Impact of Correlated Measurement Errors on Some Efficient Classes of Estimators

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Introduction
In survey research, the primary objective of any surveyor is to enhance the efciency of the estimation procedures with the help of information on the auxiliary/supplementary variables that are usually associated with the research variable.In this context, the literature contains the ratio, regression, product, exponential methods, and their modifed forms for efciently estimating the parameters of interest.Tese estimation methods are further extended using twoor multiauxiliary information under diferent sampling schemes.Tese estimation methods either consist of a supplementary variable or only on research variable, presupposing that all data are independent from ME, but this presupposition practically never happen.Te data are tainted with or have hidden ME due to diferent types of reasons (readers can refer to Murthy [1] and Cochran [2]).Te discrepancy between the true and observed values is known as ME.Many attempts have been made to examine the impact of ME on various parameters of the population such as the mean, variance, total, and distribution function.Te efect of ME has been observed over the efciency of the estimation methods by many authors.Shalabh [3] examined the ME's impact on the classical ratio estimators.Infuenced by Shalabh [3], Manisha and Singh [4] studied the ME's impact using a new class of estimators for the population mean.Subsequently, Singh and Karpe [5][6][7] examined the efect of ME over the parameters of the population using diferent sampling strategies.Te variance computation in the existence of ME was provided by Diana and Giordan [8].Hussain et al. [9] suggested the estimation of a fnite population distribution function with the dual use of supplementary information under nonresponse.Tariq et al. [10] proposed a supplementary information-based variance estimator to tackle the problem of ME.Tariq et al. [11] suggested a generalized variance estimator utilizing supplementary information in the presence and absence of ME.Zahid et al. [12] developed a generalized class of estimators for the sensitive variables in the case of ME and nonresponse.Ahmad et al. [13] discussed the estimation of the fnite population mean using a dual supplementary variable for nonresponse under SRS.Tiwari et al. [14] suggested a novel class of efcient estimators to assess the impact of nonresponse and ME.Tiwari and Sharma [15] developed an efcient estimation procedure of the population mean under the joint infuence of nonresponse and ME.Bhushan et al. [16] suggested some novel logarithmic estimators for the population mean in the presence of ME.Tese studies have all taken into account the possibility of uncorrelated measurement errors (UMEs), which can be found in both the supplementary variable and research variable.Te subject of our attention is diferent from the research described above.We assume that both the study and supplementary variables have access to ME.It may be incorrect to assume that both variables are independent of ME because often the same surveyor collects data on the supplementary variable and research variable.Tis correlation in ME can exist caused by the ulterior inherent propensity of the data.Te signifcance of CME was initially studied by Shalabh and Tsai [17] using the ratio and product estimators of the population mean utilizing the SRS framework.Recently, Bhushan et al. [18,19] assessed the efectiveness of some new classes of estimators based on the CME.
In the present article, we study the CME's impact on efcient classes of population mean estimators under SRS.Te subsequent material is divided into a few sections.Section 2 devotes to the follow-up of the literature related to the CME.Section 3 presents the recommended classes of estimators with their characteristics and conducts a comparative analysis with the class of estimators that is already in use.To support the theoretically obtained results, a numerical and simulation studies along with the important fndings are given in Section 4 and Section 5, respectively.Section 6 of the research provides conclusions.

Follow-Up of the Literature
It is assumed that F � (F 1 , F 2 , . . ., F N ) is a fnite population of size N units from which a sample of size n is chosen using simple random sampling with replacement (SRSWR).Let x i and y i be the true amount of research variable and supplementary variable for i th units of the population F, respectively.Tese amounts are unavailable, while they may easily be quantifed as (y i , x i ) having the ME's (u i , v i ) in the i th unit of the variables, respectively.Suppose we can write x i and y i as y i � Y i + u i and x i � X i + v i , i � 1, 2, . . ., n. MEs u i and v i are also unobservable such that E(u) � E(v) � 0, V(u) � σ 2 u , and V(v) � σ 2 v , respectively, and the correlation coefcient between u and v is Cor(u, v) � ρ uv .Let (y, x), (μ y , μ x ), (σ 2 y , σ 2 x ), and (C y , C x ) be the sample means, population means, population variances, and population coefcient of variations of the research variable and supplementary variable, respectively.Let ρ xy be the correlation coefcient between the supplementary variable and the research variable.
For calculating the mean square error (MSE) of several estimators in the situation of ME, we employ the notations discussed as , and . Te mean per unit estimator under CME is defned as follows: It is found that Shalabh and Tsai [17] advocated the usual ratio and product estimators in the presence of CME as given by Te estimators t r and t p have the following MSE: where

Proffered Estimators under CME and Their Properties
Te objective of this article is to provide some efcient estimates of the population mean and to study the efect of CME on some prominent classes of estimators.To the best of our knowledge, the impact of the CME has not been attempted so far over any class of estimators.We propose to study the following efcient classes of estimators of the population mean under CME as follows: 2 Journal of Mathematics where β 1 , β 2 , and β 3 are duly selected scalars to be determined.
Considering the notations described in Section 2, we write the estimator T 1 as follows: To derive the MSE (T 1 ), the square and expectation are taken on both sides of (9): By minimizing (10) in relation to β 1 , we obtain the optimum value of β 1 as follows: Replacing the value of β 1 with β 1(opt) in (10), we obtain the minimum MSE of the estimator T 1 as follows: Again, considering the notations described in Section 2, we write the estimator T 2 as follows: To derive the MSE (T 2 ), the square and expectation are considered on both sides of (13): By minimizing (14) in relation to β 2 , we obtain the optimum value of β 2 as follows: Replacing the value of β 2 with β 2(opt) in ( 14), we obtain the minimum MSE of the estimator T 2 as follows: Now, we write the estimator T 3 by using the notations described in Section 2 as follows: To derive the MSE (T 3 ), we square and take the expectation on both sides of (17): By minimizing (18) in relation to β 3 , we obtain the optimum value of β 3 as follows: Replacing the value of β 3 with β 3(opt) in (18), we obtain the minimum MSE of the estimator T 3 as follows: It is to be noted that the minimum MSE expressions of the proposed estimator T i , i � 1, 2, 3 are same.

Numerical Study
Tis section presents a numerical study using two real populations which are discussed as follows: Population 1: origin: Te book of U.S. Census Bureau (1986) X i � actual size of farm, x i � quantifed size of farm, Y i � real sale price of the item, and y i � quantifed value of the goods sold Population 2: origin: Gujarati and Sangeetha [20] X i � actual income, x i � quantifed income, Y i � actual consumption costs, and y i � quantifed consumption cost Te parameters of these populations are given in Table 1.Te percent relative efciency (PRE) is calculated by utilizing the following expression: where T * � t m , t p , t r , T 1 , T 2 , and T 3 .
Te numerical results (PREs) are given in Table 2 that demonstrate the outperformance of the proposed estimators against the existing estimators in each population.Te results in Table 2 also demonstrate that the proposed estimators T i , i � 1, 2, 3 perform equally in each population.Moreover, these results are further generalized by simulation.
Step 3: calculate the required statistics.
Step 4: using 10000 replications, calculate the PRE of several estimators regarding the usual mean estimator t m utilizing the expression given as follows:   3-10.Moreover, we have also calculated the PRE of estimators for several values of measurement errors such as 10%, 15%, 20%, and 25% and presented the results in Tables 11-18.

Main Findings.
Te computed results of the PRE for the profered estimators are presented in Tables 3-18.Te comparative studies of several estimators are presented in terms of the PRE in Tables 3-10 for several parameters σ 2 x , σ 2 y , σ 2 u , σ 2 v , ρ xy , and ρ uv .Te comparisons of diferent estimators are also presented in terms of the PRE in  for diferent amounts of ME.Te important results of the profered estimators are as follows: (1) From Table 3, concerning to the values of σ 2 y � 1, σ 2 x � 1 along with the positive correlation ρ xy � 0, 0.5, 0.9, σ 2 u � 1, and σ 2 v � 1, we can observe that (i) As the value of ρ xy rises, the percent relative efciency of the traditional ratio estimator t r also rises.In addition, the values of ρ uv afect the rate and size of this growth.Tis can also be observed from Figures 1-3.(ii) As ρ xy fuctuates between 0 and 0.9, the percent relative efciency of the traditional product estimator t p drops.Te percent relative efciency likewise reduces when ρ uv fuctuates from −0.9 to +0.9.Tis can easily be observed from Figures 1-3.(iii) When the values of ρ xy rises between 0 and 0.9, the percent relative efciency of the proposed estimators T 1 , T 2 , and T 3 rises.Te size and rate of this rise both rely on the sign and value of ρ uv , and they both drop as ρ uv values decline from 0.9 to −0.9.(iv) Te primary infuence of the correlated measurement errors over the percent relative efciency of the suggested estimators may be seen by comparing the percent relative efciency of the estimators at ρ uv � 0 and ρ uv � ± 0.9.Tis efect can easily be observed from Figures 1-3.(v) Te percent relative efciency of the suggested estimators T i , i � 1, 2, 3 is higher for positively correlated measurement errors, and the percent relative efciency decreases as the valuation of ρ uv varies from −0.9 to 0 and increases and the values of ρ uv vary from 0 to 0.9.Tis efect can easily be observed from Figures 1-3.(vi) For various combinations of σ 2 u and σ 2 v , the same trend can be seen in the percent relative efciency values of the profered estimators.
Journal of Mathematics errors are negative, and it gets less as ρ uv values increase incrementally from −0.9 to +0.9.Tis tendency can easily be observed from Figures 4-6.(iv) Te considerable infuence of the correlated measurement errors over the percent relative efciency of the suggested estimators can be seen by comparing the percent relative efciency of the estimators at ρ uv � 0 and ρ uv � ± 0.9.Tis tendency can easily be observed from Figures 4-6.(v) For diferent combinations of σ 2 u and σ 2 v , the same trend in the percent relative efciency values of the profered estimators may be seen.( 4) Te same trend in the fuctuation of percent relative efciency that is shown in Table 4 can also be seen in Tables 6, 8, and 10, consisting of diferent values of σ 2 x , σ 2 y , and negative correlation coefcient ρ xy .Te graphs for the same can be provided, if required.
(5) From Table 11, concerning to the values of σ 2 y � 1, σ 2 x � 1, the positive correlation coefcient ρ xy � 0, 0.5, 0.9, and for the level of ME � 10%, we can observe that (i) As ρ xy fuctuates from 0 to 0.9, the percent relative efciency of the traditional ratio estimator t r grows.In addition, when the value of ρ uv rises from −0.9 to +0.9, the percent relative efciency rises as well, which can also be observed from Figures 7-9.(ii) As the value of ρ xy changes between 0 and 0.9, the percent relative efciency of the traditional product estimator t p declines.Te percent relative efciency declines as well when the value of ρ uv increases incrementally from −0.9 to +0.9, which can also be observed from Figures 7-9.(iii) Te percent relative efciency of the profered estimators T i , i � 1, 2, 3 rises as ρ xy 's value changes from 0 to 0.9.Te direction and value of ρ uv also afect the size and rate of this decline.Te percent relative efciency is greater for correlated measurement errors that are negative, and it reduces as ρ uv varies from −0.9 to 0 and rises as it fuctuates from 0 to 0.9.Tis efect can easily be observed from Figures 7-9.(iv) Te considerable infuence of the correlated measurement errors over the percent relative efciency of the recommended estimators can be shown by comparing the percent relative efciency of the suggested estimators at ρ uv � 0 and ρ uv � ± 0.9.Tis pattern can easily be observed from Figures 7-9.(v) For additional levels of ME, namely, at 15%, 20%, and 25%, an analogous pattern in the percent relative efciency of the profered estimators may be seen.
(vi) As the amount of ME rises, the percent relative efciency of several estimators drops for ρ uv � 0. (vii) At the smallest amount of ME, the percent relative efciency of the ratio estimator t r is larger, and it drops as the amount of ME changes from 10% to 25% over sequential increments of 5%.However, for the product estimator t p , when the amount of ME rises, the percent relative efciency rises for ρ uv � −0.9, −0.5 and falls for ρ uv � 0, 0.5, 0.9.In addition, for the proposed estimators, the percent relative efciency rises at the upper side of ρ uv and falls for the remaining values of ρ uv as the amount of ME rises.
(6) Te same trend in the fuctuation in the percent relative efciency that is shown in Table 11 can also be seen from Tables 13, 15, and 17 for diferent combinations of σ 2 x , σ 2 y , and positive correlation coefcient ρ xy .Te graphs for the same can be provided, if required.(7) From Table 12, consisting of the values of σ 2 y � 1, σ 2 x � 1 along with negative correlation ρ xy � −0.9, −0.5, −0.1, and the level of ME as 10%, we observe that (8) Te same trend in the fuctuation in percent relative efciency that is shown in Table 12 can also be seen from Tables 14, 16, and 18 consisting of diferent amounts of σ 2 y , σ 2 x , and negative correlation coefcient ρ xy .Te graphs for the same can be provided, if required.(9) Furthermore, Tables 3-18 rely on several values of σ 2 y , σ 2 x , σ 2 u , σ 2 v , ρ xy , ρ uv , and the percentage of ME, and the percent relative efciency of the suggested estimators surpasses the percent relative efciency of the usual mean estimator t m , classical ratio, and product estimators t r and t p , respectively.

Conclusion
Tis article has introduced few efcient classes of estimators of the population mean in the existence of correlated measurement errors under simple random sampling.Te mean square error of the suggested classes of estimators is obtained with the approximation of order one.Te theoretical comparison of the suggested estimators and the existing estimators has been performed.Subsequently, numerical and simulation studies have been conducted to substantiate the theoretical fndings.Te efects of correlated measurement errors on the performance of the suggested estimators have also been looked at, and the percent relative efciency has been provided in Tables 2-18.Te numerical results given in Table 2 show that the proposed estimators perform better than the traditional estimators with higher percent relative efciency in each real population.Moreover, the simulation fndings reported in Tables 3-10 exhibit that 26 Journal of Mathematics the percent relative efciency of the profered estimators T i , i � 1, 2, 3 in case of correlated measurement errors for several values of σ 2 y , σ 2 x , σ 2 u , and σ 2 v increases as the value of ρ xy moves between 0 and 0.9 and declines as the value of ρ xy moves between −0.9 and −0.1 for the sequential increment of 0.4.Te sign and magnitude of ρ uv have an impact on the percent relative efciency as well.Likewise, a pattern in the percent relative efciency of the profered estimators can be seen from  that are concerned to the various percentage of ME, namely, 10, 15, 20, and 25.In the cases of uncorrelated and correlated measurement errors, the percent relative efciency of the suggested estimators is signifcantly diferent.Furthermore, the suggested estimators surpass the traditional estimators for various values of σ 2 y , σ 2 x , σ 2 u , σ 2 v , ρ xy , ρ uv , and various amounts of measurement errors.Since the proposed classes of estimators perform superior than their counterparts, therefore, they are enthusiastically recommended to the surveyors for computing the population mean under correlated as well as uncorrelated measurement errors.

Table 2 :
Numerical results of several estimators.