]

[0.202462, 0.1896759] [0.02465218,
0.02063136]

[0.2085363, 0.1955049] [0.2465265,
0.02063214]

[0.2618358¸0.2617713] [0.02465485,
0.02063499]



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 16 of 22

Figure 3: Showing mean squared error of all considered estimators based on simulation study



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 17 of 22

Table 6: PRE using simulated neutrosophic data

EstimatorPRE Values of
existing work

Ŷ SN (Singh es-
timator)

Ŷ propN(Proposed
work)

Ŷ UN [100,100] [129.5042,
138.3407 ]

[1062.568, 1269.65]

Ŷ RN [202.9022,
174.9006]

[125.7631,
134.1869 ]

[1062.547, 1269.603]

Ŷ PN [58.3981,
64.05912]

[129.813,
138.5941]

[1062.569, 1269.654]

Ŷ DN [1062.045,
1268.383]

[118.8312,
123.9178]

[1062.503, 1269.527]

Ŷ BTRN [129.9925,
138.7405]

[128.3673,
137.0336]

[1062.561, 1269.635]

Ŷ BTPN [75.15756,
79.11846]

[124.8543,
131.9058]

[1062.537, 1269.592]

[129.4841,
138.3207]

[1062.568, 1269.65]

[129.3802,
138.1018]

[1062.567, 1269.649]

[125.6116,
133.9843]

[1062.546,1269.601]

[100.042,
100.0667]

[1062.451, 1269.426]



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 18 of 22

Figure 4: Showing percentage relative efficiency of all considered estimators based on simulation study

7. Conclusion

Point estimates in survey sampling have the limitation of providing only a single value
for the parameter in controversy, which can vary between samples due to sampling error.
Therefore, ambiguous, indeterminate, or uncertain facts are dealt with using the neutro-
sophic strategy, which is an extension of the classical approach. In this article, we have
recommended neutrosophic general class of estimators for estimating population mean
under simple random sampling. The suggested estimators are checked with both actual
UBL data and simulated data to determine how effectively they work. The result based
on the neutrosophic data for the mean squared error and percentage relative efficiency are
given in Tables 3 and 4. On the way the mean square error and PRE based on simulated
data are given in Tables 5 and 6. It is established that the proposed estimators achieve
better than a number of alternately modified estimators. This research provides the way
for future work in developing more precise estimators for use with a wide variety of neu-
trosophic data and sampling strategies. This study adds to its creativity by incorporating
pre-existing estimators into the neutrosophic framework. This shows how adaptable and
versatile it is. The results imply that neutrosophic statistics are a strong way to look at
data that is not certain, which makes it easier to make decisions in a number of situations.
We suggest using these sophisticated estimators in the future and pressure the need for
more study to make them work better with different types of neutrosophic data and ways
of sampling. Also, future studies will include more than one sample design, like systematic,
sequential, and double sampling.

Data availability

All the data are available within the manuscript.



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6477 19 of 22

Conflict of interest

The authors declare no conflict of interest.

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