# Mean Estimators Using Robust Quantile Regression and L-Moments’ Characteristics for Complete and Partial Auxiliary Information

Authors: Malik Muhammad Anas; Zhensheng Huang; David Anekeya Alilah; Ambreen Shafqat; Sajjad Hussain
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Ratio type regression estimator is a prevalent and readily implemented heuristic under simple random sampling (SRS) and two-stage sampling for the estimation of population. But this existing method is based on the ordinary least square (OLS) regression coefficient which is not an effective approach in the presence outliers in the data. In this article, we proposed a class of estimators firstly for complete auxiliary information and, later on, for partial auxiliary information for the presence of outliers in the data. To address this problem, initially we presented a distinct class of estimators by introducing the characteristics of L-moments in the existing estimators. Later on, quantile regression estimators are defined as more robust in the presence of outliers. These techniques empowered the proposed estimators to handle the problem of outliers. To prove the better performance of the proposed estimators, numerical studies are carried out using R language. To calculate the mean square error (MSE), hypothetical equations are expressed for adapted and proposed estimators. Percentage Relative Efficiencies (PRE) are compared to justify the proposed estimators.


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## Body

## 1. Introduction

Auxiliary information is key tool in sampling survey to tackle the problem of missingness in the dataset. By using auxiliary information, precision of the estimator can be enhanced for the mean estimation. In 1814, Laplace introduced an idea and also justified the significance of using auxiliary information in a simple but concise manner. He discovered the idea of estimating the population of country by using the information given in the birth registers rather than conducting a census of the whole population of the country. But it can only be done if it has given the annual rate of birth. Furthermore, studies ordinarily report more focused on data, for example, an extent of likely electors preferring popularity-based (or conservative) political stage or an average pay of clients shopping at a general store. The researcher usually accepts zero difference, while utilizing census data; however, it is not normally a case to overlook sample inconsistency for survey-based assessment. Views of experts can be communicated in a form of a very much grounded supposition on one or a few population parameters. A few specialists can force a bunch of limitations on population parameters or a basic distribution. Overall, auxiliary information can be of the accompanying two sorts such as auxiliary information of definite nature such as census information, assumptions of experts, or else linear limitations, and another sort is information known with some level of uncertainty, for example, survey results and estimations of previous analysis. For detailed study, interested readers may refer to [1–5].Several estimators exist for the estimation of population parameters e.g., population mean, average, median, quantile, and distribution function. These estimators require the information about the auxiliary variable and also the study variable parameter. As the use of auxiliary information improves the estimation results of an estimator, it is also a very common practice in the sampling survey because it is very useful for developing sampling schemes for different situations. The relationship between the auxiliary variable and study variable is a linear relationship. For example, in case of weight and height, a linear relationship is developed between them. Another example for the linear relationship is demand and supply, as demand increases, supply also increases. In these types of circumstances, estimation outcomes of the study variable can be improved by using auxiliary information. To acquire auxiliary information, different sources can be used e.g., census data and results or findings of already collected data through experiments or expert opinions. This type of information can be used in different methods. Data collected through census can be used to get likelihood distribution for these parameters e.g., age, weight, and height. Auxiliary information may be utilized at the estimation stage or design stage, or both. In case of finding the estimates of population parameters by utilizing auxiliary information, it more efficiently enhances the effectiveness of the estimation outcomes that are grounded on traditional calibration, ratio, and regression approaches. Numerous studies published regarding these approaches for the estimation of the population parameters with the effective use of auxiliary information. For instance, see [6].The remaining sections of the article are arranged as follows: in Section2, we presented the literature review related to the article, initial definitions are presented, and also the adapted estimator details are given. Section 3 discusses the proposed estimator using L-moments’ characteristics and quantile regression. Section 4 elaborates the simulation study and performance of the proposed and adapted estimators. Section 5 explains the results, and the conclusion is drawn in Section 6.


## 2. Adapted Estimators Using L-Moments’ Characteristics

There are different techniques to eliminate the outliers from the data and make it normal for analysis. But with the developments in statistics, it is found that the when outliers are removed from the data, it may affect the extensiveness of the results as the assumptions of sufficiency are not fulfilled. According to the assumptions of sufficiency in statistics, the accuracy of the results increases by utilizing the whole data or information for analysis. But if the outliers are removed from the data to make it normal means that some part of real data or information is cut down, and it may affect the results. According to modern statistics, outliers do not always lead to bad results but sometimes they provide a very useful information as well. So, the results of the estimators are more reliable as the information concluded from such estimators is more comprehensive, absolute, and exhaustive.There are numerous estimators that are suggested on the basis of traditional moments but they are influenced by outliers. L-moments’ are more robust in this situation when extreme values or outliers’ spike occurs in the resulting curve of the data. For any random variable that exists with mean, the L-moments are defined as expectations of specific linear combinations of order statistics [7]. Hosking in 19th century formed the basis of a general theory about L-moments that is summarizing and describing theoretical probability distributions, summarizing and describing samples of observed data, estimating parameters and quantities, and hypothesizing tests of distributions. The theory includes static procedures such as the use of order statistics, and it leads to some promising innovations such as measures of kurtosis and skewness and new procedures for estimating parameters of many distributions. The L-moments’ theory parallels with traditional moment theory and is distinguished by it in that way, and it is linear functions of data and produces more efficient parameter estimates, suffering less from the effects of sample volatility, more powerful to outliers in data, and enables safer inferences made from small samples about a fundamental distribution. L‐moments provide a tool of wide‐ranging practical utility. Computation of the first few sample L‐moments and L‐moment ratios of a dataset provides a useful summary of the location, dispersion, and shape from which the sample was drawn.An idea of nontraditional measure of the location with traditional measures of the location for mean estimation is suggested by many researchers including Shahzad et al. [4]. However, we are going to incorporate summary statistics related to L-moments with the OLS regression coefficient:(1)y¯cdj=y¯+βools′X¯−x¯M1x¯+M2M1X¯+M2,forj=1,2,…,9.In the above equation,X¯,Y¯ is the population means whereas the sample mean is denoted by x¯,y¯; a sample of size n is taken from the population under the simple random sampling scheme. However, M1 and M2 can be (0, 1) or some other known population measures. We are using L-moments-based kurtosis that is represented by L-β2x, L-moments-based coefficient of variation denoted by L-CVx, L-moments-based skewness expressed as L-β1x, L-moments-based standard deviation represented by L-SDx, ∂ is the coefficient of correlation, and OLS regression coefficient presented by βools′. The mean square error (MSE) of y¯cdj can be written as(2)MSEy¯cdj=∅Sy2+wj2Sx2+2βools′wj′Sx2+βools′2Sx2−2wj′Sxy−2βools′Sxy,forj=1,2,3,…,9.Therefore,wj=M1Y¯/M1X¯+M2 and ∅=1−f/nforj=1,2,…,9. Furthermore, Sy2 and Sx2 present unbiased variances, and Sxy are the covariances of Y and X. The family members are provided in Table 1.Table 1
Family members of the adapted class.
Under complete information estimator(A1,A2)Under complete information estimatory¯cd1(1, L-β2x)y¯cd1′y¯cd2(L-CVx, L-β2x)y¯cd2′y¯cd3(ρ, L-β2x)y¯cd3′y¯cd4(1, L-β1x)y¯cd4′y¯cd5(L-CVx, L-β1x)y¯cd5′y¯cd6(ρ, L-β1x)y¯cd6′y¯cd7(1, L-SDx)y¯cd7′y¯cd8(L-CVx, L-SDx)y¯cd8′y¯cd9(ρ, L-SDx)y¯cd9′


## 3. Proposed Class of Quantile Regression-Ratio-Type Estimators

By replacing the OLS regression coefficient with the quantile regression coefficient, we propose the following class of estimators as quantile regression is more robust to outliers, and it is preferable for use in estimation purpose when the dataset is suffering with an issue of extreme values or outliers. Quantile regression gives better estimates in the presence of outliers, and when we use quantile regression technique for estimation, then there is no need to eliminate the outliers from the data:(3)y¯aj=y¯+βoq′X¯−x¯M1x¯+M2M1X¯+M2,forj=1,2,3,…,9,along with(4)βoq′=argminβo∈Rpρqv∑i=1nyj−<xj,βo>.Therefore,ρqv is a continuous piecewise linear function or loss function that belongs to quantile q∈0,1, and this is not differentiable at v=0. It is important to note that all the notations of y¯aj have same meanings as already explained in the above section. Moreover, βoq′ is the quantile regression coefficient when a=2 variables. Mean square error (MSE) of the proposed estimator is written mathematically as(5)MSEy¯cdj=∅Sy2+wj2Sx2+2βoolswjSx2+βools2Sx2−2wjSxy−2βoolsSxy,forj=1,2,3,…,9.It is important to note thatq15th=0.15,q25th=0.25, and q35th=0.35 quantiles are used for the proposed work of this research paper. According to numerical illustrations of the proposed estimator that are conducted in Section 5, we observed that the efficiency of the proposed estimator is remarkably enhanced by using quantile regression coefficients. Also, by using these three mentioned quantiles, our proposed class comprises twenty-seven affiliations. For the purpose of readability, we provided 27 affiliations of the proposed class with MSE in the compressed form is given by(6)y¯aj=y¯+βo0.15′X¯−x¯M1x¯+M2M1X¯+M2,forq=0.15=1,2,3,…,9y¯+βo0.25′X¯−x¯M1x¯+M2M1X¯+M2forq=0.25=1,2,3,…,9y¯+βo0.35′X¯−x¯M1x¯+M2M1X¯+M2forq=0.35=1,2,3,…,9,MSEy¯cdj=∅Sy2+wj2Sx2+2βo0.15wjSx2+βo0.152Sx2−2wjSxy−2βo0.15Sxy∅Sy2+wj2Sx2+2βo0.25wjSx2+βo0.252Sx2−2wjSxy−2βo0.25Sxy∅Sy2+wj2Sx2+2βo0.25wjSx2+βo0.252Sx2−2wjSxy−2βo0.25Sxy,where M1 and M2 have same meanings, as described in the previous section. For more details about usage of L-moments’ characteristics in survey sampling, see [8–11]. The family members are provided in Table 2.Table 2
Family members of the proposed class.
Under complete information estimator(A1,A2)Under complete information Estimatory¯a1(1, L-β2x)y¯a1′y¯a2(L-CVx, L-β2x)y¯a2′y¯a3(ρ, L-β2x)y¯a3′y¯a4(1, L-β1x)y¯a4′y¯a5(L-CVx, L-β1x)y¯a5′y¯a6(ρ, L-β1x)y¯a6′y¯a7(1, L-SDx)y¯a7′y¯a8(L-CVx, L-SDx)y¯a8′y¯a9(ρ, L-SDx)y¯a9′


### 3.1. Two-Stage Sampling Scheme (Partial Information)

When information about the supplementary variable is unknown for the population mean, then the two-stage sampling scheme is preferable for use. Neyman [12] is considered as a pioneer of the probability of this sampling approach for the population parameters. The two-stage sampling method is easier and more reliable as well. This sampling plan is used to find the data about the supplementary variable more efficiently by selecting a valuable sample from the first stage and an adequate size sample at second-stage sampling. To study more robust methods of estimation, interested readers are referred to [13, 14].Under the scheme of two-stage sampling, we select a first-stage sample of sizenc units from the population of size N with the help of the SRSWOR method. After that, we choose a second-stage sample i.e., nd is a subsample of the first stage.


### 3.2. Adapted and Proposed Estimators by Using Two-Stage Sampling Method

(7)y¯cdj′=y¯d+βools′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forj=1,2,…,9,y¯aj′=y¯d+βoq′x¯a−x¯bM1x¯b+M2M1x¯a+M2,forj=1,2,…,9.Therefore,x¯b−y¯b demonstrates the sample mean at 2nd stage and x¯a is the sample mean at 1st stage. Moreover, M1 and M2 have the similar meanings as explained. The family affiliations of y¯cdj′ and y¯aj′ are described in Tables 1 and 2, respectively.Using the Taylor series approach, we derived the MSE ofy¯cdj′ as follows:(8)MSEy¯cdj′=α1∑α1′.Therefore,(9)α1=∂hy¯d,x¯c,x¯d∂y¯d|Y¯,X¯∂hy¯d,x¯c,x¯d∂x¯c|Y¯,X¯∂hy¯d,x¯c,x¯d∂x¯d|Y¯,X¯,α1=1wj+βools−wj+βools,∑3∗3Vy¯dCovy¯d,x¯cCovy¯d,x¯dCovx¯c,y¯dVx¯cCovx¯d,x¯cCovx¯d,y¯dCovx¯d,x¯cVx¯d,Vy¯d=η2Sy2,Vx¯c=η1Sx2,Vx¯d=η2Sy2,Covy¯d,x¯c=Covx¯c,y¯d=η1Syx,Covy¯d,x¯d=Covx¯d,y¯d=η2Syx,Covy¯c,x¯d=Covx¯d,x¯c=η1Sx2.By using notations of covariances and variances and also by replacingα1 values and ∑3∗3 in equation (8), the MSE formula of y¯cdj′ can be written as(10)MSEy¯cdj′=η2Sy2+η2−η1wj+βools2Sx2−2wj+βoolsSyx,forj=1,2,3,…,9.By interchangingβools with βoq, it is easy to find MSE of the proposed family of the estimator as given below:(11)MSEy¯aj′=η2Sy2+η2−η1wj+βoq2Sx2−2wj+βoqSyx,forj=1,2,3,…,9.Therefore,η1=1/nc−1/N and η2=1/nd−1/N.Twenty-seven family affiliations of the proposed family with their MSE in the compressed form under partial information can be expressed as follows:(12)y¯aj′=y¯d+βo0.15′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forq=0.15=1,2,3,…,9y¯d+βo0.25′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forq=0.25=1,2,3,…,9y¯d+βo0.35′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forq=0.35=1,2,3,…,9,MSEy¯aj′=η2Sy2+η2−η1wj+βo0.152Sx2−2wj+βo0.15Syxη2Sy2+η2−η1wj+βo0.252Sx2−2wj+βo0.25Syxη2Sy2+η2−η1wj+βo0.352Sx2−2wj+βo0.35Syx.


## 3.1. Two-Stage Sampling Scheme (Partial Information)

When information about the supplementary variable is unknown for the population mean, then the two-stage sampling scheme is preferable for use. Neyman [12] is considered as a pioneer of the probability of this sampling approach for the population parameters. The two-stage sampling method is easier and more reliable as well. This sampling plan is used to find the data about the supplementary variable more efficiently by selecting a valuable sample from the first stage and an adequate size sample at second-stage sampling. To study more robust methods of estimation, interested readers are referred to [13, 14].Under the scheme of two-stage sampling, we select a first-stage sample of sizenc units from the population of size N with the help of the SRSWOR method. After that, we choose a second-stage sample i.e., nd is a subsample of the first stage.


## 3.2. Adapted and Proposed Estimators by Using Two-Stage Sampling Method

(7)y¯cdj′=y¯d+βools′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forj=1,2,…,9,y¯aj′=y¯d+βoq′x¯a−x¯bM1x¯b+M2M1x¯a+M2,forj=1,2,…,9.Therefore,x¯b−y¯b demonstrates the sample mean at 2nd stage and x¯a is the sample mean at 1st stage. Moreover, M1 and M2 have the similar meanings as explained. The family affiliations of y¯cdj′ and y¯aj′ are described in Tables 1 and 2, respectively.Using the Taylor series approach, we derived the MSE ofy¯cdj′ as follows:(8)MSEy¯cdj′=α1∑α1′.Therefore,(9)α1=∂hy¯d,x¯c,x¯d∂y¯d|Y¯,X¯∂hy¯d,x¯c,x¯d∂x¯c|Y¯,X¯∂hy¯d,x¯c,x¯d∂x¯d|Y¯,X¯,α1=1wj+βools−wj+βools,∑3∗3Vy¯dCovy¯d,x¯cCovy¯d,x¯dCovx¯c,y¯dVx¯cCovx¯d,x¯cCovx¯d,y¯dCovx¯d,x¯cVx¯d,Vy¯d=η2Sy2,Vx¯c=η1Sx2,Vx¯d=η2Sy2,Covy¯d,x¯c=Covx¯c,y¯d=η1Syx,Covy¯d,x¯d=Covx¯d,y¯d=η2Syx,Covy¯c,x¯d=Covx¯d,x¯c=η1Sx2.By using notations of covariances and variances and also by replacingα1 values and ∑3∗3 in equation (8), the MSE formula of y¯cdj′ can be written as(10)MSEy¯cdj′=η2Sy2+η2−η1wj+βools2Sx2−2wj+βoolsSyx,forj=1,2,3,…,9.By interchangingβools with βoq, it is easy to find MSE of the proposed family of the estimator as given below:(11)MSEy¯aj′=η2Sy2+η2−η1wj+βoq2Sx2−2wj+βoqSyx,forj=1,2,3,…,9.Therefore,η1=1/nc−1/N and η2=1/nd−1/N.Twenty-seven family affiliations of the proposed family with their MSE in the compressed form under partial information can be expressed as follows:(12)y¯aj′=y¯d+βo0.15′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forq=0.15=1,2,3,…,9y¯d+βo0.25′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forq=0.25=1,2,3,…,9y¯d+βo0.35′x¯a−x¯bM1x¯b+M2M1x¯c+M2,forq=0.35=1,2,3,…,9,MSEy¯aj′=η2Sy2+η2−η1wj+βo0.152Sx2−2wj+βo0.15Syxη2Sy2+η2−η1wj+βo0.252Sx2−2wj+βo0.25Syxη2Sy2+η2−η1wj+βo0.352Sx2−2wj+βo0.35Syx.


## 4. Simulation Study

To evaluate the performance of the proposed estimator by comparing it with existing estimators, we utilized three different populations.Population-1: in pop-1, a dataset of “total loan for nonreal estate farm during the year 1977” is taken as auxiliary variable (X), whereas “total of real estate loan during the same year” is taken as study variable (Y), where N=50,n=nc=20, and nd=16. The data is taken from [15].The existing estimator is related toβools, whereas their parallel proposed estimators are related to βo0.15,βo0.25, and βo0.35. Therefore, the results of every existing estimator along their corresponding proposed estimators (∅^) are given in a same row such as ∅^=βools,βo0.15,βo0.25,βo0.35 in Tables 3–5 as well as percentage relative efficiency (PRE) of the existing and proposed estimator, say ∅′^, is also presented in Tables 3–5 under partial information.Table 3
PRE of estimators for population 1.
Estimatorsβools′βo0.15′βo0.25′βo0.35′t^11927.4502431.5462369.5282351.966t^21927.8662432.1152370.0782352.512t^31927.6122431.7672369.7422352.178t^41928.2762432.6762370.6202353.048t^51929.2112433.9562371.8572354.272t^61928.6392433.1732371.1012353.524t^73779.5154974.5704829.3824788.119t^84776.3216264.6296090.6246040.767t^94187.5315514.7475355.5555310.186t^1′1927.5612431.6212369.6192351.205t^2′1927.9632432.1352370.1542352.213t^3′1927.0132431.5152369.6422352.267t^4′1928.3672432.2542370.3352353.604t^5′1929.1142433.8992371.7512354.117t^6′1928.4502433.2712371.1312353.631t^7′3779.4994974.5554829.3504788.129t^8′4776.1996264.1136090.1386040.714t^9′4187.4895514.6815355.5655310.177Population 1: in the second population, “US cereals” data is used, which consists of 65 regularly available cereals in the United States of America. This dataset was gathered from the information available on the label of the packets and also used by Venables and Ripley [16]. There are many variables in the data. But a gram of fiber in one portion is to be considered as auxiliary variable (X), whereas a gram of potassium is considered to be study variable (Y); therefore, N=65,n=nc=20, and nd=15.Table 4
PRE of estimators for population 2.
Estimatorsβools′βo0.15′βo0.25′βo0.35′t^11181.4371337.5351416.8811470.935t^21186.5591343.5311423.3211477.676t^31183.1791339.5741419.0711473.227t^41179.7061335.5081414.7041468.657t^51180.2891336.1901415.4371469.424t^61179.9041335.7401414.9531468.918t^71620.9181850.5121966.1392044.329t^82506.1642842.5553000.1323101.599t^91969.6772250.9782389.9832482.754t^1′1181.3711337.5281416.8711470.921t^2′1186.5411343.5211423.3131477.653t^3′1183.1621339.5511419.0411473.212t^4′1179.6951335.5011414.7001468.641t^5′1180.2731336.1791415.4281469.411t^6′1179.9551335.7111414.9391468.905t^7′1620.9111850.5021966.1252044.318t^8′2506.1412842.5663000.1253101.581t^9′1969.6692250.9642389.9812482.742Population 2: in the third population, we also considered the “US cereal” data [16]. As there are many variables available in the data, so we used the other variable from that dataset in the third simulation study. A gram of sodium in one portion is considered as the auxiliary variable whereas “no. of calories” is considered as study variable (Y). Therefore, N=65,n=nc=20, and nd=15.Table 5
PRE of estimators for population 3.
Estimatorsβools′βo0.15′βo0.25′βo0.35′t^11658.9332241.7841760.2471710.795t^21787.1932443.4301900.4721845.136t^31667.9872255.9101770.1321720.272t^41801.8952466.7561916.5731860.549t^52013.8412807.7312149.2942083.039t^61816.7502490.3681932.8481876.125t^73686.6355774.6344019.7013855.566t^85159.7598719.0135705.0445435.119t^93786.7385965.9634133.2383962.422t^1′1658.9212241.7751760.2391710.781t^2′1787.1842443.4191900.4671845.126t^3′1667.9772255.9011770.1251720.267t^4′1801.8842466.7491916.5681860.543t^5′2013.8332807.7272149.2882083.035t^6′1816.7442490.3621932.8411876.119t^7′3686.6275774.6294019.7003855.551t^8′5159.7528719.0115705.0375435.111t^9′3786.7325965.9584133.2323962.415


## 5. Results and Discussions

On the basis of numerical studies, the results are shown in Tables3–5 under complete and partial information setting. From these tables, it is clear that y¯aj with i=1,…,9 considered in the proposed class outperformed the corresponding estimator of the existing class i.e., y¯cdj. It is also proved that the proposed estimator is more useful than the existing estimators in situation of partial information, as the PRE of the proposed estimators is higher as compared to adapted estimators, which proves the better performance of proposed estimators.


## 6. Conclusion

In this article, a new class for estimators is proposed for mean estimation initially by using complete auxiliary information and later on the partial auxiliary information. By introducing L-moments and quantile regression methods in the existing estimator significantly enhanced the performance of the estimator as the results mentioned in Tables3–5. As the PREs of the proposed estimator have significantly increased, it is an evidence to draw the conclusion that the proposed estimator performed much better than the existing estimator. Also, the proposed estimator is more robust to outliers and provides better estimates.


### 6.1. Final Remarks

In the beginning of this article, with some continuing involvement of Shahzad et al. [4] for the estimation of mean under the sampling method, we have suggested a class of quantile regression-ratio-type estimators using L-moments for the population mean for the nonnormal dataset having extreme values or outliers. The proposed family of estimators outperformed the existing ones, where the population mean of the auxiliary variable is given. We mathematically derived the MSE expressions. We also suggested the existing and proposed classes for the situations where we have given the partial information. To prove the efficiency of the proposed estimators, three different real life datasets are used for the simulation purpose. According to the results of numerical illustrations, it is proved that the proposed estimator is more efficient for the survey conducted under the given circumstances.


## 6.1. Final Remarks

In the beginning of this article, with some continuing involvement of Shahzad et al. [4] for the estimation of mean under the sampling method, we have suggested a class of quantile regression-ratio-type estimators using L-moments for the population mean for the nonnormal dataset having extreme values or outliers. The proposed family of estimators outperformed the existing ones, where the population mean of the auxiliary variable is given. We mathematically derived the MSE expressions. We also suggested the existing and proposed classes for the situations where we have given the partial information. To prove the efficiency of the proposed estimators, three different real life datasets are used for the simulation purpose. According to the results of numerical illustrations, it is proved that the proposed estimator is more efficient for the survey conducted under the given circumstances.



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