On Enhanced Ratio-Type Estimators Using Quantile Regression for Finite Population Mean under Robustness and Empirical Validation

Abstract

When the conditions of traditional regression analysis aren't met, an alternative method called quantile regression is utilized to estimate the value of the study variable across different quantiles of the distribution. This study proposes leveraging quantile regression information to develop ratio-type estimators for the finite population mean, particularly under robust measures of auxiliary variables in simple random sampling (SRS) without replacement. The performance of these proposed families of estimators is compared with existing studies using metrics such as mean squared error (MSE) equations and percentage relative efficiency (PRE). Additionally, this article incorporates simulation studies. Moreover, various real-world datasets are considered for empirical investigation to validate the theoretical findings.

Introduction

The process of creating a sample set is acknowledged as part of sampling theory. Many research sectors nowadays, including opinion surveys, marketing research, science, engineering, health, and social sciences, use the sampling method. In practical scenarios, a majority of variables, like household income, the poverty index, taxes, consumption, and output, have extreme values ranging from high to low. These distributions are referred to as highly skewed or non-normal. Since the quantile regression can handle extreme values in non-normal distributions, it is often utilized. There is an extensive amount of literature available for calculating the proportion, variance, coefficient of variation, and mean/total of a finite population in sample surveys (Mukhtar et al. 2023; Gupta and Shabbir 2008; Hussain et al. 2024; Gulzar et al. 2022; Ahmad et al. 2023). Several efforts have been made to devise effective techniques for estimating the unknown populations mean using quantile regression with auxiliary information. In situations where real-life datasets exhibit non-normality, skewness, and extreme values, reference (Shahzad et al. 2022a) introduced a family of ratio-type estimators based on quantile regression for population mean estimation. Quantile regression proves to be a valuable and effective tool, particularly for visualizing changes in the conditional distribution of real-life applications affected by extreme values (Baur et al. 2004). For skewed or non-normal datasets with outliers in SRS without replacement, reference (Anas et al. 2021) proposed a family of ratio-type estimators utilizing quantile regression information and employing L-moments for finite population mean estimation. The proposed estimators exhibited exceptional performance relative to existing methods, as evidenced by rigorous theoretical MSE derivations and compelling empirical validation across three real-world datasets. Reference (Anas et al. 2022) introduces three advanced classes of estimators designed for missing data using robust quantile regression, evaluating their effectiveness across three distinct populations. The comprehensive numerical analysis compellingly illustrates the superior performance of these new estimators across all scenarios. According to Koc and Koc (2023), new ratio-type estimators using quantile regression, significantly enhancing the precision of finite population mean estimates in stratified random sampling. These estimators show superior performance over traditional methods, particularly by achieving lower mean square error, supported by theoretical and practical validation. The paper (Shahzad et al. 2022b) introduces a robust class of quantile regression estimators under the Särndal approach for both non-sensitive and sensitive variables in stratified random sampling. These estimators significantly improve the accuracy of population mean estimates by effectively managing extreme observations and using robust regression techniques, leading to a notable reduction in mean square error. The study (Shahzad et al. 2023) highlights that the proposed robust quantile regression mean estimator for systematic sampling is significantly more efficient, achieving over a 30% improvement in efficiency when estimating the volume of timber compared to traditional ratio estimators, particularly in the presence of outliers. The reference (Shahzad et al. 2021) introduces quantile robust regression-type estimators based on minimum covariance determinant, which offer enhanced robustness for estimating mean parameters. Their methods improve the accuracy and stability of regression estimates under the influence of outliers.

In our present investigation, we employ quantile regression to estimate the finite population mean, considering both conventional and non-traditional measures of auxiliary variables in SRS without replacement. To achieve this goal, we introduced three families of estimators.

The novelty of this work lies in several aspects:

1. Introducing the concept of researching ratio-type estimators using quantile regression for estimating an unknown population mean under robust measures of auxiliary variables.

2. Depending on the availability of auxiliary data, our proposed families of estimators have the potential to generate numerous ratio-type estimators using quantile regression.

3. Evaluating the performance of the suggested ratio-type estimators using quantile regression in the presence of outliers through a robustness study.

The subsequent sections of the article are structured as follows:

Section 2 delivers a comprehensive exposition on well-known traditional and reviewing estimators. Section 3 introduces suggested families of ratio-type estimators using quantile regression for estimating the population mean, incorporating robust measures of auxiliary information. This section also includes the provision of mean square error (MSE) expressions. In Sect. 4, the potential of novel families of ratio-type estimators using quantile regression is explored, contrasting with well-known traditional and reviewing estimators, through the analysis of real-world datasets and simulated exercises. Section 5 conducts a robustness study of the suggested families of estimators using real-life application. Finally, Sect. 6 presents concluding thoughts and additional recommendations.

Some Well-Known Traditional and Reviewing Estimators

In this section, we explore several well-known traditional and existing quantile regression ratio-type estimators proposed by different authors for estimating the population mean under SRS without replacement.

A review of Abid et al. (2016) Estimators

For mean estimation, Abid et al. (2016) integrated robust or non-traditional measures of location alongside traditional ones. They introduced the Tri-mean of the auxiliary variable (X), denoted by $T_M$, which is a weighted average of the population median and two extreme quartiles, as the first robust or non-traditional measure of location. The second robust or non-traditional measure of location is the Mid-range of the auxiliary variable (X), denoted by $M_R$, additionally, the authors utilized the Hodges-Lehmann estimator, denoted by $H_L$, as the third robust or non-traditional measure of location. According to Abid et al. (2016), these measures exhibit high sensitivity in the presence of outliers and the absence of normality. By incorporating either robust or non-traditional measures of location with traditional ones, they introduced the following class of OLS regression ratio-type estimators given as:

1

$${\widehat{Y}}_{\mathit{abk}}=\frac{\overline{y}+{\widehat{\beta }}_{\left(\mathit{ols}\right)}\left(\overline{X}-\overline{x}\right)}{\left(\overline{x}\alpha +\gamma \right)}\left(\overline{X}\alpha +\gamma \right)$$

For $k=\text{1,2},\cdots ,9$.

Here, ($\overline{Y},\overline{X}$ denotes the population means of the study and auxiliary variables, while ($\overline{y},\overline{x}$) represent the sample means. Moreover, $\alpha$ and $\gamma$ are some known traditional and robust or non-traditional measures of location of $X$ such as coefficient of variation denoted by $\left(C_x\right)$, tri-mean $\left(T_M\right)$, mid-range $\left(M_R\right)$, Hodges-Lehmann $\left(H_L\right)$. The correlation coefficient between $\left(Y,,,X\right)$ denoted by $\left(\mathrm{\rho }\right)$. All the family members of Abid et al. (2016) listed in Table 1.

2

$$MSE\left({\widehat{Y}}_{\mathit{abk}}\right)=\theta \left(S_y^2+{\mu }_i^2{S}_x^2+2{\widehat{\beta }}_{\left(\mathit{ols}\right)}{\mu }_i{S}_x^2+{\widehat{\beta }}_{\left(\mathit{ols}\right)}^2{S}_x^2-2{\mu }_i{S}_{\mathit{xy}}-2{\widehat{\beta }}_{\left(\mathit{ols}\right)}{S}_{\mathit{xy}}\right)$$where $\theta =\frac{1-f}{N},$${\mu }_i=\frac{\alpha \overline{Y}}{\alpha \overline{Y}+\gamma }$ and ${\widehat{\beta }}_{\left(o,l,s\right)}=\frac{\mathrm{\rho }{S}_y}{S_x}$. Furthermore, $S_x^2$ and $S_y^2$ represents the unbiased variances, while $S_{\mathit{xy}}$ denotes the covariance of $X$ and $Y$.

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

$k$	1	2	3	4	5	6	7	8	9
$\alpha$	1	$C_x$	ρ	1	$C_x$	ρ	1	$C_x$	ρ
$\gamma$	$T_M$	$T_M$	$T_M$	$M_R$	$M_R$	$M_R$	$H_L$	$H_L$	$H_L$

Shahzad et al. (2022a) Suggested a General Class of Ratio Type Estimators Using Quantile Regression

This study expands upon the ideas introduced in Abid et al. (2016) by introducing a general class of quantile regression ratio-type estimators specifically designed for data contaminated with outliers and exhibiting non-normality, as expressed in Eq. (3).

3

$${\widehat{Y}}_{uk\left(q\right)}=\frac{\overline{y}+{\widehat{\beta }}_{\left(q\right)}\left(\overline{X}-\overline{x}\right)}{\left(\overline{x}\alpha +\gamma \right)}\left(\overline{X}\alpha +\gamma \right)$$

For $k=\text{1,2},\cdots ,9$.

With:$${\widehat{\beta }}_{\left(q\right)}={\mathit{argmin}}_{\beta ϵR^P}{\rho }_q\left(v\right)\sum _{i=1}^n\left(y_i,-,⟨x_i,\beta ⟩\right),$$

Here ${\rho }_q\left(v\right)$ represents the asymmetric absolute loss function for quantile $qϵ$ (0, 1), which is non-differentiable at $v=0$. However, ${\widehat{\beta }}_{\left(q\right)}$ denotes the quantile regression coefficients for $p=2$ variables, for more detailed insights into quantile regression, interested readers may consult references (Koenker and Hallock 2001; Koenker and Bassett 1978; Koenker 2005). The MSE of suggested general family of ${\widehat{Y}}_u$ is given as;

4

$$MSE\left({\widehat{Y}}_{uk\left(q\right)}\right)=\theta \left(S_y^2+{\mu }_i^2{S}_x^2+2{\widehat{\beta }}_{\left(q\right)}{\mu }_i{S}_x^2+{\widehat{\beta }}_{\left(q\right)}^2{S}_x^2-2{\mu }_i{S}_{\mathit{xy}}-2{\widehat{\beta }}_{\left(q\right)}{S}_{\mathit{xy}}\right)$$

They used fifteenth quantile ($q^{15th}=0.15$), Twenty-fifth quantile ($q^{25th}=0.25$) and thirty-fifth quantile ($q^{35th}=0.35$) in this study.

A Review of Irfan et al. (2018) Proposed Class of Ordinary Least Square Regression Ratio Type Estimators

Irfan et al. (2018) Introduced a family of (OLS-R) ordinary least square regression ratio-type estimators aimed at estimating the population mean. They utilized traditional measures of location, including the coefficient of variation, correlation coefficient, and kurtosis, for mean estimation. The authors presented the following family of OLS-R ratio-type estimators, incorporating such traditional measures of location such as.

5

$${\overline{Y}}_{\mathit{IRk}}=h\left[\overline{y}{\left(\frac{\overline{X}\alpha +R\gamma }{\overline{x}\alpha +R\gamma }\right)}^{\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }}+{\widehat{\beta }}_{\left(\mathit{ols}\right)}\left(\overline{X}-\overline{x}\right)\right]$$

For $k=\text{1,2},\cdots ,9$.

Where $h$ is the unknown constant and $\left(\overline{Y},,,\overline{X}\right)$ are the population means and $\left(\overline{y},,,\overline{x}\right)$ represents the sample means. Furthermore $\alpha$ and $\gamma$ are either $(\text{0,1})$ or some known traditional population parameters, such as coefficient of variation ${(C}_x)$, correlation coefficient $\left({\rho }_{\mathit{yx}}\right)$, kurtosis $\left({\beta }_{2\left(x\right)}\right)$ and OLS regression coefficient $\left({\widehat{\beta }}_{\left(o,l,s\right)}\right)$.

6

$$MS{E}_{\mathit{min}}\left({\overline{Y}}_{\mathit{IRk}}\right)\approx \left[{\overline{Y}}^2-\frac{{\phi }_3^2}{2{\phi }_4}\right]$$where ${\delta }_s=\left(\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }\right)$$$h=\frac{{\phi }_3}{{\phi }_4}=\frac{{\overline{Y}}^2+\frac{\theta }{2}\left[\left({\delta }_s^4,+,{\delta }_s^3\right),R^2,S_x^2,-,2,{\delta }_s^2,R,S_{\mathit{xy}}\right]}{{\overline{Y}}^2+\theta \left[S_y^2,+,\left\{2,{\delta }_s^4,R^2,+,{\widehat{\beta }}_{(ols)}^2,+,{\delta }_s^3,R^2,+,2,{\delta }_s^2,{\widehat{\beta }}_{\left(o,l,s\right)},R\right\},S_x^2,-,\left\{4,{\delta }_s^2,R,+,2,{\widehat{\beta }}_{\left(o,l,s\right)}\right\},S_{\mathit{xy}}\right]}$$

Suggested General Family of Ratio Type Estimators Using Quantile Regression

The primary drawback of the proposed class of (OLS-R) ordinary least square regression ratio-type estimators for population mean by Irfan et al. (2018) is their reliance on traditional measures of location with OLS-R under simple random sampling without replacement (SRSWOR). The family of OLS-R ratio-type estimators proposed by Irfan et al. (2018) demonstrates inefficiency in handling outliers within the data. The presence of extreme values significantly affects mean estimation using OLS-R (ordinary least square regression). As a result, extreme values could significantly affect the conventional regression coefficients determined with OLS-R technique. Quantile regression is one method of solving the problem. It is not sensitive to outliers; it can be utilized as a stronger strategy when the data sets are non-normal.

Inspired by references (Irfan et al. 2018; Kadilar and Cingi 2006a, b) for ratio, regression and ratio type regression estimator and also taking inspiration from reference (Shahzad et al. 2022a) for ratio-type estimators using quantile regression.

7

$${\overline{Y}}_{\mathit{jk}}=h\left[\overline{y}{\left(\frac{\overline{X}\alpha +R\gamma }{\overline{x}\alpha +R\gamma }\right)}^{\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }}+{\widehat{\beta }}_{\left(q\right)}\left(\overline{X}-\overline{x}\right)\right]$$

For $k=\text{1,2},\cdots ,9$.

Where $\alpha$ and $\gamma$ may take on any constant values or functions of robust or non-traditional measures of location, such as Tri-Mean ($T_M$), Hodges-Lehmann estimator ($H_L$), and Mid-range ($M_R$) as well as conventional measures associated with $X$ variable. Further,${\widehat{\beta }}_{\left(q\right)}\left[q,=,0.15,,,0.25,,,0.35\right]$ represents the quantile regression coefficients for $p=2$ variables. Furthermore, for a more comprehensive understanding of conventional and robust or non-conventional measures, interested readers may refer to Irfan et al. (2021). According to Irfan et al. (2021), these measures exhibit high sensitivity in the presence of outliers and in the absence of normality.

Remark 3.1

Substituting different values of ${\widehat{\beta }}_{\left(q\right)}\left[q,=,0.15,,,0.25,,,0.35\right]$ in Eq. (7), we obtain following families of ratio estimators using quantile regression:

(i) When substituting ${\widehat{\beta }}_{\left(q\right)}$= 0.15; the suggested family of ratio estimators using quantile regression is reduced to:

8

$${{\overline{Y}}_{\mathit{jk}}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}\alpha +R\gamma }{\overline{x}\alpha +R\gamma }\right)}^{\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$$

(ii) When substituting ${\widehat{\beta }}_{\left(q\right)}$= 0.25; the suggested family of ratio estimators using quantile regression is reduced to:

9

$${{\overline{Y}}_{\mathit{jk}}}^{⊚}=h\left[\overline{y},{\left(\frac{\overline{X}\alpha +R\gamma }{\overline{x}\alpha +R\gamma }\right)}^{\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }},+,{\widehat{\beta }}_{\left(0.25\right)},\left(\overline{X},-,\overline{x}\right)\right]$$

(iii) When substituting ${\widehat{\beta }}_{\left(q\right)}$= 0.35; the suggested family of ratio estimators using quantile regression is reduced to:

10

$${{\overline{Y}}_{\mathit{jk}}}^{⊛}=h\left[\overline{y},{\left(\frac{\overline{X}\alpha +R\gamma }{\overline{x}\alpha +R\gamma }\right)}^{\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }},+,{\widehat{\beta }}_{\left(0.35\right)},\left(\overline{X},-,\overline{x}\right)\right]$$

Remark 3.2

Various optimal estimators can be derived by substituting known conventional and robust or non-conventional measures of the auxiliary variable in lieu of α and γ in Eq. (7), or by employing suitable constants.

Remark 3.3

When robust or non-conventional known parameters of auxiliary variables, such as tri-mean $\left(T_M\right)$, mid-range $\left(M_R\right)$, Hodges-Lehmann $\left(H_L\right)$ are incorporated with a linear combination of the coefficient of variation $\left(C_x\right)$ and the correlation coefficient between $\left(Y,,,X\right)$ of an auxiliary variable in Eq. (7), various series of estimators are derived, such as ${{\overline{Y}}_{\mathit{jk}}}^{\odot }$, ${{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$. Some members of the suggested class of ratio type estimators using quantile regression ${{\overline{Y}}_{\mathit{jk}}}^{\odot }$ are presented in the Table 2. Substituting the values of $\alpha$ and $\gamma$ in ${{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$, we obtain a number of estimators.

11

$${\mathit{MSE}}_{\mathit{min}}\left({\overline{Y}}_{\mathit{jk}}\right)\approx \left[{\overline{Y}}^2,-,\frac{{\phi }_3^2}{2{\phi }_4}\right]$$where ${\delta }_s=\left(\frac{\overline{X}\alpha }{\overline{X}\alpha +R\gamma }\right)$

Table 2. Some members of ${{\overline{Y}}_{\mathit{jk}}}^{\odot }$

Family members of ${{\overline{Y}}_{\mathit{si}}}^{\odot }$ quantile regression type estimators
${{\overline{Y}}_{j1}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}+R{T}_M}{\overline{x}+R{T}_M}\right)}^{\frac{\overline{X}}{\overline{X}+R{T}_M}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$	${{\overline{Y}}_{j5}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}{C}_x+R{M}_R}{\overline{x}{C}_x+R{M}_R}\right)}^{\frac{\overline{X}{C}_x}{\overline{X}{C}_x+R{M}_R}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$
${{\overline{Y}}_{j2}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}{C}_x+R{T}_M}{\overline{x}{C}_x+R{T}_M}\right)}^{\frac{\overline{X}{C}_x}{\overline{X}{C}_x+R{T}_M}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$	${{\overline{Y}}_{j6}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}\mathrm{\rho }+R{M}_R}{\overline{x}\mathrm{\rho }+R{M}_R}\right)}^{\frac{\overline{X}\mathrm{\rho }}{\overline{X}\mathrm{\rho }+R{M}_R}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$
${{\overline{Y}}_{j3}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}\mathrm{\rho }+R{T}_M}{\overline{x}\mathrm{\rho }+R{T}_M}\right)}^{\frac{\overline{X}\mathrm{\rho }}{\overline{X}\mathrm{\rho }+R{T}_M}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$	${{\overline{Y}}_{j7}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}+R{H}_L}{\overline{x}+R{H}_L}\right)}^{\frac{\overline{X}}{\overline{X}+R{H}_L}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$
${{\overline{Y}}_{j4}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}+R{M}_R}{\overline{x}+R{M}_R}\right)}^{\frac{\overline{X}}{\overline{X}+R{M}_R}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$	${{\overline{Y}}_{j8}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}{C}_x+R{H}_L}{\overline{x}{C}_x+R{H}_L}\right)}^{\frac{\overline{X}{C}_x}{\overline{X}{C}_x+R{H}_L}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$
${{\overline{Y}}_{j9}}^{\odot }=h\left[\overline{y},{\left(\frac{\overline{X}\mathrm{\rho }+R{H}_L}{\overline{x}\mathrm{\rho }+R{H}_L}\right)}^{\frac{\overline{X}\mathrm{\rho }}{\overline{X}\mathrm{\rho }+R{H}_L}},+,{\widehat{\beta }}_{\left(0.15\right)},\left(\overline{X},-,\overline{x}\right)\right]$

$$h=\frac{{\phi }_3}{{\phi }_4}=\frac{{\overline{Y}}^2+\frac{\theta }{2}\left[\left({\delta }_s^4,+,{\delta }_s^3\right),R^2,S_x^2,-,2,{\delta }_j^2,R,S_{\mathit{xy}}\right]}{{\overline{Y}}^2+\theta \left[S_y^2,+,\left\{2,{\delta }_s^4,R^2,+,{\widehat{\beta }}_{(q)}^2,+,{\delta }_s^3,R^2,+,2,{\delta }_s^2,{\widehat{\beta }}_{\left(q\right)},R\right\},S_x^2,-,\left\{4,{\delta }_s^2,R,+,2,{\widehat{\beta }}_{\left(q\right)}\right\},S_{\mathit{xy}}\right]}$$

Real-Life Applications

In this section, we compare the ${\overline{Y}}_{\mathit{jk}}$ estimators with the reviewing estimators under study using real-life applications and a simulation study. We evaluate the performance of the suggested classes of ratio-type estimators using quantile regression in terms of mean square error and percentage relative efficiency compared to other competing estimators. Two real-life datasets are considered: the first without outliers, while the second contains some outliers.

Real Life Application (Pop- I)

The first population is obtained from reference (Irfan et al. 2021). In this scenario, the study variable (Y) is "Number of inhabitants (in 1000’s) in 1930," while the auxiliary variable (X) is "Number of inhabitants (in 1000’s) in 1920".Furthermore, descriptive measures are $N=49,$$n=12$, $\overline{Y}=127.7959,$$\overline{X}=103.1429,$$T_M=72.75,$$M_R=254.5,C_y=0.9634,$$C_x=$ 1.0122, ${\rho }_{\mathit{yx}}=0.9817,$$H_L=77.25,{\widehat{\beta }}_{\left(0.15\right)}=1.082,$${\widehat{\beta }}_{\left(0.25\right)}=1.048,{\widehat{\beta }}_{\left(0.35\right)}=1.091.$

Real Life Application (Pop- II)

The quantile regression coefficient performs well in comparison to other measures of location when there are outliers in the data, as mentioned in the previous sections. Therefore, in this subsection, we assess our suggested estimators in the context of outliers. For this purpose, the second population is sourced from reference (Singh 2003). In this instance, the study variable (Y) is "Amount of real estate farm loans during 1977," and the auxiliary variable (X) is "Amount of non-real estate farm loans during 1977." Further descriptive measures are $N=50,$$n=12$, $\overline{Y}=555.4345,$$\overline{X}=878.1624,$$T_M=536.4088,$$M_R=1964.483,C_y=1.052916,$$C_x=$ 1.235168, ${\rho }_{\mathit{yx}}=0.8038,H_L=628.0575,{\widehat{\beta }}_{\left(0.15\right)}=0.3447,{\widehat{\beta }}_{\left(0.25\right)}=0.3543,{\widehat{\beta }}_{\left(0.35\right)}=0.35704.$ Scatter-plot in Fig. 1 pointing out the presence of outliers. Therefore Pop-II is suitable for the utilization.

[See PDF for image]

Fig. 1

Scatter plot of real life application (Pop-II)

This study calculated the percentage relative efficiencies (PRE’s) of both populations to prove the dominance of the suggested families of ratio type estimator’s using quantile regression, i.e. ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$ where $k=\text{1,2},3,\cdots ,9$, over ${\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk(0.15)}$,${\widehat{Y}}_{uk(0.25)}$,${\widehat{Y}}_{uk(0.35)}$ and ${\overline{Y}}_{\mathit{IRk}}$. Hence PRE is the ratio of MSE of reviewing estimators $(r)$ to the MSE of suggested quantile regression ratio-type estimators $(sqr)$ given as

12

$$PRE\left(s,q,r,,,r\right)=\frac{MSE(r)}{MSE(sqr)}\times 100$$

We computed both the mean squared error (MSE) and the percentage relative efficiency (PRE) for all the estimators. i.e.${\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk(0.15)}$,${\widehat{Y}}_{uk(0.25)}$,${\widehat{Y}}_{uk(0.35)},{\overline{Y}}_{\mathit{IRk}},{{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$ for the Pop-I and Pop-II. Expressions for mean squared error of all the reviewing and proposed estimators are given in Sects. 2 and 3 in detailed. All empirical results are summarized in Tables 3, 4, 5 and 6. Some important observations are made from Tables 3, 4, 5 and 6 as follows:

• ${\overline{Y}}_{\mathit{IRk}}$ performs better than reviewing class of estimators ${\widehat{Y}}_{\mathit{abk}}$

• All the suggested classes of quantile regression ratio type estimators i.e. ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$ have minimum MSE as compare to ${\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk(0.15)}$,${\widehat{Y}}_{uk(0.25)}$,${\widehat{Y}}_{uk(0.35)}$ and ${\overline{Y}}_{\mathit{IRk}}$.

• The suggested classes of quantile regression ratio type estimators i.e. ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$ demonstrate superior performance with significantly higher values of PRE compared to the reviewing estimators ${\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk(0.15)}$,${\widehat{Y}}_{uk(0.25)}$,${\widehat{Y}}_{uk(0.35)}$ and ${\overline{Y}}_{\mathit{IRk}}$ in this study.

Table 3. MSE of Pop-I

${\overline{Y}}_{\mathit{jk}}$	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$	${{\overline{Y}}_{\mathit{jk}}}^{\odot }$	${{\overline{Y}}_{\mathit{jk}}}^{⊚}$	${{\overline{Y}}_{\mathit{jk}}}^{⊛}$
${\overline{Y}}_{j1}$	396.672	325.1658	295.6009	333.2575	119.8975	86.9739	74.7246	90.4809
${\overline{Y}}_{j2}$	400.3092	328.4251	298.6904	336.5615	121.8498	88.5118	76.0751	92.0683
${\overline{Y}}_{j3}$	391.1526	320.2236	290.9179	328.2464	116.9839	84.6881	72.7227	88.1202
${\overline{Y}}_{j4}$	122.1716	89.0007	76.65724	92.53353	38.4703	34.5779	35.3916	34.6284
${\overline{Y}}_{j5}$	123.6935	90.20162	77.71397	93.77262	38.6137	34.5768	35.3259	34.6445
${\overline{Y}}_{j6}$	119.8935	87.20832	75.08305	90.6834	38.2607	34.5842	35.4946	34.6092
${\overline{Y}}_{j7}$	378.8328	309.2083	280.4888	317.076	110.692	79.7928	68.4588	83.0581
${\overline{Y}}_{j8}$	382.414	312.408	283.517	320.321	112.491	81.1865	69.6693	84.5002
${\overline{Y}}_{j9}$	373.4019	304.36	275.9021	312.1584	108.0103	77.7248	66.6680	80.9168

Table 4. MSE of Pop-II

${\overline{Y}}_{\mathit{jk}}$	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$	${{\overline{Y}}_{\mathit{jk}}}^{\odot }$	${{\overline{Y}}_{\mathit{jk}}}^{⊚}$	${{\overline{Y}}_{\mathit{jk}}}^{⊛}$
${\overline{Y}}_{j1}$	19,155.3	14,554.57	14,992.14	15,121.34	15,062.68	11,597.15	11,920.76	12,016.65
${\overline{Y}}_{j2}$	21,012.87	16,008.61	16,489.47	16,631.21	16,767.6	12,899.64	13,268.14	13,376.96
${\overline{Y}}_{j3}$	17,291.15	13,129.74	13,520.18	13,635.73	13,399.12	10,386.15	10,659.58	10,741.01
${\overline{Y}}_{j4}$	10,511.43	8515.855	8673.914	8722.189	8406.339	7591.754	7625.359	7637.511
${\overline{Y}}_{j5}$	11,438.86	9052.307	9252.315	9312.735	8923.406	7742.292	7815.448	7839.045
${\overline{Y}}_{j6}$	9749.391	8126.284	8244.38	8281.086	8046.391	7555.086	7553.948	7556.039
${\overline{Y}}_{j7}$	17,799.74	13,514.53	13,918.25	14,037.65	13,846.59	10,704.93	10,992.59	11,078.13
${\overline{Y}}_{j8}$	19,622.77	14,917.58	15,366.36	15,498.81	15,488.37	11,917.24	12,252.6	12,351.88
${\overline{Y}}_{j9}$	16,014.05	12,178.74	12,534.19	12,639.61	12,303.2	9632.101	9867.807	9938.364

Table 5. The percentage relative efficiencies (PREs) of the competing estimators for Pop-I are calculated

Suggested estimators	Reviewing estimators
	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$
${{\overline{Y}}_{j1}}^{\odot }$	456.0817	373.8659	339.8731	383.1693	137.8546
${{\overline{Y}}_{j2}}^{\odot }$	452.2664	371.0522	337.4582	380.2446	137.665
${{\overline{Y}}_{j3}}^{\odot }$	461.8743	378.121	343.5168	387.5944	138.135
${{\overline{Y}}_{j4}}^{\odot }$	353.3228	257.3918	221.6943	267.6089	111.2568
${{\overline{Y}}_{j5}}^{\odot }$	357.7348	260.8728	224.7572	271.2005	111.6751
${{\overline{Y}}_{j6}}^{\odot }$	346.671	252.1621	217.1019	262.2102	110.6305
${{\overline{Y}}_{j7}}^{\odot }$	474.7702	387.5137	351.5211	397.3739	138.7241
${{\overline{Y}}_{j8}}^{\odot }$	471.031	384.8024	349.2165	394.5492	138.5586
${{\overline{Y}}_{j9}}^{\odot }$	480.4154	391.5867	354.9731	401.6201	138.9651
${{\overline{Y}}_{j1}}^{⊚}$	530.8452	435.1522	395.5871	445.9806	160.4525
${{\overline{Y}}_{j2}}^{⊚}$	526.2027	431.7116	392.6257	442.4068	160.1704
${{\overline{Y}}_{j3}}^{⊚}$	537.8684	440.3349	400.0371	451.3669	160.8629
${{\overline{Y}}_{j4}}^{⊚}$	345.1989	251.4737	216.5969	261.4558	108.6987
${{\overline{Y}}_{j5}}^{⊚}$	350.1493	255.3411	219.9913	265.4498	109.3071
${{\overline{Y}}_{j6}}^{⊚}$	337.7795	245.6945	211.5336	255.485	107.793
${{\overline{Y}}_{j7}}^{⊚}$	553.3729	451.6702	409.7187	463.1628	161.6912
${{\overline{Y}}_{j8}}^{⊚}$	548.8982	448.415	406.9463	459.773	161.4641
${{\overline{Y}}_{j9}}^{⊚}$	560.0911	456.5304	413.8445	468.2278	162.0121
${{\overline{Y}}_{j1}}^{⊛}$	438.404	359.375	326.6997	368.3177	132.5114
${{\overline{Y}}_{j2}}^{⊛}$	434.7957	356.7188	324.4225	365.5561	132.3471
${{\overline{Y}}_{j3}}^{⊛}$	443.8855	363.3942	330.1378	372.4986	132.755
${{\overline{Y}}_{j4}}^{⊛}$	352.8072	257.0162	221.3708	267.2184	111.0944
${{\overline{Y}}_{j5}}^{⊛}$	357.0359	260.3631	224.3181	270.6707	111.4569
${{\overline{Y}}_{j6}}^{⊛}$	346.4215	251.9806	216.9457	262.0215	110.5509
${{\overline{Y}}_{j7}}^{⊛}$	456.1054	372.2793	337.7017	381.7518	133.2704
${{\overline{Y}}_{j8}}^{⊛}$	452.5595	369.7124	335.522	379.0769	133.1251
${{\overline{Y}}_{j9}}^{⊛}$	461.4638	376.1392	340.9699	385.7768	133.4831

Table 6. The percentage relative efficiencies (PREs) of the competing estimators for Pop-II are calculated

Suggested estimators	Reviewing estimators
	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$
${{\overline{Y}}_{j1}}^{\odot }$	162.4259	123.4143	127.1246	128.2202	130.6689
${{\overline{Y}}_{j2}}^{\odot }$	159.44	121.469	125.1177	126.1931	130.8685
${{\overline{Y}}_{j3}}^{\odot }$	164.3619	124.8053	128.5167	129.615	129.6787
${{\overline{Y}}_{j4}}^{\odot }$	137.4758	111.3763	113.4435	114.0749	110.8771
${{\overline{Y}}_{j5}}^{\odot }$	146.7407	116.1253	118.6911	119.4662	115.4759
${{\overline{Y}}_{j6}}^{\odot }$	128.0399	106.7234	108.2744	108.7564	106.5896
${{\overline{Y}}_{j7}}^{\odot }$	163.9943	124.5135	128.233	129.3331	130.0509
${{\overline{Y}}_{j8}}^{\odot }$	161.7407	122.9582	126.6573	127.749	130.7785
${{\overline{Y}}_{j9}}^{\odot }$	164.5024	125.1047	128.756	129.8389	128.3102
${{\overline{Y}}_{j1}}^{⊚}$	157.9279	119.9966	123.6042	124.6694	127.0503
${{\overline{Y}}_{j2}}^{⊚}$	154.9122	118.0195	121.5645	122.6094	127.152
${{\overline{Y}}_{j3}}^{⊚}$	160.0713	121.5474	125.1618	126.2315	126.2936
${{\overline{Y}}_{j4}}^{⊚}$	136.8625	110.8794	112.9374	113.5659	110.3825
${{\overline{Y}}_{j5}}^{⊚}$	145.3501	115.0249	117.5663	118.334	114.3816
${{\overline{Y}}_{j6}}^{⊚}$	128.0594	106.7397	108.2909	108.773	106.6059
${{\overline{Y}}_{j7}}^{⊚}$	159.6241	121.1953	124.8158	125.8865	126.5851
${{\overline{Y}}_{j8}}^{⊚}$	157.2221	119.5231	123.1188	124.1799	127.1249
${{\overline{Y}}_{j9}}^{⊚}$	160.5096	122.0681	125.6308	126.6875	125.1959
${{\overline{Y}}_{j1}}^{⊛}$	156.6415	119.0192	122.5974	123.654	126.0155
${{\overline{Y}}_{j2}}^{⊛}$	153.6228	117.0371	120.5527	121.5889	126.0937
${{\overline{Y}}_{j3}}^{⊛}$	158.8356	120.6091	124.1956	125.2571	125.3186
${{\overline{Y}}_{j4}}^{⊛}$	136.642	110.7008	112.7555	113.383	110.2047
${{\overline{Y}}_{j5}}^{⊛}$	144.9071	114.6743	117.208	117.9734	114.033
${{\overline{Y}}_{j6}}^{⊛}$	128.0235	106.7098	108.2606	108.7425	106.576
${{\overline{Y}}_{j7}}^{⊛}$	158.3683	120.2419	123.8338	124.8961	125.5893
${{\overline{Y}}_{j8}}^{⊛}$	155.9314	118.5419	122.1081	123.1605	126.0813
${{\overline{Y}}_{j9}}^{⊛}$	159.3512	121.1872	124.7241	125.7732	124.2923

Simulation Study

Using R software, we conducted Monte Carlo simulations to validate the superiority of ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$ over ${\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk(0.15)}$,${\widehat{Y}}_{uk(0.25)}{,\widehat{Y}}_{uk(0.35)}$ and ${\overline{Y}}_{\mathit{IRk}}$. To accomplish this, we utilized two real-life applications (Pop-I and Pop-II) from Sects. 4.1 and 4.2. Different sample sizes namely $n=8,14\text{and}20$ were employed for both populations. Equation (13) was utilized to compute the mean squared error (MSE) for all reviewing and suggested estimators under study.

13

$$MSE\left(p\right)=\frac{{\sum }_{i=1}^{25000}\left(p,-,\overline{Y}\right)}{25000}$$where $p={\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk(0.15)}$,${\widehat{Y}}_{uk(0.25)}$,${\widehat{Y}}_{uk(0.35)}$, ${\overline{Y}}_{\mathit{IRk}}$, ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$.

The simulation study is carried out using the five steps listed below.

(i) Select a SRS without replacement of size $n$ from the population $N$.

(ii) By using step (i), obtained MSE of all the estimators.

(iii) Repeat (i) and (ii), 25,000 times.

(iv) Get 25,000 values of MSE for of each estimator i.e.${\widehat{Y}}_{\mathit{abk}}$,${\widehat{Y}}_{uk\left(0.15\right)}$,${\widehat{Y}}_{uk(0.25)}$,${\widehat{Y}}_{uk(0.35)}$,

${\overline{Y}}_{\mathit{IRk}}$, ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$

(v) Use the step (iv), take the average of 25,000 values and get the MSE of each estimator.

Tables 7 and 8 shows the values of MSE by using different sample sizes. It is revealed from Table 7 and 8:

1. ${\overline{Y}}_{\mathit{IRk}}$ have least MSE as compare to reviewing class of estimators ${\widehat{Y}}_{\mathit{abk}}$

2. MSE of the all classes of suggested estimators ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$ is least as compare to all reviewing estimators under study.

3. As the sample size increases, the MSE of all suggested classes of estimators, such as ${{\overline{Y}}_{\mathit{jk}}}^{\odot },{{\overline{Y}}_{\mathit{jk}}}^{⊚}$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$, decreases.

4. A detailed examination of the columns of ${{\overline{Y}}_{\mathit{jk}}}^{⊚}$ reveals that it exhibits the lowest MSE among all other proposed classes, such as ${{\overline{Y}}_{\mathit{jk}}}^{\odot }$ and ${{\overline{Y}}_{\mathit{jk}}}^{⊛}$

Table 7. Simulation study results based on MSE for Pop-I

$n$	${\overline{Y}}_{\mathit{jk}}$	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$	${{\overline{Y}}_{\mathit{jk}}}^{\odot }$	${{\overline{Y}}_{\mathit{jk}}}^{⊚}$	${{\overline{Y}}_{\mathit{jk}}}^{⊛}$
$n=8$	${\overline{Y}}_{j1}$	659.3332	540.4783	491.3366	553.9276	199.1913	144.22	123.7293	150.0827
	${\overline{Y}}_{j2}$	665.3788	545.8957	496.4718	559.4197	202.4421	146.7831	125.9798	152.7282
	${\overline{Y}}_{j3}$	650.1591	532.2635	483.5528	545.5987	194.3399	140.4111	120.3939	146.1489
	${\overline{Y}}_{j4}$	203.069	147.9336	127.4168	153.8057	63.93486	57.45803	58.81228	57.54213
	${\overline{Y}}_{j5}$	205.5986	149.9297	129.1732	155.8653	64.17227	57.45456	58.70124	57.56721
	${\overline{Y}}_{j6}$	199.2825	144.9544	124.8002	150.7305	63.58809	57.47101	58.9859	57.51247
	${\overline{Y}}_{j7}$	629.6815	513.9544	466.2178	527.0317	183.8634	132.256	113.2927	137.716
	${\overline{Y}}_{j8}$	635.6341	519.2727	471.2512	532.4255	186.8589	134.5775	115.3083	140.1181
	${\overline{Y}}_{j9}$	620.6545	505.8956	458.5941	518.8579	179.3985	128.8119	110.3117	134.1497
$n=14$	${\overline{Y}}_{j1}$	321.6259	263.6479	239.6764	270.2086	97.22768	70.56684	60.65291	73.40577
	${\overline{Y}}_{j2}$	324.575	266.2906	242.1814	272.8877	98.80986	71.81288	61.7471	74.69192
	${\overline{Y}}_{j3}$	317.1508	259.6407	235.8794	266.1457	94.86649	68.71483	59.03086	71.493
	${\overline{Y}}_{j4}$	99.05806	72.16273	62.15452	75.02719	31.19334	28.03837	28.69794	28.07932
	${\overline{Y}}_{j5}$	100.292	73.13645	63.01133	76.03185	31.30982	28.03777	28.64489	28.09262
	${\overline{Y}}_{j6}$	97.21097	70.70945	60.87815	73.52708	31.02324	28.04316	28.78107	28.06335
	${\overline{Y}}_{j7}$	307.1617	250.7094	227.4233	257.0886	89.76744	64.74824	55.57559	67.3913
	${\overline{Y}}_{j8}$	310.0654	253.3038	229.8786	259.7198	91.22541	65.87761	56.55661	68.55988
	${\overline{Y}}_{j9}$	302.7583	246.7784	223.7044	253.1014	87.59418	63.07235	54.1242	65.65608
$n=20$	${\overline{Y}}_{j1}$	186.5431	152.9158	139.0123	156.721	56.40605	40.97796	35.24641	42.61977
	${\overline{Y}}_{j2}$	188.2535	154.4485	140.4652	158.2748	57.3229	41.69969	35.88022	43.36475
	${\overline{Y}}_{j3}$	183.9474	150.5916	136.8101	154.3645	55.03778	39.90516	34.30676	41.51177
	${\overline{Y}}_{j4}$	57.45367	41.85438	36.04962	43.51577	18.09343	16.26458	16.64689	16.28832
	${\overline{Y}}_{j5}$	58.16936	42.41914	36.54657	44.09847	18.16115	16.26448	16.61638	16.29627
	${\overline{Y}}_{j6}$	56.38236	41.01148	35.30933	42.64571	17.99456	16.267	16.69474	16.27871
	${\overline{Y}}_{j7}$	178.1538	145.4115	131.9055	149.1114	52.08294	37.60716	32.3047	39.13551
	${\overline{Y}}_{j8}$	179.838	146.9162	133.3296	150.6375	52.92783	38.2615	32.87319	39.81255
	${\overline{Y}}_{j9}$	175.5998	143.1315	129.7486	146.7988	50.82353	36.6361	31.46353	38.1301

Table 8. Simulation study results based on MSE for Pop-II

$n$	${\overline{Y}}_{\mathit{jk}}$	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$	${{\overline{Y}}_{\mathit{jk}}}^{\odot }$	${{\overline{Y}}_{\mathit{jk}}}^{⊚}$	${{\overline{Y}}_{\mathit{jk}}}^{⊛}$
$n=8$	${\overline{Y}}_{j1}$	31,757.48	24,129.94	24,855.39	25,069.59	24,287.69	18,835.2	19,347.71	19,499.45
	${\overline{Y}}_{j2}$	34,837.13	26,540.58	27,337.81	27,572.79	26,864.78	20,833.88	21,412.62	21,583.36
	${\overline{Y}}_{j3}$	28,666.91	21,767.73	22,415.03	22,606.6	21,729.53	16,949	17,385.34	17,515.2
	${\overline{Y}}_{j4}$	17,426.84	14,118.39	14,380.44	14,460.47	13,791.47	12,475.42	12,529.81	12,549.47
	${\overline{Y}}_{j5}$	18,964.43	15,007.77	15,339.36	15,439.53	14,634.65	12,727.47	12,845.88	12,884.07
	${\overline{Y}}_{j6}$	16,163.46	13,472.52	13,668.31	13,729.17	13,198.19	12,404.69	12,402.85	12,406.23
	${\overline{Y}}_{j7}$	29,510.09	22,405.67	23,074.99	23,272.94	22,421.83	17,448.13	17,906.29	18,042.42
	${\overline{Y}}_{j8}$	32,532.49	24,731.78	25,475.81	25,695.39	24,935.4	19,329.27	19,859.31	20,016.08
	${\overline{Y}}_{j9}$	26,549.61	20,191.07	20,780.36	20,955.14	20,021.1	15,760.78	16,138.67	16,251.72
$n=14$	${\overline{Y}}_{j1}$	15,554.68	11,818.75	12,174.07	12,278.98	12,330.65	9473.5	9739.781	9818.71
	${\overline{Y}}_{j2}$	17,063.08	12,999.47	13,389.95	13,505.04	13,752.43	10,554.85	10,858.83	10,948.62
	${\overline{Y}}_{j3}$	14,040.94	10,661.74	10,978.79	11,072.62	10,950.25	8472.428	8696.905	8763.776
	${\overline{Y}}_{j4}$	8535.596	6915.13	7043.479	7082.68	6846.791	6180.408	6207.886	6217.822
	${\overline{Y}}_{j5}$	9288.699	7350.745	7513.158	7562.221	7268.662	6302.287	6362.102	6381.397
	${\overline{Y}}_{j6}$	7916.799	6598.787	6694.684	6724.491	6554.023	6152.062	6151.132	6152.842
	${\overline{Y}}_{j7}$	14,453.92	10,974.21	11,302.04	11,398.99	11,320.9	8735.546	8971.853	9042.137
	${\overline{Y}}_{j8}$	15,934.28	12,113.53	12,477.95	12,585.5	12,684.95	9738.797	10,014.92	10,096.68
	${\overline{Y}}_{j9}$	13,003.89	9889.503	10,178.14	10,263.74	10,044.46	7851.161	8044.413	8102.273
$n=20$	${\overline{Y}}_{j1}$	9073.565	6894.268	7101.54	7162.739	7299.53	5586.281	5745.385	5792.567
	${\overline{Y}}_{j2}$	9953.465	7583.024	7810.803	7877.94	8169.813	6242.764	6425.232	6479.16
	${\overline{Y}}_{j3}$	8190.546	6219.35	6404.294	6459.029	6462.204	4983.24	5116.812	5156.618
	${\overline{Y}}_{j4}$	4979.098	4033.826	4108.696	4131.563	4015.748	3621.8	3638.03	3643.899
	${\overline{Y}}_{j5}$	5418.408	4287.935	4382.675	4411.296	4263.956	3692.506	3727.833	3739.23
	${\overline{Y}}_{j6}$	4618.133	3849.292	3905.233	3922.619	3844.466	3606.793	3606.243	3607.254
	${\overline{Y}}_{j7}$	8431.454	6401.621	6592.856	6649.412	6686.315	5141.308	5282.069	5323.953
	${\overline{Y}}_{j8}$	9294.996	7066.224	7278.803	7341.54	7515.644	5746.849	5912.017	5960.951
	${\overline{Y}}_{j9}$	7585.602	5768.877	5937.247	5987.184	5916.693	4611.241	4725.948	4760.303

Robustness of ${\overline{\mathit{Y}}}_{\mathit{s}\mathit{i}}$ Under Real Life Application (Pop-III)

As highlighted earlier, quantile regression exhibits robustness against outliers. Unlike alternative measures of location, its coefficients perform effectively even in the presence of outliers within the datasets. We proceed to evaluate our proposed families of ratio-type estimators using quantile regression within the outlier’s context. To do so, we utilize the third population derived from Venables and Ripley (2013). Here, "Number of calories" serves as the study variable (Y), while "Grams of Sodium in one portion" acts as the auxiliary variable (X). Additionally, we analyze descriptive statistics of the data N $=65,$$n=20$, $\overline{Y}=149.4083,$$\overline{X}=237.8384,$$T_M=233.5,$$M_R=393.9394,C_y=0.4177271,$$C_x$=0.549237,${\rho }_{\mathit{yx}}=0.5286,$$H_L=235.1244,{\widehat{\beta }}_{\left(0.15\right)}=0.198324,{\widehat{\beta }}_{\left(0.25\right)}=0.173771,{\widehat{\beta }}_{\left(0.35\right)}=0.157894$ for an in-depth exploration of robustness, we varied the sample sizes $(n=10and20).$ Analysis of the scatter plot confirms the existence of outliers in Pop-III, as illustrated in Fig. 2. This allows us to evaluate the robustness of the ratio-type estimators using quantile regression for Pop-III. Numerical results are presented in Table 9, derived from our robustness study. It's evident from Table 9 that the mean squared error (MSE) of all the proposed classes of ratio-type estimators using quantile regression is the lowest compared to all the estimators under review. We conclude that our proposed classes of quantile regression ratio-type estimators perform efficiently in the presence of extreme values.

[See PDF for image]

Fig. 2

Scatter plot of real life application (Pop-III)

Table 9. MSE of suggested and reviewing estimators under robustness study of Pop-III

$n$	${\overline{Y}}_{\mathit{jk}}$	${\widehat{Y}}_{\mathit{abk}}$	${\widehat{Y}}_{uk(0.15)}$	${\widehat{Y}}_{uk(0.25)}$	${\widehat{Y}}_{uk(0.35)}$	${\overline{Y}}_{\mathit{IRk}}$	${{\overline{Y}}_{\mathit{jk}}}^{\odot }$	${{\overline{Y}}_{\mathit{jk}}}^{⊚}$	${{\overline{Y}}_{\mathit{jk}}}^{⊛}$
$n=10$	${\overline{Y}}_{j1}$	384.6522	339.039	321.1496	310.5769	317.4545	284.8157	272.7071	265.8185
	${\overline{Y}}_{j2}$	312.5649	281.3075	269.9306	263.5428	263.9745	246.5805	241.4253	239.0154
	${\overline{Y}}_{j3}$	308.5575	278.2728	267.3371	261.233	261.5539	245.1197	240.4014	238.2726
	${\overline{Y}}_{j4}$	320.0263	287.0255	274.8576	267.9616	268.6713	249.5278	243.5759	240.6534
	${\overline{Y}}_{j5}$	274.3154	253.8572	247.3793	244.1397	244.3432	236.8039	236.129	236.6001
	${\overline{Y}}_{j6}$	272.0642	252.393	246.2722	243.262	243.4695	236.5675	236.182	236.8391
	${\overline{Y}}_{j7}$	383.6533	338.2107	320.3987	309.8757	316.6123	284.1631	272.1412	265.3085
	${\overline{Y}}_{j8}$	311.9113	280.8107	269.5049	263.1629	263.5747	246.3363	241.2518	238.8875
	${\overline{Y}}_{j9}$	307.9299	277.8	266.9346	260.8755	261.1815	244.899	240.2497	238.1653
$n=20$	${\overline{Y}}_{j1}$	157.3577	138.6978	131.3794	127.0542	131.1244	117.5412	112.508	109.6461
	${\overline{Y}}_{j2}$	127.8675	115.0803	110.4261	107.813	108.7112	101.5008	99.36513	98.36695
	${\overline{Y}}_{j3}$	126.2281	113.8389	109.3652	106.868	107.7021	100.8907	98.93622	98.05456
	${\overline{Y}}_{j4}$	130.9199	117.4195	112.4418	109.6207	110.6706	102.7326	100.2663	99.05558
	${\overline{Y}}_{j5}$	112.2199	103.8507	101.2006	99.87535	100.5503	97.42821	97.14879	97.34382
	${\overline{Y}}_{j6}$	111.299	103.2517	100.7477	99.51629	100.1889	97.33088	97.17128	97.4433
	${\overline{Y}}_{j7}$	156.9491	138.3589	131.0722	126.7673	130.7699	117.2665	112.2696	109.431
	${\overline{Y}}_{j8}$	127.6001	114.8771	110.252	107.6576	108.5445	101.3988	99.29247	98.31319
	${\overline{Y}}_{j9}$	125.9713	113.6455	109.2005	106.7218	107.5469	100.7986	98.87269	98.00943

Summary

This study introduces three novel families of ratio-type estimators employing quantile regression, designed to enhance robustness in SRS without replacement, particularly concerning auxiliary variables. We analyze two real-life datasets: one devoid of outliers and other containing outliers. Additionally, we conducted a simulation study incorporating both practical scenarios to assess the efficacy of the newly proposed estimator families. Furthermore, we evaluate robustness using a real-life dataset. Based on numerical results, we conclude that these new families of quantile regression ratio-type estimators yield optimal estimates. Consequently, we advocate for the adoption of these proposed estimator families in future applications.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Declarations

Competing interests

The authors have not disclosed any competing interests.

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