# A Chain Ratio Exponential-Type Compromised Imputation for Mean Estimation: Case Study on Ozone Pollution in Saraburi, Thailand

Authors: Kanisa Chodjuntug; Nuanpan Lawson
Journal: Journal of Probability and Statistics
 (2020)

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## Abstract

Due to its impact on health and quality of life, Thailand’s ozone pollution has become a major concern among public health investigators. Saraburi Province is one of the areas with high air pollution levels in Thailand as it is an important industrialized area in the country. Unfortunately, the August 2018 Pollution Control Department (PCD) report contained some missing values of the ozone concentrations in Saraburi Province. Missing data can significantly affect the data analysis process. We need to deal with missing data in a proper way before analysis using standard statistical techniques. In the presence of missing data, we focus on estimating ozone mean using an improved compromised imputation method that utilizes chain ratio exponential technique. Expressions for bias and mean square error (MSE) of an estimator obtained from the proposed imputation method are derived by Taylor series method. Theoretical finding is studied to compare the performance of the proposed estimator with existing estimators on the basis of MSE’s estimators. In this case study, the results in terms of the percent relative efficiencies indicate that the proposed estimator is the best under certain conditions, and it is then applied to the ozone mean estimation for Saraburi Province in August 2018.


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## Body

## 1. Introduction

Air pollution is a global problem which results in negative effects on both the environment and human health. Many researchers have found that air pollution is associated with mortality and morbidity from lung cancer, respiratory, cardiovascular diseases, and exacerbation of chronic respiratory conditions [1, 2]. Moultion and Yang [3] have shown that air pollution is correlated with Alzheimer’s disease and other neurodegenerative disorders. From the World Health Organization’s report in 2018, air pollution caused approximately 4.2 million deaths [4]. In fact, children are more vulnerable than adults because their lungs, heart, and brain are still growing. Therefore, air pollution is a major public health concern; monitoring and measuring the quality of air is critical.The Pollution Control Department (PCD) in Thailand is an agency that measures the amount of air pollutants such as sulphur dioxideSO2, carbon monoxide CO, nitrogen oxides NOX, particulate matter PM2.5,PM10, and ozone O3. As shown in the PCD’s Air Quality Management Division report in 2015, O3 and PM10concentrations were higher than standard levels in almost every province [5]. There is also a relationship between O3 and PM10 [6, 7]. Naphralan subdistrict, Chaloem Phra Kiat district is an industrialized area in Saraburi Province with traffic congestion and several stone and cement factories. According to the information from the Saraburi-based ground monitoring stations of PCD in August 2018 [8], we found that the O3 concentration data were missing due to equipment malfunction or errors in measurement. Missing values may cause a significant effect of data analysis process. We deal with it using standard statistical techniques. In environmental research, a number of techniques can be employed to impute missing values in air pollutants concentration data such as the mean top bottom method, mean imputation method, the multiple regression method, and artificial neural network models [9–13].In sample surveys, missing values or nonresponses often occur. There are two types of nonresponse: item nonresponse and unit nonresponse. The imputation method is used to handle item nonresponse, and the weight method is applied to deal with unit nonresponse. In addition, imputation which uses available data as a source for replacement of missing data is the most common method to solve missing data.In addition, many researchers have studied the auxiliary information in order to improve the precision of population mean estimation under a simple random sampling without replacement (SRSWOR). For example, Cochran [14] applied the auxiliary information at the estimation stage and proposed an estimator to estimate the population mean. Bahl and Tuteja [15] first proposed new ratio-type exponential method for estimating the mean of population using information on auxiliary variable, and their methods are more efficient than common methods: mean and ratio methods. Later, Singh and Pal [16] proposed a chain ratio-ratio-type exponential technique which is more efficient than the common estimators including mean, ratio, and ratio-type exponential estimators under certain condition as follows:(1)y¯CR=y¯nX¯x¯nexpX¯−x¯nX¯+x¯n,where y¯n is sample mean of interest variable Y and X¯ and x¯n are population mean and sample mean of auxiliary variable X, respectively.Similarly, Lee et al. [17] applied the auxiliary information for the purpose of imputation. Recently, Singh and Horn [18] suggested a compromised imputation method to estimate the population mean as follows:(2)y.i=αnryi+1−αb^xi;i∈R,1−αb^xi;i∈Rc,where b^=∑iɛRyi/∑iɛRxi, yi and xi are observed values of Y and X for the ith unit, R and Rc are the set of responding and nonresponding units, respectively, n and r are sizes of sample and response data, and α is a chosen constant.Under this imputation method, the point estimator of the population mean becomes(3)y¯COMP=αy¯r+1−αy¯rx¯nx¯r,where y¯r, x¯n, and x¯r are response mean of the variable of interest Y, sample mean of auxiliary variable X, and response mean of auxiliary variable X, respectively.The bias and mean square error ofy¯COMP are, respectively, given by(4)Biasy¯COMP=1−α1r−1nY¯CX21−K,(5)MSEy¯COMP=1r−1NY¯2CY2+1r−1nY¯2CX21−α2−21−αK,where K=ρCY/CX, CY, and CX are population coefficient of variations of Y and X, respectively. ρ is population correlation coefficient between Y and X. N, n, and r are sizes of population, sample, and response data, respectively. Their research showed that the suitably chosen constant is correlated with the performance of the estimator.In this study, we aim to use the imputation method to estimate the population mean in the presence of nonresponse occurring in the variable of interestY only. We propose to improve the compromised imputation method by using the chain ratio-type exponential technique and its corresponding estimator. The bias and the mean square error have been obtained to the first degree of approximation using the Taylor series method. The efficiency of the proposed estimator is compared with some existing estimators on the basis of MSE in order to obtain the certain conditions for application of proposed estimator. In this case study, we use the percent relative efficiency (PRE) as an indicator to assess the performance of the estimator. Then, the best estimator is applied to estimate the population mean of O3 as a variable of interest Y based on PM10 concentration data as a variable of auxiliary X of the Saraburi Province’s data in August 2018.


## 2. Materials and Methods

### 2.1. Basic Setup Framework

LetU=1,2,…,N be a finite population of size N,yi be the value of interest variable Y, and xi be value of auxiliary variable X. Let Y¯=∑i=1Nyi/N and X¯=∑i=1Nxi/N be the population means of Y and X, respectively, and they are unknown values. Let R and Rc be the set of responding units and nonresponding units, respectively. The value of yi is observed for every iɛR; meanwhile, the value of yi is missing for every iɛRc and imputed value with xi. Based on SRSWOR scheme, s of size n with paired variable X,Y is selected from U and contains both r responding units and n−r nonresponding units. Let x¯n=∑i=1nxi/n, x¯r=∑i=1rxi/r and y¯r=∑i=1ryi/r be the sample mean of X and the response mean of X and Y, respectively.


### 2.2. Existing Imputation Methods and Corresponding Estimation

The mean imputation method, the data after imputation, is defined as(6)y.i=yi;iɛR,y¯r;iɛRc,where y¯r=∑i=1rxi/r.Under this imputation method, the point estimator of the population mean becomes(7)y¯s=∑iɛsy.ir=y¯r.The bias and variance ofy¯S are, respectively, given by(8)biasy¯S=0,(9)vary¯S=1r−1NY¯2CY2,where CY=SY/Y¯ and SY2=∑i=1Nyi−Y¯2/N−1.The ratio imputation method, the data after imputation is defined as(10)y.i=yi;iɛR,b^xi;iɛRc,where b^=∑iɛRyi/∑iɛRxi,Under this imputation method, the point estimator of the population mean becomes(11)y¯RAT=y¯rx¯nx¯r,where x¯n=∑i=1nxi/n, x¯r=∑i=1rxi/r, and y¯r=∑i=1ryi/r be the sample mean of X and the response mean of X and Y, respectively.The bias and mean square error ofy¯RAT are, respectively, given by(12)biasy¯RAT=1r−1nY¯CX21−K,(13)MSEy¯RAT=1r−1NY¯2CY2+1r−1nY¯2CX21−2K,where SY2=∑i=1Nyi−Y¯2/N−1, SX2=∑i=1Nxi−X¯2/N−1, SXY=∑i=1Nxi−X¯yi−Y¯/N−1, CY=SY/Y¯, CX=SX/X¯, K=ρCY/CX, and ρ=SXY/SXSY.


### 2.3. Proposed Compromised Imputation Method and Corresponding Estimator

Motivated by Singh and Pal [16] and Singh and Horn [18], we propose a new compromised imputation method by using the idea of chain ratio-type exponential estimator. The data after imputation are defined as(14)y.i=αnryi+1−αxix¯ry¯rexpx¯n−x¯rx¯n+x¯r;iɛR,1−αxix¯ry¯rexpx¯n−x¯rx¯n+x¯r;iɛRc,where α denotes suitably chosen constants in order that MSE of the estimator is minimum. x¯n=∑i=1nxi/n, x¯r=∑i=1rxi/r, and y¯r=∑i=1ryi/r be the sample mean of X and the response mean of X and Y, respectively.Under the proposed imputation method, the point estimator of the population mean is given as follows:(15)y¯PE=αy¯r+1−αy¯rx¯nx¯rexpx¯n−x¯rx¯n+x¯r.Note that ifα=1, then y¯PE=y¯r, and if α=0, then the y¯PE is analogue of the estimator for population mean proposed by Singh and Pal [16].To find the properties of the proposed estimator, both bias andMSE of y¯WTP are considered up to the first degree of approximation by using the Taylor series method. We define y¯r=Y¯1+e1,x¯n=X¯1+e2,x¯r=X¯1+e3,M1=1/r−1/N,M2=1/n−1/N, and M3=M1−M2=1/r−1/n,Mi>0;i=1,2,3.Since, SRSWOR is being followed, and we haveEei=0,i=1,2,3,(16)Ee12=M1CY2,Ee22=M2CX2,Ee32=M1CX2,Ee1e2=M2ρCXCY,Ee2e3=M2CX2,Ee1e3=M1ρCXCY,where SY2=∑i=1Nyi−Y¯2/N−1, SX2=∑i=1Nxi−X¯2/N−1, SXY=∑i=1Nxi−X¯yi−Y¯/N−1, ρ=SXY/SXSY, CY=SY/Y¯, and CX=SX/X¯.Naturally, we can usesy2=∑i=1ryi−y¯r2/r−1, sx2=∑i=1rxi−x¯r2/r−1, sxy=∑i=1rxi−x¯ryi−y¯r/r−1, cy=sy/y¯r, cx=sx/xr, and ρ^=sxy/sxsy to estimate SY2, SX2, SXY, CY, CX, and ρ as population parameters when we cannot find these population parameters.Next, writingy¯PE in term of ei ’s, equation (15) takes the form as follows:(17)y¯PE=αY¯1+e1+1−αY¯1+e1X¯1+e2X¯1+e3expX¯1+e2−X¯1+e3X¯1+e2+X¯1+e3.From equation (17), we have(18)y¯PE−Y¯=αY¯1+e1+1−αY¯1+e1X¯1+e2X¯1+e3expX¯1+e2−X¯1+e3X¯1+e2+X¯1+e3−Y¯≈αY¯1+e1+1−αY¯1+e1+32e2+⋯+158e32.Taking expectation on both sides of equation (18), we get biasy¯PE as follows:(19)biasy¯PE=1−αY¯M3385CX2−4ρCXCY.Squaring both sides of equation (18), expanding the term and taking expectations, and retaining the terms up to first degree of approximations, we get MSEy¯PE as follows:(20)MSEy¯PE=Y¯2M1CY2+31−αM3CX341−αCX−ρCY.


### 2.4. Estimation of Optimum Value Constantαopt

In this section, we considerα that is optimum in order to find the minimum MSEy¯PE. Since y¯PE as given in equation (20) is a function of unknown constant α, it needs to search for α optimum values, such that MSEy¯PE becomes minimum value. To obtain the constant α, we differentiate equation (20) with respect to α and equate it to zero as follows:(21)dMSEy¯PEdα=Y¯23M3CXρCY−321−αCX=0.We solve equation (21), and we get(22)αopt=1−23ρCYCX.


## 2.1. Basic Setup Framework

LetU=1,2,…,N be a finite population of size N,yi be the value of interest variable Y, and xi be value of auxiliary variable X. Let Y¯=∑i=1Nyi/N and X¯=∑i=1Nxi/N be the population means of Y and X, respectively, and they are unknown values. Let R and Rc be the set of responding units and nonresponding units, respectively. The value of yi is observed for every iɛR; meanwhile, the value of yi is missing for every iɛRc and imputed value with xi. Based on SRSWOR scheme, s of size n with paired variable X,Y is selected from U and contains both r responding units and n−r nonresponding units. Let x¯n=∑i=1nxi/n, x¯r=∑i=1rxi/r and y¯r=∑i=1ryi/r be the sample mean of X and the response mean of X and Y, respectively.


## 2.2. Existing Imputation Methods and Corresponding Estimation

The mean imputation method, the data after imputation, is defined as(6)y.i=yi;iɛR,y¯r;iɛRc,where y¯r=∑i=1rxi/r.Under this imputation method, the point estimator of the population mean becomes(7)y¯s=∑iɛsy.ir=y¯r.The bias and variance ofy¯S are, respectively, given by(8)biasy¯S=0,(9)vary¯S=1r−1NY¯2CY2,where CY=SY/Y¯ and SY2=∑i=1Nyi−Y¯2/N−1.The ratio imputation method, the data after imputation is defined as(10)y.i=yi;iɛR,b^xi;iɛRc,where b^=∑iɛRyi/∑iɛRxi,Under this imputation method, the point estimator of the population mean becomes(11)y¯RAT=y¯rx¯nx¯r,where x¯n=∑i=1nxi/n, x¯r=∑i=1rxi/r, and y¯r=∑i=1ryi/r be the sample mean of X and the response mean of X and Y, respectively.The bias and mean square error ofy¯RAT are, respectively, given by(12)biasy¯RAT=1r−1nY¯CX21−K,(13)MSEy¯RAT=1r−1NY¯2CY2+1r−1nY¯2CX21−2K,where SY2=∑i=1Nyi−Y¯2/N−1, SX2=∑i=1Nxi−X¯2/N−1, SXY=∑i=1Nxi−X¯yi−Y¯/N−1, CY=SY/Y¯, CX=SX/X¯, K=ρCY/CX, and ρ=SXY/SXSY.


## 2.3. Proposed Compromised Imputation Method and Corresponding Estimator

Motivated by Singh and Pal [16] and Singh and Horn [18], we propose a new compromised imputation method by using the idea of chain ratio-type exponential estimator. The data after imputation are defined as(14)y.i=αnryi+1−αxix¯ry¯rexpx¯n−x¯rx¯n+x¯r;iɛR,1−αxix¯ry¯rexpx¯n−x¯rx¯n+x¯r;iɛRc,where α denotes suitably chosen constants in order that MSE of the estimator is minimum. x¯n=∑i=1nxi/n, x¯r=∑i=1rxi/r, and y¯r=∑i=1ryi/r be the sample mean of X and the response mean of X and Y, respectively.Under the proposed imputation method, the point estimator of the population mean is given as follows:(15)y¯PE=αy¯r+1−αy¯rx¯nx¯rexpx¯n−x¯rx¯n+x¯r.Note that ifα=1, then y¯PE=y¯r, and if α=0, then the y¯PE is analogue of the estimator for population mean proposed by Singh and Pal [16].To find the properties of the proposed estimator, both bias andMSE of y¯WTP are considered up to the first degree of approximation by using the Taylor series method. We define y¯r=Y¯1+e1,x¯n=X¯1+e2,x¯r=X¯1+e3,M1=1/r−1/N,M2=1/n−1/N, and M3=M1−M2=1/r−1/n,Mi>0;i=1,2,3.Since, SRSWOR is being followed, and we haveEei=0,i=1,2,3,(16)Ee12=M1CY2,Ee22=M2CX2,Ee32=M1CX2,Ee1e2=M2ρCXCY,Ee2e3=M2CX2,Ee1e3=M1ρCXCY,where SY2=∑i=1Nyi−Y¯2/N−1, SX2=∑i=1Nxi−X¯2/N−1, SXY=∑i=1Nxi−X¯yi−Y¯/N−1, ρ=SXY/SXSY, CY=SY/Y¯, and CX=SX/X¯.Naturally, we can usesy2=∑i=1ryi−y¯r2/r−1, sx2=∑i=1rxi−x¯r2/r−1, sxy=∑i=1rxi−x¯ryi−y¯r/r−1, cy=sy/y¯r, cx=sx/xr, and ρ^=sxy/sxsy to estimate SY2, SX2, SXY, CY, CX, and ρ as population parameters when we cannot find these population parameters.Next, writingy¯PE in term of ei ’s, equation (15) takes the form as follows:(17)y¯PE=αY¯1+e1+1−αY¯1+e1X¯1+e2X¯1+e3expX¯1+e2−X¯1+e3X¯1+e2+X¯1+e3.From equation (17), we have(18)y¯PE−Y¯=αY¯1+e1+1−αY¯1+e1X¯1+e2X¯1+e3expX¯1+e2−X¯1+e3X¯1+e2+X¯1+e3−Y¯≈αY¯1+e1+1−αY¯1+e1+32e2+⋯+158e32.Taking expectation on both sides of equation (18), we get biasy¯PE as follows:(19)biasy¯PE=1−αY¯M3385CX2−4ρCXCY.Squaring both sides of equation (18), expanding the term and taking expectations, and retaining the terms up to first degree of approximations, we get MSEy¯PE as follows:(20)MSEy¯PE=Y¯2M1CY2+31−αM3CX341−αCX−ρCY.


## 2.4. Estimation of Optimum Value Constantαopt

In this section, we considerα that is optimum in order to find the minimum MSEy¯PE. Since y¯PE as given in equation (20) is a function of unknown constant α, it needs to search for α optimum values, such that MSEy¯PE becomes minimum value. To obtain the constant α, we differentiate equation (20) with respect to α and equate it to zero as follows:(21)dMSEy¯PEdα=Y¯23M3CXρCY−321−αCX=0.We solve equation (21), and we get(22)αopt=1−23ρCYCX.


## 3. Efficiency Comparison of the Proposed Estimator

Under optimum value constant in equation (22), comparison of y¯PE with those estimators including y¯S, y¯RAT, and y¯COMP is carried out by using MSEand an estimator with a preferred smaller MSE value. We can observe the efficiency of the proposed estimator as follows:From equations (9) and (20), we have(23)vary¯S−MSEy¯PE=M3Y¯23CX1−αρCy−341−αCX>0,if 1−4/3ρCY/CX<α<1 or 1<α<1−4/3ρCY/CX.From equations (13) and (20), we have(24)MSEy¯RAT−MSEy¯PE=M3Y¯2CX1−941−α2CX−2−31−αρCy>0,if 1/3<α<1−2/32ρCY/CX−1 or 1−2/32ρCY/CX−1<α<1/3.From equations (5) and (20), we have(25)MSEy¯COMP−MSEy¯PE=M3Y¯2CX1−αρCy−541−αCX>0,if 1−4/5ρCY/CX<α<1 or 1<α<1−4/5ρCY/CX.When the conditions in equations (23)–(25) are satisfied, y¯PE is more efficient than y¯S, y¯RAT, and y¯COMP, respectively.


## 4. Case Study

For the case study, we obtained dataO3 level ppb and PM10 level μg/m3 on a time-scale of one per hour (hourly average) from the PCD website in August 2018. The data belong to the population of 744 units. On examination, we found that the O3 concentration data had missing values, so it was taken as a variable of interest Y, and PM10 concentration data was taken as a variable of auxiliary X. The following values were obtained for the considered variables: N=744, n=580, r=545, y¯r=13.55, x¯n=69.33, x¯r=66.29, sy=12.45, sx=32.65, cy=0.92, and cx=0.49. We identified 6.03% of the O3 data as missing.The conditions ofα for which y¯PE is better than the existing estimators are shown in Table 1. The table also presents the percent relative efficiencies PREs of each estimator y¯ with respect to y¯S which can be computed by PREy¯=Vary¯S/MSEy¯×100.Table 1
PRE⋅ of each estimator with respect to y¯S and conditions of α in which y¯PE is better than others.
EstimatorsConditions ofα in which y¯PE is better thanPREy¯y¯S1<α<2.18100.00y¯RAT0.33<α<2.8484.97y¯COMP1<α<1.71104.64y¯PEα=αopt=1.59105.25The scatter plot (Figure1) indicates that O3 and PM10 concentration data have a negative relationship. The correlation coefficient between O3 and PM10 concentration data is −0.47. From Table 1, we consider both the certain conditions of α and PREs; accordingly, y¯PE is more efficient than others. y¯PE is the most suitable for estimating the mean value of O3 concentration in this case study. The mean value of O3 concentration by using y¯PE is equal to 13.00 ppb/hr or 0.01 ppm/hr which does not exceed the average O3 standard of the PDC in Thailand (≤0.10 ppm/hr). The mean square error of y¯PE is 0.07 which is close to zero, so it indicates that the proposed estimator is effective in this case.Figure 1
The scatter plot ofO3 and PM10 data.


## 5. Conclusions

In this case study, when missing data occur in the variable of interest, we propose the improved compromised imputation method using the chain ratio-type exponential technique for the population mean estimation. The mean square error of the proposed estimator was studied under general and optimum situations. We suggest this proposed method is useful for estimating the population mean in the presence of common, real-world, nonresponse data, and it is efficient for applying to the real dataset with missing values under certain condition ofα. In fact, in this case study, we applied our proposed method to the whole process of ozone population mean estimation from real data. On the contrary, the common methods proposed by the other authors have not applied their techniques to estimate population mean from real data. In addition, when a dataset contains some missing values, our proposed method can handle this problem to complete data and can also save both time and budget for research conduction. Therefore, the proposed method is a good strategy to apply in practice in case that a missing data problem occurs. However, this study focused on missing values in the variable of interest only, but we could apply a similar method to cases where missing data happens in both the interest and auxiliary variables.



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*Source: 8864412-2020-12-05.xml*