EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6434
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

An Enhanced Class of Estimators for the Population
Mean Using Neutrosophic Statistics: A Case Study of

the Islamabad Stock Exchange

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. Point estimates have their limitations in survey sampling due to the fact that they
provide just a single value for the parameter under study, which may vary between samples due
to sampling errors. By producing interval estimates of the expected position of the parameter,
the neutrosophic approach serves as a viable alternative in sampling theory. The neutrosophic
approach optimizes the traditional strategy for effectively handling ambiguous data. To find the
mean of the population using neutrosophic information we introduce a new family of estimators
that incorporate additional information. Discovering the bias and mean square error is performed
up to the first-order approximation. These estimators are ideal for data which is logical, confused,
or ambiguous. This estimator is designed to make neutrosophic statistics (NS) in basic random
sampling easier to understand. To better show the range of possible values for our population
parameter, we show numerical findings for these estimators as intervals rather than single points.
To further assess the efficiency of our proposed neutrosophic estimator, we utilize interval data
and simulation derived from the Islamabad Stock Exchange (ISE).

2020 Mathematics Subject Classifications: 62D05, 62f10

Key Words and Phrases: Neutrosophic statistics, simulation, visualization, mean estimation,
bias, mean squared error (MSE), percentage relative efficiency (PRE)

1. Introduction

In conventional statistics, data are already established and are represented by precise fig-
ures. The estimation of population means using auxiliary data has been extensively studied
and developed by numerous scholars in traditional statistics. Typically, a specific value is
assigned to a study variable for each population unit before sample selection. A high cor-
relation between the study and auxiliary variables significantly lowers the ratio’s sampling

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6434

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), 6434 2 of 22

error when the auxiliary variable is also considered. This impacts the smallest necessary
sample size for ratio estimation methods, or how much these methods can reduce sample
size without losing precision. Classical statistics focus on fixed data, where all possible
measurement errors are preemptively addressed, leading to a scenario with no measure-
ment error margin. Consequently, new approaches are required to handle indeterminate
data. Fuzzy logic (FL) is one approach for managing data when precise measurements
of the variable of interest are unavailable. Fuzzy statistics allow for the analysis of data
with uncertain, ambiguous, or undefined characteristics, although this method does not
account for the level of data indeterminacy. Conversely, neutrosophic logic extends FL,
enabling the assessment of both conclusive and inconclusive elements of data under vague
or uncertain conditions. The utilization of uncertain logic in decision-making processes
has surged Ali and Mahmood [3] and Jan et al. [9], accompanied by an enhancement in
the complexity of its methodologies. Following the concept of fuzzy sets, the development
of a complex neutrosophic set Liu et al. [11] represents an advanced stage of evolution
in fuzzy collections. These collections and their broader applications are detailed in Li et
al. [10], which includes thorough discussions of their characteristics and functionalities,
and also examines interval-valued neutrosophic sets. When fuzzy sets are inadequate for
addressing ambiguities in decision-making contexts, neutrosophic sets emerge as a supe-
rior alternative. Neutrosophic sets are categorized into diverse types. Chakraborty et al.
[14] has introduced a trapezoidal bipolar neutrosophic quantity, specifically designed for
decision-making applications. Additionally, generalized spherical fuzzy numbers and their
analytical frameworks were discussed in a publication by Haque et al.
Different research support the use of a mobile application to handle geometric and math-
ematical operations on pentagonal neutrosophic numbers Chakraborty et al. [13]. The
obsession with neutrosophic numbers intense with time; for example, a technique has been
devised for the cylindrical neutrosophic environment Chakraborty et al. [12]. Neutrosophic
statistics provide a solution for issues involving ambiguous data display. It is imperative
to use neutrosophic statistics instead of traditional statistics when data or samples show
neutrosophic features. Statistically, neutrosophic approaches Smarandache [16] are used
where the data from populations or samples are vague, obscure, or not well-defined. This
discussion covers several uses of neutrosophic logic: by addressing uncertainties in the
opinions of survey participants, neutrosophic logic enhances survey sampling techniques.
It adopts a more intricate approach to data gathering and analysis by accepting ambigu-
ity, conflicting information, and indeterminate details. This approach leads to a deeper
understanding of diverse viewpoints, thereby improving the accuracy and reliability of
survey results.
Neutrosophic data pertains to datasets characterized by inherent uncertainties, analyzed
through neutrosophic numerical methods. In the realm of neutrosophic statistics, the ex-
act sample size may remain undefined Smarandache [16]. It has been established that
neutrosophic statistics provide substantial benefits and should be integrated into current
models dealing with uncertainty Aslam [23] and Smarandache [24]. Rock engineers uti-
lize neutrosophic numbers to investigate the effects of gauge and anisotropy on the total
roughness coefficient, which results in reduced loss of information and the generation of



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 3 of 22

an adequate number of appropriate functions Chen et al. [25]. A novel method for con-
ducting neutrosophic analysis of variance has been introduced within the framework of
neutrosophic data Aslam [30]. Contributions from Singh et al. [32] and Alsharif et al.
[33] have discussed diverse kerosene-based nano-fluids in a dual-zone vertical annulus with
electro-osmosis. Pioneers in the area of neutrosophic interval statistics, Aslam [26], Aslam
[27], Aslam [28] and Aslam [29] have led developments in their fields. Further information
on neutrosophic data is available in references Tahir et al. [17], Yadav and Smarandache
[18], Bouze and Subzar [21], Aslam [22], Das et al [36], Granados et al. [34].
In real-world applications, a variety of factors can be examined, yet data collection can
be exceedingly costly, particularly when the data is not clear. This situation makes it
expensive and risky to accurately determine a population’s value using methods designed
for uncertain data. When both primary and auxiliary variables are neutrosophic, it be-
comes unfeasible to describe the issue. Reviews of existing literature reveal that there
has been no substantial research on the use of enhanced generalized class estimators for
estimating the unknown mean of a population with auxiliary variables in survey sampling.
This observation is based on current studies in published works. The field of statistics is
yet to see significant pioneering research to be deemed complete. This research begins to
address this gap.
Neutrosophic analysis is advantageous for dealing with data that is either incomplete or
vague. This approach also accommodates multiple viewpoints. During data collection
using specific instruments, it is conceivable that some data may fall into an undefined
category. The failure of traditional analytical methods due to statistical ambiguity ne-
cessitates the adoption of neutrosophic statistics, which acts as both a substitute and a
supplement to traditional statistics in uncertain conditions. Recent progress in neutro-
sophic survey sampling is significant, however, the wider field of estimation is still largely
uncharted and requires meticulous attention due to the uncertainties involved in the data
framework. Products with minor flaws in measurements or manufacturing defects are ac-
ceptable within a certain tolerance, such as in the production of nuts or bolts. Employing
traditional statistical methods that yield a single result could result in the discarding of
otherwise usable items. Neutrosophic statistics offer the most accurate evaluation of inter-
val results with the minimal mean squared error, effectively tackling these issues Kumar
et al. [19], Raghav [20] and Granado et al. [35].

1.1. Research gap

Previously researchers who studied survey sampling only looked at data that was partic-
ular, clear, and certain. The one thing that all of these methods can reliably give you is
not always free of mistakes, overstatements, or underestimates. Using these methods only
leads to one correct result. Neutrosophic data, on the other hand, are present in many
circumstances and under certain conditions. In this case, neutrosophic statistics are used
even when more typical statistical approaches didn’t work. Neutrosophic data includes
observations that aren’t clear, reasoning that isn’t clear, and fuzzy interval values. This
means that data from trials or populations can be used as neutrosophic numbers with in-



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 4 of 22

terval values. One of the hypotheses said that the genuine observation, whose value could
not be known when the data was being collected, would fall within that range. In the
real world, there is a lot more data that is hard to define than data that is easy to define.
This shows that neutrosophic research needs more statistical methods to be devised. In
the actual world, it might be hard and expensive to get information on all the possible
study parts, especially when the proof isn’t clear. It is impossible to explain the problem
when a study and its extra variables are neutrosophic. A literature review indicated that
enhanced generalized class estimators have not been studied while trying to use regression
cum-exponential estimation methods to find a way to estimate the unknown population
mean with extra variables in neutrosophic data. This is the conclusion that comes from the
review’s results. Therefore, there is a demand for additional statistical methods tailored
to neutrosophic research. This is clear from a thorough look of Alomair and Ahmad [1].
More research has to be done in this area of statistics. This study gives a basic overview
of the topic.

Table 1: LIST OF ACRONYMS

MSE Mean Square Error

PRE Percentage Relative Efficiency

SRSWOR Simple Random Sampling Without Replace-
ment

ISE Islamabad Stock Exchange

NS Neutrosophic statistics

FL Fuzzy Logic

2. Methods and materials

Choose a neutrosophic sample of size nN∈ (nL, nU ), from a population of NN units
(Ω1, Ω2, . . . , ΩN ) with the help of SRSWOR. Allow yNi be the ith sample observa-
tion of neutrosophic data, of the form yN ∈ (yL, yU ) and similarly for auxiliary variable
xN∈ (xL, xU ). As Y N∈ (YL, YU ), and XN∈ (XL, XU ) XU is the population mean of
neutrosophic study and auxiliary variable. The neutrosophic coefficient of variation for
yN and xN are denoted by CyN∈ (CyNL, CyNU ), and CxN∈ (CxNL, CxNU ). Let s2yN

=
∑n

i=1 (yiN−yN )2

nN−1 , s2xN =
∑n

i=1 (xiN−xN )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 , of YN and XN

respectively.
Let CyN and CxN , denote the population coefficient of variation of yN and xN , where

CyN=
SyN

Y
, and CXN=SxN

X
.

Where SyN =

√∑N
i=1 (yiN−Y N )

2

NN−1 and SxN=

√∑N
i=1 (xiN−XN)

2

NN−1 . The correlation between



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 5 of 22

yN and xN is denoted by ρyxN . i.e. ρyxN=
SyxN

SyNSxN
,

Syx =

∑N
i=1

(
yiN − Y N

) (
xiN −XN

)
NN − 1

, λN =

(
1

nN
− 1

NN

)
, λN∈ (ξL, ξU ) .

3. Review of existing estimators

In this section, we have presented some prevailing counterparts.

(i) The conventional estimator is given in equation (1):

Ŷ UN = yN (1)

The variance of Ŷ UN is given in equation (2):

V (Ŷ UN ) = λNY
2
NC2

yN (2)

(ii) The adopted ratio estimator recommended by Cochran [42] is given in equation (3):

Ŷ RN = yN

(
XN

xN

)
(3)

The bias of Ŷ RN is given by:

Bias(Ŷ RN ) ∼= λNY N

(
C2
yN − ρyxNCyNCxN

)
,

The MSE of Ŷ RN is given in equation (4):

MSE(Ŷ RN ) ∼= λNY
2
N

(
C2
yN + C2

xN − 2ρyxNCyNCxN

)
(4)

(iii) The product estimator suggested by Murthy [7], given in equation (5):

Ŷ PN = yN

(
xN

XN

)
(5)

The bias of Ŷ PN is given by:

Bias (Ŷ PN ) ∼= λNY N ρyxNCyNCxN ,

The MSE of Ŷ PN is given in equation (6):

MSE(Ŷ PN ) ∼= λNY
2
N

(
C2
yN + C2

xN + 2ρyxNCyNCxN

)
(6)



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 6 of 22

(iv) The usual difference estimator is given in equation (7):

Ŷ DN = yN +Q
(
XN − xN

)
, (7)

where Q is constant.

Q= ρyxN

(
SyN

SxN

)
The variance of Ŷ DN is given in equation (8):

Var(Ŷ DN ) min
∼= λNY

2
NC2

yN (1− ρ2yxN ) = MSE(P̂DN ) min (8)

(v) Bahl and Tuteja [6] recommended the following two estimators, which are given in
Equation (9) and Equation (10):

Ŷ BTRN = yNexp

(
XN − xN

XN + xN

)
(9)

Ŷ BTPN = yNexp

(
xN −XN

XN + xN

)
(10)

The bias of Ŷ BTN is given by:

Bias
(
Ŷ BTRN

)
∼= λN Y N

[
3
8 C2

xN − 1
2ρyxNCyNCxN

]
,

The MSE of Ŷ BTRN is given in equation (11):

MSE
(
Ŷ BTRN

)
∼= λNY

2
N

[
C2
yN +

1

4
C2
xN − ρyxNCyNCxN

]
(11)

The bias of Ŷ BTPN is given by:

Bias
(
Ŷ BTPN

)
∼= λN Y N

[
3
8 C2

xN + 1
2ρyxNCyNCxN

]
,

The MSE of Ŷ BTPN is given in equation (12):

MSE
(
Ŷ BTPN

)
∼= λNY

2
N

[
C2
yN +

1

4
C2
xN + ρyxNCyNCxN

]
(12)

(vi) Singh et al. [31] recommended a generalized exponential-type estimator, which is
given in equation (13):

Ŷ SN = yNexp

(
a
(
XN − xN

)
a
(
XN + xN

)
+ 2b

)
(13)



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 7 of 22

Bias
(
Ŷ SN

)
∼= λN Y N

(
3
8 θ2NC

2
xN − 1

2θNρyxNCyNCxN

)
,

The mean square error of Ŷ SN is given in equation (14):

MSE
(
Ŷ SN

)
= λNY

2
N

(
C2
yN +

1

4
θ2NC2

xN + θNρyxNCyNCxN

)
, (14)

where θN= aXN

aXN+b
.

4. Suggested estimator:

By taking Inspiration from Alomair and Ahmad [1], Ahmad et al. [2] and Ahmad et al. [4],
we propose enhanced generalized neutrosophic estimators for calculating the population
mean with simple random sampling and auxiliary data. These neutrosophic estimators
could outperform traditional estimators when the observations of the study variable are
nondeterministic, but might not be as effective when the observations are deterministic.
The proposed estimator is presented in equation (15):

Ŷ
(∗)
Prop, N =

[
Q3yN + Q4

(
XN − xN

)]
exp

(
a
(
XN − xN

)
a
(
XN + xN

)
+ 2b

)
(15)

where θN= aXN

aXN+b
Some members of the generalized class of estimators are given in Table 2.

Table 2: 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



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 8 of 22

Simplifying equation (15), we got:

Ŷ
(∗)
Prop, N =

{
Q3Y N (1 + ξ0)− Q4XNξ1

}
exp

(
1− θNξ1

2
+

3θ2Nξ21
8

+ . . .

)
(16)

By expanding equation (16):

(
Ŷ

(∗)
Prop, N − Y N

)
=

− Y N+Y NQ3+Y Nξ0 Q3 −
1

2
Y N θNξ1 Q3 −XNξ1 Q4

− 1

2
Y N θNξ0ξ1 Q3+

3

8
Y Nθ2Nξ

2
1 Q3+

1

2
XN θNξ21 Q4

 (17)

From equation (17), the bias and MSE of Ŷ
(∗)
Prop, N , are given by:

Bias

(
Ŷ

(∗)
Prop, N

)
∼= 1

8

[
−8Y N + 4λNθNCxN (XNCxN Q4) + Y N Q3 {8 + λNθNCxN (3θNCxN − 4CyNρyxN )}

]
The MSE of Ŷ

(∗)
Prop, N , is given in equation (18):

MSE

(
Ŷ

(∗)
Prop, N

)
∼=



Y
2
N + λNXNC2

xN Q4

(
−Y NθN +XN Q4

)
+ Y

2
NQ2

4

[
1 + λN

{
C2
yN + θNCxN (θN − 2CuρyxN )

}]
+

1

4
Y N Q3

[
−8Y N + CxN

{
θNCxN

(
−3Y NθN + 8XN Q4

)}]
1

4
Y N Q3

[
+4CyN

(
Y NθN − 2XN Q4

)
ρyxN

]


(18)

The values of Q3 and Q4 are:

Q3(opt) =
8− λNθ2NC2

xN

8
{
1 + λNC2

yN (1− ρyxN )
} ,

and

Q4(opt) =
Y N

[
λNθ2NC3

xN + 8CyNρyxN − λNθ2NC2
xNCyNρyxN − 4θNCxN (1− λNC2

yN{1− ρ2yxN})
]

8XNCxN{1− λNC2
yN{1− ρ2yxN}}

Using values of Q3(opt), and Q4(opt) in equation (18), we got the minimum mean square
error, which is given in equation (19):

MSE

(
Ŷ

(∗)
Prop, N

)
min

∼=
λNY

2
N

(
64C2

yN

(
1− ρ2yxN

)
− λNθ4NC4

xN − 16λNθ2NC2
xNC2

yN

(
1− ρ2yxN

))
64
{
1 + λNC2

yN

(
1− ρ2yxN

)}
(19)

Figure 1:



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 9 of 22

Figure 1: Showing Bias of Estimator

5. Numerical study

In this section, we utilize real data to compare the proposed and current estimators numer-
ically. A numerical study was performed using interval data containing uncertain values
from the ISE, focusing on United Bank Limited (UBL), taken from Alomair and Ahmad
[1]. In Table 2 we see the data sets and their explanations. Estimators are compared with

one another based on their PRE. The PRE of Ŷ UN with due regard to Ŷ UN is expressed
below:

PRE= V ar(Ŷ UN )

MSE
(
Ŷ i,NN

) ×100,

where i = Ŷ RN , Ŷ PN , Ŷ DN , Ŷ RDN , Ŷ BTRN , Ŷ BTPN , Ŷ SN ,

Ŷ
(∗)
Prop, N (∗ = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 10 of 22

Table 3: Summary statistics using actual data

Parameter Values Parameter Values

NN [239,239] XN [149.6231,153.5848]

nN [35,35] S2
yN [2968.468,3131.635]

λN [0.02438733, 0.02438733] S2
xN [3110.931,3156.542]

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

Table 4: Biases using actual data

Estimator Bias Values

Ŷ UN -

Ŷ RN [-0.002514721, -0.001244049]

Ŷ PN [0.003125356, 0.001836447]

Ŷ DN -

Ŷ BTRN [0.006398227, 0.006869877]

Ŷ BTPN [0.009523583, 0.008706324]

Ŷ BTPN [0.006390427, 0.00686232]

Ŷ
(∗)
Prop, N [-0.00273157, -0.0175342]

Figure 2: Showing MSEs and PREs of estimators based on real data set



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 11 of 22

Table 5: MSE using actual data

Estimator Values For values of a
and b

Ŷ SN Ŷ
(∗)
Prop, N

Ŷ UN [0.004771434,
0.004771434]

[For a=1,
b=CxN ]

[0.002051822,
0.003495624]

[0.001176135,
0.003463482]

Ŷ RN [0.001176741,
0.004079918]

[For a=1,
b=β2(xN)]

[0.002077862,
0.003500653]

[0.001176136,
0.003463481]

Ŷ PN [0.01574454,
0.01290342]

[For a=β2(xN),
b=CxN ]

[0.002051802,
0.00349562]

[0.001176135,
0.003463482]

Ŷ DN [0.002050780,
0.001895328]

[For a=CxN ,
b=ρyxN ]

[0.004184703,
0.004426577]

[0.001176135,
0.003463482]

Ŷ BTRN [0.002051786,
0.003495617]

[For a=1,
b=ρyxN ]

[0.002062184,
0.003496803]

[0.001176135,
0.003463482]

Ŷ BTPN [0.009335684,
0.007907368]

[For a=CxN ,
b=ρyxN ]

[ 0.003647159,
0.003956309]

[0.001176135,
0.003463482]

[For a=ρyxN ,
b=CxN ]

[0.002051827,
0.00349563]

[0.001176135,
0.003463482]

[For a=β2(xN),
b=ρyxN ]

[0.002056549,
0.003496158]

[0.001176135,
0.003463482]

[For a=ρyxN ,
b=β2(xN)]

[0.002081793,
0.003505421]

[0.001176135,
0.003463482]

[For a=1,
b=NNXN ]

[0.004756275,
0.004771342]

[0.001176135,
0.003463482]

5.1. Theoretical Analysis of MSE and PRE Using Actual Data

Mean Squared Errors (MSEs)

(i) The MSE gauges the average squared difference between the actual and projected
values. Better estimate performance is shown by minimum MSE values.

(ii) Table 5 shows changing MSE values for several estimators and parameter combina-



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 12 of 22

Table 6: PRE using actual data

Estimator Values For values of a
and b

Ŷ SN Ŷ
(∗)
Prop, N

Ŷ UN [100,100] [For a=1, b=CxN ] [136.4973,
232.5462]

[137.7641,
405.6875]

Ŷ RN [116.9493,
405.4787]

[For a=1,
b=β2(xN)]

[136.3012,
229.6319]

[137.7642,
405.6875]

Ŷ PN [30.30533,
36.97806]

[For a=β2(xN),
b=CxN ]

[136.4975,
232.5484]

[137.7631,
405.6865]

Ŷ DN [138.28,
251.7471]

[For a=CxN ,
b=ρyxN ]

[107.7906,
114.0208]

[137.7631,
405.6864]

Ŷ BTRN [136.4976,
232.5503]

[For a=1,
b=ρyxN ]

[136.4513,
231.3777]

[137.7661,
405.6895]

Ŷ BTPN [51.10963,
60.34162]

[For a=CxN ,
b=ρyxN ]

[120.6031,
130.8261]

[137.7641,
405.6874]

[For a=ρyxN ,
b=CxN ]

[136.4971,
232.5456]

[137.7641,
405.6875]

[For a=β2(xN),
b=ρyxN ]

[136.4765,
232.0116]

[137.7641,
405.6875]

[For a=ρyxN ,
b=β2(xN)]

[136.1159,
229.1983]

[137.7641,
405.6875]

[For a=1,
b=NNXN ]

[100.0019,
100.3187]

[137.7641,
405.6874]

tions. Estimator Ŷ ARN , for instance, indicates that it offers reliable estimates with
less variability by showing somewhat low MSE values compared to other estimators.

With somewhat high MSE values, estimator Ŷ PN indicates poorer accuracy than
other estimators.

(iii) The choice of parameters (a and b) influences the performance of the estimators.



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 13 of 22

For instance, estimators with a=b=CxN tend to have lower MSE values compared to
other parameter combinations. This suggests that considering the mean of X values
(CxN ) as both a and b yields more accurate estimates.

(iv) To minimize MSE, it is advisable to choose parameter combinations that lead to
lower MSE values, such as a=b=CxN .

Percentage Relative Efficiency (PREs)

(i) The PRE measures the relative prediction error compared to the actual values.
Higher PRE values indicate greater prediction errors.

(ii) Table 6 shows different PRE values for several estimators and parameter combina-

tions. for instance: With rather high PRE values, estimator Ŷ ARN indicates more
prediction errors than other estimators. Relatively accurate predictions are shown

by estimator Ŷ
(∗)
Prop, N whose PRE values cross to 100.

(iii) As compared to all existing considered estimators, the performance of Ŷ
(∗)
Prop, N is

more accurate in terms of minimum MSE and higher percentage relative efficiency.

(iv) Similar to MSE, the choice of parameters (a and b) influences the PRE. Parameter
combinations that involve the mean of X values (CxN .) tend to yield lower PRE
values, suggesting better prediction accuracy.

(v) To minimize prediction errors, it is advisable to choose parameter combinations that
lead to lower PRE values, such as a=b=CxN .

Overall Analysis

The suggested class of estimators Ŷ
(∗)
Prop, Ngenerally performs better in terms of minimum

MSE and higher PRE compared to other estimators across different parameter combina-
tions. Choosing parameter combinations where both a and b are equal to the mean of X
values (CxN ) tends to lead to more accurate estimates and predictions.

6. Simulation study

This section illustrates the algebraic procedure for comparing Ŷ
(∗)
Prop, N to its existing

estimators, Ŷ UN , Ŷ RN , Ŷ PN , Ŷ DN , Ŷ BTRN , Ŷ BTPN , and Ŷ SN .
Numerical results are validated by means of a simulation research. We simulated neu-
trosophic data and performed MSE and PRE based comparison. Neutrosophic random
variables, XN and YN follow the neutrosophic normal distribution. With mean µXN and
variance σ2

XN exhibits a neutrosophic normal distribution. YN likewise has mean µY N and
variance σ2

Y N . When XN ∈ (XL, XU ) and YN ∈ (YL, YU ). Comparatively mean and
variance of neutrosophic random variables XN and YN have lower and upper limits. We



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 14 of 22

have produced 5000 random normal variate from the 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 )).

The simulation study algorithm, which demonstrates the performance of different Ŷ UN

estimators, is as follows:

(i) The Average mean square error of the estimators is defined by:

Average MSE
(
Ŷ UN

)
=

1

500

500∑
i=1

E(Ŷ i − Ŷ UN )
2

(ii) The PRE of estimators as compared to the usual estimator Ŷ UN is defined by:

PRE (i) =
V ar(Ŷ UN )

MSE(Ŷ i)
× 100

(iii) The Average percentage relative efficiency of the estimators is given by:

Average PRE (i) =
1

500

500∑
i=1

PRE(Ŷ i)



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 15 of 22

Table 7: Biases using simulated data

Estimator Bias Values

Ŷ UN -

Ŷ RN [-0.001244049, -0.001244049]

Ŷ PN [0.001836447, 0.002023558]

Ŷ DN -

Ŷ BTRN [0.006869877, 0.006934565]

Ŷ BTPN [0.008706324, 0.009213423]

Ŷ BTPN [0.00686232, 0.00694341]

Ŷ
(∗)
Prop, N [-0.0175342,-0.0163563]

Figure 3: Showing MSEs and PREs of estimators based on simulation study



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 16 of 22

Table 8: MSE using a simulation study

Estimator Values For values
of a and b

Ŷ SN Ŷ
(∗)
Prop, N

Ŷ UN [0.2619458,
0.2619458]

[For a=1,
b=CxN ]

[0.2022682,
0.1893483]

[0.02465215,
0.02063134]

Ŷ RN [0.1290996,
0.1497684]

[For a=1,
b=β2(xN)]

[0.2082852,
0.1952096]

[0.02465263,
0.02063211]

Ŷ PN [0.448552,
0.4089126]

[For
a=β2(xN),
b=CxN ]

[0.2017871,
0.1890021]

[0.02465212,
0.02063127]

Ŷ DN [0.02566218,
0.02260200]

[For
a=CxN ,
b=ρyxN ]

[0.2204352¸
0.2113867]

[0.02465366,
0.02063334]

Ŷ BTRN [0.1888027,
0.2015085]

[For a=1,
b=ρyxN ]

[0.2040597,
0.1911544]

[0.0246523,
0.02063158]

Ŷ BTPN [0.3310805,
0.3485289]

[For
a=CxN ,
b=ρyxN ]

[0.2098011,
0.1985855]

[0.02465288,
0.02063229]

[For
a=ρyxN ,
b=CxN ]

[0.2022997,
0.1893758]

[0.02465215,
0.02063134]

[For
a=β2(xN),
b=ρyxN ]

[0.202462,
0.1896759]

[0.02465218,
0.02063136]

[For
a=ρyxN ,
b=β2(xN)]

[0.2085363,
0.1955049]

[0.2465265,
0.02063214]

[For a=1,
b=NNXN ]

[0.2618358,
0.2617713]

[0.02465485,
0.02063499]

6.1. Theoretical Analysis of MSE and PRE Using Simulation Data

Mean Squared Errors (MSEs)

(i) The MSE indicates the average squared difference between the actual and projected
values. Greater MSE values point to more estimate mistakes.

(ii) Table 8 shows changing MSE values for several estimators and parameter combina-



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 17 of 22

Table 9: PRE using simulation

Estimator Values For values of a
and b

Ŷ SN Ŷ
(∗)
Prop, N

Ŷ UN [100,100] [For a=1,
b=CxN ]

[129.5042,
138.3407 ]

[1062.568, 1269.65]

Ŷ RN [202.9022,
174.9006]

[For a=1,
b=β2(xN)]

[125.7631,
134.1869 ]

[1062.547, 1269.603]

Ŷ PN [58.3981,
64.05912]

[For a=β2(xN),
b=CxN ]

[129.813,
138.5941 ]

[1062.569, 1269.654]

Ŷ DN [1022.347,
1158.95]

[For a=CxN ,
b=ρyxN ]

[118.8312,
123.9178 ]

[1062.503, 1269.527]

Ŷ BTRN [129.9925,
138.7405]

[For a=1,
b=ρyxN ]

[128.3673,
137.0336 ]

[1062.561, 1269.635]

Ŷ BTPN [75.15756,
79.11846]

[For a=CxN ,
b=ρyxN ]

[124.8543,
131.9058 ]

[1062.537, 1269.592]

[For a=ρyxN ,
b=CxN ]

[129.4841,
138.3207]

[1062.568, 1269.65]

[For a=β2(xN),
b=ρyxN ]

[129.3802,
138.1018 ]

[1062.567, 1269.649]

[For a=ρyxN ,
b=β2(xN)]

[125.6116,
133.9843 ]

[1062.546,1269.601]

[For a=1,
b=NNXN ]

[100.042,
100.0667 ]

[1062.451, 1269.426]

tions. As one illustration: With somewhat high MSE values, estimator Ŷ PN indicates
more estimating error than other estimators. With rather low MSE values, estimator

Ŷ DN suggests improved accuracy performance.

(iii) The performance of the estimators depends on the parameters a and b selected.



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 18 of 22

For example, parameter combinations including the mean of X values CxNor other
pertinent statistics usually produce smaller MSE values, hence suggesting improved
estimate accuracy.

(iv) To minimize MSE, it is advisable to choose parameter combinations that lead to
lower MSE values, such as those involving relevant statistics like CxN or β2(xN).

Percentage Relative Efficiency (PREs)

(i) The PRE measures the relative prediction error compared to the actual values.
Higher PRE values indicate larger prediction errors.

(ii) Table 9 shows us different PRE values for several estimators and parameter combi-

nations. As an instance: With rather high PRE values, Estimator Ŷ RN indicates
more prediction errors than other estimators. With PRE values crossing to 100,

estimator Ŷ
(∗)
Prop, N suggests rather accurate predictions.

(iii) Similar to MSE, the choice of parameters (a and b) influences the PRE. Parameter
combinations that involve relevant statistics like CxN or β2(xN) tend to yield lower
PRE values, suggesting better prediction accuracy.

(iv) To minimize prediction errors, choosing parameter combinations that lead to lower
PRE values, such as those involving relevant statistics like CxN or β2(xN) is advis-
able.

Overall Analysis

(i) The suggested class of estimaotrs Ŷ
(∗)
Prop, N generally have better MSE and PRE

than other estimators over several parameter settings.

(ii) The performance of estimators is significantly influenced by the parameters cho-
sen; so, combinations of appropriate statistics produce superior accuracy in both
estimation and prediction.

(iii) Theoretical study indicates that choosing the best parameter combinations depends
on knowing the link between parameters and their influence on estimator perfor-
mance.

7. Discussion

In table 1, we presented a list of acronyms. In Table 2, we present the generalized class
of estimators. Table 3 consist the overall statistics for the real data sets. Table 4 shows
the detailed bias from estimators used on the real data set. Tables 5 and Table 6 record
the results from the real data set. Table 5 details the mean square errors for both existing



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 19 of 22

and proposed estimators, while Table 6 shows the percentage relative efficiency results
for all considered estimators in this analysis. By varying the parameters ’a’ and ’b’, the
proposed generalized class of estimators achieves different minimum mean squared errors,
as shown in Tables 5 and 8. Based on the numerical result it is found that the MSE of our
generalized class of estimators is minimal as compared to all considered estimators in this
paper. The Biases of estimators using simulated data sets are given in Table 7. Similarly,
the MSE and PRE using the simulated data sets are given in Table 8 and Table 9. Data
presented in Table 6 and Table 9 show that changing a and b changes the PREs for any
variety of estimators. Furthermore, it is possible to notice that PREs are higher when a =
1 and b = ρyxN , a = ρyxN and b = CxN , a = ρyxN and b = β2(xN), and somewhat lower

when a = 1 and b = NNXN . The numerical results are shown with the help of graphs.
The Figure 1, shown the biases of all considered estimators. Figure 2 and Figure 3, shown
the MSEs and PREs for the real data set and a simulation study. It is demonstrated that
the suggested estimators achieve minimum MSE demonstrating improvement in efficiency
as compared to existing estimators.

8. Conclusion

Point estimates in survey sampling are limited by providing only a single value for the pa-
rameter being studied, which can differ across samples due to sampling error. Conversely,
parameter estimates in sampling theory are significantly enhanced by the neutrosophic
approach, which offers credible interval estimates with a high probability of encompass-
ing the parameter. Thus, the neutrosophic strategy, an advancement of the traditional
method, addresses ambiguous, indeterminate, or uncertain elements. This paper proposes
a new general class of estimators for the population mean using neutrosophic data and
simple random sampling. By varying the values of ’a’ and ’b’, this new class of estimators
yields different minimum values for each case. The numerical analysis shows that the Mean
Squared Error (MSE) of these new estimators is the lowest when compared to all other
estimators discussed in this study. The results confirm that these estimators have the
lowest MSE, indicating enhanced efficiency over several other modified estimators. This
study paves the way for future research in creating more accurate estimators suitable for
a broad range of neutrosophic data and sampling techniques, including stratified ranked
set sampling, systematic sampling, and predictive methods in simple random sampling.

Data availability
All the data are available within the article.
Conflict of interest
The authors declare no conflict of interest.
Authors contribution
Sanaa Al-Marzouki: Wrote the main manuscript, analysis and visualization
Sohaib Ahmad: Writing original draft and supervision



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 20 of 22

References

[1] Alomair, A.M.; Ahmad, S. New comprehensive mean estimation using regression-
cum-exponential type estimator: Application with neutrosophic data. Kuwait J. Sci.
2025, 52, 100346.

[2] Ahmad, S.; Masood, S.; Alomair, A.M.; Alomair, M.A. Improved modified estimator
for estimation of median using auxiliary information under simple random sampling.
Sci. Rep. 2024, 14, 16504.

[3] Ali, Z.; Mahmood, T. Complex neutrosophic generalized dice similarity measures and
their application to decision making. CAAI Trans. Intell. Technol. 2020, 5, 78–87.

[4] Ahmad, S.; Hussain, S.; Shabbir, J.; Aamir, M.; El-Morshedy, M.; Ahmad, Z.; Al-
rajhi, S. Improved generalized family of estimators in estimating the finite population
mean using two auxiliary variables under two-stage sampling. AIMS Math. 2022, 7,
10609–10624.

[5] Azeem, M.; Hanif, M. The joint influence of measurement error and non-response on
estimation of population mean. Commun. Stat. Theory Methods 2017, 46, 1679–1693.

[6] Bahl, S.; Tuteja, R. Ratio and product type exponential estimators. J. Inf. Optim. Sci.
1991, 12, 159–164.

[7] Murthy, M.N. Product method of estimation. Sankhya Indian J. Stat. Ser. A 1964,
26, 69–74.

[8] Singh, H.P.; Kumar, S. Combination of regression and ratio estimate in the presence
of nonresponse. Braz. J. Probab. Stat. 2011, 25, 205–217.

[9] Jan, N.; Zedam, L.; Mahmood, T.; Ullah, K.; Ali, Z. Multiple attribute decision-
making method under linguistic cubic information. J. Intell. Fuzzy Syst. 2019, 36,
253–269.

[10] Li, D.F.; Mahmood, T.; Ali, Z.; Dong, Y. Decision-making based on interval-valued
complex single-valued neutrosophic hesitant fuzzy generalized hybrid weighted averag-
ing operators. J. Intell. Fuzzy Syst. 2020, 38, 4359–4401.

[11] Liu, P.; Ali, Z.; Mahmood, T. The distance measures and cross-entropy based on
complex fuzzy sets and their application in decision-making. J. Intell. Fuzzy Syst.
2020, 39, 3351–3374.

[12] Chakraborty, A.; Mondal, S.P.; Alam, S.; Mahata, A. Cylindrical neutrosophic single-
valued number and its application in networking problems, multi-criterion group
decision-making problems, and graph theory. CAAI Trans. Intell. Technol. 2020, 5,
68–77.

[13] Chakraborty, A.; Banik, B.; Mondal, S.P.; Alam, S. Arithmetic and geometric opera-
tors of pentagonal neutrosophic number and its application in mobile communication
service based MCGDM problem. Neutrosophic Sets Syst. 2020, 32, 61–79.

[14] Chakraborty, A.; Mondal, S.P.; Alam, S.; Dey, A. Classification of trapezoidal bipolar
neutrosophic number, de-polarization technique and its execution in cloud service-
based MCGDM problem. Complex Intell. Syst. 2021, 7, 145–162.

[15] Haque, T.S.; Chakraborty, A.; Mondal, S.P.; Alam, S. Approach to solve multi-criteria
group decision-making problems by exponential operational law in generalized spherical



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 21 of 22

fuzzy environment. CAAI Trans. Intell. Technol. 2020, 5, 106–114.
[16] Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic

Synthesis & Synthetic Analysis; American Research Press: Rehoboth, NM, USA,
1998.

[17] Tahir, Z.; Khan, H.; Aslam, M.; Shabbir, J.; Mahmood, Y.; Smarandache, F. Neu-
trosophic ratio-type estimators for estimating the population mean. Complex Intell.
Syst. 2021, 7, 2991–3001.

[18] Yadav, S.K.; Smarandache, F. Generalized neutrosophic sampling strategy for elevated
estimation of population mean. Neutrosophic Sets Syst. 2023, 53, 14.

[19] Kumar, S.; Kour, S.P.; Choudhary, M.; Sharma, V. Determination of population
mean using neutrosophic, exponential-type estimator. Lobachevskii J. Math. 2022,
43, 3359–3367.

[20] Raghav, Y.S. Neutrosophic generalized exponential robust ratio type estimators. Int.
J. Anal. Appl. 2023, 21, 41.

[21] Bouza-Herrera, C.N.; Subzar, M. Estimating the ratio of a crisp variable and a neu-
trosophic variable. Int. J. Neutrosophic Sci. 2020, 11, 9–21.

[22] Aslam, M. A variable acceptance sampling plan under the neutrosophic statistical
interval method. Symmetry 2019, 11, 114.

[23] Aslam, M. A new sampling plan using neutrosophic process loss consideration. Sym-
metry 2018, 10, 132.

[24] Smarandache, F. Introduction to Neutrosophic Statistics; Infinite Study: Columbus,
OH, USA, 2014.

[25] Chen, J.; Ye, J.; Du, S. Scale effect and anisotropy analyzed for neutrosophic numbers
of rock joint roughness coefficient based on neutrosophic statistics. Symmetry 2017,
9, 208.

[26] Aslam, M. Monitoring the road traffic crashes using NEWMA chart and repetitive
sampling. Int. J. Inj. Control Saf. Promot. 2020, 28, 39–45.

[27] Aslam, M. A new goodness of fit test in the presence of uncertain parameters. Complex
Intell. Syst. 2021, 7, 359–365.

[28] Aslam, M. A study on skewness and kurtosis estimators of wind speed distribution
under indeterminacy. Theor. Appl. Climatol. 2021, 143, 1227–1234.

[29] Aslam, M.; Arif, O.H.; Sherwani, R.A.K. New diagnosis test under the neutrosophic
statistics: An application to diabetic patients. BioMed Res. Int. 2020, 2020, 2083167.

[30] Aslam, M. Neutrosophic analysis of variance: Application to university students.
Complex Intell. Syst. 2019, 5, 403–407.

[31] Singh, R.; Chauhan, P.; Sawan, N.; Smarandache, F. Improvement in estimating the
population mean using exponential estimator in simple random sampling. In Auxiliary
Information and a priori Values in Construction of Improved Estimators; Infinite
Study: Columbus, OH, USA, 2007; p. 33.

[32] Singh, A.; Kulkarni, H.; Smarandache, F.; Vishwakarma, G.K. Computation of sepa-
rate ratio and regression estimator under neutrosophic stratified sampling: An appli-
cation to climate data. J. Fuzzy Ext. Appl. 2024, 5, 605–621.

[33] Alsharif, A.M.; Abdellateef, A.I.; Elmaboud, Y.A.; Abdelsalam, S.I. Performance



S. Al-Marzouki, S. Ahmad / Eur. J. Pure Appl. Math, 18 (3) (2025), 6434 22 of 22

enhancement of a DC-operated micropump with electroosmosis in a hybrid nanofluid:
Fractional Cattaneo heat flux problem. Appl. Math. Mech. 2022, 43, 931–944.

[34] Granados, C.; Das, A.K.; Das, B. Some continuous neutrosophic distributions with
neutrosophic parameters based on neutrosophic random variables. Adv. Theory Non-
linear Anal. Appl. 2022, 6, 380–389.

[35] Granados, C.; Das, A.K.; Osu, B.O. Weighted neutrosophic soft multiset and its
application to decision making. YUJOR 2023, 33, 293–308.

[36] Das, A.K.; Das, R.; Das, S.; Debnath, B.K.; Granados, C.; Shil, B.; Das, R. A
comprehensive study of neutrosophic superhyper BCI-semigroups and their algebraic
significance. Trans. Fuzzy Sets Syst. 2025, 8, 80.