# Poisson Regression-Based Mean Estimator

Authors: Usman Shahzad; Shabnam Shahzadi; Noureen Afshan; Nadia H. Al-Noor; David Anekeya Alilah; Muhammad Hanif; Malik Muhammad Anas
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The most frequent method for modeling count responses in numerous investigations is the Poisson regression model. Under simple random sampling, this paper offers utilizing Poisson regression-based mean estimator and discovers its associated formula of the mean square error (MSE). The MSE of the proposed estimator is compared to the MSE of traditional ratio estimators in theory. As a result of these evaluations, the proposed estimator has been proven to be more efficient than traditional estimators. Furthermore, the practical results corroborated the theoretical findings.


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## Body

## 1. Introduction

The emphasis on using supplemental/auxiliary data to improve estimate precision might be the criterion that distinguishes sample survey theory from other statistical theories. Auxiliary data are used in almost every important phase of a sample survey, including stratification, selection probability creation, and the formula for the population parameter estimator of interest. The assessment of the total or mean of a population is a common goal of most surveys. Utilizing auxiliary data to improve the estimation of the mean population has been considered by numerous authors, such as Upadhyaya et al. [1], Upadhyaya and Singh [2], Koyuncu [3], Shahzad [4], Abid et al. [5], Shahzad et al. [6–8], Zaman and Bulut [9, 10], Ali et al. [11], and Zaman [12].When a positive correlation between values of an auxiliary variable with the study variable is available, the ratio-type and regression estimators are good choices for mean estimation. In sampling theory, auxiliary variable’s population information, like the coefficient of variation or kurtosis, is frequently employed to improve the efficiency of the estimate for a population mean for ratio estimators. On the other hand, the presence of outliers or extreme values in the data degrades the efficiency of traditional estimation methods. For minimizing the impact of outliers in ratio-type mean estimators, Kadilar et al. [13] presented a robust regression strategy based on the Huber-M approach. For the mean estimation with a simple random sampling (SRS) technique, Oral and Kadilar [14, 15] used two approaches that are modified maximum likelihood and its integrated method. Through combining the ratio estimators given in Zaman and Bulut [9], Zaman [12] created a new class of robust ratio-type estimators. Using robust regression estimates as well as robust covariance matrices under stratified random sampling, Zaman and Bulut [10] proposed novel regression-type estimators. More recently, with the case of sensitive research under SRS, Ali et al. [11] proposed a class of robust regression-type estimators.Furthermore, with the count data, when the mean is large enough, it is inconvenient to apply a linear regression model to such data, even though the Poisson distribution converges to the normal distribution. Negative prediction values are possible because the linear model links the predicted value with auxiliary (or explanatory or independent) variables. Also, in linear regression, the validity of hypothesis tests is contingent on the assumption of constant variance of the study (or response or dependent) variable. For count data, these assumptions are invalid. As a result, the Poisson regression model is the most commonly employed approach for modeling count data in the applied sciences.In Poisson regression, the study variableYiyi=0,1,2,… is the number of events that occur at a particular period, with a Poisson distribution given by(1)Pyi;λi=e−λiλiyiyi!,λ>0,and its mean and variance are both the same, EYi=varYi=λi.The natural log-likelihood function is defined as follows:(2)ℒλ;y=∑i=1nyilnλi−λi−lnyi!.LetX be the explanatory variable matrix of order n×k+1. Then, the relationship between Yi and ith row of matrix X, xi associating with gλi, is(3)lnλi=ξi=xiTA,where A=A0,A1,…,Ak are the coefficients of regression parameters. Such a model is well known as the Poisson regression model. A^ represents the maximum likelihood estimator of A, and it may be obtained by differentiating (3) with respect to A.(4)∑i=1nyi−expxiTAxi=0.Iterative approaches such as algorithms of Fisher scoring and Newton–Raphson are employed to solve thesek equations (see Cameron and Trivedi [16], Montgomery et al. [17], and Koҫ [18]).This article focused on utilizing Poisson regression to estimate the parameter mean of the study variable underSRS. The rest of the article is constructed as follows. First, we introduce the notations and review some existing ratio-type mean estimators with their MSE. Second, we deﬁne a novel mean estimator based on Poisson regression and its MSE. Furthermore, the proposed estimator’s condition efficiency is inspected theoretically. Then, using numerical illustrations based on two real datasets, we examine the relative efficiency of the proposed estimator over the adapted estimators. Finally, some concluding remarks are introduced.


## 2. Some Existing Ratio-Type Estimators of Mean with Their MSE

This section outlines some of the existing population mean estimators that employed SRS and relied on known information on the auxiliary variable’s conventional parameters to improve the mean estimators’ efficiency. Before delving into the specifics of the existing population mean estimators, the notations used are as follows.N: population size; n: sample size.f: sample ratio, f=n/N.Y¯,X¯: population means associated with study and auxiliary variables Y,X, respectively.Y¯,X¯: sample means associated with Y and X, respectively.Cy,Cx: population coefficients of variation associated with Y and X, respectively.δx: population coefficient of the kurtosis associated with X.Sy2,Sx2: population variance associated with Y and X, respectively.sy2,sx2: sample variance associated with Y and X, respectively.Sxy,sxy: population and sample covariance between X and Y, respectively.ρxy: correlation coefficient between X and Y.Bℓ,bℓ: coefficient of slope attained by the least squares method, Bℓ=Sxy/Sx2 and bℓ=sxy/sx2.As one of the important estimators for the population meanY¯in SRS, with X¯assumed known, Kadilar and Cingi [19] proposed the following ratio estimators inspired by Sisodia and Dwivedi [20] and Upadhyaya et al. [1]:(5)y¯KC1=y¯+bℓX¯−x¯x¯X¯,y¯KC2=y¯+bℓX¯−x¯x¯+CxX¯+Cx,y¯KC3=y¯+bℓX¯−x¯x¯+δxX¯+δx,y¯KC4=y¯+bℓX¯−x¯x¯δx+CxX¯δx+Cx,y¯KC5=y¯+bℓX¯−x¯x¯Cx+δxX¯Cx+δx.Kadilar and Cingi [19] also provided the following formula for the MSE associated with their estimators:(6)MSEy¯KCi=1−fnℛKCi2Sx2+2BℓℛKCiSx2+Bℓ2Sx2−2ℛKCiSxy−2BℓSxy+Sy2,i=1,2,…,5,where the population ratios ℛKCi,i=1,2,…,5can be gained as follows:(7)ℛKC1=ℛ=Y¯X¯,ℛKC2=Y¯X¯+Cx,ℛKC3=Y¯X¯+δx,ℛKC4=Y¯δxX¯δx+Cx,ℛKC5=Y¯CxX¯Cx+δx.Recently, Koҫ [18] offered his idea to improve the aforementioned estimators, introducing new ratio estimators based on Poisson regression as follows:(8)y¯H1=y¯+bPX¯−x¯x¯X¯,y¯H2=y¯+bPX¯−x¯x¯+CxX¯+Cx,y¯H3=y¯+bPX¯−x¯x¯+δxX¯+δx,y¯H4=y¯+bPX¯−x¯x¯δx+CxX¯δx+Cx,y¯H5=y¯+bPX¯−x¯x¯Cx+δxX¯Cx+δx,where bP represents the coefficient of slopes attained by Poisson regression.Koҫ [18] also provided the following formula for the MSE associated with his suggested estimators that is the same as the MSE equation in (6), but it is evident that Bℓ should be changed by BP, whose value is acquired from the Poisson regression model.(9)MSEy¯Hi≅1−fnℛKCi2Sx2+2BPℛKCiSx2+BP2Sx2−2ℛKCiSxy−2BPSxy+Sy2,i=1,2,…,5.In addition, Koҫ [18] demonstrated that his estimators are more efficient than Kadilar and Cingi [19] estimators if any of the following conditions are satisfied:(10)BP−Bℓ<−2ℛKCi,BP−Bℓ>−2ℛKCi.


### 2.1. Proposed Estimator Based on Poisson Regression and Its MSE

The newly constructed estimator of the mean population can be arranged in the frameworks of Zaman and Bulut [10] and Ali et al. [11]. But we are implementing their frameworks based on Poisson regression as(11)y¯U=y¯+bPX¯−x¯.Furthermore, taking advantage of established results, with some basic algebra, and eliminating tedious or futile calculations, we mention the MSE expression of the proposed mean estimator as(12)MSEy¯U=1−fnY¯2Cy2−2BPX¯Y¯ρxyCxCy+BP2X¯2Cx2.


### 2.2. Efficiency Comparisons

The efficiency condition for the proposed mean estimator may be found by comparing the MSE of the proposed estimator in (12) to the MSE of Haydar’s estimators in (9):(13)MSEy¯U<MSEy¯Hi,i=1,2,…,5,Y¯2Cy2−2BpX¯Y¯ρxyCxCy+Bp2X¯2Cx2<ℛKCi2Sx2+2BPℛKCiSx2+BP2Sx2−2ℛKCiSxy−2BPSxy+Sy2.Now(14)Sy2−2BpSxy+Bp2Sx2<ℛKCi2Sx2+2BPℛKCiSx2+BP2Sx2−2ℛKCiSxy−2BPSxy+Sy2,2ℛKCiSxy−ℛKCi2Sx2<2BPℛKCiSx2,2ℛKCiSxy−ℛKCi2Sx22ℛKCiSx2<BP,SxySx2−12ℛKCi<BP.The proposed Poisson mean estimatory¯U=y¯+bPX¯−x¯ is more efficient than Haydar’s estimators, if condition BP>Sxy/Sx2−1/2ℛKCi is satisfied.


### 2.3. Numerical Illustrations

Here, we use two real datasets to evaluate the performance of proposed and existing estimators.Population I (Pop-I). We consider the dataset collected between 2006 and 2010 from the Afyon Respiratory Disease Hospital and the Afyon Environmental Department Air Pollution Unit, which was used in [18]. The number of patients admitted to the hospital on a weekly basis was taken as the dependent variable Y, and PM10 was taken as the explanatory variable X.Population II (Pop-II). We consider the dataset obtained from TUIK of 81 provinces in 2019 used in [18]. The number of people who died due to traffic accidents was taken as a dependent variable Y, and the number of motor vehicles was taken as explanatory variable X.For these datasets, using SRS, we considern=40 and n=20. The characteristics of the two populations, as well as the values of population ratios, are given in Tables 1 and 2.Table 1
Characteristics of two populations (Pop-I and Pop-II).
NotationPop-IPop-IIN21381n4020Y¯4.67605633882.40740741X¯116.1915274308Cy0.6335551591.025504149Cx0.4398461.90331δx−0.690835.77509Sy2.96253961684.50913823Sx51.10636522093Sxy53.5170638124520.6ρxy0.3530.864077723Bℓ0.020490020.02049002BP0.0046.131E − 07Table 2
Values of population ratios.
RatioPop-IPop-IIℛKC10.0402440.000300419ℛKC20.0400930.000300417ℛKC30.0404850.000300380ℛKC40.0404660.000300419ℛKC50.0407960.000300399Based on the MSE values of the proposed estimator and reviewed estimators, we calculate the relative efficiency (RE) values of the proposed estimators with respect to reviewed estimators, sayθi=y¯KCi,y¯Hi,i=1,2,…,5, as follows:(15)Ry¯U=MSEy¯UMSEθi,i=1,2,…,5.Tables3 and 4 give the outcomes for the relative efficiencies.Table 3
Relative efficiencies with respect to Kadilar and Cingi estimators.
EstimatorPop-IPop-IIθ^1141.71711624.092θ^2141.34031624.081θ^3142.32131624.086θ^4142.27361624.091θ^5143.10641624.083Table 4
Relative efficiencies with respect to Haydar estimators.
EstimatorPop-IPop-IIθ^1109.0927125.2937θ^2108.8701125.2912θ^3109.4509125.2454θ^4109.4225125.2921θ^5109.9185125.2689After assessing the relative efficiency values, in Tables3 and 4, it is observed that all values exceed 100. As a result, the proposed Poisson regression estimator is more efficient than the reviewed estimators. On the other hand, it is observed that all relative efficiency values of proposed estimator with respect to Kadilar and Cingi [19] estimators are greater than corresponding values with respect to Haydar estimators, and thus(16)Ry¯U>Ry¯Hi>Ry¯KCi,i=1,2,…,5.The proposed estimator and Haydar’s estimators outperform Kadilar and Cingi [19] estimators, which is an expected result due to the employment of the count data in this analysis, and the proposed estimator has the best performance. The consequences of relative efficiency values are also provided graphically in Figures 1–3 .Figure 1
Relative efficiencies with respect to Kadilar and Cingi estimators.Figure 2
Relative efficiencies with respect to Haydar estimators.Figure 3
Relative efficiencies with respect to reviewed estimators.Furthermore, we investigated the efficiency condition of the proposed estimator as follows.For Pop-I:(17)SxySx2−12ℛKC1=0.000368,SxySx2−12ℛKC2=0.0004435,SxySx2−12ℛKC3=0.0002475,SxySx2−12ℛKC4=0.000257,SxySx2−12ℛKC5=0.000092.Therefore, the conditionBP=0.004>Sxy/Sx2−1/2ℛKCi,i=1,2,…,5is fulfilled.For Pop-II:(18)SxySx2−12ℛKC1=−0.0000103495,SxySx2−12ℛKC2=−0.0000103485,SxySx2−12ℛKC3=−0.00001033,SxySx2−12ℛKC4=−0.0000103495,SxySx2−12ℛKC5=−0.0000103395.Therefore, the conditionBP=0.0000006131>Sxy/Sx2−1/2ℛKCi,i=1,2,…,5 is fulfilled.


## 2.1. Proposed Estimator Based on Poisson Regression and Its MSE

The newly constructed estimator of the mean population can be arranged in the frameworks of Zaman and Bulut [10] and Ali et al. [11]. But we are implementing their frameworks based on Poisson regression as(11)y¯U=y¯+bPX¯−x¯.Furthermore, taking advantage of established results, with some basic algebra, and eliminating tedious or futile calculations, we mention the MSE expression of the proposed mean estimator as(12)MSEy¯U=1−fnY¯2Cy2−2BPX¯Y¯ρxyCxCy+BP2X¯2Cx2.


## 2.2. Efficiency Comparisons

The efficiency condition for the proposed mean estimator may be found by comparing the MSE of the proposed estimator in (12) to the MSE of Haydar’s estimators in (9):(13)MSEy¯U<MSEy¯Hi,i=1,2,…,5,Y¯2Cy2−2BpX¯Y¯ρxyCxCy+Bp2X¯2Cx2<ℛKCi2Sx2+2BPℛKCiSx2+BP2Sx2−2ℛKCiSxy−2BPSxy+Sy2.Now(14)Sy2−2BpSxy+Bp2Sx2<ℛKCi2Sx2+2BPℛKCiSx2+BP2Sx2−2ℛKCiSxy−2BPSxy+Sy2,2ℛKCiSxy−ℛKCi2Sx2<2BPℛKCiSx2,2ℛKCiSxy−ℛKCi2Sx22ℛKCiSx2<BP,SxySx2−12ℛKCi<BP.The proposed Poisson mean estimatory¯U=y¯+bPX¯−x¯ is more efficient than Haydar’s estimators, if condition BP>Sxy/Sx2−1/2ℛKCi is satisfied.


## 2.3. Numerical Illustrations

Here, we use two real datasets to evaluate the performance of proposed and existing estimators.Population I (Pop-I). We consider the dataset collected between 2006 and 2010 from the Afyon Respiratory Disease Hospital and the Afyon Environmental Department Air Pollution Unit, which was used in [18]. The number of patients admitted to the hospital on a weekly basis was taken as the dependent variable Y, and PM10 was taken as the explanatory variable X.Population II (Pop-II). We consider the dataset obtained from TUIK of 81 provinces in 2019 used in [18]. The number of people who died due to traffic accidents was taken as a dependent variable Y, and the number of motor vehicles was taken as explanatory variable X.For these datasets, using SRS, we considern=40 and n=20. The characteristics of the two populations, as well as the values of population ratios, are given in Tables 1 and 2.Table 1
Characteristics of two populations (Pop-I and Pop-II).
NotationPop-IPop-IIN21381n4020Y¯4.67605633882.40740741X¯116.1915274308Cy0.6335551591.025504149Cx0.4398461.90331δx−0.690835.77509Sy2.96253961684.50913823Sx51.10636522093Sxy53.5170638124520.6ρxy0.3530.864077723Bℓ0.020490020.02049002BP0.0046.131E − 07Table 2
Values of population ratios.
RatioPop-IPop-IIℛKC10.0402440.000300419ℛKC20.0400930.000300417ℛKC30.0404850.000300380ℛKC40.0404660.000300419ℛKC50.0407960.000300399Based on the MSE values of the proposed estimator and reviewed estimators, we calculate the relative efficiency (RE) values of the proposed estimators with respect to reviewed estimators, sayθi=y¯KCi,y¯Hi,i=1,2,…,5, as follows:(15)Ry¯U=MSEy¯UMSEθi,i=1,2,…,5.Tables3 and 4 give the outcomes for the relative efficiencies.Table 3
Relative efficiencies with respect to Kadilar and Cingi estimators.
EstimatorPop-IPop-IIθ^1141.71711624.092θ^2141.34031624.081θ^3142.32131624.086θ^4142.27361624.091θ^5143.10641624.083Table 4
Relative efficiencies with respect to Haydar estimators.
EstimatorPop-IPop-IIθ^1109.0927125.2937θ^2108.8701125.2912θ^3109.4509125.2454θ^4109.4225125.2921θ^5109.9185125.2689After assessing the relative efficiency values, in Tables3 and 4, it is observed that all values exceed 100. As a result, the proposed Poisson regression estimator is more efficient than the reviewed estimators. On the other hand, it is observed that all relative efficiency values of proposed estimator with respect to Kadilar and Cingi [19] estimators are greater than corresponding values with respect to Haydar estimators, and thus(16)Ry¯U>Ry¯Hi>Ry¯KCi,i=1,2,…,5.The proposed estimator and Haydar’s estimators outperform Kadilar and Cingi [19] estimators, which is an expected result due to the employment of the count data in this analysis, and the proposed estimator has the best performance. The consequences of relative efficiency values are also provided graphically in Figures 1–3 .Figure 1
Relative efficiencies with respect to Kadilar and Cingi estimators.Figure 2
Relative efficiencies with respect to Haydar estimators.Figure 3
Relative efficiencies with respect to reviewed estimators.Furthermore, we investigated the efficiency condition of the proposed estimator as follows.For Pop-I:(17)SxySx2−12ℛKC1=0.000368,SxySx2−12ℛKC2=0.0004435,SxySx2−12ℛKC3=0.0002475,SxySx2−12ℛKC4=0.000257,SxySx2−12ℛKC5=0.000092.Therefore, the conditionBP=0.004>Sxy/Sx2−1/2ℛKCi,i=1,2,…,5is fulfilled.For Pop-II:(18)SxySx2−12ℛKC1=−0.0000103495,SxySx2−12ℛKC2=−0.0000103485,SxySx2−12ℛKC3=−0.00001033,SxySx2−12ℛKC4=−0.0000103495,SxySx2−12ℛKC5=−0.0000103395.Therefore, the conditionBP=0.0000006131>Sxy/Sx2−1/2ℛKCi,i=1,2,…,5 is fulfilled.


## 3. Concluding Remarks

In this article, the mean estimator based on the Poisson regression model has been proposed under simple random sampling. It is found that the proposed estimator produced more efficient results than the ratio estimators proposed in Koҫ [18] and Kadilar and Cingi [19]. For the numerical investigation, the performances of these estimators are compared based on two real populations, and it is seen that the new proposed estimator in terms of relative efficiency is more proficient than the reviewed estimators (see Figures 1–3), as all values surpass 100. As a result, we recommend emphatically using the proposed mean estimator over the other estimators considered in this study for such count data analysis. Furthermore, the estimator developed here can be utilized to calculate new estimates in other count models. The proposed estimator can also be derived in future studies using the concepts of Zaman and Bulut and Ali et al. [10, 11].



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