# Assessing the Effect of Nonresponse and Measurement Error Using a Novel Class of Efficient Estimators

Authors: Kuldeep Kumar Tiwari; Sandeep Bhougal; Sunil Kumar; Ronald Onyango
Journal: Journal of Mathematics
 (2022)

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## Abstract

The purpose of any statistical analysis is to uncover the trend using real situations and provide an accurate model for future policies. In order to fulfill this aim, it is important to collect and use the data carefully and minimize the possibility of various types of errors. Nonresponse (NR) and measurement error (ME) are the two major types of nonsampling error that occur in almost every survey. We propose an estimator of the population mean and study its efficacy in the combined and separate effects of NR and ME. Some existing estimators are also considered for comparison. A simulation study was performed to see its efficiency numerically.


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## Body

## 1. Introduction

We make inferences about the population by using samples. There is, therefore, a possibility of error that occurs due to the limited nature of sampling, which inevitably leaves out a major part of the population from its representation. This error is called a sampling error. An error that is different from the sampling error is called a nonsampling error. Sampling error tends to decrease when the sample size increases, but nonsampling error increases when the sample size increases. The two main factors that contribute to nonsampling error are nonresponse (NR) and measurement error (ME). The unavailability of data due to NR is a common problem in the survey. At present, most of the surveys are performed online (through e-mail, social media, etc.). Online surveys are cost-effective, but the chances of NR are also high. The likelihood of NR is at its minimum when the sample is collected by directly interviewing the respondents. Hansen and Hurwitz [1] present a method to deal with NR. We first send the questionnaire to selected units in the sample through online mode, and subsequently take a subsample from the group of nonrespondents and conduct a direct interview. Cochran [2] applies the method of Hansen and Hurwitz [1] to ratio and regression estimator. Moreover, the input through the auxiliary information is expected to increase the efficiency of the estimators. Khare and Srivastava [3] study ratio estimator when complete information is available about the auxiliary variable and NR is present only on the study variable. Okafor and Lee [4] consider the ratio and regression estimator under NR in the double sampling. Some other conspicuous works on the estimation under NR are Singh and Kumar [5], Kreuter et al. [6], Unal and Kadilar [7], Bii et al. [8], and Pandey et al. [9].Another nonsampling error is measurement error (ME), which is recognized as a difference between the observed value and the true value. There are many sources that can lead to ME, such as equipment error, error in data entry, bias in the survey questionnaire, biased processing, false information, and respondent error. Hansen et al. [10] present a study on ME in the census and show that ignoring ME would generate unreliable results. Chandhok [11] points out various perspectives on ME in his thesis. Fuller [12] presents various models for ME. Shalabh [13] studies the classical ratio estimator in the presence of ME. Some more works on ME are by Shukla et al. [14], Shalabh and Tsai [15], and Kumar et al. [16].In the majority of the work, the study is either on NR or ME. But in real practice, both appear together, usually. Azeem [17] works on the simultaneous presence of NR and ME. In his thesis, he studies various well-known estimators and also suggests some new estimators under NR and ME. Further, Kumar et al. [18] propose a transformation-based exponential type estimator of the population mean in the presence of NR and ME. Some recent contributions on the topic are Singh and Sharma [19], Azeem and Hanif [20], Kumar, Trehan, and Joorel [21], Singh, Bhattacharyya, and Bandyopadhyay [22], and Audu et al. [23].Continuing the work, we propose an estimator of the population mean and study it in situations such as when there is no presence of NR and ME or it is negligible, when there is only NR, when there is only ME, and when there is both NR and ME together. It will be interesting to see the results of all four cases together and the comparison of their effects.


## 2. Sampling Strategy and Notations

Let∪ be a finite population of size N. Let ∪s be a sample of the size n taken from ∪ using simple random sampling without replacement (SRSWOR) scheme. Let y and x be the study and the auxiliary variables, respectively. With this initial information, we use the following sampling strategies.Some usual notations arey¯, x¯ be the sample means and Y¯, X¯ be the population means of the study variable y and the auxiliary variable x, respectively. Sy2 and Sx2 be the population variances for y and x. ρ is correlation coefficient between y and x. λ=1/n−1/N, R=Y¯/X¯.


### 2.1. Sampling StrategyΩ

First, we consider the default sampling strategy. That is, there is no existence of NR and ME or it is minute enough to be ignored. Letyi and xi be the observed values of y and x at ith;i=1,2,…,n units in the sample. We denote this sampling strategy by Ω. AΩ=λSy2, BΩ=λSx2, and CΩ=λρSySx.


### 2.2. Sampling StrategyΩ°

Let measurement error (ME) be present in the sample. That is, the sample is selected, recorded, or processed with some unknown ME. Letu be the ME on y and v be the ME on x. We use ° to mention the presence of ME. So, in the presence of ME, the ith observed unit for study variable y and auxiliary variable x in the sample is y°i and x°i, i=1,2,…,n. y°i=Yi+ui and x°i=Xi+vi, where Yi and Xi be the true values of y and x at ith unit in the sample and ui and vi be the corresponding ME. Since ME present on y and x are independent of each other, so u and v are uncorrelated. Also, we assume that the average of ui and vi are zero as the ME caused by both under-reporting and over-reporting. Su2 and Sv2 be the population variance for u and v, respectively. We denote this sampling strategy by Ω°. AΩ°=λSy2+Su2, BΩ°=λSx2+Sv2, CΩ°=λρSySx.


### 2.3. Sampling StrategyΩ∗

Let there is a nonresponse (NR). LetN1 and N2 be the group of respondents and nonrespondents in the population such that N1+N2=N and n1 and 2 be the group of respondent and nonrespondent in the sample such that n1+n2=n. We use Hansen and Hurwitz’s [1] technique to handle the problem of NR. In this method, we take a sample of n units using SRSWOR and mail a questionnaire. Among n units, n1 units respond to the questionnaire while from n2=n−n1 nonresponding units, we select a subsample of size r=n2/k;k≥1 and arrange a direct interview to them by assuming they will respond this time. With this method, Hansen and Hurwitz [1] redefined the sample mean per unit estimator in the presence of NR as y¯∗=w1y¯n1+w2y¯r; where w1=n1/n, w2=n2/n, y¯n1=1/n1∑i=1n1yi, and y¯r=1/r∑i=1ryi. The variance of y¯∗ is vary¯∗=λSy2+θSy22; where θ=W2k−1/n, W2=N2/N. Sy22 is population variance for y for the group of nonrespondent units.Similarly,x¯∗=w1x¯n1+w2x¯r; where x¯n1=1/n1∑i=1n1xi and x¯r=1/r∑i=1rxi. The variance of x¯∗ is varx¯∗=λSx2+θSx22; Sx22 is population variance of x for the group of nonrespondent units. We denote this sampling strategy by Ω∗. AΩ∗=λSy2+θSy22, BΩ∗=λSx2+θSx22, and CΩ∗=λρSySx+θρ2Sy2Sx2; ρ2 is the correlation between x and y for the group of nonrespondents.


### 2.4. Sampling StrategyΩ⊗

Let there is both NR and ME. That is, both the sampling strategiesΩ° and Ω∗ are combined. We use ⊗ to mention the presence of NR and ME. In the presence of NR and ME, the ith observed unit for the study variable y and the auxiliary variable x in the sample is yi⊗ and xi⊗, i=1,2,..,n, respectively. yi⊗=Yi+ui∗ and xi⊗=Xi+vi∗, where Yi and Xi be the true values of y and x at ith unit in the sample and ui∗ and vi∗ be the corresponding ME. y¯⊗=y¯∗, x¯⊗=x¯∗, vary¯⊗=λSy2+Su2+θSy22+Su22, and varx¯⊗=λSx2+Sv2+θSx22+Sv22. Su22 and Sv22 be population variances of u and v for the group of nonrespondents. We denote this sampling strategy by Ω⊗. AΩ⊗=λSy2+Su2+θSy22+Su22, BΩ⊗=λSx2+Sv2+θSx22+Sv22, and CΩ⊗=λρSySx+θρ2Sy2Sx2.


## 2.1. Sampling StrategyΩ

First, we consider the default sampling strategy. That is, there is no existence of NR and ME or it is minute enough to be ignored. Letyi and xi be the observed values of y and x at ith;i=1,2,…,n units in the sample. We denote this sampling strategy by Ω. AΩ=λSy2, BΩ=λSx2, and CΩ=λρSySx.


## 2.2. Sampling StrategyΩ°

Let measurement error (ME) be present in the sample. That is, the sample is selected, recorded, or processed with some unknown ME. Letu be the ME on y and v be the ME on x. We use ° to mention the presence of ME. So, in the presence of ME, the ith observed unit for study variable y and auxiliary variable x in the sample is y°i and x°i, i=1,2,…,n. y°i=Yi+ui and x°i=Xi+vi, where Yi and Xi be the true values of y and x at ith unit in the sample and ui and vi be the corresponding ME. Since ME present on y and x are independent of each other, so u and v are uncorrelated. Also, we assume that the average of ui and vi are zero as the ME caused by both under-reporting and over-reporting. Su2 and Sv2 be the population variance for u and v, respectively. We denote this sampling strategy by Ω°. AΩ°=λSy2+Su2, BΩ°=λSx2+Sv2, CΩ°=λρSySx.


## 2.3. Sampling StrategyΩ∗

Let there is a nonresponse (NR). LetN1 and N2 be the group of respondents and nonrespondents in the population such that N1+N2=N and n1 and 2 be the group of respondent and nonrespondent in the sample such that n1+n2=n. We use Hansen and Hurwitz’s [1] technique to handle the problem of NR. In this method, we take a sample of n units using SRSWOR and mail a questionnaire. Among n units, n1 units respond to the questionnaire while from n2=n−n1 nonresponding units, we select a subsample of size r=n2/k;k≥1 and arrange a direct interview to them by assuming they will respond this time. With this method, Hansen and Hurwitz [1] redefined the sample mean per unit estimator in the presence of NR as y¯∗=w1y¯n1+w2y¯r; where w1=n1/n, w2=n2/n, y¯n1=1/n1∑i=1n1yi, and y¯r=1/r∑i=1ryi. The variance of y¯∗ is vary¯∗=λSy2+θSy22; where θ=W2k−1/n, W2=N2/N. Sy22 is population variance for y for the group of nonrespondent units.Similarly,x¯∗=w1x¯n1+w2x¯r; where x¯n1=1/n1∑i=1n1xi and x¯r=1/r∑i=1rxi. The variance of x¯∗ is varx¯∗=λSx2+θSx22; Sx22 is population variance of x for the group of nonrespondent units. We denote this sampling strategy by Ω∗. AΩ∗=λSy2+θSy22, BΩ∗=λSx2+θSx22, and CΩ∗=λρSySx+θρ2Sy2Sx2; ρ2 is the correlation between x and y for the group of nonrespondents.


## 2.4. Sampling StrategyΩ⊗

Let there is both NR and ME. That is, both the sampling strategiesΩ° and Ω∗ are combined. We use ⊗ to mention the presence of NR and ME. In the presence of NR and ME, the ith observed unit for the study variable y and the auxiliary variable x in the sample is yi⊗ and xi⊗, i=1,2,..,n, respectively. yi⊗=Yi+ui∗ and xi⊗=Xi+vi∗, where Yi and Xi be the true values of y and x at ith unit in the sample and ui∗ and vi∗ be the corresponding ME. y¯⊗=y¯∗, x¯⊗=x¯∗, vary¯⊗=λSy2+Su2+θSy22+Su22, and varx¯⊗=λSx2+Sv2+θSx22+Sv22. Su22 and Sv22 be population variances of u and v for the group of nonrespondents. We denote this sampling strategy by Ω⊗. AΩ⊗=λSy2+Su2+θSy22+Su22, BΩ⊗=λSx2+Sv2+θSx22+Sv22, and CΩ⊗=λρSySx+θρ2Sy2Sx2.


## 3. Some Existing Estimators

The usual unbiased estimatorty¯=y¯ is the most basic estimator of the population mean. Hansen and Hurwitz [1] modify it for situation of NR. Grover and Kaur [24] propose an estimator under sampling strategy Ω as tgk=k1y¯+k2X¯−x¯expX¯−x¯/X¯+x¯, where k1,k2 are suitable constants. They study various properties of the estimator and show that this estimator performs better than the classical ratio, product and regression estimator, Bahl and Tuteja’s [25] exponential estimator, and some more. To increase the efficiency of the mean estimators, Ekpenyong and Enang [26] suggest an estimator under sampling strategy Ω as tee=d1y¯+d2X¯−x¯expX¯−x¯/X¯+x¯, where d1,d2 are suitable constants. This estimator proves its worth over Gupta and Shabbir [27] and Singh and Solanki [28] estimators. Singh and Pal [29] propose an exponential ratio type estimator as tsp=y¯X¯/x¯expX¯−x¯/X¯+x¯. Study shows that tsp performs efficiently than the usual unbiased estimator, classical ratio estimator, Kadilar and Cingi [30], and Bahl and Tuteja [25] estimator. Gupta and Yadav [31] propose an estimator as tgy=y¯k+1−kX¯+n/x¯+n, where k is a suitable constant. For the optimum value of k, tgy perform similar to regression estimator.These estimators are defined under sampling strategyΩ. We can redefine them in the sampling strategy Ωi,i=°,∗,and⊗ as(1)ty¯=y¯i,tgk=k1y¯i+k2X¯−x¯iexpX¯−x¯iX¯+x¯i,tee=d1y¯i+d2X¯−x¯iexpX¯−x¯iX¯+x¯i,tsp=y¯iX¯x¯iexpX¯−x¯iX¯+x¯i,tgy=y¯ik+1−kX¯+nx¯i+n,where i=°,∗,and⊗.The mean squared error (MSE) of the estimatorsty¯, tgk, tee, tsp, and tsp up to the first order of approximation under the sampling strategy j; j=Ω,Ω°,Ω∗,Ω⊗ are(2)MSEty¯=Aj,(3)MSEtgk=Y¯2−4Y¯4Bj−R2BjCj2+R2Bj2Aj+1/16R4Bj34Y¯2Bj+4AjBj−4Cj2,(4)MSEtee=Y¯2−4Y¯4Bj−R2Y¯2Bj2+R2Bj2Aj+4RY¯2BjCj4Y¯2Bj+4AjBj−RBj−2Cj2,(5)MSEtsp=Aj+94R2Bj−3RCj,(6)MSEtgy=Aj−Cj2B.


## 4. Proposed Class of Estimators and Their MSE

We propose the following class of estimators to estimate the population mean of the study variabley under sampling strategy Ω.(7)tδ,η=αy¯+βX¯−x¯δX¯+ηδx¯+ηexp3X¯−x¯X¯+x¯,where α, β, δ, and η are either constants or known parameters of the auxiliary variable.Under sampling strategyΩi,i=°,∗,⊗, the proposed estimator tδ,η is presented as(8)tδ,η=αy¯i+βX¯−x¯iδX¯+ηδx¯i+ηexp3X¯−x¯iX¯+x¯i.The MSE of the proposed estimator in various defined sampling strategies is presented in the following theorem.Theorem 1.
The bias and minimum MSE oftδ,η up to the first order of approximation for the optimum values of α and β under the sampling strategy j; j=Ω,Ω°,Ω∗,Ω⊗ is(9)Biastδ,η=αopt−1Y¯−αoptY¯R1+32RCj+βoptY¯R1+32R+αoptY¯R12+32R1R+158R2Bj,MSEmintδ,η=Y¯2−M1j2M4j+M2jM3j2−M1jM3jM5j4M2jM4j−M5j2,where αopt=M3jM5j−2M1jM4j/4M2jM4j−M5j2, βopt=M1jM5j−2M2jM3j/4M2jM4j−M5j2; M1j=R1+3/2RCj−2R12+15/4R2+3RR1Bj−2Y¯2, M2j=Y¯2+Aj−3R1+9/2RCj+3R12+6R2+6RR1Bj, M3j=−2R1+3RBj, M4j=Bj, M5j=4R1+6RBj−Cj; and R1=δY¯/δX¯+η.Proof.
The proof is provided in the appendix section.


## 5. Theoretical Comparison

IfY¯^1 and Y¯^2 are the estimators of Y¯, then Y¯^1 is said to be more efficient than Y¯^2 whenever MSEY¯^1<MSEY¯^2.Here, we have to find the conditions under which the proposed estimator is more efficient than the existing estimators. Under the sampling strategyj; j=Ω,Ω°,Ω∗,Ω⊗, the proposed estimator tδ,η will be more efficient than ty¯, tgk, tee, tsp, and tgy whenever the following respective condition satisfies.(1)
MSEmintδ,η<MSEty¯, when(10)Y¯2<Aj+M1j2M4j+M2jM3j2−M1jM3jM5j4M2jM4j−M5j2.(2)
MSEmintδ,η<MSEtgk, when(11)4Y¯4Bj−R2BjCj2+R2Bj2Aj+1/16R4Bj34Y¯2Bj+4AjBj−4Cj2<M1j2M4j+M2jM3j2−M1jM3jM5j4M2jM4j−M5j2.(3)
MSEmintδ,η<MSEtee, when(12)4Y¯4Bj−R2Y¯2Bj2+R2Bj2Aj+4RY¯2BjCj4Y¯2Bj+4AjBj−RBj−2Cj2<M1j2M4j+M2jM3j2−M1jM3jM5j4M2jM4j−M5j2.(4)
MSEmintδ,η<MSEtsp, when(13)Y¯2+3RCj<Aj+94R2Bj+M1j2M4j+M2jM3j2−M1jM3jM5j4M2jM4j−M5j2.(5)
MSEmintδ,η<MSEtgy, when(14)Y¯2+Cj2Bj<Aj+M1j2M4j+M2jM3j2−M1jM3jM5j4M2jM4j−M5j2.


## 6. Simulation Study

To study the performance of the estimators numerically, we have generated an artificial population using R software. The bivariate normal population is derived by input dataN=200, the population mean vector μ=X¯/Y¯, and the variance-covariance matrix ∑ as(15)μ=15080,∑=8006506501100.When required, we employ a MEu on study variable y and v on auxiliary variable x. u and v have been taken as normal distribution with mean zero and standard deviation 5. For the situation of NR, we assume there is 20 percent nonresponse, i.e., W2=0.2 and among the nonrespondents units in the sample, 40 percent are selected in the subsample for callback, i.e., k=2.5. The MSE of the estimators are calculated using equations (1)–(5) and (9). MSEs are calculated for three different samples of sizes n=40, n=80, and n=120. We have repeated the process 10000 times to get stable result. Here, we are presenting the proposed estimator for limited values of δ,η. We have taken δ,η=a,a,a,2a,a,3a,a,4a,2a,a,2a,3a,3a,a,3a,2a,3a,4a,4a,a,4a,3a, where a≠0 can be any real number. For δ,η=a,a, the member estimator will be ta,a. For any real number a≠0, the estimator ta,a will perform similar. That is, t1,1, t2,2, t3,3 etc., will give same result. Similarly, t1,2, t2,4, t3,6 etc., will give same result as ta,2a for any real number a≠0. Infinitely many combinations of δ,η can be made that will give efficient results.The percent relative efficiency (PRE) of the estimatorti is calculated with respect to ty¯ using the following equation:(16)PREti=MSEty¯MSEti×100;i=y¯,gk,ee,sp,gy,δ,η.


### 6.1. Interpretations from the Table

From Tables1–3, we can conclude that(1)
The MSEs of the proposed estimators are sufficiently smaller than the MSEs of the existing estimators in every situation. This proves that the proposed class of estimators is more efficient than the existing estimators.(2)
MSEs of all the estimators increase when we approach from sampling strategyΩ⟶Ω°⟶Ω∗⟶Ω⊗.(3)
All the estimators have a minimum MSE under the sampling strategyΩ and a maximum MSE under the sampling strategy Ω⊗. That is, MSE is the minimum when there is no presence of NR and ME, and MSE is the maximum when there is a presence of both NR and ME.(4)
The MSEs in the sampling strategyΩ⊗ is significantly larger than the MSEs in the sampling strategy Ω. This shows that ignoring NR and ME can lead to unreliable results.(5)
MSEs of all the estimators decrease as sample size increases in all the sampling strategies. Forn=40 in Table 1 all the estimators have the highest MSE, while in Table 3, it is the lowest and MSEs for n=80 lie in between them.(6)
It is interesting to see that the MSEs of the proposed estimators in Table1 are almost similar to or better than the MSEs of existing estimators in Table 3. That is, the proposed estimator with sample size n=40 performs almost similar or better than the existing estimators with sample size n=120. Hence, the proposed estimator is more efficient as well as cost-effective than the existing estimators, as the proposed estimators give better results with a small sample than the existing estimators with a large sample.Table 1
MSE and PRE of the estimators forn=40.
EstimatorΩΩ°Ω∗Ω⊗MSEPREMSEPREMSEPREMSEPREty¯22.007 2100.0022.506 8100.0030.276 3100.0030.961 1100.00tgk11.111 0198.0611.937 1188.5415.278 4198.1616.410 2188.66tee11.098 9198.2811.924 3188.7415.255 5198.4616.386 0188.94tsp11.458 2192.0612.278 7183.2915.744 0192.3016.870 8183.51tgy11.133 7197.6611.963 0188.1315.321 2197.6116.459 1188.10ta,a2.008 761095.52.591 42868.512.722 141112.23.521 34879.24ta,2a2.054 271071.22.636 92853.522.784 961087.13.584 15863.83ta,3a2.099 181048.32.681 83839.232.846 951063.43.646 12849.15ta,4a2.143 501026.62.726 14825.592.908 121041.03.707 29835.14t2a,a1.985 781108.22.568 44876.282.690 421125.33.489 62887.23t2a,3a2.031 591083.22.614 25860.922.753 651099.43.552 85871.44t3a,a1.978 081112.52.560 75878.912.679 791129.83.479 00889.94t3a,2a1.993 451103.92.576 12873.672.701 011120.93.500 22884.54t3a,4a2.024 001087.32.606 66863.432.743 171103.63.542 37874.02t4a,a1.974 231114.72.556 89880.232.674 471132.03.473 68891.30t4a,3a1.997 291101.82.579 95872.372.706 311118.73.505 51883.21Table 2
MSE and PRE of the estimators forn=80.
EstimatorΩΩ°Ω∗Ω⊗MSEPREMSEPREMSEPREMSEPREty¯8.262 03100.008.449 56100.0012.388 5100.0012.669 8100.00tgk4.234 79195.094.543 51185.966.349 45195.116.812 02185.99tee4.233 07195.174.541 70186.046.345 60195.236.807 95183.38tsp4.296 90192.274.604 51183.506.447 62192.146.908 83186.10tgy4.237 98194.954.547 16185.826.356 62194.896.820 22185.76ta,a0.764 791080.20.982 82859.721.150 291076.91.477 39857.58ta,2a0.781 831056.70.999 86845.061.175 861053.51.502 95842.99t3a,a0.798 651034.41.016 69831.081.201 091031.41.528 18829.08ta,4a0.815 261013.41.033 29817.731.225 991010.41.553 08815.78t2a,a0.756 181092.50.974 21867.311.137 381089.21.464 48865.14t2a,3a0.773 341068.30.991 37852.301.163 121065.11.490 21850.20t3a,a0.753 301096.70.971 33869.891.133 061093.31.460 15867.70t3a,2a0.759 061088.40.977 09864.761.141 701085.01.468 79862.60t3a,4a0.770 501072.20.988 53854.751.158 851069.01.485 95852.64t4a,a0.751 861098.80.969 89871.181.130 891095.41.457 99868.99t4a,3a0.760 491086.30.978 52863.491.143 851083.01.470 94861.34Table 3
MSE and PRE of the estimators forn=120.
EstimatorΩΩ°Ω∗Ω⊗MSEPREMSEPREMSEPREMSEPREty¯3.669 60100.003.752 98100.006.398 66100.006.544 65100.00tgk1.887 28194.432.024 58185.373.288 57194.573.528 52185.47tee1.886 94194.472.024 22185.403.287 54194.633.527 44185.53tsp1.905 80192.542.042 60183.733.328 84192.213.568 17183.41tgy1.887 91194.372.025 30185.303.290 48194.453.530 71185.36ta,a0.339 451081.00.436 35860.070.592 051080.70.761 66859.25ta,2a0.347 021057.40.443 92845.400.605 261057.0.774 86844.61ta,3a0.354 501035.10.451 39831.410.618 291034.80.787 89830.64ta,4a0.361 871014.00.458 77818.050.631 151013.70.800 76817.30t2a,a0.335 631093.30.432 53867.670.585 391093.00.754 99866.84t2a,3a0.343 251069.00.440 15852.650.598 681068.70.768 28851.85t3a,a0.334 351097.50.431 25870.240.583 151097.20.752 76869.41t3a,2a0.336 911089.10.433 81865.110.587 611088.90.757 22864.29t3a,4a0.341 991073.00.438 89855.100.596 481072.70.766 08854.30t4a,a0.333 711099.60.430 61871.540.582 031099.30.751 64870.71t4a,3a0.337 551087.10.434 44863.850.588 731086.80.758 3863.03


## 6.1. Interpretations from the Table

From Tables1–3, we can conclude that(1)
The MSEs of the proposed estimators are sufficiently smaller than the MSEs of the existing estimators in every situation. This proves that the proposed class of estimators is more efficient than the existing estimators.(2)
MSEs of all the estimators increase when we approach from sampling strategyΩ⟶Ω°⟶Ω∗⟶Ω⊗.(3)
All the estimators have a minimum MSE under the sampling strategyΩ and a maximum MSE under the sampling strategy Ω⊗. That is, MSE is the minimum when there is no presence of NR and ME, and MSE is the maximum when there is a presence of both NR and ME.(4)
The MSEs in the sampling strategyΩ⊗ is significantly larger than the MSEs in the sampling strategy Ω. This shows that ignoring NR and ME can lead to unreliable results.(5)
MSEs of all the estimators decrease as sample size increases in all the sampling strategies. Forn=40 in Table 1 all the estimators have the highest MSE, while in Table 3, it is the lowest and MSEs for n=80 lie in between them.(6)
It is interesting to see that the MSEs of the proposed estimators in Table1 are almost similar to or better than the MSEs of existing estimators in Table 3. That is, the proposed estimator with sample size n=40 performs almost similar or better than the existing estimators with sample size n=120. Hence, the proposed estimator is more efficient as well as cost-effective than the existing estimators, as the proposed estimators give better results with a small sample than the existing estimators with a large sample.Table 1
MSE and PRE of the estimators forn=40.
EstimatorΩΩ°Ω∗Ω⊗MSEPREMSEPREMSEPREMSEPREty¯22.007 2100.0022.506 8100.0030.276 3100.0030.961 1100.00tgk11.111 0198.0611.937 1188.5415.278 4198.1616.410 2188.66tee11.098 9198.2811.924 3188.7415.255 5198.4616.386 0188.94tsp11.458 2192.0612.278 7183.2915.744 0192.3016.870 8183.51tgy11.133 7197.6611.963 0188.1315.321 2197.6116.459 1188.10ta,a2.008 761095.52.591 42868.512.722 141112.23.521 34879.24ta,2a2.054 271071.22.636 92853.522.784 961087.13.584 15863.83ta,3a2.099 181048.32.681 83839.232.846 951063.43.646 12849.15ta,4a2.143 501026.62.726 14825.592.908 121041.03.707 29835.14t2a,a1.985 781108.22.568 44876.282.690 421125.33.489 62887.23t2a,3a2.031 591083.22.614 25860.922.753 651099.43.552 85871.44t3a,a1.978 081112.52.560 75878.912.679 791129.83.479 00889.94t3a,2a1.993 451103.92.576 12873.672.701 011120.93.500 22884.54t3a,4a2.024 001087.32.606 66863.432.743 171103.63.542 37874.02t4a,a1.974 231114.72.556 89880.232.674 471132.03.473 68891.30t4a,3a1.997 291101.82.579 95872.372.706 311118.73.505 51883.21Table 2
MSE and PRE of the estimators forn=80.
EstimatorΩΩ°Ω∗Ω⊗MSEPREMSEPREMSEPREMSEPREty¯8.262 03100.008.449 56100.0012.388 5100.0012.669 8100.00tgk4.234 79195.094.543 51185.966.349 45195.116.812 02185.99tee4.233 07195.174.541 70186.046.345 60195.236.807 95183.38tsp4.296 90192.274.604 51183.506.447 62192.146.908 83186.10tgy4.237 98194.954.547 16185.826.356 62194.896.820 22185.76ta,a0.764 791080.20.982 82859.721.150 291076.91.477 39857.58ta,2a0.781 831056.70.999 86845.061.175 861053.51.502 95842.99t3a,a0.798 651034.41.016 69831.081.201 091031.41.528 18829.08ta,4a0.815 261013.41.033 29817.731.225 991010.41.553 08815.78t2a,a0.756 181092.50.974 21867.311.137 381089.21.464 48865.14t2a,3a0.773 341068.30.991 37852.301.163 121065.11.490 21850.20t3a,a0.753 301096.70.971 33869.891.133 061093.31.460 15867.70t3a,2a0.759 061088.40.977 09864.761.141 701085.01.468 79862.60t3a,4a0.770 501072.20.988 53854.751.158 851069.01.485 95852.64t4a,a0.751 861098.80.969 89871.181.130 891095.41.457 99868.99t4a,3a0.760 491086.30.978 52863.491.143 851083.01.470 94861.34Table 3
MSE and PRE of the estimators forn=120.
EstimatorΩΩ°Ω∗Ω⊗MSEPREMSEPREMSEPREMSEPREty¯3.669 60100.003.752 98100.006.398 66100.006.544 65100.00tgk1.887 28194.432.024 58185.373.288 57194.573.528 52185.47tee1.886 94194.472.024 22185.403.287 54194.633.527 44185.53tsp1.905 80192.542.042 60183.733.328 84192.213.568 17183.41tgy1.887 91194.372.025 30185.303.290 48194.453.530 71185.36ta,a0.339 451081.00.436 35860.070.592 051080.70.761 66859.25ta,2a0.347 021057.40.443 92845.400.605 261057.0.774 86844.61ta,3a0.354 501035.10.451 39831.410.618 291034.80.787 89830.64ta,4a0.361 871014.00.458 77818.050.631 151013.70.800 76817.30t2a,a0.335 631093.30.432 53867.670.585 391093.00.754 99866.84t2a,3a0.343 251069.00.440 15852.650.598 681068.70.768 28851.85t3a,a0.334 351097.50.431 25870.240.583 151097.20.752 76869.41t3a,2a0.336 911089.10.433 81865.110.587 611088.90.757 22864.29t3a,4a0.341 991073.00.438 89855.100.596 481072.70.766 08854.30t4a,a0.333 711099.60.430 61871.540.582 031099.30.751 64870.71t4a,3a0.337 551087.10.434 44863.850.588 731086.80.758 3863.03


## 7. Conclusion

The present article discusses the problem of estimation to reduce the effect of nonsampling error. An estimator of the population mean is proposed, and its properties are discussed. The proposed estimator is studied for the separate and combined effects of NR and ME. A simulation study shows that the proposed estimator is far better than the existing estimators of Grover and Kaur [24], Ekpenyong and Enang [26], Singh and Pal [29], and Gupta and Yadav [31]. The study also shows that the presence of NR and ME had a huge impact on the result by contributing to MSE. And so, ignoring the presence of NR and ME can lead to an adverse conclusion.



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*Source: 4946265-2022-04-15.xml*