Int. J. Anal. Appl. (2023), 21:41 

 

 

Received: Feb. 11, 2023. 

2020 Mathematics Subject Classification. 62P05. 

Key words and phrases. neutrosophic statistics; classical statistics; simulation; robust type estimators; indeterminacy 

intervals. 

 

https://doi.org/10.28924/2291-8639-21-2023-41 © 2023 the author(s) 

ISSN: 2291-8639  

1 

 

Neutrosophic Generalized Exponential Robust Ratio Type Estimators 

Yashpal Singh Raghav* 

Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia 

*Corresponding author: yraghav@jazanu.edu.sa 

ABSTRACT. Estimators proposed under classical statistics fail if data are vague or indeterminate. Neutrosophic Statistics 

are the only alternative because its deal with indeterminacy. Extensive reserch has been conducted in this field because of 

its wide applicability. This study aimed to further develop the theory of neutosophic simple random sampling without 

replacement. In this study, a generalized neutrosophic exponential robust ratio-type estimator was proposed, and five of its 

member neutrosophic estimators were developed. Derivations of  the bias and Mean Square Error were provided up to the 

first-order approximation. To demonstrate the high efficiency of the proposed neutrosophic estimators an empirical study 

on the stock price of Moderna and four simulation studies have been conducted, and the results show that the proposed 

neutrosophic estimators are more efficient than similar existing ratio type estimators discussed in this paper in neutrosophic 

as well as classical forms. 

 

1. INTRODUCTION 

Classical statistics and its methods deal with randomness but there are cases where the data at hand 

is indeterminate or vague or ambiguous or imprecise rather than random. In such situations estimation 

using classical statistical methods does not yield promising results. Fuzzy logic [1, 2] is one solution 

to tackle such a problem but still, it ignores indeterminacy. In such cases, neutrosophic methods are 

much more reliable. They deal with both randomness and more importantly with indeterminacy. 

Neutrosophic statistics refers to a set of data such that the data or a part of it is indeterminate and 

methods to analyze such a data [3]. 

Neutrosophic statistics is an extension of classical statistics and when the indeterminacy is zero, 

neutrosophic statistics coincides with classical statistics [3]. Estimation through neutrosophic 

https://doi.org/10.28924/2291-8639-21-2023-41


2 Int. J. Anal. Appl. (2023), 21:41 

 

methods is a new field and therefore it is unexplored unlike estimation problems in classical probability 

sampling designs where the data is determinate [4-8]. But, due to its wide applicability, it has gained 

much more importance than classical statistics and as a results it is being applied in various fileds for 

instance in decision making [9]. [10] developed a new sampling plan using neutrosophic process. [11] 

proposed neutrosophic analysis of variance. [12] used neutrosophic statistics in analyzing road traffic 

accidents. [13] proposed goodness of fit test in neutrosophic statistics. As a result filed of 

neutrosophic sampling has been developed and some neutrosophic ratio-type estimators has been 

proposed [14] and this paper is the second paper aimed at further developing the theory of 

neutrosophic SRSWOR or NSRSWOR sampling.  

It has been observed in some sample surveys that the data collected containes some vagueness due 

to many factors like methodolgy used (observing blood pressure multiple times within an interval in 

NFHS 4 [16]) , observing daily stock price [15, 19] or daily temperature of a city [14]. All these are 

examples where the data contains some indeterminacy and clssical statistical measures like mean, 

median or standard deviation might not give results which are useful for decision making. 

Thus the aim of this paper is to further develop neutrosophic probability sampling theory particulary 

NSRSWOR by developing various generalized neutrosophic exponential robust ratio type estimators. 

In Section 2, the paper presents the terminologies of neutrosophic statistics for new readers. In 

Section 3, existing related neutrosophic ratio-type estimators have been presented.  

In Section 4, the proposed generalized neutrosophic exponential robust ratio type estimator and the 

five developed estimators along with their derivations of biases and MSEs are presented. In order to 

demonstrate the high efficiecny of the developed neutrosophic generalized neutrosophic exponential 

robust ratio type estimators four simulation studies have been conducted in Section 5. The results 

are compared with their classical MSE values as well. Results and concluding remarks on this paper 

are provided in Section 6 along with some future fruitful areas of research. 

 

2. TERMINOLOGY 

A simple random neutrosophic sample of size n from a classical or neutrosophic population is a 

sample of n individuals such that at least one of them has some indeterminacy [3, 14]. 

As presented in [14], a neutrosophic observation is of the form  

𝑍𝑁 = 𝑍𝐿 + 𝑍𝑈 𝐼𝑁, where 𝐼𝑁 ∈ [𝐼𝐿 , 𝐼𝑈 ] and 𝑍𝑁 ∈ [𝑍𝑙 , 𝑍𝑢 ]. 

Now consider a simple random neutrosophic sample of size 𝑛𝑁 ∈ [𝑛𝐿 , 𝑛𝑈 ] drawn from a finite 

population of size N and 𝑦𝑁 (𝑖) ∈ [𝑦𝐿 , 𝑦𝑈 ] and 𝑥𝑁 (𝑖) are 𝑖
𝑡ℎ ∈ [𝑥𝐿 , 𝑥𝑈 ] neutrosophic sample 



3 Int. J. Anal. Appl. (2023), 21:41 

 

observation. Here the population mean of neutrosophic survey and auxiliary variable are �̅�𝑁 ∈ [𝑌𝐿 , 𝑌𝑈 ] 

and �̅�𝑁 ∈ [𝑋𝐿 , 𝑋𝑈 ] respectively. 

𝐶𝑦𝑁 ∈ [𝐶𝑦𝑁𝐿 , 𝐶𝑦𝑁𝑈 ] and 𝐶𝑥𝑁 ∈ [𝐶𝑥𝑁𝐿 , 𝐶𝑥𝑁𝑈 ] are population coefficient of variation of neutrosophic 

survey and auxiliary variables respectively. In addition, 𝜌𝑥𝑦𝑁 ∈ [𝜌𝑥𝑦𝑁𝐿 , 𝜌𝑥𝑦𝑁𝑈 ], 𝛽1(𝑥𝑁 ) ∈

[𝛽1(𝑥𝑁𝐿 ), 𝛽1(𝑥𝑁𝑈 )] and 𝛽2(𝑥𝑁 ) ∈ [𝛽2(𝑥𝑁𝐿 ), 𝛽2(𝑥𝑁𝑈 )] are the correlation coefficient between the 

neutrosophic survey and auxiliary variables, coefficient of skewness and coefficient of kurtosis of the 

neutrosophic auxiliary variable respectively. 

The MSE of a neutrosophic estimator is of the form, 𝑀𝑆𝐸(�̅�𝑁 ) ∈ [𝑀𝑆𝐸𝐿 , 𝑀𝑆𝐸𝑈 ]. 

The error terms in neutrosophic statistics are: 

�̅�𝑦𝑁 = �̅�𝑁 − �̅�𝑁, 

�̅�𝑥𝑁 = �̅�𝑁 − �̅�𝑁, 

𝐸(�̅�𝑦𝑁 ) = 𝐸(�̅�𝑥𝑁) = 0,  𝐸(�̅�𝑦𝑁
2 ) =

𝑁−𝑛

𝑁𝑛

𝑆𝑦𝑁
2

�̅�𝑁
2 = ξ20 

  𝐸(�̅�𝑥𝑁
2 ) =

𝑁−𝑛

𝑁𝑛

𝑆𝑥𝑁
2

�̅�𝑁
2 = 𝜉02 

𝐸(�̅�𝑥𝑁 �̅�𝑦𝑁) =
𝑁−𝑛

𝑁𝑛

𝑆𝑦𝑁𝑆𝑥𝑁

�̅�𝑁�̅�𝑁
= 𝜉11, 

where �̅�𝑦𝑁 ∈ [�̅�𝑦𝑁𝐿 , �̅�𝑦𝑁𝑈 ], 

  �̅�𝑥𝑁 ∈ [�̅�𝑥𝑁𝐿 , �̅�𝑥𝑁𝑈 ], 

  �̅�𝑦𝑁
2 ∈ [�̅�𝑦𝑁𝐿

2 , �̅�𝑦𝑁𝑈
2 ], 

  �̅�𝑥𝑁
2 ∈ [�̅�𝑥𝑁𝐿

2 , �̅�𝑥𝑁𝑈
2 ]. 

 

3. SOME RELATED NEUTROSOPHIC ESTIMATORS 

Since neutrosophic probability sampling is a new area of research handful of ratio type estimators 

are proposed in this Neutrosophic Simple Random Sampling Without Replacement (NSRSWOR). 

Tahir et al. [14] proposed the following ratio-type estimators given by 

  �̅�𝑅𝑁 =
�̅�𝑁

�̅�𝑁
�̅�𝑁 ,                                                                              (3.1) 

  �̅�𝑆𝐷𝑟𝑁 = �̅�𝑁
�̅�𝑁 +𝐶𝑥𝑁

�̅�𝑁+𝐶𝑥𝑁
,                                                                        (3.2) 

  �̅�𝑆𝐾𝑟𝑁 = �̅�𝑁
�̅�𝑁+𝛽2(𝑥𝑁)

�̅�𝑁+𝛽2(𝑥𝑁)
,                                                                     (3.3) 

 �̅�𝑈𝑆𝑟𝑁 = �̅�𝑁
�̅�𝑁𝛽2(𝑥𝑁)+𝐶𝑥𝑁

�̅�𝑁𝛽2(𝑥𝑁)+𝐶𝑥𝑁
,                                                                 (3.4) 



4 Int. J. Anal. Appl. (2023), 21:41 

 

where �̅�𝑁 ∈ [�̅�𝑁𝐿 , �̅�𝑁𝑈 ] and 𝑦𝑅𝑁 ∈ [𝑦𝑅𝐿 , 𝑦𝑅𝑈 ], �̅�𝑈𝑆𝑟𝑁 ∈ [�̅�𝑆𝐷𝑟𝐿 , �̅�𝑆𝐷𝑟𝑈 ], �̅�𝑆𝐾𝑟𝑁 ∈ [�̅�𝑆𝐾𝑟𝐿 , �̅�𝑆𝐾𝑟𝑈 ], and 

�̅�𝑈𝑆𝑟𝑁 ∈ [�̅�𝑈𝑆𝑟𝐿 , �̅�𝑈𝑆𝑟𝑈 ]. 

Their expressions of MSEs are:  

𝑀𝑆𝐸(�̅�𝑅 ) =
𝑁−𝑛

𝑁𝑛
�̅�𝑁

2[𝐶𝑦𝑁
2 + 𝐶𝑥𝑁

2 − 2𝐶𝑥𝑁 𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ],                                              (3.5) 

  𝑀𝑆𝐸(�̅�𝑆𝐷𝑟𝑁 ) =
𝑁−𝑛

𝑁𝑛
�̅�𝑁

2 [𝐶𝑦𝑁
2 + (

�̅�𝑁

�̅�𝑁 +𝐶𝑥𝑁
) 𝐶𝑥𝑁

2 − 2 (
�̅�𝑁

�̅�𝑁+𝐶𝑥𝑁
) 𝐶𝑥𝑁𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ],                    (3.6) 

  𝑀𝑆𝐸(�̅�𝑆𝐾𝑟𝑁 ) =
𝑁−𝑛

𝑁𝑛
�̅�𝑁

2 [𝐶𝑦𝑁
2 + (

�̅�𝑁

�̅�𝑁+𝛽2(𝑥𝑁)
) 𝐶𝑥𝑁

2 − 2 (
�̅�𝑁

�̅�𝑁+𝛽2(𝑥𝑁)
) 𝐶𝑥𝑁 𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ]              (3.7) 

and 

  𝑀𝑆𝐸(�̅�𝑈𝑆𝑟 ) =
𝑁−𝑛

𝑁𝑛
�̅�𝑁

2 [𝐶𝑦𝑁
2 + (

�̅�𝑁𝛽2(𝑥𝑁)

�̅�𝑁 𝛽2(𝑥𝑁)+𝐶𝑥𝑁
) 𝐶𝑥𝑁

2 − 2 (
�̅�𝑁𝛽2(𝑥𝑁)

�̅�𝑁𝛽2(𝑥𝑁)+𝐶𝑥𝑁
) 𝐶𝑥𝑁 𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ]      (3.8) 

where 𝐶𝑦𝑁
2 ∈ [𝐶𝑦𝑁𝐿

2 , 𝐶𝑦𝑁𝑈
2 ], 𝐶𝑥𝑁

2 ∈ [𝐶𝑥𝑁𝐿
2 , 𝐶𝑥𝑁𝑈

2 ] and 𝜌𝑥𝑦𝑁 ∈ [𝜌𝑥𝑦𝑁𝐿 , 𝜌𝑥𝑦𝑁𝑈 ]. 

 

4. PROPOSED NEUTROSOPHIC GENERALIZED ESTIMATORS 

The aim of this article is to propose a generalized neutrosophic exponential robust ratio type 

estimator of finite neutrosophic population mean.  

Motivated by [14], [17] and [18] we propose the following generalized neutrosophic exponential robust 

ratio type estimator 

                 𝑡𝑝𝑁
G = (o1�̅�𝑁 + 𝑜2(X̅N − x̅N))𝑒𝑥𝑝(

�̅�𝑁Ω +Ψ

𝛼(�̅�𝑁Ω+Ψ)+(1−𝛼)(�̅�𝑁 Ω+Ψ)
− 1),                     (4.1) 

where, 𝑜1 and 𝑜2 are scalars which minimizes the MSE of the proposed generalizedneutrosophic 

estimator 𝑡𝑝
𝐺 . Further, Ω and Ψ are scalaras which would assume different known population 

parameter values of neutrosophic auxiliary variable precisely Hodges Lehmann, Tri-mea, Mid range 

and coefficient of variation. It should be noted that 𝑡𝑝𝑁
G ∈ [𝑡𝑝

G
𝑁 𝐿

, 𝑡𝑝
G

𝑁 𝑈
], 𝑜1 ∈ [𝑜1L , 𝑜1U ], 𝑜2 ∈

[𝑜2L , 𝑜2U ], �̅�𝑁 ∈ [�̅�𝑁𝐿 , �̅�𝑁𝑈 ].  In order to obtain the expression of bias and Mean squared error of the 

proposed generalized neutrosophic estimator 𝑡𝑝
𝐺 , we re-write it usingerror terms defined in Section 2 

and using Taylor series obtain the expression as follows 

𝐵𝑖𝑎𝑠(𝑡𝑝𝑁
𝐺 ) = −�̅�𝑁 + �̅�𝑁 θ𝑜2𝜉02 + 𝑜1(�̅�𝑁 +

3

2
�̅�𝑁 𝜃

2𝜉02 − �̅�𝑁 𝜃𝜉11),                                 (4.2) 

𝑀𝑆𝐸 (𝑡𝑝𝑁  
𝐺 ) = −�̅�𝑁

2 + �̅�𝑁 𝑜2(−2�̅�𝑁 𝜃 + �̅�𝑁 𝑜2)𝜉02 + �̅�𝑁 𝑜1(−2�̅�𝑁 + 𝜃(−3�̅�𝑁 𝜃 + 4�̅�𝑁 𝑜2)𝜉02 + 2(�̅�𝑁 𝜃 +

 �̅�𝑁 𝑜2)𝜉11) + �̅�𝑁
2𝑜1

2(1+4𝜃2𝜉2
02

− 4𝜃𝜉11 + ξ20).   (4.3) 

Partially differentiating 𝑀𝑆𝐸 (𝑡𝑝𝑁  
𝐺 )  𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑜1 and 𝑜2to find their optimum values we get 

  𝑜1𝑜𝑝𝑡 =
𝜉02(2−𝜃

2𝜉02)

2(−𝜉211+𝜉02(1+𝜉20))
                                                             (4.4) 



5 Int. J. Anal. Appl. (2023), 21:41 

 

                   𝑜2𝑜𝑝𝑡 =
�̅�𝑁{2𝜃

3𝜉202−2𝜉11(−1+𝜃𝜉11)−𝜃𝜉02(2+𝜃𝜉11−2𝜉20))

2�̅�𝑁(−𝜉
2

11+𝜉02(1+𝜉20))
                               (4.5) 

Using these optimum values we get 

                    𝑀𝑆𝐸 (𝑡𝑝𝑁    𝑜𝑝𝑡
𝐺 ) =

�̅�𝑁
2{4𝜉211+𝜉02{𝜃

4𝜉202−4𝜃
2𝜉211+4(−1+𝜃

2𝜉02)𝜉20}}

4{𝜉211−𝜉02(1+𝜉20)}
,                     (4.6) 

where 𝜃 = α
�̅�𝑁 Ω

�̅�𝑁Ω+Ψ
. 

From the proposed generalized neutrosophic exponential robust ratio type estimator 𝑡𝑝
𝐺  we have 

developed five generalized neutrosophic exponential robust ratio type estimators. 

(i) 𝑡𝑝𝑁
𝐺1 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝(

�̅�𝑁HL +TM

�̅�𝑁HL+TM
− 1)                                             (4.7) 

The bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are 

                    𝐵𝑖𝑎𝑠(𝑡𝑝𝑁
𝐺1 ) = −�̅�𝑁 + �̅�𝑁 𝜃1𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 +

3

2
�̅�𝑁 𝜃1

2𝜉02 − �̅�𝑁 𝜃1𝜉11),              (4.8) 

                    𝑀𝑆𝐸 (𝑡𝑝𝑁    𝑜𝑝𝑡
𝐺1 ) =

�̅�𝑁
24𝜉211+𝜉02𝜃1

4𝜉202−4𝜃1
2𝜉211+4(−1+𝜃1

2𝜉02)𝜉20

4𝜉211−𝜉02(1+𝜉20)
,                                (4.9) 

where 

                    𝑜1𝑜𝑝𝑡 =
𝜉02(2−𝜃1

2𝜉02)

2(−𝜉211+𝜉02(1+𝜉20))
,                                                                                 (4.10) 

                    𝑜2𝑜𝑝𝑡 =
�̅�𝑁{2𝜃

3𝜉202−2𝜉11(−1+𝜃1𝜉11)−𝜃1𝜉02(2+𝜃1𝜉11−2𝜉20))

2�̅�𝑁(−𝜉
2

11+𝜉02(1+𝜉20))
,                                           (4.11) 

 and  𝜃1 =
�̅�𝑁HL

�̅�𝑁HL+TM
,   

where,   𝑡𝑝𝑁
𝐺1 ∈ [𝑡𝑝

𝐺1

𝐿
, 𝑡𝑝

𝐺1

𝑈
], 𝜃1 ∈ [𝜃1𝐿 , 𝜃1𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿

, 𝑜1𝑜𝑝𝑡 𝑈
] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2𝑜𝑝𝑡 𝐿

, 𝑜2𝑜𝑝𝑡 𝑈
]. 

(ii)  𝑡𝑝𝑁
𝐺2 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝(

�̅�𝑁 TM +MR

�̅�𝑁TM+MR
− 1)                                          (4.12) 

The bias and MSE_opt are  

                     𝐵𝑖𝑎𝑠(𝑡𝑝𝑁
𝐺2 ) = −�̅�𝑁 + �̅�𝑁 𝜃2𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 +

3

2
�̅�𝑁 𝜃2

2𝜉02 − �̅�𝑁 𝜃2𝜉11),           (4.13) 

                  𝑀𝑆𝐸 (𝑡𝑝𝑁    𝑜𝑝𝑡
𝐺2 ) =

�̅�𝑁
24𝜉211+𝜉02𝜃2

4𝜉202−4𝜃2
2𝜉211+4(−1+𝜃2

2𝜉02)𝜉20

4𝜉211−𝜉02(1+𝜉20)
,                               (4.14) 

where 

                   𝑜1𝑜𝑝𝑡 =
𝜉02(2−𝜃2

2𝜉02)

2(−𝜉211+𝜉02(1+𝜉20))
,                                                            (4.15) 

                  𝑜2𝑜𝑝𝑡 =
�̅�𝑁{2𝜃

3 𝜉202−2𝜉11(−1+𝜃2𝜉11)−𝜃1𝜉02(2+𝜃2𝜉11−2𝜉20))

2�̅�𝑁(−𝜉
2

11+𝜉02(1+𝜉20))
,                                (4.16) 

and  𝜃2 =
�̅�𝑁TM

�̅�𝑁TM+MR
  

where,  𝑡𝑝𝑁
𝐺2 ∈ [𝑡𝑝

𝐺2

𝐿
, 𝑡𝑝

𝐺2

𝑈
], 𝜃2 ∈ [𝜃2𝐿 , 𝜃2𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿

, 𝑜1𝑜𝑝𝑡 𝑈
] and  𝑜2𝑜𝑝𝑡 ∈ [𝑜2𝑜𝑝𝑡 𝐿

, 𝑜2𝑜𝑝𝑡 𝑈
]. 



6 Int. J. Anal. Appl. (2023), 21:41 

 

(iii) 𝑡𝑝𝑁
𝐺3 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝(

�̅�𝑁HL +MR

�̅�𝑁HL+MR
− 1)                                             (4.17) 

The bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are  

                    𝐵𝑖𝑎𝑠(𝑡𝑝𝑁
𝐺3 ) = −�̅�𝑁 + �̅�𝑁 𝜃3𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 +

3

2
�̅�𝑁 𝜃3

2𝜉02 − �̅�𝑁 𝜃3𝜉11),               (4.18) 

                    𝑀𝑆𝐸 (𝑡𝑝𝑁    𝑜𝑝𝑡
𝐺3 ) =

�̅�𝑁
24𝜉211+𝜉02𝜃3

4𝜉202−4𝜃3
2𝜉211+4(−1+𝜃3

2𝜉02)𝜉20

4𝜉211−𝜉02(1+𝜉20)
,                                  (4.19) 

where 

                    𝑜1𝑜𝑝𝑡 =
𝜉02(2−𝜃3

2𝜉02)

2(−𝜉211+𝜉02(1+𝜉20))
,                                                                             (4.20) 

                    𝑜2𝑜𝑝𝑡 =
�̅�𝑁{2𝜃

3 𝜉202−2𝜉11(−1+𝜃3𝜉11)−𝜃3𝜉02 (2+𝜃3𝜉11−2𝜉20))

2�̅�𝑁(−𝜉
2

11+𝜉02(1+𝜉20))
,                                    (4.21) 

and 𝜃3 =
�̅�𝑁 HL

�̅�𝑁 HL+MR
  

where, 𝑡𝑝𝑁
𝐺3 ∈ [𝑡𝑝

𝐺3

𝐿
, 𝑡𝑝

𝐺3

𝑈
], 𝜃3 ∈ [𝜃3𝐿 , 𝜃3𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿

, 𝑜1𝑜𝑝𝑡 𝑈
] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2 𝑜𝑝𝑡 𝐿

, 𝑜2𝑜𝑝𝑡 𝑈
]. 

(iv)  𝑡𝑝𝑁
𝐺4 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝(

�̅�𝑁 CxN +𝐻𝐿

�̅�𝑁CxN +HL
− 1)                                            (4.22) 

The bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are  

                    𝐵𝑖𝑎𝑠(𝑡𝑝𝑁
𝐺4 ) = −�̅�𝑁 + �̅�𝑁 𝜃4𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 +

3

2
�̅�𝑁 𝜃4

2𝜉02 − �̅�𝑁 𝜃4𝜉11),            (4.23) 

                    𝑀𝑆𝐸 (𝑡𝑝𝑁    𝑜𝑝𝑡
𝐺4 ) =

�̅�𝑁
24𝜉211+𝜉02𝜃4

4𝜉202−4𝜃4
2𝜉211+4(−1+𝜃4

2𝜉02)𝜉20

4𝜉211−𝜉02(1+𝜉20)
                                     (4.24) 

where, 

                   𝑜1𝑜𝑝𝑡 =
𝜉02(2−𝜃4

2𝜉02)

2(−𝜉211+𝜉02(1+𝜉20))
,                                                                                       (4.25) 

                   𝑜2𝑜𝑝𝑡 =
�̅�𝑁{2𝜃

3 𝜉202−2𝜉11(−1+𝜃4𝜉11)−𝜃4𝜉02(2+𝜃4𝜉11−2𝜉20))

2�̅�𝑁(−𝜉
2

11+𝜉02(1+𝜉20))
,                                         (4.26) 

and  𝜃4 =
�̅�𝑁CxN

�̅�𝑁CxN +HL
  

where,  𝑡𝑝𝑁
𝐺4 ∈ [𝑡𝑝

𝐺4

𝐿
, 𝑡𝑝

𝐺4

𝑈
], 𝜃4 ∈ [𝜃4𝐿 , 𝜃4𝑈], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿

, 𝑜1𝑜𝑝𝑡 𝑈
] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2 𝑜𝑝𝑡 𝐿

, 𝑜2𝑜𝑝𝑡 𝑈
]. 

(v) 𝑡𝑝𝑁
𝐺5 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝(

�̅�𝑁CxN
+𝑇𝑀

�̅�𝑁CxN +TM
− 1)                                        (4.27) 

 

The bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are 

                    𝐵𝑖𝑎𝑠(𝑡𝑝𝑁
𝐺5 ) = −�̅�𝑁 + �̅�𝑁 𝜃5𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 +

3

2
�̅�𝑁 𝜃5

2𝜉02 − �̅�𝑁 𝜃5𝜉11),             (4.28) 

                    𝑀𝑆𝐸 (𝑡𝑝𝑁    𝑜𝑝𝑡
𝐺5 ) =

�̅�𝑁
24𝜉211+𝜉02𝜃5

4𝜉202−4𝜃5
2𝜉211+4(−1+𝜃5

2𝜉02)𝜉20

4𝜉211−𝜉02(1+𝜉20)
                                    (4.29) 



7 Int. J. Anal. Appl. (2023), 21:41 

 

                  𝑜1𝑜𝑝𝑡 =
𝜉02(2−𝜃5

2𝜉02)

2(−𝜉211+𝜉02(1+𝜉20))
,                                                                                 (4.30) 

                  𝑜2𝑜𝑝𝑡 =
�̅�𝑁{2𝜃

3 𝜉202−2𝜉11(−1+𝜃5𝜉11)−𝜃5𝜉02(2+𝜃5𝜉11−2𝜉20))

2�̅�𝑁(−𝜉
2

11+𝜉02(1+𝜉20))
,                                      (4.31) 

and 𝜃5 =
�̅�𝑁CxN

�̅�𝑁 CxN +TM
  

where,  𝑡𝑝𝑁
𝐺5 ∈ [𝑡𝑝

𝐺5

𝐿
, 𝑡𝑝

𝐺5

𝑈
], 𝜃5 ∈ [𝜃5𝐿 , 𝜃5𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿

, 𝑜1𝑜𝑝𝑡 𝑈
] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2𝑜𝑝𝑡 𝐿

, 𝑜2𝑜𝑝𝑡 𝑈
]. 

 

5. EMPIRICAL STUDY 

In this section we have conducted an empirical study to demonstrate the high efficiency of the 

developed estimators. This study, is conducted using daily stock price of Moderna. The rationale 

behind taking the stock price as a neutrosophic data is the fact that the daily stock price ranges 

between a high and a low values each day. Pin pointing the point estimate of the daily stock price 

will not give a reliable estimate. Thus, we have taken it as a  neutrosophic dataset. In this empirical 

study, daily stock price of Moderna has been considered form 1-September-2020 to 1-September-

2021 [20] (N=253). The neutrosophic survey variable 𝑦𝑁 i.e., varying price of the stock on each day 

where 𝑦𝑁 ∈ [𝑦𝑙 , 𝑦𝑢 ]  ( 𝑦𝑢 is the highest price of the stock on each day and  𝑦𝑙 is the lowest price of 

the stock each day). 

 

6. SIMULATION STUDY 

In this section we have conducted four simulation studies to demonstrate the high efficiency of the 

proposed generalized neutrosophic robust type exponential ratio estimator over similar existing ratio 

estimators discussed in this article. The comparison has been made on the basis of neutrosophic 

MSEs and neutrosophic REs.  

 

6.1 Simulation study-1 

The following algorithm is used in R language to perform the simulation study: 

(i) Nutrosophic auxiliary variable 𝑥𝑁 has been generated from Neutrosophic normal 

distribution NN([0.7, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single 

indetermincay where population mean 𝜇𝑋 is indeterminate. Thus 𝑥𝑁  ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. 

(ii) Neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 7𝑒 

                    such that 𝑦𝑁  ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). 



8 Int. J. Anal. Appl. (2023), 21:41 

 

(iii) For sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various 

values of neutrosophic estimates are obtained with 20000 iterations. 

(iv) For each neutrosophic sample size used, neutrosophic MSEs and RES have been obtained 

and presented in Tables. 

(v) Values of estimates have been calculated under classical statistics as well and their MSEs 

and REs are tabulated in Tables 1-4. 

 

Table 1: Data statistics for empirical study 

 

 𝑆𝑦𝑈
2 = 9624, 𝑆𝑦𝐿

2 = 8111, 𝐶𝑦𝑈
2 = 0.3124, 𝐶𝑦𝐿

2 = 0.3022, 𝑆𝑥𝑈
2 = 8743, 𝑆𝑥𝐿

2 = 8965,  

 𝐶𝑥𝑈
2 = 0.3055, 𝐶𝑥𝐿

2 = 0.3092, 𝑁 = 253, 𝑛 = 160, 𝑇𝑀𝑈 = 150, 𝐻𝐿𝑈 = 153.44,  

 𝑀𝑅𝑈 = 270.76, β2𝑥𝑈 = 1.06, 𝑇𝑀𝐿 = 153, 𝐻𝐿𝐿 = 153.96, 𝑀𝑅𝐿 = 269.4, 𝛽2(𝑥𝐿 ) = 1.01 

 ρ𝑦𝑈𝑥𝑈 = 0.99, 𝜌𝑦𝐿𝑥𝐿 = 0.99. 

 

Table 2: Neutrosophic MSE of the estimators 

Estimators MSE 

𝑴𝑺𝑬[�̅�∗𝐿 , �̅�∗𝑈 ] 

Relative Eficiency 

𝑹𝑬[�̅�∗𝐿 , �̅�∗𝑈 ] 

�̅�𝑅 𝑁 0.1038 0.1209 1 1 

�̅�𝑆𝐷𝑟𝑁  0.1021 0.1223 1.01665 0.988553 

�̅�𝑆𝐾𝑟𝑁  0.1009 0.124 1.028741 0.975 

�̅�𝑈𝑆𝑟𝑁  0.1222 0.1021 0.849427 1.184133 

𝑡𝑝𝑁
𝐺2  0.0835 0.116 1.243114 1.042241 

𝑡𝑝𝑁
𝐺2  0.0836 0.116 1.241627 1.042241 

𝑡𝑝𝑁
𝐺3  0.0836 0.1156 1.241627 1.045848 

𝑡𝑝𝑁
𝐺4  0.0868 0.1193 1.195853 1.013412 

𝑡𝑝𝑁
𝐺5  0.0869 0.119 1.194476 1.015966 

 

*Denotes appropriate estimator 

 

  



9 Int. J. Anal. Appl. (2023), 21:41 

 

Table 3: Neutrosophic MSEs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑀𝑆𝐸(𝐿, 𝑈) 

[60, 60] 0.79376 0.71468 0.68268 0.69029 0.92986 0.75644 0.67749 0.71476 0.36048 0.49557 0.37457 0.49494 0.49492 0.49492 0.38379 0.49608 0.38321 0.49603 

[65, 65] 0.70738 0.64959 0.62434 0.63017 0.82481 0.68538 0.62051 0.65438 0.33174 0.46315 0.34235 0.46272 0.46271 0.46271 0.34892 0.46352 0.34851 0.46349 

[70, 70] 0.6408 0.58546 0.56655 0.57157 1.02672 0.61103 0.56208 0.59378 0.30331 0.43304 0.31307 0.43287 0.43287 0.43287 0.31948 0.43323 0.31906 0.43321 

[75, 75] 0.58023 0.53854 0.52302 0.52725 0.64351 0.55999 0.52060 0.54858 0.28057 0.40533 0.28759 0.40525 0.40525 0.40525 0.29178 0.40546 0.29152 0.40545 

 

 

Table 4: Neutrosophic REs of all the neutrosophic estimators 

Sample Size �̅�𝑅 𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁  

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁  

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑅𝐸(𝐿, 𝑈) 

[60, 60] 1 1 1.162712 1.035319 0.085363 0.944794 1.171619 0.999888 2.201953 1.442137 2.119123 1.443973 2.121842 1.444031 2.068214 1.440655 2.071345 1.440800 

[65, 65] 1 1 1.133004 1.030817 0.857628 0.947781 1.139998 0.99268 2.132333 1.402548 2.066248 1.403851 2.068363 1.403881 2.027342 1.401428 2.029727 1.401519 

[70, 70] 1 1 1.131056 1.024301 0.624123 0.958153 1.140051 0.985988 2.11269 1.351977 2.046827 1.352508 2.048921 1.352508 2.005759 1.351384 2.0084 1.351446 

[75, 75] 1 1 1.109384 1.021413 0.901664 0.961696 1.114541 0.981698 2.06804 1.328646 2.01756 1.328908 2.019104 1.328908 1.988587 1.32822 1.990361 1.328253 

  

 



10 Int. J. Anal. Appl. (2023), 21:41 

 

Table 5: Classical MSEs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁  

 

�̅�𝑆𝐾𝑟𝑁  

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 0.70002 0.67756 0.68675 0.67328 0.39037 0.38957 0.38955 0.39044 0.39042 

65 0.63624 0.61897 0.62596 0.61597 0.34903 0.34853 0.34853 0.34907 0.34906 

70 0.57653 0.56183 0.56783 0.55915 0.32683 0.32636 0.32635 0.32688 0.57341 

75 0.53054 0.51869 0.52345 0.5168 0.31015 0.30982 0.30982 0.31018 0.31017 

 

Table 6: Classical REs of all the neutrosophic estimators 

Sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁  

 

�̅�𝑆𝐾𝑟𝑁  

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 1 1.03314 1.01932 1.03971 1.79322 1.79690 1.79699 1.79290 1.79299 

65 1 1.02790 1.01642 1.03290 1.82288 1.82549 1.82549 1.82267 1.82272 

70 1 1.02616 1.01532 1.03108 1.76400 1.76654 1.76660 1.76373 1.00544 

75 1 1.02284 1.01354 1.02658 1.71059 1.71241 1.71241 1.71043 1.71048 

 

6.2 Simulation study-2 

The following algorithm is used in R language to perform the simulation study: 

 

(i) Nutrosophic auxiliary variable 𝑥𝑁 has been generated from Neutrosophic normal 

distribution NN([0.7, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single 

indetermincay where population mean 𝜇𝑋 is indeterminate. Thus 𝑥𝑁  ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. 

(ii) Neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 6𝑒 

                    such that 𝑦𝑁  ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). 

(iii) For sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various 

values of neutrosophic estimates are obtained with 20000 iterations. 

(iv) For each neutrosophic sample size used, neutrosophic MSEs and RES have been obtained 

and presented in Tables. 

(v) Values of estimates have been calculated under classical statistics as well and their MSEs 

and REs are tabulated in Tables 5-8. 

  



11 Int. J. Anal. Appl. (2023), 21:41 

 

Table 7: Neutrosophic MSEs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑀𝑆𝐸(𝐿, 𝑈) 

[60, 60] 0.58317 0.52507 0.50408 0.50940 7.06050 0.55537 0.50234 0.53718 0.28861 0.42083 0.29898 0.42056 0.29863 0.42056 0.30594 0.42109 0.30549 0.42107 

[65, 65] 0.51971 0.47725 0.4612 0.46516 0.60191 0.50290 0.46063 0.49209 0.26721 0.39468 0.27514 0.39456 0.27487 0.39455 0.28036 0.39484 0.28003 0.39483 

[70, 70] 0.47079 0.43013 0.41825 0.42180 0.74261 0.44856 0.41663 0.44617 0.24573 0.37008 0.25288 0.37014 0.25264 0.37014 0.25790 0.70110 0.25756 0.37011 

[75, 75] 0.42629 0.39566 0.38630 0.38914 0.47267 0.41108 0.38624 0.41245 0.22828 0.34735 0.23318 0.34747 0.23302 0.34748 0.23623 0.34735 0.23604 0.34734 

 

Table 8: Neutrosophic REs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑅𝐸(𝐿, 𝑈) 

[60, 60] 1 1 1.156900 1.030760 0.082596 0.945442 1.160907 0.977456 2.020616 1.247701 1.950532 1.248502 1.952818 1.248502 1.906158 1.246931 1.908966 1.24699 

[65, 65] 1 1 1.126865 1.025991 0.863435 0.948996 1.128259 0.969843 1.94495 1.209207 1.888893 1.209575 1.890748 1.209606 1.853724 1.208717 1.855908 1.208748 

[70, 70] 1 1 1.125619 1.019749 0.633967 0.958913 1.129995 0.964050 1.915883 1.162262 1.861713 1.162074 1.863482 1.16E-05 1.825475 0.613507 1.827885 1.162168 

[75, 75] 1 1 1.103521 1.016755 0.901877 0.962489 1.103692 0.959292 1.867400 1.139082 1.828159 1.138688 1.829414 1.138655 1.804555 1.139082 1.806007 1.139114 

 

  



12 Int. J. Anal. Appl. (2023), 21:41 

 

Table 9: Classical MSEs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁  

 

�̅�𝑆𝐾𝑟𝑁  

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 0.51430 0.49854 0.50479 0.49612 0.33024 0.32978 0.32978 0.33028 0.33027 

65 0.46744 0.45557 0.46018 0.45409 0.30102 0.30078 0.30077 0.30104 0.30103 

70 0.42357 0.41335 0.41736 0.41196 0.2837 0.28345 0.28345 0.28373 0.28372 

75 0.38978 0.38173 0.3848 0.38095 0.26904 0.26891 0.26891 0.26905 0.26905 

Table 10: Classical REs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 

 

�̅�𝑆𝐾𝑟𝑁 

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 1 1.031612 1.01884 1.036644 1.557352 1.559525 1.559525 1.557164 1.557211 

65 1 1.026055 1.015776 1.029399 1.552854 1.554093 1.554144 1.55275 1.552802 

70 1 1.024725 1.014879 1.028182 1.493021 1.494338 1.494338 1.492863 1.492916 

75 1 1.021088 1.012942 1.023179 1.448781 1.449481 1.449481 1.448727 1.448727 

 

6.3 Simulation study-3 

The following algorithm is used in R language to perform the simulation study: 

 

(i) Nutrosophic auxiliary variable 𝑥𝑁 has been generated from Neutrosophic normal 

distribution NN([0.75, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single 

indetermincay where population mean 𝜇𝑋 is indeterminate. Thus 𝑥𝑁  ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. 

(ii) Neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 7𝑒 

                  such that 𝑦𝑁  ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). 

(iii) For sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various 

values of neutrosophic estimates are obtained with 20000 iterations. 

(iv) For each neutrosophic sample size used, neutrosophic MSEs and RES have been obtained 

and presented in Tables . 

(v) Values of estimates have been calculated under classical statistics as well and their MSEs 

and REs are tabulated in Tables 9-1.



13 Int. J. Anal. Appl. (2023), 21:41 

 

Table 11: Neutrosophic MSEs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑀𝑆𝐸(𝐿, 𝑈) 

[60, 60] 0.56715 0.52507 0.50377 0.5094 0.64524 0.55537 0.50262 0.53718 0.30052 0.42083 0.30811 0.42056 0.30785 0.42056 0.31202 0.42109 0.31173 0.42107 

[65, 65] 0.50815 0.47725 0.46082 0.46516 0.55396 0.5029 0.46063 0.49209 0.2788 0.39468 0.28427 0.39456 0.28405 0.39455 0.28699 0.39484 0.28679 0.39483 

[70, 70] 0.46026 0.43013 0.41799 0.4218 5.3525 0.44856 0.41683 0.44617 0.25763 0.37008 0.26279 0.37014 0.26262 37014 0.26538 0.37011 0.26519 0.37011 

[75, 75] 0.41836 0.39566 0.38598 0.38914 0.5705 0.41108 0.38646 0.41245 0.23982 0.34735 0.24343 0.34747 0.24331 0.34748 0.24543 0.34735 0.2451 0.34734 

 

Table 12: Neutrosophic REs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑅𝐸(𝐿, 𝑈) 

[60, 60]  1 1 1.125811 1.030762 0.878975 0.945442 1.128387 0.977456 1.887229 1.247701 1.840739 1.248502 1.842293 1.248502 1.817672 1.246931 1.819363 1.24699 

[65, 65] 1 1 1.102708 1.025991 0.917304 0.948996 1.103163 0.969843 1.822633 1.209207 1.787561 1.209575 1.788946 1.209606 1.770619 1.208717 1.771854 1.208748 

[70, 70] 1 1 1.101127 1.019749 0.08599 0.958913 1.104191 0.96405 1.786516 1.162262 1.751437 1.162074 1.75257 1.16E-05 1.734343 1.162168 1.735586 1.162168 

[75, 75] 1 1 1.08389 1.016755 0.733322 0.962489 1.082544 0.959292 1.744475 1.139082 1.718605 1.138688 1.719453 1.138655 1.7046 1.139082 1.706895 1.139114 

 



14 Int. J. Anal. Appl. (2023), 21:41 

 

Table 13: Classical MSEs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁  

 

�̅�𝑆𝐾𝑟𝑁  

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 0.51298 0.4984 0.50418 49604 0.33895 0.33853 0.33852 0.33888 0.33887 

65 0.46636 0.45541 0.45966 0.454 0.30994 0.30979 0.30972 0.3099 0.30989 

70 0.4227 0.41324 0.41695 0.41189 0.2924 0.29217 0.29216 0.29236 0.29235 

75 0.38903 0.38161 0.38443 0.38087 0.27718 0.27707 0.27707 0.27716 0.27716 

 

Table 14: Classical REs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 

 

�̅�𝑆𝐾𝑟𝑁 

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 1 1.029254 1.017454 1.03E-05 1.513439 1.515316 1.515361 1.513751 1.513796 

65 1 1.024044 1.014576 1.027225 1.504678 1.505407 1.505747 1.504873 1.504921 

70 1 1.022892 1.013791 1.026245 1.445622 1.44676 1.44681 1.44582 1.44587 

75 1 1.019444 1.011966 1.021425 1.403528 1.404086 1.404086 1.40363 1.40363 

 

6.4 Simulation study-4 

The following algorithm is used in R language to perform the simulation study: 

(i) Nutrosophic auxiliary variable 𝑥𝑁 has been generated from Neutrosophic normal 

distribution NN([0.65, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single 

indetermincay where population mean 𝜇𝑋 is indeterminate. Thus 𝑥𝑁  ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. 

(ii) Neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 6𝑒 

                  such that 𝑦𝑁  ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). 

(iii) For sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various 

values of neutrosophic estimates are obtained with 20000 iterations. 

(iv) For each neutrosophic sample size used, neutrosophic MSEs and RES have been obtained 

and presented in Tables. 

(v) Values of estimates have been calculated under classical statistics as well and their MSEs 

and REs are tabulated in Tables 13-16



15 Int. J. Anal. Appl. (2023), 21:41 

 

Table 15: Neutrosophic MSEs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑀𝑆𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑀𝑆𝐸(𝐿, 𝑈) 

[60, 60] 0.60649 0.52507 0.50434 0.50940 160621 0.55537 0.50209 0.53718 0.28066 0.42083 0.29939 0.42056 0.29870 0.42056 0.31687 0.42109 0.31575 0.42107 

[65, 65] 0.53563 0.47725 0.46151 0.46516 0.95151 0.50290 0.46012 0.49209 0.26131 0.39468 0.28742 0.39456 0.28619 0.39455 0.32693 0.39484 0.32387 0.39483 

[70, 70] 0.48835 0.43013 0.41848 0.42180 0.58047 0.44856 0.41644 0.44617 0.23605 0.37008 0.25087 0.37014 0.25018 0.37014 0.27917 0.37011 0.27641 0.37011 

[75, 75] 0.43770 0.39566 0.38660 0.38914 3.32296 0.41108 0.38605 0.41245 0.21814 0.34735 0.22565 0.34747 0.22539 0.34748 0.23187 0.34735 0.23148 0.34734 

 

Table 16: Neutrosophic REs of all the neutrosophic estimators 

Sample 

Size 

�̅�𝑅 𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐷𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑆𝐾𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

�̅�𝑈𝑆𝑟𝑁 

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺1  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺2  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺3  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺4  

𝑅𝐸(𝐿, 𝑈) 

𝑡𝑝𝑁
𝐺5  

𝑅𝐸(𝐿, 𝑈) 

[60, 60] 1 1 1.202542 1.030762 3.78E-06 0.945442 1.207931 0.977456 2.160942 1.247701 2.025752 1.248502 2.030432 1.248502 1.914003 1.246931 1.920792 1.24699 

[65, 65] 1 1 1.160603 1.025991 0.562926 0.948996 1.164109 0.969843 2.049788 1.209207 1.863579 1.209575 1.871589 1.209606 1.638363 1.208717 1.653843 1.208748 

[70, 70] 1 1 1.166961 1.019749 0.841301 0.958913 1.172678 0.96405 2.068841 1.162262 1.946626 1.162074 1.951995 1.162074 1.749293 1.162168 1.76676 1.162168 

[75, 75] 1 1 1.132178 1.016755 0.13172 0.962489 1.133791 0.959292 2.00651 1.139082 1.93973 1.138688 1.941967 1.138655 1.887696 1.139082 1.890876 1.139114 

 



16 Int. J. Anal. Appl. (2023), 21:41 

 

Table 17: Classical MSEs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁  

 

�̅�𝑆𝐾𝑟𝑁  

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 0.51574 0.49868 0.50544 0.4962 0.3217 0.32121 0.3212 0.32189 0.32188 

65 0.46836 0.45572 0.46072 0.45418 0.29226 0.292 0.29199 0.29238 0.29237 

70 0.42453 0.41346 0.4187 0.41203 0.27515 0.27487 0.27486 0.27526 0.27525 

75 0.3906 0.38186 0.3852 0.38103 0.26101 0.26086 0.26085 0.26108 0.26107 

 

Table 18: Classical REs of all the neutrosophic estimators 

Sample 

size 

�̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 

 

�̅�𝑆𝐾𝑟𝑁 

 

�̅�𝑈𝑆𝑟𝑁 

 

𝑡𝑝𝑁
𝐺1  

 

𝑡𝑝𝑁
𝐺2  

 

𝑡𝑝𝑁
𝐺3  

 

𝑡𝑝𝑁
𝐺4  

 

𝑡𝑝𝑁
𝐺5  

 

60 1 1.03421 1.020378 1.039379 1.603171 1.605616 1.605666 1.602224 1.602274 

65 1 1.027736 1.016583 1.031221 1.602546 1.603973 1.604028 1.601888 1.601943 

70 1 1.026774 1.013924 1.030338 1.542904 1.544476 1.544532 1.542287 1.542343 

75 1 1.022888 1.014019 1.025116 1.496494 1.497355 1.497412 1.496093 1.49615 

 

7. DISCUSSION AND CONCLUSION 

Vagueness or indetermincacy is usually observed in the collected data. Instead of using Fuzzy logic 

to deal with such a data set it would be more easier and cost resource efficient to use neutrosophic 

statistical tools. Due to its need and wide applicability research in neutrosophic statistics has been 

regorously carried out. This paper aims at further developing the existing theory of Neutrosophic 

Simple Random Sampling Without Replacement (NSRSWOR). In this paper, a generalized 

neutrosophic exponential robust ratio type estimator 𝑡𝑝𝑁
G  has been presented using some known 

population parameters of neutrosophic auxiliary variables.  

From the proposed generalized neutrosophic exponential robust ratio type estimator, five generalized 

neutrosophic exponential robust ratio type estimators 𝑡𝑝𝑁
𝐺1 -𝑡𝑝𝑁

𝐺5   have been developed using known 

population parameter values of auxiliary variables viz., Hodges Lehmann, Tri mean, Mid range and 

coefficient of variation. The high efficiency of the developed neutrosophic estimators have been 

demonstrated using an empirical and four simulation studies the results of which are presented in 

Tables 1-18. 

 



17 Int. J. Anal. Appl. (2023), 21:41 

 

In the empirical study on daily stock price, we can see that the proposed estimators provide a lower 

MSE indicating high efficiency that the similar existing neutrosophic estimators. In simulation studies, 

it is clear from the results that the developed neutrosophic estimators 𝑡𝑝𝑁
𝐺1 -𝑡𝑝𝑁

𝐺5  𝑓𝑟𝑜𝑚 the proposed 

generalized neutrosphic estimator 𝑡𝑝𝑁
G  provide a much lower MSE as compared to the similar existing 

neutrosophic ratio type estimators discussed in this paper (Table [3-4], Table [7-8], Table [11-12] 

and Table [15-16]). The results of the neutrosophic estimators estimators have also been compared 

with their classical values (Table [5-6], Table [9-10], Table [13-14] and Table [17-18]). It can be 

seen that  the classical values of MSE falls in the indetermancy  neutrosophic MSE intervals implying 

that when the data contains some indetermincay, neutrosophic estimators should be used. Further, 

it can be seen that, proposed neutrosophic estimators 𝑡𝑝𝑁
𝐺1 -𝑡𝑝𝑁

𝐺5  provide lowest MSE in neutrosophic 

as well as classical form and thus it is advised to use the proposed neutrosophic estimators 𝑡𝑝𝑁
𝐺1 -𝑡𝑝𝑁

𝐺5  

when the data at hand is neutrosophic.  

 

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

publication of this paper. 

 

References 

[1] N. Jan, L. Zedam, T. Mahmood, K. Ullah, Z. Ali, Multiple Attribute Decision Making Method Under 

Linguistic Cubic Information, J. Intell. Fuzzy Syst. 36 (2019), 253–269. https://doi.org/10.3233/jifs-

181253. 

[2] D.F. Li, T. Mahmood, Z. Ali, Y. Dong, Decision Making Based on Interval-Valued Complex Single-Valued 

Neutrosophic Hesitant Fuzzy Generalized Hybrid Weighted Averaging Operators, J. Intell. Fuzzy Syst. 38 

(2020), 4359–4401. https://doi.org/10.3233/jifs-191005. 

[3] F. Smarandache, Introduction to Neutrosophic Statistics, arXiv. (2014).  

https://doi.org/10.48550/ARXIV.1406.2000. 

[4] R. Varshney, A. Pal, Mradula, I. Ali, Optimum Allocation in the Multivariate Cluster Sampling Design Under 

Gamma Cost Function, J. Stat. Comput. Simul. 93 (2022), 312–323. 

https://doi.org/10.1080/00949655.2022.2104845. 

[5] N. Gupta, I. Ali, Shafiullah, A. Bari, A Fuzzy Goal Programming Approach in Stochastic Multivariate 

Stratified Sample Surveys, South Pac. J. Nat. App. Sci. 31 (2013), 80-88. 

https://doi.org/10.1071/sp13009. 

https://doi.org/10.3233/jifs-181253
https://doi.org/10.3233/jifs-181253
https://doi.org/10.3233/jifs-191005
https://doi.org/10.48550/ARXIV.1406.2000
https://doi.org/10.1080/00949655.2022.2104845
https://doi.org/10.1071/sp13009


18 Int. J. Anal. Appl. (2023), 21:41 

 

[6] N. Kumar Adichwal, A. Ali H. Ahmadini, Y. Singh Raghav, R. Singh, I. Ali, Estimation of General 

Parameters Using Auxiliary Information in Simple Random Sampling Without Replacement, J. King Saud 

Univ. – Sci. 34 (2022), 101754. https://doi.org/10.1016/j.jksus.2021.101754. 

[7] R. Singh, R. Mishra, Ratio-cum-product Type Estimators for Rare and Hidden Clustered Population, 

Sankhya B. (2022). https://doi.org/10.1007/s13571-022-00298-x. 

[8] A. Haq, J. Shabbir, Improved Family of Ratio Estimators in Simple and Stratified Random Sampling, 

Commun. Stat. – Theory Methods. 42 (2013), 782–799. https://doi.org/10.1080/03610926.2011.579377. 

[9] Z. Ali, T. Mahmood, Complex Neutrosophic Generalised Dice Similarity Measures and Their Application to 

Decision Making, CAAI Trans. Intell. Technol. 5 (2020), 78–87. https://doi.org/10.1049/trit.2019.0084. 

[10] M. Aslam, A New Sampling Plan Using Neutrosophic Process Loss Consideration, Symmetry. 10 (2018), 

132. https://doi.org/10.3390/sym10050132. 

[11] M. Aslam, Neutrosophic Analysis of Variance: Application to University Students, Complex Intell. Syst. 5 

(2019), 403–407. https://doi.org/10.1007/s40747-019-0107-2. 

[12] M. Aslam, Monitoring the Road Traffic Crashes Using NEWMA Chart and Repetitive Sampling, Int. J. 

Injury Control Safe. Promotion. 28 (2020), 39–45. https://doi.org/10.1080/17457300.2020.1835990. 

[13] M. Aslam, A new goodness of fit test in the presence of uncertain parameters, Complex Intell. Syst. 7 

(2020), 359–365. https://doi.org/10.1007/s40747-020-00214-8. 

[14] Z. Tahir, H. Khan, M. Aslam, J. Shabbir, Y. Mahmood, F. Smarandache, Neutrosophic Ratio-Type 

Estimators for Estimating the Population Mean, Complex Intell. Syst. 7 (2021), 2991–3001. 

https://doi.org/10.1007/s40747-021-00439-1. 

[15] Yahoo Finance: TESLA. https://finance.yahoo.com/quote/TSLA/history/. Accessed 2021-09-13. 

[16] National Family Health Survey (NFHS-4), (2015-2016). http://rchiips.org/nfhs/factsheet_nfhs-4.shtml. 

[17] R. Singh, R. Mishra, Improved Exponential Ratio Estimators in Adaptive Cluster Sampling, J. Stat. Appl. 

Probab. Lett. 9 (2022), 19–29. https://doi.org/10.18576/jsapl/090103. 

[18] Z. Yan, B. Tian, Ratio Method to the Mean Estimation Using Coefficient of Skewness of Auxiliary 

Variable, in: R. Zhu, Y. Zhang, B. Liu, C. Liu (Eds.), Information Computing and Applications, Springer 

Berlin Heidelberg, Berlin, Heidelberg, 2010: pp. 103–110. https://doi.org/10.1007/978-3-642-16339-

5_14. 

[19] R. Mishra, B. Ram, Portfolio Selection Using R, Yugoslav J. Oper. Res. 30 (2020), 137–146. 

https://doi.org/10.2298/yjor181115002m. 

[20] Yahoo Finance: MRNA. https://finance.yahoo.com/quote/MRNA/history/. Accessed 2021-09-13. 

 

https://doi.org/10.1016/j.jksus.2021.101754
https://doi.org/10.1007/s13571-022-00298-x
https://doi.org/10.1080/03610926.2011.579377
https://doi.org/10.1049/trit.2019.0084
https://doi.org/10.3390/sym10050132
https://doi.org/10.1007/s40747-019-0107-2
https://doi.org/10.1080/17457300.2020.1835990
https://doi.org/10.1007/s40747-020-00214-8
https://doi.org/10.1007/s40747-021-00439-1
https://finance.yahoo.com/quote/TSLA/history/
http://rchiips.org/nfhs/factsheet_nfhs-4.shtml
https://doi.org/10.18576/jsapl/090103
https://doi.org/10.1007/978-3-642-16339-5_14
https://doi.org/10.1007/978-3-642-16339-5_14
https://doi.org/10.2298/yjor181115002m
https://finance.yahoo.com/quote/MRNA/history/