Finite Population Mean Estimation Under Systematic Sampling Scheme inPresence ofMaximumandMinimumValuesUsingTwo Auxiliary Variables

Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan Foundation Foundation University Medical College, Foundation University School of Health Sciences Islamabad, DHA-I Islamabad 44000, Pakistan Department of Statistics, University of Peshawar, Peshawar, Pakistan Department of Statistics, University of Wah, Wah Cantt, Pakistan Department of Biology, Yazd University, P. O. Box 89175-741, Yazd, Iran


Introduction
It is common practise in sampling surveys to employ auxiliary data during the estimation stage to improve the precision of population parameter estimates. In this context, several ratios, products, and regression estimators are appropriate examples. In the literature, a larger number of consistent estimators using auxiliary information for estimating the nite population mean or a total of the study variable, as well as their properties in simple random sampling, have been addressed, for example, see [1,2] and the references cited therein. For estimating the population characteristics, there are some populations, such as forest areas for estimating timber volume and areas under various forms of cover [3], where simple random sampling or other sampling schemes are di cult to apply. In such a setting, systematic sampling produces precise results for selecting a sample from a population. e advantages of systematic sampling include the ability to select the entire sample with just one random start.
Also sampling theory, appropriate use of the auxiliary information may increase the precision of the estimators. But unfortunately, many real data sets contain values that are suddenly maximum or minimum. As a result, if any unexpected values are chosen in the sample, the estimator may generate misleading results. To handle such situations, we proposed mean estimation under a systematic sampling design in the presence of maximum and minimum values.
Assume that the population units N are sequentially numbered from 1 to N. We choose a unit at random from the first k units and every k th unit to create a sample of n units. As an example, if k is 12 and the first unit drawn is number 10, the following units are numbers 22, 34, 44, and so on. e whole number is determined by the first unit chosen. A simple random sample appears to be less exact than systematic sampling. It divides the population into n strata, each of which includes the first k units, second k units, and so on.References [4,5] discovered that systematic sampling is efficient and convenient in sampling. It gives estimators that are more efficient than those offered by basic random sampling under certain actual situations, in addition to its simplicity, which is very important in large-scale sampling work. e ratio and product estimators for estimating the finite population mean Y of the study variable y were built by [6,7]. References [7][8][9][10][11][12][13][14][15] and Javaid et al. [16], all go into great length about systematic sampling.

Notation and Symbols
Consider the study variable Y and the auxiliary variables X, Z for a finite population with U � U 1 , U 2 . . . , U N units. We choose a systemic sample of size n, starting with a random selection of the first unit, and subsequently selecting every kth unit after each interval of k. We'll use the formula N � nk, with n and k being positive integers. For selected systematic random sample, say y ij , x ij , where i � 1, 2, . . . n, j � 1, 2, . . . , k, are the values of j th unit in the i th selected sample for Y, X, and Z variables correspondingly. For y, x, and z, the sample mean in systematic random sampling is those are unbiased estimators for population means for the variables Y,X, and Z, respectively. e bias and mean squared error were calculated, let us define where ρ * y � 1 +(n − 1)ρ y , e coefficients of y, x, and z are C y , C x , and C z , respectively, and are interclass correlation in the systematic sample for the research variable y as well as the auxiliary variables x and z, is the correlation between the study variable y and auxiliary variables (x, z). Many genuine data sets have unexpectedly large or tiny (y max ) or (y min ) values. When such values appear in the estimation of a finite population mean, the results are vulnerable. e findings will be either inflated or underestimated if y max and y min exist. To deal with such a situation, [17] suggested the following unbiased estimator for the estimation of finite population mean using maximum and minimum values: for all other samples.
e variance of Y s , is calculated as follows: the population variance is S 2 y , and the constant is c. c has a minimum value of e variance of Y s , is calculated as follows: this is always less than the variance of Y. Under a systematic sampling approach, the usual ratio estimator is e bias and mean squared error of Y Rsys are provided by the following up to the first order of approximation: and where, R 1 � Y/X and R 2 � Y/ZUnder systematic sampling, the product estimator is as follows: e bias and MSE of Y P(sys) are provided by the following up to the first order of approximation: and Under a systematic sampling procedure, the standard regression estimator for predicting the unknown population mean is as follows: e sample regression coefficients are b 1 and b 2 , respectively. If b 1 and b 2 are the least square estimators of B 1 and B 2 , respectively, then up to the first order of approximation, the variance of the estimator Y lr(sys) is as follows:

Suggested Estimators
We provide a ratio, product, and regression type estimator using auxiliary variables and the study variable in a systematic sampling method, based on [18]. We also take into account the study's minimum and maximum values, as well as the two auxiliary variables.

First Situation.
When the study variable and the auxiliary variable have a positive connection, a bigger value of the auxiliary variable should be chosen, as should a greater value of the study variable. A smaller study variable value should also be chosen, as well as a smaller value of the auxiliary variable. To make use of these type of data, using auxiliary variables and the research variable, we recommend using a ratio type estimator. or for all the rest the samples e estimator of regression type is: where, If the samples contain y max and (x max , z max ), then Y c 11 � Y, X c 21 � X, Z c 31 � Z, for all other samples.

Second Situation.
When the study variable and the auxiliary variable have a negative correlation, choose the auxiliary variable with the bigger value and the study variable with the smallest value. e smaller value of the auxiliary variable is chosen, while the larger value of the study variable is chosen. e proposed product type estimator using the auxiliary variables (x, z) with the study variable y is given by for all other samples.
e estimator of regression type is as follows: where If the samples contain y min and (x max , z max ), and also .If the samples contain y max and (x min , z min ) and for all types of samples. Also c 1 , c 2 , and c 3 are unknown constants. e following relative error terms and their expectations are used to generate biases and mean squared errors.
such that Expressing (19) in terms of e ′ s, we have Expressing (27) we have the following results up to the first order of approximation:

(28)
Taking both sides of (28) into consideration, we have Mathematical Problems in Engineering 5 We have after squaring (28) and taking expectations Differentiate (30) we have, c 1 , c 2 , and c 3 We derive the minimum MSE by substituting the optimum values of c 1 , c 2 , and c 3 in (30) MSE of (Y R(sys) ), given by where Similarly Bias Y P suggested(sys) � λ and where In the case of positive correlation, the minimum MSE of the regression estimator is provided by where e population regression coefficients are β 1 and β 2 . Similarly, in the case of negative correlation, the minimal MSE of Y lr suggested(sys) is In the case of both positive and negative correlation between the study and the auxiliary variable, a general form for MSE is (40)

Comparison of Estimators
In this part, we use a systematic sampling approach to compare the suggested estimators against classic ratios, products, and regression estimators.

Empirical Study
For numerical comparisons, we utilize two distinct data sets: Population 1: (Source: [19]). y � Imports of merchandise in millions.
x � In billions, the gross national GDP.   From the results of MSEs, which are available in Table 1, the mean squared errors of the suggested estimators are lower than the existing estimators. We can also see from Table 2, that PREs of all suggested estimators are surpassing all the existing estimators. Of all the suggested estimators, the regression estimator has the best performance.

Conclusion
We proposed several standard ratios, products, and regression estimators using auxiliary variables in a systematic sampling method in the presence of maximum and minimum values. Under some conditions, the proposed estimators are more efficient than traditional mean ratios, products, and regression estimators. Table 1 shows that the suggested estimators outperform the standard estimators in both populations. e numerical study also supports the superiority of our proposed estimators. It is found that the new suggested estimators of the finite population mean are more precise than some of the existing estimators.

Data Availability
e data used to support the numerical findings of this study are available from the corresponding author upon request. e data can also be obtained upon searching the given sources of data.

Conflicts of Interest
e authors declare that they have no conflicts of interest.