Abstract
It is well-known that the appearance of measurement errors spoils the traditional results in survey sampling. The concept of correlated measurement errors (CMEs) is true in various practical situations, but this has been seldom considered by researchers in survey sampling. In this article, the influence of the CME under simple random sampling (SRS) has been considered over some prominent classes of estimators for the population mean. The first-order approximated formulae of the mean square error of the introduced estimators are reported, and a comparative analysis has also been conducted with traditional estimators. The theoretical findings are extended by a broad spectrum computational study using real and artificial data.
1. Introduction
In survey research, the primary objective of any surveyor is to enhance the efficiency 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 modified forms for efficiently estimating the parameters of interest. These estimation methods are further extended using two- or multiauxiliary information under different sampling schemes. These 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. The data are tainted with or have hidden ME due to different types of reasons (readers can refer to Murthy [1] and Cochran [2]). The 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. The effect of ME has been observed over the efficiency of the estimation methods by many authors. Shalabh [3] examined the ME’s impact on the classical ratio estimators. Influenced 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 [5–7] examined the effect of ME over the parameters of the population using different sampling strategies. The variance computation in the existence of ME was provided by Diana and Giordan [8]. Hussain et al. [9] suggested the estimation of a finite 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 finite population mean using a dual supplementary variable for nonresponse under SRS. Tiwari et al. [14] suggested a novel class of efficient estimators to assess the impact of nonresponse and ME. Tiwari and Sharma [15] developed an efficient estimation procedure of the population mean under the joint influence of nonresponse and ME. Bhushan et al. [16] suggested some novel logarithmic estimators for the population mean in the presence of ME. These 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. The subject of our attention is different 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. This correlation in ME can exist caused by the ulterior inherent propensity of the data. The significance 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 effectiveness of some new classes of estimators based on the CME.
In the present article, we study the CME’s impact on efficient classes of population mean estimators under SRS. The 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 findings are given in Section 4 and Section 5, respectively. Section 6 of the research provides conclusions.
2. Follow-Up of the Literature
It is assumed that is a finite population of size units from which a sample of size is chosen using simple random sampling with replacement (SRSWR). Let and be the true amount of research variable and supplementary variable for units of the population , respectively. These amounts are unavailable, while they may easily be quantified as (, ) having the ME’s (, ) in the unit of the variables, respectively. Suppose we can write and as and , . MEs and are also unobservable such that , , and , respectively, and the correlation coefficient between and is . Let , (, ), (, ), and (, ) be the sample means, population means, population variances, and population coefficient of variations of the research variable and supplementary variable, respectively. Let be the correlation coefficient 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 . It is very appealing to observe that and .
The mean per unit estimator under CME is defined 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
The estimators and have the following :where
3. Proffered Estimators under CME and Their Properties
The objective of this article is to provide some efficient estimates of the population mean and to study the effect 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 efficient classes of estimators of the population mean under CME as follows:where , , and are duly selected scalars to be determined.
Considering the notations described in Section 2, we write the estimator as follows:
To derive the MSE , the square and expectation are taken on both sides of (9):
By minimizing (10) in relation to , we obtain the optimum value of as follows:
Replacing the value of with in (10), we obtain the minimum MSE of the estimator as follows:
Again, considering the notations described in Section 2, we write the estimator as follows:
To derive the MSE , the square and expectation are considered on both sides of (13):
By minimizing (14) in relation to , we obtain the optimum value of as follows:
Replacing the value of with in (14), we obtain the minimum MSE of the estimator as follows:
Now, we write the estimator by using the notations described in Section 2 as follows:
To derive the MSE , we square and take the expectation on both sides of (17):
By minimizing (18) in relation to , we obtain the optimum value of as follows:
Replacing the value of with in (18), we obtain the minimum MSE of the estimator as follows:
It is to be noted that the minimum MSE expressions of the proposed estimator are same.
Furthermore, we present the theoretical comparisons of the proffered estimators with other existing estimators in the presence of CME and obtain the following conditions:(i)Comparing (3) with (12), (16), and (20), we get(ii)Comparing (5) with (12), (16), and (20), we get(iii)Comparing (6) with (12), (16), and (20), we get
The suggested estimators will surpass the existing estimators under the above conditions.
4. Numerical Study
This section presents a numerical study using two real populations which are discussed as follows: Population 1: origin: The book of U.S. Census Bureau (1986) = actual size of farm, = quantified size of farm, = real sale price of the item, and = quantified value of the goods sold Population 2: origin: Gujarati and Sangeetha [20] = actual income, = quantified income, = actual consumption costs, and = quantified consumption cost
The parameters of these populations are given in Table 1.
The percent relative efficiency (PRE) is calculated by utilizing the following expression:where , , , and .
The numerical results (PREs) are given in Table 2 that demonstrate the outperformance of the proposed estimators against the existing estimators in each population. The results in Table 2 also demonstrate that the proposed estimators perform equally in each population. Moreover, these results are further generalized by simulation.
5. Simulation
To assess the cogency of the theoretical results in practice, we consider an extensive simulation study over an artificially drawn population of size from the normal distribution with population parameters = 30, = 20, = 1 and 3, = 1 and 3, = −0.9, −0.5, −0.1, 0, 0.5, and 0.9, = 1 and 3, = 1 and 3, and = −0.9, −0.5, 0, 0.5, and 0.9. The steps used in the simulation study are as follows: Step 1: draw the data hypothetically from R software with , , , and utilizing a 4-variate multivariate normal distribution as with the mean vector and the covariance matrix: Step 2: quantify a sample of size from the above population utilizing SRSWR. Step 3: calculate the required statistics. Step 4: using 10000 replications, calculate the PRE of several estimators regarding the usual mean estimator utilizing the expression given as follows:
Subsequently, we have computed the PREs of the estimators for several values of parameters , , , , , and . The simulation results are presented in Tables 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.
5.1. Main Findings
The computed results of the PRE for the proffered estimators are presented in Tables 3–18. The comparative studies of several estimators are presented in terms of the PRE in Tables 3–10 for several parameters , , , , , and . The comparisons of different estimators are also presented in terms of the PRE in Tables 11–18 for different amounts of ME. The important results of the proffered estimators are as follows:(1)From Table 3, concerning to the values of , along with the positive correlation , , and , we can observe that(i)As the value of rises, the percent relative efficiency of the traditional ratio estimator also rises. In addition, the values of affect the rate and size of this growth. This can also be observed from Figures 1–3.(ii)As fluctuates between 0 and 0.9, the percent relative efficiency of the traditional product estimator drops. The percent relative efficiency likewise reduces when fluctuates from −0.9 to +0.9. This can easily be observed from Figures 1–3.(iii)When the values of rises between 0 and 0.9, the percent relative efficiency of the proposed estimators , , and rises. The size and rate of this rise both rely on the sign and value of , and they both drop as values decline from 0.9 to −0.9.(iv)The primary influence of the correlated measurement errors over the percent relative efficiency of the suggested estimators may be seen by comparing the percent relative efficiency of the estimators at and . This effect can easily be observed from Figures 1–3.(v)The percent relative efficiency of the suggested estimators is higher for positively correlated measurement errors, and the percent relative efficiency decreases as the valuation of varies from −0.9 to 0 and increases and the values of vary from 0 to 0.9. This effect can easily be observed from Figures 1–3.(vi)For various combinations of and , the same trend can be seen in the percent relative efficiency values of the proffered estimators.(2)The same pattern in the percent relative efficiency like Table 3 can also be observed from Tables 5, 7, and 9 consisting of different combinations of and and the positive correlation coefficient . The graphs for the same can be provided, if required.(3)From Table 4, concerning to the values of , with negative correlation , and , we can observe that(i)As fluctuates from −0.1 to −0.9 over sequential decrements of 0.4, the percent relative efficiency of the traditional ratio estimator declines. The percent relative efficiency also drops as the value of is successively decreased from +0.9 to −0.9. This can easily be observed from Figures 4–6.(ii)As fluctuates from −0.1 to −0.9 across sequential decrements of 0.4, the percent relative efficiency of the traditional product estimator rises. The percent relative efficiency also rises when the value of is successively decreased from +0.9 to −0.9. This can easily be observed from Figures 4–6.(iii)As fluctuates from −0.9 to −0.1 over the sequential increase of 0.4, the percent relative efficiency of the suggested estimators declines. The direction and value of also affect the size and rate of this decline. It is greater when the correlated measurement errors are negative, and it gets less as values increase incrementally from −0.9 to +0.9. This tendency can easily be observed from Figures 4–6.(iv)The considerable influence of the correlated measurement errors over the percent relative efficiency of the suggested estimators can be seen by comparing the percent relative efficiency of the estimators at and . This tendency can easily be observed from Figures 4–6.(v)For different combinations of and , the same trend in the percent relative efficiency values of the proffered estimators may be seen.(4)The same trend in the fluctuation of percent relative efficiency that is shown in Table 4 can also be seen in Tables 6, 8, and 10, consisting of different values of , , and negative correlation coefficient . The graphs for the same can be provided, if required.(5)From Table 11, concerning to the values of , , the positive correlation coefficient , and for the level of ME = , we can observe that(i)As fluctuates from 0 to 0.9, the percent relative efficiency of the traditional ratio estimator grows. In addition, when the value of rises from −0.9 to +0.9, the percent relative efficiency rises as well, which can also be observed from Figures 7–9.(ii)As the value of changes between 0 and 0.9, the percent relative efficiency of the traditional product estimator declines. The percent relative efficiency declines as well when the value of increases incrementally from −0.9 to +0.9, which can also be observed from Figures 7–9.(iii)The percent relative efficiency of the proffered estimators rises as ’s value changes from 0 to 0.9. The direction and value of also affect the size and rate of this decline. The percent relative efficiency is greater for correlated measurement errors that are negative, and it reduces as varies from −0.9 to 0 and rises as it fluctuates from 0 to 0.9. This effect can easily be observed from Figures 7–9.(iv)The considerable influence of the correlated measurement errors over the percent relative efficiency of the recommended estimators can be shown by comparing the percent relative efficiency of the suggested estimators at and . This 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 efficiency of the proffered estimators may be seen.(vi)As the amount of ME rises, the percent relative efficiency of several estimators drops for .(vii)At the smallest amount of ME, the percent relative efficiency of the ratio estimator 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 , when the amount of ME rises, the percent relative efficiency rises for and falls for . In addition, for the proposed estimators, the percent relative efficiency rises at the upper side of and falls for the remaining values of as the amount of ME rises.(6)The same trend in the fluctuation in the percent relative efficiency that is shown in Table 11 can also be seen from Tables 13, 15, and 17 for different combinations of , , and positive correlation coefficient . The graphs for the same can be provided, if required.(7)From Table 12, consisting of the values of , along with negative correlation , and the level of ME as , we observe that(8)The same trend in the fluctuation in percent relative efficiency that is shown in Table 12 can also be seen from Tables 14, 16, and 18 consisting of different amounts of , , and negative correlation coefficient . The graphs for the same can be provided, if required.(9)Furthermore, Tables 3–18 rely on several values of , , , , , , and the percentage of ME, and the percent relative efficiency of the suggested estimators surpasses the percent relative efficiency of the usual mean estimator , classical ratio, and product estimators and , respectively.
6. Conclusion
This article has introduced few efficient classes of estimators of the population mean in the existence of correlated measurement errors under simple random sampling. The mean square error of the suggested classes of estimators is obtained with the approximation of order one. The 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 findings. The effects of correlated measurement errors on the performance of the suggested estimators have also been looked at, and the percent relative efficiency has been provided in Tables 2–18. The numerical results given in Table 2 show that the proposed estimators perform better than the traditional estimators with higher percent relative efficiency in each real population. Moreover, the simulation findings reported in Tables 3–10 exhibit that the percent relative efficiency of the proffered estimators in case of correlated measurement errors for several values of , , , and increases as the value of moves between 0 and 0.9 and declines as the value of moves between −0.9 and −0.1 for the sequential increment of 0.4. The sign and magnitude of have an impact on the percent relative efficiency as well. Likewise, a pattern in the percent relative efficiency of the proffered estimators can be seen from Tables 11–18 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 efficiency of the suggested estimators is significantly different. Furthermore, the suggested estimators surpass the traditional estimators for various values of , , , , , , 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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia, for funding this research work through the project no. IFKSUOR3-216-3.