Quantile regression-ratio-type estimators for mean...

Quantile regression-ratio-type estimators for mean

estimation under complete and partial auxiliary

information

1,2∗Usman Shahzad, 2Muhammad Hanif, 3Irsa Sajjad, 4Malik Muhammad Anas

1Department of Mathematics and Statistics, International Islamic University, Islamabad, 46000, Pakistan.2Department of Mathematics and Statistics - PMAS-Arid Agriculture University, Rawalpindi, 46300, Pakistan.

3Department of Lahore Business School - University of Lahore, Islamabad, 46000, Pakistan.4School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, P.R. China.

Email Corressponding Author: [email protected] (Shahzad, U.), Cell: 00923315995587

Email: [email protected] (Hanif, M.)

Email: [email protected] (Sajjad, I.)

Email: [email protected] (Anas, M.)

Abstract

Traditional ordinary least square (OLS) regression is commonly utilized to develop regression-ratio-

type estimators with traditional measures of location. Abid et al. [1] extended this idea and developed

regression-ratio-type estimators with traditional and non-traditional measures of location. In this article,

the quantile regression with traditional and non-traditional measures of location is utilized and a class

of ratio type mean estimators are proposed. The theoretical mean square error (MSE) expressions are

also derived. The work is also extended for two phase sampling (partial information). The pertinence of

the proposed and existing group of estimators is shown by considering real data collections originating

from different sources. The discoveries are empowering and prevalent execution of the proposed group

of estimators is witnessed and documented throughout the article.

Keywords: Quantile regression; OLS regression; Ratio-type estimators; Simple random sampling;

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Two stage sampling.

1 Introduction

It is a commonly known fact that this is the era of information. This fact not only highlights the volume

and pace of information but also underlines the necessity of accurate flow of it. This is all linked with

the true urge of collecting data. It also enables ourselves to profile the surroundings precisely which

in turn helps in making an optimal decision. Till date, the sampling theory and methods are still con-

sidered as a core of literature of multidisciplinary research. Utility of surveys and sampling methods

in applied research can be clearly seen in the surveys for instance, Eurostat on European Union Labor

Force giving periodical statistics of labor participation. Another example depicting the importance of

sampling method is market profiling-segmentation survey providing significant evidence to encourage

market share investigation and National Co-morbidity survey evaluating the substance disorder and anx-

iety levels.

The most philosophical goal of sampling methods is the valuation of the occurrence of patterns or at-

tributes present in inhabitants or population of research. Auxiliary information is widely used to bring

more precision in sample estimates. Laplace in 1814, gave the justification and significance of practicing

auxiliary information in easy but succinct manner. For instance, he revealed that the births register pro-

viding the conditions of the residents, can help in estimating the population of a large country without

holding a census of its residents. But this is only possible if the ratio of masses to annual birth is known.

In addition, McClellan et al. [2] practiced distinction of separations of heart assault injured individual’s

home to closest cardiovascular catheterization medical clinic and the separation to the closest emergency

clinic of any kind, as auxiliary information to help the estimation of impact of catheterization on patients

health.

An alternative method for situations in which abundance of auxiliary information is available is ranked

set sampling (RSS) method due to McIntyre [3], which is shown is far more cost-efficient than simple

random sampling method. See Adel Rastkhiz et al. [4], Zamanzade and Vock [5], Zamanzade and Wang

[6], Zamanzade and Mahdizadeh [7], Zamanzade and Wang [8] and Mahdizadeh and Zamanzade [9] for

more information about this.

A vast amount of literature is available on ratio-type estimators for mean estimation. Such as Koyuncu

[10], Shahzad el al. [11], Hanif and Shahzad [12] have developed some classes of estimators utilizing

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supplementary information under simple random sampling scheme. However, for positive correlation,

traditional regression-ratio-type estimator is better for the estimation of population parameters (see, e.g.,

Abid et al. [13]; Irfan et al. [14]; Naz et al. [15], Abid et al. [16]). Note that traditional regression-type-

ratio estimators based on conventional regression coefficient i.e. known as OLS regression coefficient.

However, the OLS estimate become inappropriate when data are contaminated by outliers. For solving

this issue, there are some modifications available in literature (see, e.g., Zaman et al. [17]; Zaman and

Bulut [18]; Zaman and Bulut [19]; Zaman [20]; Ali et al. [21]). Abid et al. [1] first time introduced

the idea of utilizing non-traditional measures of location, and enhance the estimates of population mean.

They found enhanced estimates of population mean utilizing non-traditional measures. The current study

is based on an extension of Abid et al. [1] work, by incorporating quantile regression.

Outliers have a negative effect on accuracy of the estimators obtained by OLS technique. The presence

of outliers disregard OLS model assumption (Hao et al., [22]). The outliers can change the regression

parameters to be smaller or bigger than the parameters evaluated when the outliers were excluded in

the data (Moore, [23]; Pedhazur, [24]). It was recommended that the outliers can be barred from the

examination if they were, from exhaustive examinations, demonstrated to be non-legitimate perceptions.

Yet, when the outliers are substantial, they can give new bits of knowledge about the nature of the data

(Pedhazur, [24]). It implies that a measurable procedure is required that will capture the outliers in the

investigation but is less impacted by their quality. An elective procedure that has greater ability to settle

a few issues referenced before is called quantile regression. It was created by Boscovich in the eigh-

teenth century, even before the possibility of least squares regression estimators developed (Koenker,

[25]; Koenker and Bassett, [26]). In olden times, utilizations of quantile regression were restricted to fi-

nancial matters or natural investigations, however now it is used in almost all the fields of social sciences

as a data analysis device. In light of preceding lines, we are introducing quantile regression coefficient

rather than OLS, in ratio type mean estimators, by extending the idea of Abid et al. [1].

Inspired by the above documented developments, in this article, we propose a new family of estimators

for estimating population mean using more scrupulous use of auxiliary variable. The objective is met by

utilizing the quantile regression, traditional and non-traditional measures of location of auxiliary vari-

able. The applicability of the scheme is further demonstrated in simple and two stage random sampling

frameworks by employing on three diverse data sets coming from various fields of inquiries. Moreover,

keenly persuaded comparative investigation between Abid et al. [1] and proposed family, by means of

numerical evaluations. The rest of the article is arranged in seven major parts. In section 2, we present

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review of estimators constructed in the study conducted by Abid et al. [1]. The section 3 proposes a

class of quantile regression-ratio-type estimators. Whereas, in section 4, Two stage sampling scheme

version of reviewed and proposed estimators are provided. Numerical illustration and general findings

are documented in section 5. Some final remarks are also given in section 6.

2 A review of Abid et al. estimators

Abid et al. [1] incorporated non-traditional measures of location with traditional measures of location

for mean estimation. They used mid-range of X , denoted by MR, as a first non-traditional measure of

location. Hodges-Lehmann estimator, denoted by HL, as a second non-traditional measure of location.

They also used the weighted average of the population median and two extreme quartiles, known as

Tri-mean, denoted by TM , as a third non-traditional measure of location. According to Abid et al. [1],

these measures are highly sensitive in absence of normality and in presence of outliers. They introduced

the following class of OLS regression-ratio-type estimators utilizing traditional and non-traditional mea-

sures of location as given by

yabi =y + β(ols)(X − x)

(A1x+A2)(A1X +A2) for i = 1, 2, ..., 9

where (X, Y ) be the population means and (x, y) with be the sample means when a simple random

sample of size n is drawn from the population. Further, A1 and A2 are either (0,1) or some known

population measures namely, HL, the Hodges Lemon, MR, the Mid Range, TM, the Tri Mean, Cx, the

coefficient of variation, ρ, the coefficient of correlation, and β(ols) is the OLS regression coefficient. The

family members of yabi are provided in Table 1. The MSE of yabi is given below

MSE(yabi) = θ[S2y+u2

iS2x+2β(ols)uiS

2x+β2

(ols)S2x−2uiSxy−2β(ols)Sxy] for i = 1, 2, ..., 9 (2.1)

where ui =A1Y

A1X +A2and θ = (

1− fn

) for i=1,2,...,9. Further, S2y and S2

x are the unbiased variances,

and Sxy be the co-variance of Y and X .

[Table 1 Here]

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3 Proposed class of quantile regression-ratio-type estimators

Outliers are the observations in a data set which appear to be inconsistent with the rest of that data set.

Presence of outliers significantly effect mean estimation which is one of the most important measure of

central tendency. Mean estimators using OLS regression coefficient are the most ideal choices for the

estimation of population mean i.e. (Y ). However, outliers may have significant impact on the traditional

regression coefficient calculated from OLS tool. Hence the estimate of population mean i.e. (y), based

upon OLS may indicate poor performance. One of the solution is to utilize quantile regression. It can be

used as a robust approach in such circumstances i.e. whenever data is non-normal and contaminated with

outliers. Because it is not sensitive to outliers (Hao et al., [22]; Koenker, [25]). Quantile regression is

like customary OLS regression in a sense that both of them explore connections among endogenous and

exogenous variables. The primary distinction is that OLS regression picks parameter esteems that have

the least squared deviation from the regression line as the parameter estimates, while quantile regression

picks parameter esteems that have the least absolute deviation from the regression line as the parameter

estimates. So we are proposing a group of quantile regression-ratio-type estimators by extending the

idea of abid et al. [1], as given below:

ypi =y + β(q)(X − x)

(A1x+A2)(A1X +A2) for i = 1, 2, ..., 9,

with

β(q) = argminβεRp ρq(v)

n∑i=1

(yi − 〈xi, β〉),

where ρq(v) is a continuous piecewise linear function (or asymmetric absolute loss function), for quantile

qε (0, 1), but nondifferentiable at v = 0. Note that all the notations of ypi have usual meanings as discuss

in previous section. However, β(q) is the quantile regression coefficient for p = 2 variables. For deep

study of quantile regression, interested readers may refer to Koenker and Hallock [27]. The family

members of ypi are provided in Table 2. The MSE of proposed family of estimators is given below

MSE(ypi) = θ[S2y + u2

iS2x + 2β(q)uiS

2x + β2

(q)S2x − 2uiSxy − 2β(q)Sxy] for i = 1, 2, ..., 9. (3.1)

It is worth mentioning that we are using q15th = 0.15, q25th = 0.25 and q35th = 0.35 qunatiles for the

purposes of current article. We see from the consequences of the numerical study conducted in Sec. 5 that

utilizing the quantile regression coefficients, based on these referenced qunatiles, incredibly enhance the

efficiencies of proposed estimators. Note that utilizing these three referenced quantiles, proposed class

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contain twenty seven members. For sack of readability, let us provide twenty seven members of proposed

class with their MSE in compact form, as follows

ypi =

y + β(0.15)(X − x)

(A1x+A2)(A1X +A2) for (q) = (0.15) = 1, 2, ..., 9

y + β(0.25)(X − x)

(A1x+A2)(A1X +A2) for (q) = (0.25) = 1, 2, ..., 9

y + β(0.35)(X − x)

(A1x+A2)(A1X +A2) for (q) = (0.35) = 1, 2, ..., 9,

(3.2)

MSE(ypi) =

θ[S2y + u2

iS2x + 2β(0.15)uiS

2x + β2

(0.15)S2x − 2uiSxy − 2β(0.15)Sxy]

θ[S2y + u2

iS2x + 2β(0.25)uiS

2x + β2

(0.25)S2x − 2uiSxy − 2β(0.25)Sxy]

θ[S2y + u2

iS2x + 2β(0.35)uiS

2x + β2

(0.35)S2x − 2uiSxy − 2β(0.35)Sxy]

(3.3)

[Table 2 Here]

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4 Two stage sampling scheme (partial information)

One can use the two-stage sampling, exactly when the information on population mean of supplemen-

tary variable isn’t available. Neyman [28] was the pioneer who gave the possibility of this sampling

scheme in assessing the population parameters. The two-stage sampling is monetarily insightful and

more straightforward as well. This sampling plan is used to get the data about supplementary variable

productively by choosing a more noteworthy sample from first stage and moderate size sample at the

second stage. Sukhatme [29] used two-stage sampling plan to propose a general ratio type estimator.

Cochran [30] is another reference for more comprehensive study about two-stage sampling.

Under two-stage sampling plan, we select a first stage sample of size na units from the population of size

N with the assistance of SRSWOR plan. After that we select a second stage sample i.e. nb, a sub-sample

of the first stage sample i.e. na.

4.1 Reviewed and proposed estimators in two stage sampling scheme

In this sub-section, we are presenting Abid et al. [1] and proposed family of estimators under two-stage

sampling scheme as given by

y′abi

=yb + β(ols)(xa − xb)

(A1xb +A2)(A1xa +A2) for i = 1, 2, ..., 9

y′pi =

yb + β(q)(xa − xb)(A1xb +A2)

(A1xa +A2) for i = 1, 2, ..., 9

where (xb, yb) representing sample means at second stage and xa be the sample mean at first stage.

Further, A1 and A2 have the same meanings as described in previous section. The family members of

y′abi

and y′pi are available in Table 1 and 2, respectively.

Utilizing Taylor series method we are obtaining MSE for y′abi

as follows

MSE(y′abi

) = d1Σd′1, (4.1)

where

d1 =

[δh(yb, xa, xb)

δyb|Y ,X

δh(yb, xa, xb)

δxa|Y ,X

δh(yb, xa, xb)

δxb|Y ,X

],

d1 =[1 (ui + β(ols)) − (ui + β(ols))

].

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∑=

V (yb) Cov(yb, xa) Cov(yb, xb)

Cov(xa, yb) V (xa) Cov(xb, xa)

Cov(xb, yb) Cov(xb, xa) V (xb)

where

V (yb) = λ2S2y ,

V (xa) = λ1S2x,

V (xb) = λ2S2y ,

Cov(yb, xa) = Cov(xa, yb) = λ1Syx,

Cov(yb, xb) = Cov(xb, yb) = λ2Syx,

Cov(xa, xb) = Cov(xb, xa) = λ1S2x.

Utilizing these defined notations of variances and co-variances, and hence substituting the values of d1

and Σ in equation (4.1), MSE expressions of y′abi

as follows

MSE(y′abi

) = λ2S2y + (λ2 − λ1)[(ui + β(ols))

2S2x − 2(ui + β(ols))Syx], for i = 1, 2, ..., 9.

By replacing β(ols) with β(q), we can easily find MSE of proposed class of estimators as follows

MSE(y′pi) = λ2S

2y + (λ2 − λ1)[(ui + β(q))

2S2x − 2(ui + β(q))Syx],

where λ1 =

(1

na− 1

N

)and λ2 =

(1

nb− 1

N

).

The twenty seven family members of proposed class with their MSE in compact form, under partial

information, as follows

y′pi =

yb + β(0.15)(xa − xb)(A1xb +A2)

(A1xa +A2) for (q) = (0.15); i = 1, 2, ..., 9

yb + β(0.25)(xa − xb)(A1xb +A2)

(A1xa +A2) for (q) = (0.25); i = 1, 2, ..., 9

yb + β(0.35)(xa − xb)(A1xb +A2)

(A1xa +A2) for (q) = (0.35); i = 1, 2, ..., 9.

(4.2)

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MSE(y′pi) =

λ2S2y + (λ2 − λ1)[(ui + β(0.15))

2S2x − 2(ui + β(0.15))Syx]

λ2S2y + (λ2 − λ1)[(ui + β(0.25))

2S2x − 2(ui + β(0.25))Syx]

λ2S2y + (λ2 − λ1)[(ui + β(0.35))

2S2x − 2(ui + β(0.35))Syx]

(4.3)

5 Numerical illustration

In this section, we are assessing the performance of proposed and existing estimators, using three real

life data sets as given below:

Population-1 (Pop-1): We use the data set of Singh [31]. In this data set, “amount of non-real estate

farm loans during 1977” is taken as auxiliary variable (X), while “amount of real estate farm loans dur-

ing 1977” is taken as study variable (Y). Further N = 50, n = na = 20 and nb = 16.

Population-2 (Pop-2): We use “UScereals” data set. Which describes 65 commonly available breakfast

cereals in the USA, based on the information available on the mandatory food label on the packet. The

measurements are normalized here to a serving size of one American cup. The data come from ASA

Statistical Graphics Exposition and used by Venables and Ripley [32]. As the data set contains a num-

ber of variables. So, “grams of fibre in one portion” is taken as auxiliary variable (X), while “grams of

potassium” is taken as study variable (Y). Further N = 65, n = na = 20 and nb = 15.

Population-3 (Pop-3): Third population is also considered from “UScereals” data set. Where, “grams

of sodium in one portion” is taken as auxiliary variable (X), while “Number of calories” is taken as study

variable (Y). Further N = 65, n = na = 20 and nb = 15.

The remaining characteristics of all the three populations are provided in Table 3. Further, we draw His-

togram, Box-plot and Scatter-plot for all the three referenced populations, see figures 1-9. These figures

show non-normality and presence of outliers, hence suitable for utilization of non-conventional mea-

sures of location as Abid et al. [1], and for quantile regression too.

[Table 3 Here]

[Figures 1− 9 Here]

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As each existing estimator based on β(ols) while their corresponding proposed estimators based

on β(0.15),β(0.25), and β(0.35). So results of each existing estimator with their corresponding proposed

estimators (say) θ, are provided in a same row, such as θ = (β(ols), β(0.15), β(0.25), β(0.35)), in Tables 4

- 6, under complete information. Similarly, PRE of existing and proposed estimators (say) θ′

are also

available in Tables 4 - 6, under partial information. Further, MSE of (θ, θ′) presented graphically in

figures 10-12.

[Tables 4− 6 Here]

[Figures 10− 12 Here]

5.1 Results

The consequences of the numerical study are given in Tables 4-6 under complete and partial information

setting. From Tables 4-6, we see that every one of the estimator ypi, with i = 1, . . . , 9, considered in

the proposed class, beat their corresponding estimator of existing class i.e. yabi. Also, we see that the

proposed estimators are more productive than the existing estimators in case of partial information. From

figures 10-12, we see that, proposed estimators have small MSE as compare to their corresponding ones.

Hence , in both cases, the proficiency increase yielded by the new estimators is striking in populations

1-3.

6 Final remarks

In this paper, beginning from some ongoing contribution of Abid et. al [1] for mean estimation under

design/configuration based sampling, we have constructed a class of quantile regression-ratio-type esti-

mators for the population mean whenever the data is non-normal and contaminated with outliers. The

proposed class beats existing ones. In case of complete information, where population mean of auxil-

iary variable is known, we have determined the MSE expressions, theoretically. Further, we have also

introduced the existing and proposed classes to the case of partial information. In the article, three real

life populations have been considered. The numerical computations well underscore the predominance

of the proposed class in both the complete and partial information setting, at least for the considered

experimental circumstances in the article. Hence, survey practitioners should use proposed class since it

might expand the odds of getting progressively proficient evaluations of obscure objective parameters.

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List of table captions

Table 1: Family members of Abid et al. (2016b)

Table 2: Family members of proposed class

Table 3: Characteristics of Populations

Table 4: PRE of proposed and existing estimators for Pop-1



List of figure captions

Figure 1: Pop-1

Figure 2: Boxplot Pop-1

Figure 3: Scatter-plot Pop-1

Figure 4: Pop-2



Figure 7: Pop-3



Figure 10: MSE of Population-1



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Authors’ Biographies

Usman Shahzad obtained his M.Sc. degree in statistics from International Islamic University, Islam-

abad, Pakistan. He completed his M.Phil. in statistics from Pir Mehr Ali Shah Arid Agriculture Uni-

versity Rawalpinid, Pakistan. Currently he is studying his Ph.D. in statistics from International Islamic

University, Islamabad, Pakistan. He served as a lecturer in Pir Mehr Ali Shah Arid Agriculture Uni-

versity Rawalpinid, Pakistan. He has published more than 25 research papers in research journals. His

research interests include Survey Sampling, Extreme value theory, Stochastic Process, Probability and

Non-parametric statistics.

Muhammad Hanif obtained his M.Sc. degree in statistics from University of Agriculture, Faisalabad,

Pakistan. He completed his M.Phil. in statistics from Government College University, Lahore, Pakistan.

He completed his Ph.D. degree in Statistics from Zhejiang University, Hangzhou, China. Currently he

is an Associate Prof. and serving as a chairman of the Department of Mathematics and Statistics in Pir

Mehr Ali Shah Arid Agriculture University Rawalpinid, Pakistan. He has published more than 50 re-

search papers in research journals. His main research interest is in Stochastic Process, Probability and

Mathematical Statistics, Survey Sampling and Non Parametric Estimation.

Irsa Sajjad has completed her M.Sc. and M.Phil. in statistics from Pir Mehr Ali Shah Arid agriculture

university Rawalpindi, Pakistan. She is serving as a lecturer in Department of Lahore Business School-

University of Lahore, Islamabad, Pakistan. Her research interests include Sampling, Stochastic Process,

Methods and Probability.

Malik Muhammad Anas has completed his M.Sc. and M.Phil. in statistics from Pir Mehr Ali Shah

Arid Agriculture University Rawalpindi, Pakistan. Currently he is studying his Ph.D. in statistics from

Nanjing University of Science and Technology, Nanjing, China. He served as a lecturer in Pir Mehr Ali

Shah Arid Agriculture University Rawalpindi, Pakistan. His research interests include Non-parametric

regression, Stochastic Process, Kernel Smoothing and Sampling Methods.

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Table 1: Family members of Abid et al. (2016b)Under complete information Under partial information

Estimators (A1, A2) Estimators

yab1 (1, TM) y′

ab1

yab2 (Cx, TM) y′

ab2

yab3 (ρ, TM) y′

ab3

yab4 (1,MR) y′

ab4

yab5 (Cx,MR) y′

ab5

yab6 (ρ,MR) y′

ab6

yab7 (1, HL) y′

ab7

yab8 (Cx, HL) y′

ab8

yab9 (ρ,HL) y′

ab9

Table 2: Family members of proposed classUnder complete information Under partial information

Estimators (A1, A2) Estimators

yp1 (1, TM) y′p1

yp2 (Cx, TM) y′p2

yp3 (ρ, TM) y′p3

yp4 (1,MR) y′p4

yp5 (Cx,MR) y′p5

yp6 (ρ,MR) y′p6

yp7 (1, HL) y′p7

yp8 (Cx, HL) y′p8

yp9 (ρ,HL) y′p9

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Table 3: Characteristics of PopulationsY = 555.4345 X = 878.1624 HL = 628.0575

Sy = 584.826 Sx = 1084.678 MR = 1964.483

(Pop-1) ρ = 0.8038 Sxy = 509910.4 TM = 536.4088

Cx = 1.235168 β(ols) = 0.4334034 β(0.25) = 0.3542911

Cy = 1.052916 β(0.15) = 0.3447828 β(0.35) = 0.3570439

Y = 159.1197 X = 3.870844 HL = 2.5

Sy = 180.2886 Sx = 6.133404 MR = 15.15151

(Pop-2) ρ = 0.9638 Sxy = 1065.827 TM = 2.119403

Cx = 1.584513 β(ols) = 28.3324 β(0.25) = 27.175

Cy = 1.133037 β(0.15) = 22.77778 β(0.35) = 27.72727

Y = 149.4083 X = 237.8384 HL = 235.1244

Sy = 62.41187 Sx = 130.6296 MR = 393.9394

(Pop-3) ρ = 0.5286 Sxy = 4310.041 TM = 233.5

Cx = 0.549237 β(ols) = 0.2525795 β(0.25) = 0.1737113

Cy = 0.4177271 β(0.15) = 0.198324 β(0.35) = 0.1578947

Table 4: PRE of proposed and existing estimators for Pop-1Under complete information Under partial information

θ β(ols)) β(0.15)) β(0.25)) β(0.35)) θ′

β(ols)) β(0.15)) β(0.25)) β(0.35))

θ1 3769.87 4961.64 4816.81 4775.65 θ′1 2435.90 2604.31 2587.30 2582.32

θ2 3436.58 4510.95 4379.39 4342.06 θ′2 2373.92 2548.62 2530.73 2525.50

θ3 4176.24 5500.01 5341.16 5295.90 θ′3 2501.43 2661.29 2645.43 2640.77

θ4 6870.27 8480.31 8325.77 8269.79 θ′4 2772.78 2864.23 2856.77 2854.50

θ5 6313.21 7977.74 7805.28 7754.63 θ′5 2732.24 2839.06 2829.79 2827.01

θ6 7407.26 8886.77 8759.47 8720.64 θ′6 2807.01 2882.79 2877.13 2875.38

θ7 4056.99 5343.50 5188.49 5144.36 θ′7 2483.21 2645.66 2629.46 2624.70

θ8 3680.05 4840.89 4699.50 4659.34 θ′8 2420.02 2590.18 2572.93 2567.88

θ9 4509.32 5929.52 5761.35 5713.30 θ′9 2548.41 2700.73 2685.85 2681.47

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θ β(ols)) β(0.15)) β(0.25)) β(0.35)) θ′

β(ols)) β(0.15)) β(0.25)) β(0.35))

θ1 3254.74 4965.69 3531.74 3395.46 θ′1 2023.94 2256.74 2072.61 2049.36

θ2 2509.54 3638.07 2697.71 2605.44 θ′2 1858.68 2089.86 1906.10 1883.40

θ3 3334.54 5113.07 3621.65 3480.35 θ′3 2038.54 2271.03 2087.25 2063.99

θ4 19012.92 36065.05 22038.47 20539.70 θ′4 2691.95 2781.61 2717.38 2705.66

θ5 12379.90 24741.64 14219.13 13301.50 θ′5 2597.09 735.12 2631.47 2615.39

θ6 19610.21 36757.77 22728.23 21185.38 θ′6 2697.55 2783.55 2722.82 2710.90

θ7 3643.28 5692.21 3970.58 3809.31 θ′7 2090.69 2321.58 2139.42 2116.18

θ8 2730.46 4022.14 2943.84 2839.09 θ′8 1913.91 2146.56 1961.90 1938.94

θ9 3741.34 5879.02 4081.73 3913.95 θ′9 2105.94 2336.16 2154.64 2131.47


θ β(ols)) β(0.15)) β(0.25)) β(0.35)) θ′

β(ols)) β(0.15)) β(0.25)) β(0.35))

θ1 2488.90 2824.16 2981.56 3083.08 θ′1 1853.20 1935.58 1969.90 1990.75

θ2 3063.47 3403.58 3546.67 3632.31 θ′2 1986.79 2050.79 2075.07 2088.95

θ3 3098.65 3436.36 3577.02 3660.63 θ′3 1993.86 2056.48 2080.05 2093.43

θ4 2992.03 3335.88 3483.27 3572.64 θ′4 1972.09 2038.78 2064.49 2079.33

θ5 3490.14 3770.11 3867.94 3918.58 θ′5 2065.65 2110.31 2124.79 2132.08

θ6 3515.64 3789.35 3883.19 3930.94 θ′6 2069.93 2113.20 2127.01 2133.83

θ7 2495.39 2831.08 2988.55 3090.05 θ′7 1854.93 1937.14 1971.36 1992.15

θ8 3069.89 3409.59 3552.25 3637.53 θ′8 1988.09 2051.83 2075.99 2089.78

θ9 3104.98 3442.22 3582.43 3665.65 θ′9 1995.12 2057.49 2080.93 2094.22

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Figure 1: Pop-1 Figure 2: Boxplot Pop-1 Figure 3: Scatter-plot Pop-1



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