Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1450 
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A New Efficient Difference-Type Estimator for Estimating Population 

Mean using Dual Auxiliary Information under Non-Response 

Udita Gupta, Prof. Manoj Kumar Srivastava, Prof. Namita Srivastava, Anjali Bhardwaj 

Research Scholar, Department of Statistics, I.S.S., Dr. Bhimrao Ambedkar University, Agra. Mail Id: 

drudita0108@gmail.com 

Vice-Chancellor, Shahid Mahendra Karma Vishwavidalaya, Bastar, Mail Id: mksiss87@gmail.com 

Professor and Head of Department of Statistics, St. John’s College, Agra, Mail Id: drnamita.sjc@gmail.com 

Research Scholar, Department of Statistics, I.S.S., Dr. Bhimrao Ambedkar University, Agra, Mail Id: 

anjalibharadwaj32@gmail.com 

Department of Statistics, Dr. Bhimrao Ambedkar University, Agra 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

In this paper, the problem of estimating the finite population mean by using dual auxiliary 

information under non-response. This paper proposed a difference-type estimator of 

population mean under different cases. The expressions of the bias and mean square error 

(MSE) of the proposed estimator have been obtained up to the first-degree approximation. 

The proposed estimator has been compared with usual unbiased estimator, ratio estimator, 

product estimator, difference-cum-ratio estimator, difference-cum-product estimator and 

other existing estimator and the cases obtained to show the efficiency of the proposed 

estimator over other considered estimators. Numerical Illustration is carried out to 

support the theoretical findings. 

Keywords: Study Variable, Auxiliary Variable, Bias, MSE, Ranked Auxiliary Variable, 

Non-Response. 

 

1. INTRODUCTION 

One of the sample survey objectives is to estimate the unknown population parameters of the study 

variable such as population total, mean, proportion, ratio and variances etc. A procedure is desirable 

that provides a precise estimator of the parameter of interest by surveying a suitably chosen sample of 

individuals. Supplementary/ additional information provided by an auxiliary variable which is 

correlated with the study variable enhances the precision of the estimators. Survey statisticians take 

advantage of this information whenever it is available to explore the efficient estimators. Ratio, 

product, regression and their modified estimators are best examples in this regard.  

mailto:drudita0108@gmail.com
mailto:mksiss87@gmail.com
mailto:drnamita.sjc@gmail.com


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In practice almost all surveys suffer from non-response. Problems with no response are often caused 

by rejection of topics, absences and sometimes due to the lack of information. The pioneering work of 

Hansen and Hurwitz (1946), states that “A sub-sample of initially non-responsive individuals is 

recontacted using a more expensive method, suggesting the first attempt by mail-based questionnaire 

and the second attempt by a face-to-face interview”. When estimating population parameters such as 

the mean, total or ratio, sampling experts sometimes use auxiliary information to improve efficiency 

of the estimates. 

It is known that the efficiency of an estimator of the population mean of the study variable y can be 

increased by using auxiliary information which is highly correlated with the study variable y. Rao 

(1986), Khare and Srivastava (1995, 1997), Okafor and Lee (2000) and Singh and Kumar (2008, 2009, 

2010) have proposed some estimator for population mean of the study variable y using auxiliary 

information in presence of non-response. 

Recently, Haq et al. used an additional information of the auxiliary variable called ranked auxiliary 

variable to develop efficient estimators for the estimation of mean. These estimators are developed 

only to cope with the simple random sampling scheme.  

Here, a new challenge/idea arises to search for a more optimal estimator using dual auxiliary 

information to deal with non-response scheme. This challenge is successfully accomplished and new 

optimal estimators for finite population mean are developed under non-response scheme in this paper. 

The remaining part of the paper is organized as follows: In section 2, notations under non-response are 

introduced. In section 3, existing estimator under non-response. In section 4, proposed estimator for 

estimating finite population mean using the original and ranked auxiliary information are defined. In 

section 5, theoretical comparison is done between existing and proposed estimator. An empirical study 

is carried out to evaluate the performance of the proposed estimators which validate the theoretical 

results in section 6. Conclusions are enclosed in the last section. 

2. NOTATION 

For a finite population 𝑈 =  (𝑈1, 𝑈2, 𝑈3 … 𝑈𝑁) of size N and a random sample of size 𝑛 is drawn 

without replacement. Let the characteristics under study, 𝑦 (say) takes value 𝑦𝑖 on the unit 𝑈𝑖 , (𝑖 =

1, 2, 3 … 𝑁). In survey on human population, it is often the case that 𝑛𝑖 unit respond on the first attempt 

while 𝑛2 (= 𝑛 − 𝑛1) units do not provide any response. In the case of non-response at the initial stage, 

Hansen and Hurwitz (1946) proposed a double sampling plan for estimating the population mean 

having the steps given below: 



Communications on Applied Nonlinear Analysis 

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a) A simple random sample of size 𝑛 is drawn and the questionnaire is mailed to the sample units 

b) A sub-sample of size 𝑟 = (𝑛2 𝑘⁄ ), (𝑘 ≥ 1) from the 𝑛2 non-responding units in the initial 

attempt is contacted through personal interviews. 

In the Hansen and Hurwitz method the population is supposed to be consisting of response stratum of 

size 𝑁1 and the non-response stratum of size 𝑁2 = (𝑁 − 𝑁1). Let �̅� =
1

𝑁
∑ 𝑦𝑖

𝑁
𝑖=1  and 𝑆𝑦

2 =

1

𝑁−1
∑ (𝑦𝑖 − �̅�)2𝑁

𝑖=1  denote the population mean and variance of the study variable y. Let �̅�1 =

1

𝑁1
∑ 𝑦𝑖

𝑁1
𝑖=1  and 𝑆𝑦(1)

2 =
1

𝑁1−1
∑ (𝑦𝑖 − �̅�1)2𝑁1

𝑖=1  denote the mean and variance of the respondent group (or 

strata). Similarly, let �̅�2 =
1

𝑁2
∑ 𝑦𝑖

𝑁2
𝑖=1  and 𝑆𝑦(2)

2 =
1

𝑁2−1
∑ (𝑦𝑖 − �̅�2)2𝑁2

𝑖=1  denote the mean and variance 

of the non-respondent group (or strata). The population mean can be written as      

�̅� = 𝑊1�̅�1 + 𝑊2�̅�2 

where 𝑊1 = (
𝑁1

𝑁
) and 𝑊2 = (

𝑁2

𝑁
). The sample mean �̅�1 =

1

𝑛1
∑ 𝑦𝑖

𝑛1
𝑖=1  denote the mean of the 𝑛1 

responding units and �̅�2 =
1

𝑛2
∑ 𝑦𝑖

𝑛2
𝑖=1  denote the mean of the 𝑛2 non-responding units.  

Let �̅�2𝑟 =
1

𝑟
∑ 𝑦𝑖

𝑟
𝑖=1  denote the mean of the 𝑟 sub-sampled units where 𝑟 =

𝑛2

𝑘
.Hansen and Hurwitz 

(1946) suggested an unbiased estimator for the population mean �̅� of the study variable 𝑦 is given as: 

                                                         �̅�∗ = 𝑤1�̅�1 + 𝑤2�̅�2𝑟                                                                    (2.1) 

where 𝑤1 =
𝑛1

𝑛
 and 𝑤2 =

𝑛2

𝑛
 are responding proportions and non-responding proportions of the sample. 

The variance �̅�∗ is given below: 

                                      𝑉(�̅�∗) = �̅�2 [(
1 − 𝑓

𝑛
) 𝐶𝑦

2 +
𝑊2(𝑘 − 1)

𝑛
𝐶𝑦(2)

2 ]                                       (2.2) 

where 𝐶𝑦
2 =

𝑆𝑦
2

�̅�2 and 𝐶𝑦(2)
2 =

𝑆𝑦(2)
2

�̅�2 . 

Let 𝑥𝑖(𝑖 = 1,2, … 𝑁) denote the auxiliary variable correlated with the study variable           𝑦𝑖(𝑖 =

1,2, … , 𝑁). Let �̅� =
1

𝑁
∑ 𝑥𝑖

𝑁
𝑖=1  and 𝑆𝑥

2 =
1

𝑁−1
∑ (𝑥𝑖 − �̅�)2𝑁

𝑖=1  denote the population mean and variance 

of the auxiliary variable y. Let �̅�1 =
1

𝑁1
∑ 𝑥𝑖

𝑁1
𝑖=1  and 𝑆𝑥(1)

2 =
1

𝑁1−1
∑ (𝑥𝑖 − �̅�1)2𝑁1

𝑖=1  denote the mean and 

variance of the respondent group (or strata). Similarly, let �̅�2 =
1

𝑁2
∑ 𝑦𝑖

𝑁2
𝑖=1  and 𝑆𝑥(2)

2 =

1

𝑁2−1
∑ (𝑥𝑖 − �̅�2)2𝑁2

𝑖=1  denote the mean and variance of the non-respondent group (or strata). Let �̅� =



Communications on Applied Nonlinear Analysis 

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1

𝑛
∑ 𝑥𝑖

𝑛
𝑖=1  denote the mean of all the n units. Let �̅�1 =

1

𝑛1
∑ 𝑥𝑖

𝑛1
𝑖=1  denote the mean of the 𝑛1 responding 

units and �̅�2 =
1

𝑛2
∑ 𝑥𝑖

𝑛2
𝑖=1  denote the mean of the 𝑛2 non-responding units.  

Let �̅�2𝑟 =
1

𝑟
∑ 𝑥𝑖

𝑟
𝑖=1  denote the mean of the 𝑟 sub-sampled units where 𝑟 =

𝑛2

𝑘
. With this background, 

define an unbiased estimator of population mean �̅� is given as: 

                                                          �̅�∗ = 𝑤1�̅�1 + 𝑤2�̅�2𝑟                                                                  (2.3) 

The variance of �̅�∗ is given below: 

                                   𝑉(�̅�∗) = �̅�2 [(
1 − 𝑓

𝑛
) 𝐶𝑥

2 +
𝑊2(𝑘 − 1)

𝑛
𝐶𝑥(2)

2 ]                                          (2.4) 

where 𝐶𝑥
2 =

𝑆𝑥
2

�̅�2 and 𝐶𝑥(2)
2 =

𝑆𝑥(2)
2

�̅�2 . 

2.1 Bias and Mean Square Error (MSE) of the Proposed Estimators 

 

Let us define the following terms: 

�̅�∗ = �̅�(1 + 𝜀0) 

�̅�∗ = �̅�(1 + 𝜀1) 

�̅�𝑥
∗ = �̅�𝑥(1 + 𝜀2) 

𝐸(𝜀0) = 𝐸(𝜀1) = E(𝜀2) = 0                                                                                                            (2.5)                                                                                                                                      

𝐸(𝜀0
2) = [(

1 − 𝑓

𝑛
) 𝐶𝑦

2 +
𝑊2(𝑘 − 1)

𝑛
𝐶𝑦(2)

2 ] = 𝐵 (𝑠𝑎𝑦)                                                               (2.6) 

𝐸(𝜀1
2) = [(

1 − 𝑓

𝑛
) 𝐶𝑥

2 +
𝑊2(𝑘 − 1)

𝑛
𝐶𝑥(2)

2 ] = 𝐴 (𝑠𝑎𝑦)                                                               (2.7) 

𝐸(𝜀0𝜀1) = [(
1 − 𝑓

𝑛
) 𝜌𝑦𝑥𝐶𝑦𝐶𝑥 +

𝑊2(𝑘 − 1)

𝑛
𝜌𝑦𝑥(2)𝐶𝑦(2)𝐶𝑥(2)] = 𝐶 (𝑠𝑎𝑦)                            (2.8) 

𝐸(𝜀2
2) = [(

1 − 𝑓

𝑛
) 𝐶𝑟𝑥

2 +
𝑊2(𝑘 − 1)

𝑛
𝐶𝑟𝑥(2)

2 ] = 𝐷 (𝑠𝑎𝑦) 

𝐸(𝜀0𝜀2) = [(
1 − 𝑓

𝑛
) 𝜌𝑦𝑟𝑥

𝐶𝑦𝐶𝑟𝑥
+

𝑊2(𝑘 − 1)

𝑛
𝜌𝑦𝑟𝑥(2)𝐶𝑦(2)𝐶𝑟𝑥(2)] = E (𝑠𝑎𝑦)                                  



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𝐸(𝜀1𝜀2) = [(
1 − 𝑓

𝑛
) 𝜌𝑥𝑟𝑥

𝐶𝑥𝐶𝑟𝑥
+

𝑊2(𝑘 − 1)

𝑛
𝜌𝑥𝑟𝑥(2)𝐶𝑥(2)𝐶𝑟𝑥(2)] = F (𝑠𝑎𝑦)                                   

where  

𝜌𝑦𝑥 = 𝑆𝑦𝑥/𝑆𝑥𝑆𝑦 

 𝜌𝑦𝑥(2) =
𝑆𝑦𝑥(2)

𝑆𝑥(2)𝑆𝑦(2)
 

𝑆𝑦𝑥 =
1

𝑁 − 1
∑(𝑦𝑖 − �̅�)(𝑥𝑖 − �̅�)

𝑁

𝑖=1

 

𝑆𝑦𝑥(2) =
1

𝑁2 − 1
∑(𝑦𝑖 − �̅�2)(𝑥𝑖 − �̅�2)

𝑁2

𝑖=1

 

𝜌𝑦𝑟𝑥
= 𝑆𝑦𝑟𝑥

/𝑆𝑟𝑥
𝑆𝑦 

 𝜌𝑦𝑟𝑥(2) =
𝑆𝑦𝑟𝑥(2)

𝑆𝑟𝑥(2)𝑆𝑦(2)
 

𝑆𝑦𝑟𝑥
=

1

𝑁 − 1
∑(𝑦𝑖 − �̅�)(𝑟𝑥(𝑖) − �̅�𝑥)

𝑁

𝑖=1

 

𝑆𝑦𝑟𝑥(2) =
1

𝑁2 − 1
∑(𝑦𝑖 − �̅�2)(𝑟𝑥(𝑖) − �̅�𝑥(2))

𝑁2

𝑖=1

 

𝜌𝑥𝑟𝑥
= 𝑆𝑥𝑟𝑥

/𝑆𝑟𝑥
𝑆𝑥 

𝜌𝑥𝑟𝑥(2) =
𝑆𝑥𝑟𝑥(2)

𝑆𝑟𝑥(2)𝑆𝑥(2)
 

𝑆𝑥𝑟𝑥
=

1

𝑁 − 1
∑(𝑥𝑖 − �̅�)(𝑟𝑥(𝑖) − �̅�𝑥)

𝑁

𝑖=1

 

𝑆𝑥𝑟𝑥(2) =
1

𝑁2 − 1
∑(𝑥𝑖 − �̅�2)(𝑟𝑥(𝑖) − �̅�𝑥(2))

𝑁2

𝑖=1

 

𝑅 =
�̅�

�̅�
 

𝑹∗ =
�̅�

�̅�𝒙

 

 



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3. EXISTING ESTIMATOR 

When few observations are missing in the sample. Initial, estimators for estimating population mean 

�̅� was suggested by Hansen and Hurwitz (1946). Many other authors work for the similar situation. 

Here, we are giving few of the existing estimators for population mean �̅� when some of the 

observations are missing. 

(i) Rao (1986) suggested a ratio estimator for the population mean �̅� of the study variable 𝑦 is 

given as 

                                                                𝑡𝑟
∗ = �̅�∗ (

�̅�

�̅�∗
)                                                                       (3.1) 

𝐵(𝑡𝑟
∗) =  �̅�(𝐴 − 𝐶)                   

𝑀𝑆𝐸(𝑡𝑟
∗) = �̅�2(𝐵 + 𝐴 − 2𝐶)               

𝑀𝑆𝐸(𝑡𝑟
∗) = �̅�2 [(

1 − 𝑓

𝑛
) {𝐶𝑦

2 + 𝐶𝑥
2 − 2𝜌𝑦𝑥𝐶𝑦𝐶𝑥} +

𝑊2(𝑘 − 1)

𝑛
[𝐶𝑦(2)

2 + 𝐶𝑥(2)
2 − 2𝜌𝑦𝑥(2)𝐶𝑦(2)𝐶𝑥(2)]] 

(ii) Khare and Srivastava (1993) suggested a product estimator for the population mean �̅� of 

the study variable 𝒚 is given as 

                                                                𝑡𝑘𝑠
∗ = �̅�∗ (

�̅�∗

�̅�
)                                                                      (3.2) 

𝐵(𝑡𝑘𝑠
∗ ) =  �̅�𝐶                  

𝑀𝑆𝐸(𝑡𝑘𝑠
∗ ) = �̅�2(𝐴 + 𝐵 + 2𝐶)             

𝑀𝑆𝐸(𝑡𝑘𝑠
∗ ) = �̅�2 [(

1 − 𝑓

𝑛
) {𝐶𝑦

2 + 𝐶𝑥
2 + 2𝜌𝑦𝑥𝐶𝑦𝐶𝑥} +

𝑊2(𝑘 − 1)

𝑛
[𝐶𝑦(2)

2 + 𝐶𝑥(2)
2 + 2𝜌𝑦𝑥(2)𝐶𝑦(2)𝐶𝑥(2)]] 

(iii) Singh et. al. (2008) suggested an exponential ratio type estimators for the population mean 

�̅� of the study variable 𝑦 are  

                                                            𝑡𝑒𝑟
∗ = �̅�∗𝑒𝑥𝑝 [

�̅� − �̅�∗

�̅� + �̅�∗
]                                                            (3.3) 

𝐵(𝑡𝑒𝑟
∗ ) =

�̅�

8
(3𝐴 − 4𝐶)           

𝑀𝑆𝐸(𝑡𝑒𝑟
∗ ) = �̅�2 (𝐵 +

𝐴

4
− 𝐶)         

𝑀𝑆𝐸(𝑡𝑒𝑟
∗ ) = �̅�2 [(

1 − 𝑓

𝑛
) {𝐶𝑦

2 +
𝐶𝑥

2

4
− 𝜌𝑦𝑥𝐶𝑦𝐶𝑥} +

𝑊2(𝑘 − 1)

𝑛
[𝐶𝑦(2)

2 +
𝐶𝑥(2)

2

4
− 𝜌𝑦𝑥(2)𝐶𝑦(2)𝐶𝑥(2)]] 



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(iv) Singh et. al. (2008) suggested an exponential product type estimators for the population 

mean �̅� of the study variable 𝑦 are 

                                                            𝑡𝑒𝑝
∗ = �̅�∗𝑒𝑥𝑝 [

�̅�∗ − �̅�

�̅�∗ + �̅�
]                                                           (3.4) 

𝐵(𝑡𝑒𝑝
∗ ) =

�̅�

8
(4𝐶 − 𝐴)            

𝑀𝑆𝐸(𝑡𝑒𝑝
∗ ) = �̅�2 (𝐵 +

𝐴

4
+ 𝐶)        

𝑀𝑆𝐸(𝑡𝑒𝑝
∗ ) = �̅�2 [(

1 − 𝑓

𝑛
) {𝐶𝑦

2 +
𝐶𝑥

2

4
+ 𝜌𝑦𝑥𝐶𝑦𝐶𝑥} +

𝑊2(𝑘 − 1)

𝑛
[𝐶𝑦(2)

2 +
𝐶𝑥(2)

2

4
+ 𝜌𝑦𝑥(2)𝐶𝑦(2)𝐶𝑥(2)]] 

(v) Sunil Kumar and Sandeep Bhougal (2011) suggested a modified ratio-product type 

exponential estimator for the population mean �̅� of the study variable 𝑦 

𝑡𝑠𝑠
∗ = �̅�∗ {𝛼𝑒𝑥𝑝 (

�̅� − �̅�∗

�̅� + �̅�∗
) + (1 − 𝛼)𝑒𝑥𝑝 (

�̅�∗ − �̅�

�̅�∗ + �̅�
)} 

The corrected bias of this estimator is  

𝐵(𝑡𝑠𝑠
∗ ) = (

𝐴

8
−

𝐶

2
) (4𝛼 − 1) 

𝑀𝑆𝐸(𝑡𝑠𝑠
∗ ) = �̅�2 [𝐵 −

𝐶2

𝐴
] 

 

4. PROPOSED ESTIMATOR 

�̅�∗, �̅�∗, �̅� 𝑎𝑛𝑑 �̅�𝑥 are used. The non-response occurs on both study variable 𝑦 as well as auxiliary 

variable 𝑥, the population mean �̅� of the auxiliary variable and the rank of the auxiliary variable 𝑥 are 

known. In this, we proposed a difference type estimator and the estimator is  

                                              𝑡𝑟𝑝
∗ = 𝜔1�̅�∗ + 𝜔2(�̅� − �̅�∗) + 𝜔3(�̅�𝑥 − �̅�𝑥

∗)                                     (4.1) 

where 𝜔1, 𝜔2 & 𝜔3 is a real constant to be determined such that the MSE of 𝑡𝑟𝑝
∗  is minimum. 

Now, expressing 𝑡𝑟𝑝
∗  in terms of 𝜀′𝑠 we have 

𝑡𝑟𝑝
∗ = 𝜔1�̅�(1 + 𝜀0) + 𝜔2(�̅� − �̅�(1 + 𝜀1)) + 𝜔3(�̅�𝑥 − �̅�𝑥(1 + 𝜀2)) 

                              = 𝜔1�̅�(1 + 𝜀0) − 𝜔2�̅�𝜀1 − 𝜔3�̅�𝑥𝜀2                                                                 (4.2) 

Subtracting  �̅� on both sides of (4.2), we get 

                                      𝑡𝑟𝑝
∗ − �̅� = (𝜔1 − 1)�̅� + 𝜔1�̅�𝜀0 − 𝜔2�̅�𝜀1 − 𝜔3�̅�𝑥𝜀2                            (4.3) 



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Taking Expectation on both sides of (4.3), we get the bias of the estimator 𝑡𝑟𝑝
∗  as 

𝐸(𝑡𝑟𝑝
∗ − �̅�) = (𝜔1 − 1)�̅� + 𝜔1�̅�𝐸(𝜀0) − 𝜔2�̅�𝐸(𝜀1) − 𝜔3�̅�𝑥𝐸(𝜀2) 

                                𝐵(𝑡𝑟𝑝
∗ ) = (𝜔1 − 1)�̅�                                                                                          (4.4) 

Squaring of (4.3) on both sides, we have 

(𝑡𝑟𝑝
∗ − �̅�)2 = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝜀0
2 + 𝜔2

2�̅�2𝜀1
2 + 𝜔3

2�̅�𝑥
2𝜀2

2 + 2𝜔1(𝜔1 − 1)�̅�2𝜀0

− 2𝜔2(𝜔1 − 1)�̅��̅�𝜀1 − 2𝜔3(𝜔1 − 1)�̅��̅�𝑥𝜀2 − 2𝜔1𝜔2�̅��̅�𝜀0𝜀1 − 2𝜔1𝜔3�̅��̅�𝑥𝜀0𝜀2

+ 2𝜔2𝜔3�̅��̅�𝑥𝜀1𝜀2                                                                 (4.5) 

Taking Expectation on both sides of (4.5), we get the exact mean square error (MSE) of 𝑡𝑟𝑝
∗ , as 

𝐸(𝑡𝑟𝑝
∗ − �̅�)2 = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝐸(𝜀0
2) + 𝜔2

2�̅�2𝐸(𝜀1
2) + 𝜔3

2�̅�𝑥
2𝐸(𝜀2

2) + 2𝜔1(𝜔1 − 1)�̅�2𝐸(𝜀0)

− 2𝜔2(𝜔1 − 1)�̅��̅�𝐸(𝜀1) − 2𝜔3(𝜔1 − 1)�̅��̅�𝑥𝐸(𝜀2) − 2𝜔1𝜔2�̅��̅�𝐸(𝜀0𝜀1)

− 2𝜔1𝜔3�̅��̅�𝑥𝐸(𝜀0𝜀2) + 2𝜔2𝜔3�̅��̅�𝑥𝐸(𝜀1𝜀2) 

𝑀𝑆𝐸(𝑡𝑟𝑝
∗ ) = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝐵 + 𝜔2
2�̅�2𝐴 + 𝜔3

2�̅�𝑥
2𝐷 − 2𝜔1𝜔2�̅��̅�𝐶 − 2𝜔1𝜔3�̅��̅�𝑥𝐸

+ 2𝜔2𝜔3�̅��̅�𝑥𝐹                                                                                                         (4.6) 

4.1 Optimum Choice of 𝝎𝟏, 𝝎𝟐 & 𝝎𝟑 and the Minimum MSE of Proposed Class of Estimator 

‘𝒕𝒓𝒑
∗ ’ 

Differentiating equation (4.6) partially w.r.to 𝜔1, 𝜔2 & 𝜔3 and equating to zero for obtaining the 

optimum value of 𝜔1, 𝜔2 & 𝜔3. The optimum value of 𝜔1, 𝜔2 & 𝜔3 which makes the MSE minimum 

of equation (4.6) is given by  

                                              [
1 + 𝐵 −𝐶 𝑅⁄ − 𝐸 𝑅∗⁄

𝐶 − 𝐴 𝑅⁄ − 𝐹 𝑅∗⁄

𝐸 −𝐹 𝑅⁄ − 𝐷 𝑅∗⁄
] [

𝜔1

𝜔2

𝜔3

] = [
1
0
0

]                                         (4.7) 

Solving (4.7), we get the optimum values of 𝜔1, 𝜔2 & 𝜔3 as 

                      𝜔1 =
𝐴𝐷 − 𝐹2

(1 + 𝐵)(𝐴𝐷 − 𝐹2) + 𝐶(𝐸𝐹 − 𝐶𝐷) − 𝐸(𝐴𝐸 − 𝐶𝐹)
= 𝑉1(𝑠𝑎𝑦)          (4.8𝑎) 

                   𝜔2 =
𝑅(𝐶𝐷 − 𝐸𝐹)

(1 + 𝐵)(𝐴𝐷 − 𝐹2) + 𝐶(𝐸𝐹 − 𝐶𝐷) − 𝐸(𝐴𝐸 − 𝐶𝐹)
  = 𝑉2(𝑠𝑎𝑦)           (4.8𝑏) 

             𝜔3 = 𝑅∗ (
𝐴𝐸 − 𝐶𝐹

(1 + 𝐵)(𝐴𝐷 − 𝐹2) + 𝐶(𝐸𝐹 − 𝐶𝐷) − 𝐸(𝐴𝐸 − 𝐶𝐹)
) = 𝑉3(𝑠𝑎𝑦)        (4.8𝑐) 

Thus, the resulting minimum MSE of the proposed estimator ‘𝑡𝑟𝑝
∗ ’ is given by 

𝑚𝑖𝑛. 𝑀𝑆𝐸 (𝑡𝑟𝑝
∗ ) = (𝑉1 − 1)2�̅�2 + 𝑉1

2�̅�2𝐵 + 𝑉2
2�̅�2𝐴 + 𝑉3

2�̅�𝑥
2𝐷 − 2𝑉1𝑉2�̅��̅�𝐶 − 2𝑉1𝑉3�̅��̅�𝑥𝐸 + 2𝑉2𝑉3�̅��̅�𝑥𝐹(𝟒. 𝟗) 

Case 1: If 𝜔2 = 0, the proposed estimator ‘𝑡𝑟𝑝
∗ ’ will be reduced as 



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                                                    𝑡𝑟𝑝1
∗ = 𝜔1�̅�∗ + 𝜔3(�̅�𝑥 − �̅�𝑥

∗)                                                      (4.10) 

where 𝜔1 & 𝜔3 is a real constant to be determined such that the MSE of 𝑡𝑟𝑝1
∗  is minimum. 

Now, expressing 𝑡𝑟𝑝1
∗  in terms of 𝜀′𝑠 we have 

𝑡𝑟𝑝1
∗ = 𝜔1�̅�(1 + 𝜀0) + 𝜔3(�̅�𝑥 − �̅�𝑥(1 + 𝜀2)) 

                                                    = 𝜔1�̅�(1 + 𝜀0) − 𝜔3�̅�𝑥𝜀2                                                          (4.11) 

                                    𝑡𝑟𝑝1
∗ − �̅� = (𝜔1 − 1)�̅� + 𝜔1�̅�𝜀0 − 𝜔3�̅�𝑥𝜀2                                           (4.12) 

Taking Expectation on both sides of (4.12), we get the bias of the estimator 𝑡𝑟𝑝1
∗  as 

𝐸(𝑡𝑟𝑝1
∗ − �̅�) = (𝜔1 − 1)�̅� + 𝜔1�̅�𝐸(𝜀0) − 𝜔3�̅�𝑥𝐸(𝜀2) 

                                           𝐵(𝑡𝑟𝑝1
∗ ) = (𝜔1 − 1)�̅�                                                                           (4.13) 

(𝑡𝑟𝑝1
∗ − �̅�)2 = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝜀0
2 + 𝜔3

2�̅�𝑥
2𝜀2

2 + 2𝜔1(𝜔1 − 1)�̅�2𝜀0 − 2𝜔3(𝜔1 − 1)�̅��̅�𝑥𝜀2

− 2𝜔1𝜔3�̅��̅�𝑥𝜀0𝜀2                                                       (4.14) 

Taking Expectation on both sides of (4.14), we get the exact mean square error (MSE) of 𝑡𝑟𝑝1
∗ , as 

𝐸(𝑡𝑟𝑝1
∗ − �̅�)2 = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝐸(𝜀0
2) + 𝜔3

2�̅�𝑥
2𝐸(𝜀2

2) + 2𝜔1(𝜔1 − 1)�̅�2𝐸(𝜀0)

− 2𝜔3(𝜔1 − 1)�̅��̅�𝑥𝐸(𝜀2) − 2𝜔1𝜔3�̅��̅�𝑥𝐸(𝜀0𝜀2) 

𝑀𝑆𝐸(𝑡𝑟𝑝1
∗ ) = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝐵 + 𝜔3
2�̅�𝑥

2𝐷 − 2𝜔1𝜔3�̅��̅�𝑥𝐸                                        (4.15) 

Differentiating equation (4.15) partially w.r.to 𝜔1 & 𝜔3 and equating to zero for obtaining the optimum 

value of 𝜔1 & 𝜔3. The optimum value of 𝜔1 & 𝜔3 which makes the MSE minimum of equation (4.15) 

is given by 

                                                  𝜔1 =
𝐷

𝐷 + 𝐵𝐷 − 𝐸2
= 𝑉11(𝑠𝑎𝑦)                                              (4.16𝑎) 

                                                 𝜔3 = 𝑅∗ (
𝐸

𝐷 + 𝐵𝐷 − 𝐸2
) = 𝑉13(𝑠𝑎𝑦)                                    (4.16𝑏) 

Thus, the resulting minimum MSE of the proposed estimator ‘𝑡𝑟𝑝1
∗ ’ is given by 

           𝑚𝑖𝑛. 𝑀𝑆𝐸 (𝑡𝑟𝑝1
∗ ) = (𝑉11 − 1)2�̅�2 + 𝑉11

2 �̅�2𝐵 + 𝑉13
2 �̅�𝑥

2𝐷 − 2𝑉11𝑉13�̅��̅�𝑥𝐸              (4.17) 

Case 2: If 𝜔3 = 0, the proposed estimator ‘𝑡𝑟𝑝
∗ ’ will be reduced as 

                                                    𝑡𝑟𝑝2
∗ = 𝜔1�̅�∗ + 𝜔2(�̅� − �̅�∗)                                                       (4.18) 

where 𝜔1 & 𝜔2 is a real constant to be determined such that the MSE of 𝑡𝑟𝑝2
∗  is minimum. 

Now, expressing 𝑡𝑟𝑝2
∗  in terms of 𝜀′𝑠 we have 



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𝑡𝑟𝑝2
∗ = 𝜔1�̅�(1 + 𝜀0) + 𝜔2(�̅� − �̅�(1 + 𝜀1)) 

                                                     = 𝜔1�̅�(1 + 𝜀0) − 𝜔2�̅�𝜀1                                                           (4.19) 

                                    𝑡𝑟𝑝2
∗ − �̅� = (𝜔1 − 1)�̅� + 𝜔1�̅�𝜀0 − 𝜔2�̅�𝜀1                                             (4.20) 

Taking Expectation on both sides of (4.20), we get the bias of the estimator 𝑡𝑟𝑝2
∗  as 

𝐸(𝑡𝑟𝑝2
∗ − �̅�) = (𝜔1 − 1)�̅� + 𝜔1�̅�𝐸(𝜀0) − 𝜔2�̅�𝐸(𝜀1)                            

                               𝐵(𝑡𝑟𝑝2
∗ ) = (𝜔1 − 1)�̅�                                                                                       (4.21) 

Squaring of (4.20) on both sides, we have 

(𝑡𝑟𝑝2
∗ − �̅�)2 = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝜀0
2 + 𝜔2

2�̅�2𝜀1
2 + 2𝜔1(𝜔1 − 1)�̅�2𝜀0 − 2𝜔2(𝜔1 − 1)�̅��̅�𝜀1

− 2𝜔1𝜔2�̅��̅�𝜀0𝜀1                                                           (4.22) 

Taking Expectation on both sides of (4.22), we get the exact mean square error (MSE) of 𝑡𝑟𝑝2
∗ , as 

𝐸(𝑡𝑟𝑝2
∗ − �̅�)2 = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝐸(𝜀0
2) + 𝜔2

2�̅�2𝐸(𝜀1
2) + 2𝜔1(𝜔1 − 1)�̅�2𝐸(𝜀0)

− 2𝜔2(𝜔1 − 1)�̅��̅�𝐸(𝜀1) − 2𝜔1𝜔2�̅��̅�𝐸(𝜀0𝜀1)            

             𝑀𝑆𝐸(𝑡𝑟𝑝2
∗ ) = (𝜔1 − 1)2�̅�2 + 𝜔1

2�̅�2𝐵 + 𝜔2
2�̅�2𝐴 − 2𝜔1𝜔2�̅��̅�𝐶                             (4.23) 

Differentiating equation (4.23) partially w.r.to 𝜔1 & 𝜔2 and equating to zero for obtaining the optimum 

value of 𝜔1 & 𝜔2. The optimum value of 𝜔1 & 𝜔2 which makes the MSE minimum of equation (4.23) 

is given by 

                                            𝜔1 = −
𝐴

𝐶2 − 𝐴(1 + 𝐵)
= 𝑉21(𝑠𝑎𝑦)                                             (4.24𝑎) 

                                           𝜔2 = −
𝐶𝑅

𝐶2 − 𝐴(1 + 𝐵)
= 𝑉22(𝑠𝑎𝑦)                                              (4.24𝑏) 

Thus, the resulting minimum MSE of the proposed estimator ‘𝑡𝑟𝑝2
∗ ’ is given by 

           𝑚𝑖𝑛. 𝑀𝑆𝐸 (𝑡𝑟𝑝2
∗ ) = (𝑉21 − 1)2�̅�2 + 𝑉21

2 �̅�2𝐵 + 𝑉22
2 �̅�2𝐴 − 2𝑉21𝑉22�̅��̅�𝐶                (4.25) 

 

 

5.  THEORETICAL EFFICIENCY COMPARISON 

The MSE’s of the existing estimators to the first degree of approximation are derived as: 

 𝑉𝑎𝑟 (�̅�∗) =  �̅�2𝐵                                                                                                                                (5.1) 

𝑀𝑆𝐸(𝑡𝑟
∗) = �̅�2(𝐵 + 𝐴 − 2𝐶)                                                                                                           (5.2) 



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𝑀𝑆𝐸(𝑡𝑘𝑠
∗ ) = �̅�2(𝐴 + 𝐵 + 2𝐶)                                                                                                         (5.3) 

𝑀𝑆𝐸(𝑡𝑒𝑟
∗ ) = �̅�2 (𝐵 +

𝐴

4
− 𝐶)                                                                                                         (5.4) 

𝑀𝑆𝐸(𝑡𝑒𝑝
∗ ) = �̅�2 (𝐵 +

𝐴

4
+ 𝐶)                                                                                                         (5.5) 

𝑀𝑆𝐸(𝑡𝑠𝑠
∗ ) = �̅�2 [𝐵 −

𝐶2

𝐴
]                                                                                                                 (5.6) 

Below is the comparison between proposed estimators with another existing estimator. Efficiency 

condition over some related existing estimators.  

1. 𝑉𝑎𝑟 (�̅�∗) − 𝑀𝑆𝐸(𝑡𝑟𝑝
∗ ) =  �̅�2𝐵 − (𝑉1 − 1)2�̅�2 − 𝑉1

2�̅�2𝐵 − 𝑉2
2�̅�2𝐴 − 𝑉3

2�̅�𝑥
2𝐷 +

                                                2𝑉1𝑉2�̅��̅�𝐶 + 2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

 

2. 𝑀𝑆𝐸(𝑡𝑟
∗) − 𝑀𝑆𝐸 (𝑡𝑟𝑝

∗ ) = �̅�2(𝐵 + 𝐴 − 2𝐶) − (𝑉1 − 1)2�̅�2 − 𝑉1
2�̅�2𝐵 − 𝑉2

2�̅�2𝐴 −

                                                𝑉3
2�̅�𝑥

2𝐷 + 2𝑉1𝑉2�̅��̅�𝐶 + 2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

 

3. 𝑀𝑆𝐸(𝑡𝑘𝑠
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝

∗ ) = �̅�2(𝐵 + 𝐴 + 2𝐶) − (𝑉1 − 1)2�̅�2 − 𝑉1
2�̅�2𝐵 − 𝑉2

2�̅�2𝐴 −

                                                  𝑉3
2�̅�𝑥

2𝐷 + 2𝑉1𝑉2�̅��̅�𝐶 + 2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

 

4. 𝑀𝑆𝐸(𝑡𝑒𝑟
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝

∗ ) = �̅�2 (𝐵 +
𝐴

4
− 𝐶) − (𝑉1 − 1)2�̅�2 − 𝑉1

2�̅�2𝐵 − 𝑉2
2�̅�2𝐴 −

                                               𝑉3
2�̅�𝑥

2𝐷 + 2𝑉1𝑉2�̅��̅�𝐶 + 2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

 

5. 𝑀𝑆𝐸(𝑡𝑒𝑝
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝

∗ ) = �̅�2 (𝐵 +
𝐴

4
+ 𝐶) − (𝑉1 − 1)2�̅�2 − 𝑉1

2�̅�2𝐵 − 𝑉2
2�̅�2𝐴 −

                                                𝑉3
2�̅�𝑥

2𝐷 + 2𝑉1𝑉2�̅��̅�𝐶 + 2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

 

6. 𝑉𝑎𝑟(�̅�∗) − 𝑀𝑆𝐸(𝑡𝑟𝑝1
∗ ) = �̅�2𝐵 − (𝑉11 − 1)2�̅�2 − 𝑉11

2 �̅�2𝐵 − 𝑉13
2 �̅�𝑥

2𝐷 +

                                                2𝑉11𝑉13�̅��̅�𝑥𝐸 ≥ 0 

 

7. 𝑀𝑆𝐸(𝑡𝑟
∗) − 𝑀𝑆𝐸 (𝑡𝑟𝑝1

∗ ) = �̅�2(𝐵 + 𝐴 − 2𝐶) − (𝑉11 − 1)2�̅�2 − 𝑉11
2 �̅�2𝐵 − 𝑉13

2 �̅�𝑥
2𝐷 +

                                                  2𝑉11𝑉13�̅��̅�𝑥𝐸 ≥ 0 

 

8. 𝑀𝑆𝐸(𝑡𝑘𝑠
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝1

∗ ) = �̅�2(𝐵 + 𝐴 + 2𝐶) − (𝑉11 − 1)2�̅�2 − 𝑉11
2 �̅�2𝐵 − 𝑉13

2 �̅�𝑥
2𝐷 +

                                                   2𝑉11𝑉13�̅��̅�𝑥𝐸 ≥ 0 

 

9. 𝑀𝑆𝐸(𝑡𝑒𝑟
∗ ) − 𝑀𝑆𝐸(𝑡𝑟𝑝1

∗ ) = �̅�2 (𝐵 +
𝐴

4
− 𝐶) − (𝑉11 − 1)2�̅�2 − 𝑉11

2 �̅�2𝐵 − 𝑉13
2 �̅�𝑥

2𝐷 +

                                                   2𝑉11𝑉13�̅��̅�𝑥𝐸 ≥ 0 

 

10. 𝑀𝑆𝐸(𝑡𝑒𝑝
∗ ) − 𝑀𝑆𝐸(𝑡𝑟𝑝1

∗ ) = �̅�2 (𝐵 +
𝐴

4
+ 𝐶) − (𝑉11 − 1)2�̅�2 − 𝑉11

2 �̅�2𝐵 − 𝑉13
2 �̅�𝑥

2𝐷 +

                                                   2𝑉11𝑉13�̅��̅�𝑥𝐸 ≥ 0 

 



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11. 𝑉𝑎𝑟(�̅�∗) − 𝑀𝑆𝐸(𝑡𝑟𝑝2
∗ ) = �̅�2𝐵 − (𝑉21 − 1)2�̅�2 − 𝑉21

2 �̅�2𝐵 − 𝑉22
2 �̅�2𝐴 +

                                                2𝑉21𝑉22�̅��̅�𝐶 ≥ 0 

 

12. 𝑀𝑆𝐸(𝑡𝑟
∗) − 𝑀𝑆𝐸 (𝑡𝑟𝑝2

∗ ) = �̅�2(𝐵 + 𝐴 − 2𝐶) − (𝑉21 − 1)2�̅�2 − 𝑉21
2 �̅�2𝐵 − 𝑉22

2 �̅�2𝐴 +

                                                  2𝑉21𝑉22�̅��̅�𝐶 ≥ 0 

 

13. 𝑀𝑆𝐸(𝑡𝑘𝑠
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝2

∗ ) = �̅�2(𝐵 + 𝐴 + 2𝐶) − (𝑉21 − 1)2�̅�2 − 𝑉21
2 �̅�2𝐵 − 𝑉22

2 �̅�2𝐴 +

                                                    2𝑉21𝑉22�̅��̅�𝐶 ≥ 0 

 

14. 𝑀𝑆𝐸(𝑡𝑒𝑟
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝2

∗ ) = �̅�2 (𝐵 +
𝐴

4
− 𝐶) − (𝑉21 − 1)2�̅�2 − 𝑉21

2 �̅�2𝐵 − 𝑉22
2 �̅�2𝐴 +

                                                   2𝑉21𝑉22�̅��̅�𝐶 ≥ 0 

 

15. 𝑀𝑆𝐸(𝑡𝑒𝑝
∗ ) − 𝑀𝑆𝐸 (𝑡𝑟𝑝2

∗ ) = �̅�2 (𝐵 +
𝐴

4
+ 𝐶) − (𝑉21 − 1)2�̅�2 − 𝑉21

2 �̅�2𝐵 − 𝑉22
2 �̅�2𝐴 +

                                                    2𝑉21𝑉22�̅��̅�𝐶 ≥ 0 

 

COMPARISON WITHIN CASES 

Below is the comparison between proposed estimator and its cases: 

a) 𝑀𝑆𝐸 (𝑡𝑟𝑝1
∗ ) − 𝑀𝑆𝐸(𝑡𝑟𝑝

∗ )    = (𝑉11 − 1)2�̅�2 + 𝑉11
2 �̅�2𝐵 + 𝑉13

2 �̅�𝑥
2𝐷 − 2𝑉11𝑉13�̅��̅�𝑥𝐸 −

                                                        𝑉1
2�̅�2𝐵 − 𝑉2

2�̅�2𝐴 − 𝑉3
2�̅�𝑥

2𝐷 + 2𝑉1𝑉2�̅��̅�𝐶 +

                                                        2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

 

b) 𝑀𝑆𝐸 (𝑡𝑟𝑝2
∗ ) − 𝑀𝑆𝐸(𝑡𝑟𝑝

∗ ) = (𝑉21 − 1)2�̅�2 + 𝑉21
2 �̅�2𝐵 + 𝑉22

2 �̅�2𝐴 − 2𝑉21𝑉22�̅��̅�𝐶 −

                                                     (𝑉1 − 1)2�̅�2 − 𝑉1
2�̅�2𝐵 − 𝑉2

2�̅�2𝐴 − 𝑉3
2�̅�𝑥

2𝐷 +

                                                     2𝑉1𝑉2�̅��̅�𝐶 + 2𝑉1𝑉3�̅��̅�𝑥𝐸 − 2𝑉2𝑉3�̅��̅�𝑥𝐹 ≥ 0 

6. NUMERICAL ILLUSTRATION 

To illustrate numerical meaning of the theoretical results, consider a real dataset given in Sample 

Survey by Daroga Singh and F.S. Chaudhary. The description of the dataset is given below: 

A list of 70 villages in India along their population in 1981 and cultivated area (in acres) in the same 

year is considered (Singh and Choudhary, 1986). Here, the cultivated area (in acres) is taken as the 

main study variable and the population of the village is taken as the auxiliary variable. We treat first 

25% values as non-response units. The parameters of the population are as follows (using R Software): 

�̅� = 982.71, �̅� = 1755.53, �̅�𝑥 = 35.5, 𝐶𝑦 = 0.6235, 𝐶𝑥 = 0.8035, 

  𝐶𝑟𝑥
= 0.5732,   𝐶𝑦(2) = 0.3723, 𝐶𝑥(2) = 0.7824, 𝐶𝑟𝑥(2) = 0.5605,   

𝜌𝑦𝑥 = 0.7776,   𝜌𝑦𝑥(2) = 0.8223, 𝜌𝑥𝑟𝑥
= 0.8498, 𝜌𝑥𝑟𝑥(2) = 0.8937,   

𝜌𝑦𝑟𝑥
= 0.7578, 𝜌𝑦𝑟𝑥(2) = 0.9012, 𝑊2 = 0.25, 𝑁 = 70, 𝑛 = 17 

We computed the percent-relative efficiency (PRE’s) of various existing estimators with respect to the 

usual unbiased estimator �̅�∗ for different values of 𝑘, by using the formulae 



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                                                            𝑃𝑅𝐸(𝑡∗, �̅�∗) =
𝑉𝑎𝑟 (�̅�∗)

𝑀𝑆𝐸 (∗)
× 100                                                     

where 𝑡∗ = 𝑡𝑟
∗, 𝑡𝑘𝑠

∗ , 𝑡𝑒𝑟
∗ , 𝑡𝑒𝑝

∗ , 𝑡𝑠𝑠
∗ , 𝑡𝑟𝑝

∗ , 𝑡𝑟𝑝1
∗ , 𝑡𝑟𝑝2

∗  

Table 1: Percent-relative efficiency (PRE) of the various estimators 

𝑷𝑹𝑬(𝒕∗, �̅�∗) (𝟏/𝒌) 

(𝟏/𝟐) (𝟏/𝟑) (𝟏/𝟒) (𝟏/𝟓) 

𝑃𝑅𝐸(𝑡𝑟
∗, �̅�∗) 123.077 105.797 95.35 89.32 

𝑃𝑅𝐸(𝑡𝑘𝑠
∗ , �̅�∗) 21.62 21.92 22.16 22.60 

𝑃𝑅𝐸(𝑡𝑒𝑟
∗ , �̅�∗) 204.8 182.50 169.95 162.12 

𝑃𝑅𝐸(𝑡𝑒𝑝
∗ , �̅�∗) 41.83 42.44 43.16 44.072 

𝑃𝑅𝐸(𝑡𝑠𝑠
∗ , �̅�∗) 214.32 183.73 173.602 162.65 

𝑷𝑹𝑬(𝒕𝒓𝒑
∗ , �̅�∗) 277.093 281.785 288.356 295.5968 

𝑷𝑹𝑬(𝒕𝒓𝒑𝟏
∗ , �̅�∗) 245.202 255.389 265.3895 275.097 

𝑷𝑹𝑬(𝒕𝒓𝒑𝟐
∗ , �̅�∗) 242.714 248.095 249.239 251.117 

 

7. CONCLUSION 

In this paper, we have suggested an estimator of the finite population mean that use ranks of the 

auxiliary variable. Based on both theoretical and numerical findings, it turns out that the proposed 

estimator 𝑡𝑟𝑝
∗  is more efficient than the usual mean, ratio, product, exponential-ratio and exponential-

product estimators. Thus, the suggested estimator 𝑡𝑟𝑝
∗  is to be recommended for efficiently estimating 

the finite population mean. 

CONFLICT OF INTEREST 

The authors declare that they have no conflicts of interest to report regarding the present study. 

ACKNOWLEDGEMENT 

The authors are grateful to the experienced referees for their useful comments and suggestions. 

ORCID ID 

Udita Gupta  https://orcid.org/0009-0000-1754-1418 

 

 

https://orcid.org/0009-0000-1754-1418


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APPENDIX 

install.packages("xlsx") 

install.packages("readxl") 

library(xlsx) 

library(readxl) 

mydata<-read_excel("C:/Users/DELL/OneDrive/Desktop/daroga singh.xlsx") 

head(mydata) 

Rx<-rank(mydata$X) 

mydata11<-cbind(mydata$Y,mydata$X,Rx) 

mydata1<-as.data.frame(mydata11) 

head(mydata1) 

colnames(mydata1)<-c("Y","X","Rx") 

write.csv(mydata1,"mydata1.csv") 

getwd() 

N<-70 

N1<-0.25*N 

nonres<-mydata1[1:N1,] 

res<-mydata1[N1+1:N,] 

#======Population======= 

My<-mean(mydata1$Y) 

Mx<-mean(mydata1$X) 

Mr<-mean(mydata1$Rx) 

Vy<-var(mydata1$Y) 

Vx<-var(mydata1$X) 

Vr<-var(mydata1$Rx) 

Cyx<-cor(mydata1$X,mydata1$Y) 

Cxr<-cor(mydata1$X,mydata1$Rx) 

Cyr<-cor(mydata1$Y,mydata1$Rx) 



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#======Non-response======= 

my2<-mean(nonres$Y) 

mx2<-mean(nonres$X) 

mr2<-mean(nonres$Rx) 

Vy2<-var(nonres$Y) 

Vx2<-var(nonres$X) 

Vr2<-var(nonres$Rx) 

Cyx2<-cor(nonres$X,nonres$Y) 

Cxr2<-cor(nonres$X,nonres$Rx) 

Cyr2<-cor(nonres$Y,nonres$Rx) 

Cy2<-(Vy/(My^2)) 

Cx2<-(Vx/(Mx^2)) 

Cr2<-(Vr/(Mr^2)) 

Cy22<-(Vy2/(My^2)) 

Cx22<-(Vx2/(Mx^2)) 

Cr22<-(Vr2/(Mr^2)) 

#========k=2/3/4/5======= 

k<-2/3/4/5 

N<-70 

N2<-0.25*N 

n<-17 

f<-n/N 

A11<-(1-f)/n 

W2<-N2/N 

A12<-W2*(k-1)/n 

A<-(A11*Cx2)+(A12*Cx22) 

B<-(A11*Cy2)+(A12*Cy22) 

C<-(A11*(sqrt(Cy2))*(sqrt(Cx2))*Cyx)+(A12*(sqrt(Cy22))*(sqrt(Cx22))*Cyx2) 

D<-(A11*Cr2)+(A12*Cr22)   

E<-(A11*(sqrt(Cy2))*(sqrt(Cr2))*Cyr)+(A12*(sqrt(Cy22))*(sqrt(Cr22))*Cyr2) 

F<-(A11*(sqrt(Cx2))*(sqrt(Cr2))*Cxr)+(A12*(sqrt(Cx22))*(sqrt(Cr22))*Cxr2) 

Rat<-My/Mx 

a11<-(Mr/(Mx*(Rat^2)))*((A*D)-(F^2)) 

a12<-(Mr/(Mx*Rat))*((C*D)-(E*F)) 



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a13<-(1/Rat)*((A*E)-(C*F)) 

V01<-(Mr/(Mx*(Rat^2))) 

V021<-((1+B)*((A*D)-(F^2))) 

V022<-(C*((E*F)-(C*D))) 

V023<-(E*((A*E)-(C*F))) 

V02<-V021+V022-V023 

V0<-V01*V02 

W1<-a11/V0 

W2<-a12/V0 

W3<-a13/V0 

Mt1<-(((W1-1)^2)* (My^2)) + ((W1^2)*(My^2)*B) + ((W2^2)*(Mx^2)*A) + ((W3^2)*(Mr^2)*D) - 

(2*W1*W2*My*Mx*C) -(2*W1*W3*My*Mr*E) + (2*W2*W3*Mx*Mr*F) 

Vay<-((Y^2)*B) 

PRE<-((Vay*100)/Mt1) 

W11<-(D/(D+(B*D)-(E^2))) 

W13<-((Mx*Rat)/Mr)*(E/(D+(B*D)-(E^2))) 

Mt2<-(((W11-1)^2)*(My^2))+((W11^2)*(My^2)*B)+((W13^2)*(Mr^2)*D)-

(2*W11*W13*My*Mr*E) 

PRE2<-((Vay*100)/Mt2) 

V1<-(1/Rat)*((C^2)-(A*(1+B))) 

W31<-(-A/(Rat*V1)) 

W32<-(-C/V1) 

Mt3<-(((W31-1)^2)*(My^2))+((W31^2)*(My^2)*B)+((W32^2)*(Mx^2)*A)-

(2*W31*W32*My*Mx*C) 

PRE3<-((Vay*100)/Mt3)