Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

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An Exponential Class Estimator of Mean in the Presence of Correlated 

Measurement 

Error under Systematic Sampling Technique 

 

Dr. Jamohan Singh Dhakar1, Dr Sanjay Jain2, Dr. Neha Singh3, Awadhesh Pandey4 

1Assistant Professor (Statistics), Department of Community Medicine, Virendra Kumar Sakhlecha Government Medical 

College, Neemuch, MP, India 

2Professor, Department of Statistics, St. John’s college, Agra, UP, India 

3Assistant Professor, Department of Mathematics, Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India 

4Department of Applied Sciences (Mathematics), Ananad Enginnering College, Agra, India 

Corresponding Author: Dr. Jamohan Singh Dhakar,Mail id : jagmohansinghdhakar@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

In this manuscript, an  exponential class of estimators is proposed to estimate the 

average of the population by using auxiliary variables in the presence of measurement 

error (me ) as well as correlated measurement error (cme).  The estimation for the 

average is done for the systematic sampling technique. The impact me and cme on the 

mse of the estimators is obtained in terms of mean square error (mse)  and bias. The 

mean  square error is also obtained for the  ratio, product and regression estimator under 

correlated measurement error. To  validate the results of the theoretical findings  

simulation studies is done by using R programming.  

Keywords: Mean, Mean Square Error, Bias, Systematic Sampling, Measurement Error, 

Correlated Measurement Error. 

 

Introduction  

Cochran (1977), Murthy (1967), Sukhatme et. al. (1984)  have described all the literatures and theory 

about sampling techniques.  Systematic sampling can be used where the population are natural 

population, for instances selection of every nth person visiting to a shopping mall, selection of a 

person from the list of any city or place, selection of fields from any geographical location. 

Cochran(1946) provided  certain situations where systematic sampling is more efficient in 

comparison to  simple random sampling and stratified sampling. Gautschi (1957), Meadow (1949, 

1953) are legends in the literature of systematic sampling. Lahiri (1954), and Williams (1956) have 

done remarkable work in the systematic sampling.   

Auxiliary information is used to obtain better efficiency in terms of precision and to reduce the cost 

of survey.  For instance, in the estimation of the crop, temperature fertilizer, irrigation can be 

considered as amount auxiliary information. To estimate the income of any place, the expenditure 

and different sources of income can be considered as auxiliary variable. Murthy (1964) addressed the  

conditions under which the  estimations techniques ratio, product and unbiased estimators can be 



Communications on Applied Nonlinear Analysis 

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efficiently used. Use of auxiliary variable in context of systematic sampling was done by Swain 

(1964). As he shown that systematic sampling is very efficient for estimation of volume of timber 

where auxiliary information can be taken as area of the leaf or the tree’s girth as suggested by in 

introducing ratio estimator for systematic sampling.  The product estimator for systematic sampling 

was introduce by Shukla (1971).  

Measurement error is the contamination commingled in data during the survey or during the 

compilation of the data. Shalabh (1997) addressed the impact of measurement error in ratio 

estimation for simple random sampling. Singh et. al. (2019) proposed an efficient variant of the ratio, 

product and mean estimator in the presence of measurement error.  The assumption of  measurement 

error in the previous literature of the study variable 𝑌 and auxiliary variable 𝑋 are uncorrelated. The 

measurement errors of the study variable 𝑌 and auxiliary variable 𝑋 may be correlated with each 

other as the same person or same instrument are used to collect the information for both variables. 

Shalabh and Tsai (2017) first introduce the correlated measurement error  with reference to ratio, 

product regression estimator. Further Singh and Vishwakarma (2019) studied about measurement 

error.   

Singh and Vishwakarma (2020) introduce measurement error in the context of systematic sampling.  

They provided mean square error for ratio estimator, product estimator and regression estimator in 

the presence of measurement error for systematic sampling. In pre- existing literature, only Singh 

and Vishwakarma (2020) have addressed the measurement error in context of systematic sampling.  

Also, no any studied is done for  correlated measurement error in context of systematic sampling.  

By considering, the wide applicability of systematic sampling  and following above literature, in this 

manuscript, a well-known exponential estimator is proposed to obtain the effect of measurement 

error as well as correlated measurement error on mean square error.  The mean square error is also 

derived for well-known namely ratio and product estimator  in the presence of correlated 

measurement error under systematic sampling.  

For systematic sampling, the population of size 𝑁 is divided into 𝑘 intervals such that 𝑁 = 𝑛𝑘.  

Sample of size 𝑛 is selected through systematic sampling. As first unit is selected at random from the 

first 𝑘 units. If the first unit is the 𝑖th unit of the first 𝑘 units the second unit of the sample is the 𝑖 +

𝑘th unit of the second 𝑘 units. Similarly, other units of the samples are  selected. After sampling of 𝑛 

units are done through systematic sampling we observe the study and auxiliary variables. It is  

considered that  a situation where each data of the study and auxiliary variable are observed with 

error. Let us assume (𝑥𝑖𝑗 , 𝑦𝑖𝑗) be the observed values and their true values are (𝑋𝑖𝑗, 𝑌𝑖𝑗) where the 

subscript 𝑖𝑗 represents 𝑗th unit of the 𝑖th interval and 𝑖 = 1,2, … , 𝑘 and 𝑗 = 1,2, … , 𝑛. As the observed 

values are with errors they can be represented in the following form, 𝑥𝑖𝑗 = 𝑋𝑖𝑗 + 𝑉𝑖𝑗 and 𝑦𝑖𝑗 = 𝑌𝑖𝑗 +

𝑈𝑖𝑗 and (𝑈, 𝑉) represent the errors. The errors (𝑈, 𝑉) are normally distributed with mean zero and 

variance (𝜎𝑈
2, 𝜎𝑉

2). 

    Let us assume that the error variables 𝑈 and 𝑉 are correlated to each other and a they are 

uncorrelated to all the combination with the study and auxiliary variables 𝑋 and 𝑌. From the before 

mentioned assumptions we obtain 𝐶𝑜𝑣(𝑋, 𝑈) = 𝐶𝑜𝑣(𝑌, 𝑈) = 𝐶𝑜𝑣(𝑋, 𝑉) = 𝐶𝑜𝑣(𝑌, 𝑉) = 0 and  



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𝐶𝑜𝑣(𝑈, 𝑉) = 𝐶𝑜𝑣(𝑋, 𝑌) ≠ 0. Let 𝜇𝑋𝑠𝑦, 𝜇𝑌𝑠𝑦 be the population mean of the auxiliary and study 

variable respectively and let 𝜎𝑌𝑠𝑦
2 , 𝜎𝑋𝑠𝑦

2  be the population variance of the study and auxiliary variable 

respectively. Let 𝜌 be the correlation coefficient between the study and auxiliary variable. Let the 

sample means of the true values of the study auxiliary and study variable be  �̅�𝑠𝑦, �̅�𝑠𝑦 respectively. 

The unbiased estimators of the population of mean of the study and auxiliary variables are the 

sample means of the observed data. 

  The unbiased estimators of the population are as follows, 

                                   �̂�𝑌𝑠𝑦 = �̅�𝑘𝑠𝑦 =
1

𝑛
∑ 𝑦𝑖𝑗

𝑛
𝑖=1 ,    𝑖 = 1,2, … , 𝑘.                                     (1.1) 

                                   �̂�𝑋𝑠𝑦 = �̅�𝑘𝑠𝑦 =
1

𝑛
∑ 𝑥𝑖𝑗

𝑛
𝑖=1 ,   𝑖 = 1,2, … , 𝑘.                                      (1.2) 

   To obtain the bias and variance we can write the error term 𝜀𝑋 ,  𝜀𝑌 as follows, 

                                   𝜀𝑋 =
�̅�𝑠𝑦−𝜇𝑥𝑠𝑦

𝜇𝑥𝑠𝑦
  ,      𝜀𝑌 =

�̅�𝑠𝑦−𝜇𝑌𝑠𝑦

𝜇𝑌𝑠𝑦
 

As  �̅�𝑠𝑦 and �̅�𝑠𝑦 are unbiased estimators of �̂�𝑌𝑠𝑦 and  �̂�𝑋𝑠𝑦 respectively we can write 

 𝐸(𝜀𝑌) = 𝐸(𝜀𝑋) = 0,          �̅�𝑖. = �̅�𝑠𝑦 =
1

𝑛
∑ 𝑈𝑛

𝑗=1 𝑖𝑗
, 

𝐸(𝜀𝑋
2) =

1

𝜇𝑋𝑠𝑦
2 {𝜎𝑋𝑠𝑦

2 + 𝜎𝑉𝑠𝑦
2 },      𝐸(𝜀𝑌

2) =
1

𝜇𝑌𝑠𝑦
2 {𝜎𝑌𝑠𝑦

2 + 𝜎𝑈𝑠𝑦
2 }, 

𝐸[𝜀𝑋𝜀𝑌] =
𝜌𝜎𝑋𝑠𝑦𝜎𝑌𝑠𝑦

𝜇𝑌𝑠𝑦𝜇𝑋𝑠𝑦
    ,         𝑅 =

𝜇𝑌𝑠𝑦

𝜇𝑋𝑠𝑦
  ,      𝐸(𝜀𝑋

∗ 𝜀𝑌
∗) =

(𝜌𝑌𝑋𝜎𝑋𝑠𝑦𝜎𝑌𝑠𝑦+𝜌𝑈𝑉𝜎𝑈𝑠𝑦𝜎𝑉𝑠𝑦)

𝜇𝑌𝑠𝑦𝜇𝑋𝑠𝑦
, 

𝜎𝑌𝑠𝑦
2 =

1

𝑘
∑ (�̅�𝑘𝑠𝑦

𝑘
𝑖=1 −𝜇𝑌𝑠𝑦)2,             𝜎𝑋𝑠𝑦

2 =
1

𝑘
∑ (�̅�𝑘𝑠𝑦

𝑘
𝑖=1 −𝜇𝑋𝑠𝑦)2, 

𝜎𝑈𝑠𝑦
2 =

1

𝑘
∑ (�̅�𝑖.)

2𝑘
𝑖=1  ,                      𝜎𝑉𝑠𝑦

2 =
1

𝑘
∑ (�̅�𝑖.)

2𝑘
𝑖=1 ,         

 

2. Pre-Existing Estimators in the Presence of measurement error 

The mean estimator in the presence of measurement error is  �̅�𝑠𝑦. The variance in the presence of 

measurement error is given as, 

𝑉(�̅�𝑠𝑦𝑚) = 𝜎𝑌𝑠𝑦𝑚
2  = 

1

𝑘
∑ (�̅�𝑠𝑦 − 𝜇𝑌𝑠𝑦)

2𝑘
𝑖=1  = 

1

𝑘
∑ (�̅�𝑠𝑦 + �̅�𝑠𝑦 − 𝜇𝑌𝑠𝑦)

2𝑘
𝑖=1                                (2.1) 

𝑉(�̅�𝑠𝑦𝑚) = 𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2                                                                                                            (2.2) 

Ratio estimator in the presence of measurement error is defined as, 

�̅�𝑅𝑠𝑦𝑚 =  �̅�𝑠𝑦
𝜇𝑋𝑠𝑦

�̅�𝑠𝑦
                                                                      (2.3) 

The mean square error of ratio estimator in the presence of measurement error is obtained as, 

𝑏𝑖𝑎𝑠( �̅�𝑅𝑠𝑦𝑚) =  
1

𝜇𝑌𝑠𝑦
{𝑅2(𝜎𝑋𝑠𝑦

2 + 𝜎𝑉𝑠𝑦
2 ) − 𝜌𝑅𝜎𝑋𝑠𝑦𝜎𝑌𝑠𝑦}                                                        (2.4) 



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(�̅�𝑅𝑠𝑦𝑚 − 𝜇𝑌𝑠𝑦)2 = 𝜇𝑌𝑠𝑦
2  (𝜀𝑌

2 + 𝜀𝑋
2 − 2𝜀𝑌𝜀𝑋)                                                                          (2.5) 

𝑀𝑆𝐸(�̅�𝑅𝑠𝑦𝑚) = [𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 + 𝑅2(𝜎𝑋𝑠𝑦
2 + 𝜎𝑉𝑠𝑦

2 ) − 2𝜌𝑅𝜎𝑌𝑠𝑦𝜎𝑋𝑠𝑦].                                  (2.6) 

 

The bias in the presence of correlated measurement error is derived as, 

𝑏𝑖𝑎𝑠 (𝑦∗̅̅ ̅
𝑅𝑠𝑦𝑚

) =
1

𝜇𝑌𝑠𝑦
{𝑅2(𝜎𝑋𝑠𝑦

2 + 𝜎𝑉𝑠𝑦
2 ) − 𝑅(𝜌𝜎𝑋𝜎𝑌 + 𝜌𝑈𝑉𝜎𝑈𝜎𝑉}                                       (2.7) 

The mean square error in the presence of correlated measurement error is derived as 

𝑀𝑆𝐸 (𝑦∗̅̅ ̅
𝑅𝑠𝑦𝑚

) = [𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 + 𝑅2(𝜎𝑋𝑠𝑦
2 + 𝜎𝑉𝑠𝑦

2 ) − 2𝑅(𝜌𝜎𝑋𝑠𝑦𝜎𝑌𝑠𝑦 + 𝜌𝑈𝑉𝜎𝑈𝑠𝑦𝜎𝑉𝑠𝑦)]       (2.8)  

                                                                                                                   

The product estimator in the presence of measurement error is  

�̅�𝑃𝑠𝑦𝑚 = �̅�𝑠𝑦
�̅�𝑠𝑦

𝜇𝑋𝑠𝑦
                                                                                                                           (2.9) 

The bias of the estimator is obtained as 

𝑏𝑖𝑎𝑠(�̅�𝑃𝑠𝑦𝑚) = (
𝜌𝜎𝑌𝑠𝑦𝜎𝑋𝑠𝑦

𝜇𝑋𝑠𝑦
)                                                                                                  (2.10) 

The mean square error in the presence of uncorrelated measurement error is derived as 

𝑀𝑆𝐸(�̅�𝑃𝑠𝑦𝑚) = (𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 + 𝑅2(𝜎𝑋𝑠𝑦
2 + 𝜎𝑉𝑠𝑦

2 ) + 2𝜌𝑅𝜎𝑌𝑠𝑦𝜎𝑋𝑠𝑦).                                    (2.11) 

The mean square error in the presence of correlated measurement error is derived as 

𝑀𝑆𝐸 (𝑦∗̅̅ ̅
𝑃𝑠𝑦𝑚

) = (𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 + 𝑅2(𝜎𝑋𝑠𝑦
2 + 𝜎𝑉𝑠𝑦

2 ) + 2𝑅(𝜌𝜎𝑋𝑠𝑦𝜎𝑌𝑠𝑦 + 𝜌𝑈𝑉𝜎𝑈𝑠𝑦𝜎𝑉𝑠𝑦).     

                                                                                                                                            (2.12) 

The difference estimator in the presence of measurement error is 

�̅�𝑑𝑠𝑦𝑚 = �̅�𝑠𝑦 + 𝑏(𝜇𝑋𝑠𝑦 − �̅�𝑠𝑦)                                                                                                (2.13) 

𝑉(�̅�𝑑𝑠𝑦𝑚) = [(𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 ) + 𝑏2(𝜎𝑋𝑠𝑦
2 + 𝜎𝑉𝑠𝑦

2 ) − 2𝑏𝜌𝜎𝑌𝑠𝑦𝜎𝑋𝑠𝑦].                                    (2.14) 

After differentiating the above equation we get, 

𝑑(𝑉(�̅�𝑑𝑠𝑦𝑚))

𝑑𝑏
= [2𝑏(𝜎𝑋𝑠𝑦

2 + 𝜎𝑉𝑠𝑦
2 ) − 2𝜌𝜎𝑌𝑠𝑦𝜎𝑋𝑠𝑦]                                                                 (2.15) 

Equating the above equation to zero gives,  

𝑏 =
𝜌𝜎𝑌𝑠𝑦𝜎𝑋𝑠𝑦

(𝜎𝑋𝑠𝑦
2 +𝜎𝑉𝑠𝑦

2 )
                                                                                                                     (2.16) 

Now substituting the value of 𝑏, the minimum variance is obtained as 

𝑉(�̅�𝑑𝑠𝑦𝑚) = [(𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 ) −
𝜌2𝜎𝑌𝑠𝑦

2 𝜎𝑋𝑠𝑦
2

(𝜎𝑋𝑠𝑦
2 +𝜎𝑌𝑠𝑦

2 )
]                                                                         (2.17) 



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3. 1. Proposed Class of Estimators  

The proposed class of estimators in the presence of measurement error is  

 �̂�𝑌𝑅𝐸𝑠𝑦𝑚 = �̅�𝑠𝑦𝑒𝑥𝑝 ( 
𝜇𝑋𝑠𝑦𝑚 −�̅�𝑠𝑦

𝜇𝑋𝑠𝑦𝑚+�̅�𝑠𝑦+2𝐾
)                                                                                                     

(3.1.1) 

𝑀𝑆𝐸(�̂�𝑌𝑅𝑠𝑦𝑚) = [(𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 ) +
(𝜎𝑌𝑠𝑦

2 +𝜎𝑈𝑠𝑦
2 )

4(𝜇𝑋𝑠𝑦𝑚+𝐾2)
−

𝜌2𝜎𝑌𝑠𝑦
2 𝜎𝑋𝑠𝑦

2

(𝜇𝑋𝑠𝑦𝑚+𝐾2)(𝜎𝑋𝑠𝑦
2 +𝜎𝑉𝑠𝑦

2 )
]                        (3.1.2) 

     In order to obtain the min. MSE differentiate  (3.1.2) w.r.t  𝐾  and equate it to zero        

𝑑

𝑑𝐾
(𝜇

̂
𝑌𝑅𝐸𝑠𝑦𝑚) = 0                                                                                                               (3.1.3)   

𝐾 =  −𝜇𝑋𝑠𝑦𝑚 +
𝜇𝑌𝑠𝑦

2
− (𝜎𝑌𝑠𝑦

2 + 𝜎𝑈𝑠𝑦
2 )/𝑆𝑥𝑦                  (3.1.4)   

After substituting the value of 𝐾 mean square error of the estimators we can write from equation 

(3.1.2)  as 

𝑚𝑖𝑛. 𝑀𝑆𝐸(�̂�𝑌𝑠𝑦𝑚) = (𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 ) −
𝜌2𝜎𝑌𝑠𝑦

2 𝜎𝑋𝑠𝑦
2

(𝜎𝑋𝑠𝑦
2 +𝜎𝑉𝑠𝑦

2 )
                                                              (3.1.5)   

                                       

3.2. Bias and Mean Square Error for the correlated measurement error 

The minimum mean square error of the proposed class of estimator  in the presence  of correlated 

measurement error is obtained as  

𝑀𝑆𝐸(�̂�𝑌𝑠𝑦𝑚) = (𝜎𝑌𝑠𝑦
2 + 𝜎𝑈𝑠𝑦

2 ) −
(𝜌𝜎𝑋𝑠𝑦𝜎𝑌𝑠𝑦+𝜌𝑈𝑉𝜎𝑉𝑠𝑦𝜎𝑈𝑠𝑦)

2

(𝜎𝑋𝑠𝑦
2 +𝜎𝑉𝑠𝑦

2 )
                                          (3.2.1) 

 

5.1. Simulation Study 

In order to show the efficiency of the estimators and to show the impact of uncorrelated and 

correlated measurement error on the mse simulation study is conducted by using  R studio. A data 

matrix is generated using multivariate normal distribution on the auxiliary variable 𝑋, study variable 

𝑌 and the error variables 𝑈 and 𝑉 and the four variables 𝑋, 𝑌, 𝑈 and 𝑉 are with mean 

(𝜇𝑋𝑠𝑦, 𝜇𝑌𝑠𝑦, 0,0). 

The covariance matrix is: 

(

     𝑆𝑌
2

  𝜌𝑆𝑋𝑆𝑌
       

0
0

𝜌𝑆𝑋𝑆𝑌

𝑆𝑋
2 

0
0

        

0
0

     𝑆𝑈
2       

𝜌𝑈𝑉𝑆𝑈𝑆𝑉

     

0
0

𝜌𝑈𝑉𝑆𝑈𝑆𝑉

𝑆𝑉
2

)   

Here  

𝑆𝑌
2 and 𝑆𝑋

2  is the standard deviation of the study variable 𝑌  and 𝑋.. 



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𝑆𝑈
2 and 𝑆𝑉

2 is the standard deviation of the measurement error 𝑈 associated with  study variable 𝑌 

sample and 𝑉 is associated with the auxiliary variable 𝑋. 

𝜌 is the correlation coefficient between the study variable 𝑌 and auxiliary variable 𝑋. 

𝜌𝑈𝑉 is the correlation coefficient between the measurement errors of  𝑈 and 𝑉 the study variable 𝑌 

and auxiliary variable 𝑋.  

From the simulation by using R studio , data is generated for multivariate normal distribution with 

mean vector for (𝜇𝑋𝑠𝑦, 𝜇𝑌𝑠𝑦, 0, 0) = (50,60,0,0)  and 𝑆𝑋 = 35, 𝑆𝑌 = 25. From the generated data 10, 

systematic samples has been taken.  

The statistics is calculated for the generated data set.  The Mean square error is defined as the 

expectation of the squared difference between the value that is estimated and true value. The mean 

squared error for the proposed class of estimators is obtained and compared with the mean square 

error for the ratio and product estimator for different combinations of  

𝜌𝑋𝑌 = (0.95, 0.55, −0.95, −0.55) and  𝜌𝑈𝑉 = (0.95, 0.55, 0.0, −0.95, −0.55).  

To illustrate the performance of the estimator under measurement error and correlated measurement 

error 6 tables is created.  

Table 1: MSE and PRE of various estimators of 𝝁𝒀𝒔𝒚 for 𝝆𝑼𝑽 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, 𝟎. 𝟎, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) 

and 𝝆𝑿𝒀 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) when (𝝈𝑼
𝟐 , 𝝈𝑽

𝟐) = (𝟎, 𝟎) and (𝝈𝒀
𝟐 , 𝝈𝑿

𝟐 ) = (𝟏𝟓, 𝟐𝟎) and 

𝑵 = 𝟏𝟎𝟎𝟎 and 𝒏 = 𝟏𝟎. 

σU
2  σV

2  ρXY ρUV MSE 

 

var 

(μ̂sym) μ̂Rsym μ̂Psym y̅sym 

0 0 .95  0.95 

 0.55 

 0.0 

-.95 

-.55 

0.791    

 0.705    

 0.703     

1.908    

 0.742 

1.038  

 0.844   

0.836  

 2.405   

1.143 

 

24.263  

31.948  

23.633  

 61.18  

35.259 

 

8.187  

7.74 

7.313 

19.565 

7.611 

 

  .55  0.95 

 0.55 

 0.0 

-.95 

-

  6.82   

10.63 

16.29

12.82

13.32 

 

11.467 

18.439 

16.449 

14.865 

19.251 

 

38.284 

60.889 

48.846 

50.588 

65.774 

9.964 

15.25 

23.367 

18.387 

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Table 2: MSE and PRE of various estimators of 𝝁𝒀𝒔𝒚 for 𝝆𝑼𝑽 (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, 𝟎. 𝟎, −𝟎, 𝟗𝟓, −𝟎. 𝟓𝟓) 

and 𝝆𝑿𝒀 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) when (𝝈𝑼
𝟐 , 𝝈𝑽

𝟐) = (𝟐, 𝟐) and (𝝈𝒀
𝟐 , 𝝈𝑿

𝟐 ) = (𝟏𝟓, 𝟐𝟎) and 

𝑵 = 𝟏𝟎𝟎𝟎 and 𝒏 = 𝟏𝟎. 

σU
2  σV

2  ρXY ρUV MSE Var 

μ̂sym 

 

μ̂Rsym 

 

μ̂Psym 

 

y̅sym 

2 2 0.95 .95 

 .55 

 0.0 

-.95 

-.55 

2.965    

3.092    

4.693     

6.66    

4.039 

 

3.036  

3.544  

4.913  

7.346  

4.313 

 

107.252 

66.777 

75.321 

62.727 

35.695 

 

30.411 

21.80 

24.37 

17.74 

10.54 

 

  0.55 0.95 

0.55 

 0.0 

-.95 

11.651   

11.436   

10.895    

9.606    

8.599 

11.707 

12.912 

12.726 

15.486 

15.453 

38.672 

43.573 

35.056 

35.752 

40.151 

17.807 

16.396 

14.067 

11.413 

10.847 

0.55 

  -.95  0.95 

 0.55 

 0.0 

-.95 

-.55 

0.791    

1.688     

1.08    

1.473    

1.417 

 

32.482 

62.741 

40.936 

55.217 

49.023 

 

0.838  

1.697  

1.081  

1.475  

1.534 

 

8.112 

17.315 

11.081 

15.103 

14.53 

 

  -.55  0.95 

 0.55 

 0.0 

-.95 

-.55 

11.02 

5.724   

15.76   

7.554    

5.466 

 

34.034 

 20.841  

66.635  

32.194  

31.118 

 

11.203  

6.267 

19.386   

9.36  

9.408 

 

15.803 

8.206 

22.607 

10.83 

7.837 

 



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-.55     

  -.95 0.95 

0.55 

 0.0 

-.95 

-.55 

6.633    

4.687    

3.442    

2.591    

2.243 

 

53.846 

56.463 

53.616 

90.902 

44.812 

 

6.648  

5.388   

3.56  

2.686  

2.247 

 

18.864 

14.792 

14.799 

26.138 

13.082 

 

  -.55 0.95 

0.55 

 0.0 

-.95 

-.55 

16.095   

13.339    

15.91    

7.796   

17.332 

 

61.465 

58.549 

84.222 

45.465 

88.096 

 

22.308 

20.043 

27.213 

12.467 

25.926 

 

20.596 

17.613 

21.92 

11.749 

24.841 

 

 

Table 3: MSE and PRE of various estimators of 𝝁𝒀𝒔𝒚 for 𝝆𝑼𝑽 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, 𝟎. 𝟎, −𝟎, 𝟗𝟓, −𝟎. 𝟓𝟓) 

and 𝝆𝑿𝒀 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) when (𝝈𝑼
𝟐 , 𝝈𝑽

𝟐) = (𝟒, 𝟒) and (𝝈𝒀
𝟐 , 𝝈𝑿

𝟐 ) = (𝟏𝟓, 𝟐𝟎) and 

𝑵 = 𝟏𝟎𝟎𝟎 and 𝒏 = 𝟏𝟎 

σU
2  σV

2  ρXY ρUV MSE var 

μ̂sym μ̂Rsym 

 

μ̂Psym 

 

y̅sym 

4 4 0.95 0.95 

0.55 

 0.0 

-.95 

-.55 

2.353    

4.445    

7.395    

7.514    

8.056 

 

2.643  

4.465  

8.427  

8.225 

10.334 

 

77.652 

61.691 

33.187 

56.347 

22.252 

 

23.588 

19.296 

11.521 

16.952 

9.331 

 

  0.55 0.95 

0.55 

 0.0 

-.95 

-.55 

22.598   

10.359   

19.769   

19.429   

22.739 

 

26.067 

12.593 

19.964 

28.441 

33.259 

 

94.832  

42.14  

45.97 

48.519 

62.502 

 

33.61 

14.65 

25.237 

20.859 

25.083 

 

  -.95 0.95 

0.55 

17.371   

16.009   

11.014    

62.62 

65.209 

63.253 

20.624 

18.873 

11.674  

23.431 

23.091 

21.304 



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 0.0 

-.95 

-.55 

1.594    

3.777 

 

56.864 

42.463 

 

1.644  

3.934 

 

16.255 

12.257 

 

  -.55 0.95 

0.55 

 0.0 

-.95 

-.55 

20.826   

15.327    

8.238    

8.888   

12.416 

 

53.655 

39.353 

36.754 

53.181 

60.757 

 

30.339 

30.452 

15.683 

11.712 

18.679 

 

22.576 

15.583 

9.943 

15.076 

17.367 

 

 

Table 4: MSE and PRE of various estimators of 𝝁𝒀𝒔𝒚 for 𝝆𝑼𝑽 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, 𝟎. 𝟎, −𝟎, 𝟗𝟓, −𝟎. 𝟓𝟓) 

and𝝆𝑿𝒀 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) when (𝝈𝑼
𝟐 , 𝝈𝑽

𝟐) = (𝟎, 𝟎) and (𝝈𝒀
𝟐 , 𝝈𝑿

𝟐 ) = (𝟐𝟓, 𝟑𝟓) and 

𝑵 = 𝟏𝟎𝟎𝟎 and 𝒏 = 𝟏𝟎. 

σU
2  σV

2  ρXY ρUV MSE var 

μ̂sym μ̂Rsym 

 

μ̂Psym 

 

y̅sym 

0 0 .95 .95 

 .55 

 0.0 

-.95 

-.55 

1.99     

1.14    

3.206    

2.117    

1.752 

 

2.105  

1.187  

3.393  

2.202  

1.904 

 

81.594 

46.199 

131.50 

85.741 

73.051 

 

20.40 

11.69 

32.87 

21.71 

17.96 

 

  0.55 .95 

 .55 

 0.0 

-.95 

-.55 

14.18 

15.74 

18.69 

77.203   

24.151 

 

17.988 

19.387 

18.115 

10.442 

36.072 

 

61.942 

66.648 

57.044 

35.662 

122.664 

 

20.34 

22.56 

12.46 

34.625 

 

  -

0.95 

.95 

.55 

0.0 

-.95 

4.102    

2.013     

1.27    

1.607    

4.866 

143.92 

671.11 

453.32 

863.693 

4.352  

2.115  

1.398  

1.634  

4.889 

42.07 

20.64 

13.02 

16.47 



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-.55  189.115 

 

 49.90 

 

  -.55 .95 

.55 

 0.0 

-.95 

-.55 

8.069   

14.068    

7.148    

24.73    

8.322 

 

37.133 

52.502 

51.253 

107.91 

644.93 

 

10.791 

15.654 

16.881 

31.339 

13.389 

 

11.56 

20.16 

10.24 

35.45 

11.93 

 

 

Table 5: MSE and PRE of various estimators of 𝝁𝒀𝒔𝒚 for 𝝆𝑼𝑽 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, 𝟎. 𝟎, −𝟎, 𝟗𝟓, −𝟎. 𝟓𝟓) 

and 𝝆𝑿𝒀 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) when (𝝈𝑼
𝟐 , 𝝈𝑽

𝟐) = (𝟐, 𝟐) and (𝝈𝒀
𝟐 , 𝝈𝑿

𝟐 ) = (𝟐𝟓, 𝟑𝟓) and 

𝑵 = 𝟏𝟎𝟎𝟎 and 𝒏 = 𝟏𝟎. 

σU
2  σV

2  ρXY ρUV MSE var 

μ̂sym μ̂Rsym 

 

μ̂Psym 

 

y̅sym 

2 2 0.95 0.95 

0.55 

 0.0 

-.95 

-.55 

1.987    

4.224    

6.209    

8.832    

5.389 

 

2.298  

4.474  

6.763  

8.997  

7.071 

 

85.301 

78.166 

121.738 

127.646 

108.037 

 

20.348 

24.923 

31.228 

40.796 

24.903 

 

  0.55 0.95 

0.55 

 0.0 

-.95 

-.55 

6.515   

12.925    

5.413   

13.497   

23.983 

 

12.478 

14.524 

11.423  

33.11 

26.047 

 

44.475 

48.501 

33.232 

80.573 

54.299 

 

9.974 

18.447 

7.405 

17.034 

28.124 

 

  -.95 0.95 

0.55 

 0.0 

-.95 

-.55 

10.439    

7.815    

6.019    

2.819    

5.886 

 

85.623 

144.45 

103.521 

119.038 

160.013 

 

10.506  

8.982  

6.103  

3.808  

5.886 

 

30.379 

35.953 

28.986 

26.76 

44.336 

 



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  -.55 0.95 

0.55 

 0.0 

-.95 

-.55 

12.365   

22.751   

13.301   

18.897   

22.275 

 

40.978 

84.452 

53.519 

82.443 

74.537 

 

21.946 

27.465 

17.381 

22.246 

23.301 

 

13.634 

30.828 

17.971 

28.326 

31.935 

 

 

Table 6: MSE and PRE of various estimators of 𝝁𝒀𝒔𝒚 for 𝝆𝑼𝑽 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, 𝟎. 𝟎, −𝟎, 𝟗𝟓, −𝟎. 𝟓𝟓) 

and 𝝆𝑿𝒀 = (𝟎. 𝟗𝟓, 𝟎. 𝟓𝟓, −𝟎. 𝟗𝟓, −𝟎. 𝟓𝟓) when (𝝈𝑼
𝟐 , 𝝈𝑽

𝟐) = (𝟒, 𝟒) and (𝝈𝒀
𝟐 , 𝝈𝑿

𝟐 ) = (𝟐𝟓, 𝟑𝟓) and 

𝑵 = 𝟏𝟎𝟎𝟎 and 𝒏 = 𝟏𝟎. 

σU
2  σV

2  ρXY ρUV MSE var 

μ̂sym μ̂Rsym 

 

μ̂Psym 

 

y̅sym 

4 4 0.95 0.95 

0.55 

 0.0 

-.95 

-.55 

2.179    

2.738   

14.226   

14.163   

13.767 

 

2.191  

3.255 

15.702 

17.719 

15.059 

 

79.529 

48.267 

153.798 

55.637 

61.066 

 

21.996 

11.824 

42.31 

19.349 

22.006 

 

  0.55 0.95 

0.55 

 0.0 

-.95 

-.55 

23.383   

20.011   

26.354   

31.146   

14.102 

 

34.379 

24.771 

32.019 

40.656  

22.02 

 

130.356 

84.505  

92.99 

83.963 

48.613 

 

35.727 

28.564 

34.714 

35.522 

16.444 

 

  -0.95 0.95 

0.55 

 0.0 

-.95 

-.55 

15.811   

11.338    

5.359    

2.522    

4.189 

 

58.113 

79.016 

74.701 

82.203 

101.412 

 

18.213 

12.265  

6.132  

3.128  

4.311 

 

21.947 

24.529 

19.226 

19.12 

26.803 

 

  -0.55 0.95 

0.55 

 0.0 

-.95 

-.55 

17.646   

14.695   

17.007   

19.215    

8.308 

 

62.357 

70.549 

70.806 

113.737 

54.996 

 

39.282 

28.031 

22.684 

29.443 

19.39 

 

18.682 

18.347 

23.137 

29.856 

11.378 

 

 

 



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Results from the tables: 

From the table 1 for 𝜌𝑋𝑌 = 0.95 and 𝜌𝑈𝑉 = (0.95, 0.55, 0.0, −0.95, −0.55) as the correlation 

between the study variable 𝑌 and auxiliary variable 𝑋 is positively correlated the ratio estimator 

performs well. As observed from the table when 𝜌𝑋𝑌 = 0.95  the mse of the ratio estimator is very 

less as compared to the mse of product estimator but the mean square error for the proposed class of 

estimators is less than that of the ratio estimator. From the table 1, for 𝜌𝑋𝑌 = 0.55 and 𝜌𝑈𝑉 =

(0.95, 0.55, 0.0, −0.95, −0.55) as the correlation between the study variable 𝑌 and auxiliary variable 

𝑋 is positively correlated the ratio estimator performs well. As observed from the table when 𝜌𝑋𝑌 =

0.55 the mse of the ratio estimator is very less as compared to the product estimator but the mse for 

the proposed class of estimators is less than that of the ratio estimator.  

From the table 1 for 𝜌𝑋𝑌 = −0.95 and  𝜌𝑈𝑉 = (0.95, 0.55, 0.0, −0.95, −0.55) as the correlation 

between the study variable 𝑌 and auxiliary variable 𝑋 is negatively correlated the product estimator 

performs well. As observed from the table when 𝜌𝑋𝑌 = −0.95  the mean square error of the product 

estimator is very less as compared to mse of the ratio estimator but the mse for the proposed class of 

estimators is less than that of the product estimator.  

From the table 1 for 𝜌𝑋𝑌 = −0.55 and 𝜌𝑈𝑉 = (0.95, 0.55, 0.0, −0.95, −0.55) as the correlation 

between the study variable 𝑌 and auxiliary variable 𝑋 is negatively correlated the product estimator 

performs well. As observed from the table when 𝜌𝑋𝑌 = −0.55  the mse of the product estimator is 

very less as compared to the mean square error of ratio estimator but the mean square error for the 

proposed class of estimators is less than that of the product estimator. The above results are observed 

when auxiliary variable 𝑋, study variable 𝑌 and the error variables 𝑈 and 𝑉 are with mean 

(𝜇𝑋𝑠𝑦, 𝜇𝑌𝑠𝑦, 0, 0) and the population variance of the study variable and auxiliary variable 𝑋, 𝑌are 

(15, 20) and the variance of the measurement errors 𝑈 and 𝑉 are taken as (0,0), (2,2) and (4,4).  A 

similar pattern is observed as described above is followed when the population variance of the study 

variable and auxiliary variable 𝑋, 𝑌 are increased to (25,35) and for the same values of variance of 

the measurement errors 𝑈 and 𝑉.  

Conclusions 

In this manuscript,  the effect of measurement error and correlated measurement error on the 

exponential estimator, under systematic sampling technique to estimate the population mean of the 

study variable 𝑌 by using the auxiliary information has been studied. The observation is assumed to 

be recorded with some errors. Also, the error present in study and auxiliary variable are assumed to 

be correlated.  From the simulation study, it is observed that proposed method of estimation is more 

efficient than ratio and product and mean estimator . From simulation study, it is concluded that the 

mse in the presence of measurement error as well as in the presence of correlated measurement error 

is always high. Due to simplicity, systematic sampling has wide applicability and  thus, the proposed 

method of estimation can be applied to many real-life events under systematic sampling survey 

where the data are prone to recorded with measurement error and as well as correlated measurement 

error. The limitation of the study is that the method of estimation is derived for systematic sampling.  

 



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[7] Hasel, H. A. (1942). Estimation of volume of timber stands by strip sampling. The Annals of 

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[9] Madow, W. G. (1949). On the theory of systematic sampling, II. The Annals of Mathematical 

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