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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: usman.stat@yahoo.com (Shahzad, U.), Cell: 00923315995587
Email: mhpuno@hotmail.com (Hanif, M.)
Email: irsasajjad@yahoo.com (Sajjad, I.)
Email: anas.uaar@gmail.com (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;
1
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]
4
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
5
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)
8
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]
9
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.
10
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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
Table 5: PRE of proposed and existing estimators for Pop-2
Table 6: PRE of proposed and existing estimators for Pop-3
List of figure captions
Figure 1: Pop-1
Figure 2: Boxplot Pop-1
Figure 3: Scatter-plot Pop-1
Figure 4: Pop-2
Figure 5: Boxplot Pop-2
Figure 6: Scatter-plot Pop-2
Figure 7: Pop-3
Figure 8: Boxplot Pop-3
Figure 9: Scatter-plot Pop-3
Figure 10: MSE of Population-1
Figure 11: MSE of Population-2
Figure 12: MSE of Population-3
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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.
15
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
16
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
17
Table 5: PRE of proposed and existing estimators for Pop-2Under 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 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
Table 6: PRE of proposed and existing estimators for Pop-3Under 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 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
18
Figure 1: Pop-1 Figure 2: Boxplot Pop-1 Figure 3: Scatter-plot Pop-1
Figure 4: Pop-2 Figure 5: Boxplot Pop-2 Figure 6: Scatter-plot Pop-2
Figure 7: Pop-3 Figure 8: Boxplot Pop-3 Figure 9: Scatter-plot Pop-3
19
Figure 10: MSE of Population-1
Figure 11: MSE of Population-2
Figure 12: MSE of Population-320
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