Regression-in-Ratio Estimators for Population Mean by Using Robust Regression in Two Phase Sampling

Muhammad Noor-ul-Amin¹ and Aamir Raza^2,*

¹COMSATS University Islamabad-Lahore Campus, Pakistan

²Department of Statistics, Graduate College, Mandi Bahauddin, Pakistan

E-mail: nooramin.stats@gmail.com; amir.vu@gmail.com

${}^{*}$Corresponding Author

Received 11 March 2021; Accepted 29 August 2021; Publication 08 October 2021

Abstract

The estimation of population mean is not meaningful using ordinary least square method when data contains some outliers. In the current study, we proposed efficient estimators of population mean using robust regression in two phase sampling. An extensive simulation study is conduct to examine the efficiency of proposed estimators in terms of mean square error (MSE). Real life example and extensive simulation study are cited to demonstrate the performance of the proposed estimators. Theoretical example and simulation studies showed that the suggested estimators are more efficient than the considered estimators in the presence of outliers.

Keywords: Auxiliary information, M-estimator, outliers, robust regression, two phase sampling.

1 Introduction

In the presence of outliers, OLS method fails to produce efficient results as it is highly sensitive toward the outliers. Robust regression using redescending M-estimator is used as an alternate tool to obtain efficient estimates when data contain outliers. A numbers of robust regression techniques are available in literature but most commonly used technique given by Huber (1964) i.e. M-estimator to remove the effect of outliers.

Two phase sampling is normally applied to obtain auxiliary information about population parameters. Neyman (1938) introduced the two phase sampling to obtain the information on strata sizes. Chand (1975), Kiregyera (1984), Singh and Vishwakarma (2007), Vishwakarma and Gangele (2014), Noor-ul-Amin et al. (2016), Misra (2018), Raza et al. (2019), Sabzar et al. (2020), de Menezes et al. (2021), Ahuja et al. (2021) and Anas et al. (2021) incorporated auxiliary information in considered sampling design to obtain efficient estimates.

Huber (1964) developed the following objective function for robust regression by using $y_i=a+b{x}_i+r_i$.

$${\rho }_2(r_i)=\{\begin{array}{cc}\frac{r^2}{2}\mathit{\hspace{1em}}\hfill & |r|\le v\hfill \\ v|r|-\frac{v^2}{2}\mathit{\hspace{1em}}\hfill & |r|>v\hfill \end{array}$$

(1)

where ‘$r_i$’ is the residual associated with ith observation and v is tuning constant, which is used to controls the robustness of the M-estimator. Huber (1964) advised that $v=1.5s$, where “s” is the estimate of population standard deviation of error (residuals) terms.

The Huber’s (1964) M-estimator is not flexible to weight the larger residuals. To overcome this deficiency an efficient redescending M-estimator is proposed by Raza et al. (2019) that can control the robustness of estimators up to a desire level to detect the outliers. The objective function of M-estimator proposed by Raza et al. (2019) is given as

$${\rho }_3(r_i)=\frac{v^2}{2a}\left[1-{\left\{1+{\left(\frac{r}{v}\right)}^2\right\}}^{-a}\right]\mathit{\hspace{1em}}|r|\ge 0$$

(2)

where ‘v’ and ‘a’ are tuning constants. The optimum values of ‘a’ are 5 and 7. For more details, one can read Raza et al. (2019).

1.1 Notations

Let us consider a finite population of size N having observations $x_1,x_2\mathrm{\dots }{x}_N$. In first phase, sample of size $n_1(n_1<N)$ is obtained to collect the auxiliary information. In second phase, sample of size $n_2(n_2<n_1)$ is selected from $n_1$, to obtain information regarding study variable and auxiliary variable. The sample mean of auxiliary variable is represented by ${\overline{x}}_1$ in first phase. In second phase, the sample mean of auxiliary variable and study variables are denoted by ${\overline{x}}_2$ and ${\overline{y}}_2$ respectively. The simple random sampling (SRS) without replacement is utilized in both phases. The population mean of study variable and auxiliary variables are represented as ${\mu }_y$ and ${\mu }_x$ respectively. Following notations are used to develop expression of bias and means square errors:

${\overline{e}}_{y_2}$	$={\displaystyle \frac{{\overline{y}}_2-{\mu }_y}{{\mu }_y}},{\overline{e}}_{x_i}={\displaystyle \frac{{\overline{x}}_i-{\mu }_x}{{\mu }_x}},\text{where}i=1,2$	(3)
$E({\overline{e}}_{y_2})$	$=E({\overline{e}}_{x_1})=E({\overline{e}}_{x_2})=0,$
$E({\overline{e}}_{y_2}^2)$	$={\theta }_2{C}_y^2,E({\overline{e}}_{x_i}^2)={\theta }_i{C}_x^2\mathit{\hspace{1em}}\text{where}i=1,2$	(4)
$E({\overline{e}}_{y_2}{\overline{e}}_{x_i})$	$={\theta }_i{\rho }_{yx}{C}_x{C}_y,\text{where},{\theta }_i={\displaystyle \frac{1}{n_i}}-{\displaystyle \frac{1}{N}},$
$C_x$	$={\displaystyle \frac{{\sigma }_x}{{\mu }_x}},C_y={\displaystyle \frac{{\sigma }_y}{{\mu }_y}},$
$H_{yx}$	$={\rho }_{yx}{\displaystyle \frac{C_y}{C_x}}\mathit{\hspace{1em}}\text{where}{\rho }_{yx}=cor(x,y)$	(5)

2 Regression-in-Ratio Estimators in Two Phase Sampling

Kadilar et al. (2004) suggested that information regarding auxiliary variable like coefficient of variation ($C_x$) and coefficient of kurtosis ($B_2(x)$) can also be used to obtain efficient estimators of population mean. Following Kadilar (2004), Noor-ul-Amin et al. (2016) proposed following regression-in-ratio estimators for population mean in two phase sampling

$${\overline{y}}_{\mathit{OLSl}}=\frac{{\overline{y}}_2+b_1({\overline{x}}_1-{\overline{x}}_2)}{({\lambda }_l{\overline{x}}_2+{\phi }_l)}({\lambda }_l{\overline{x}}_1+{\phi }_l)\mathit{\hspace{1em}}l=1,2,3,4,5$$

(6)

where $b_1$ is the OLS estimator of regression coefficient between study variable and auxiliary variable and

$${\lambda }_1=1\&{\phi }_1=0,{\lambda }_2=1\&{\phi }_2=C_x,{\lambda }_3=1\&{\phi }_3=B_2(x),$$
$${\lambda }_4=B_2(x)\&{\phi }_4=C_x,{\lambda }_5=C_x\&{\phi }_5=B_2(x)$$	(7)

The following expression of MSE of estimators given in (8) is obtained using Taylor series approximation up to order one.

$MSE({\overline{y}}_{\mathit{OLSl}})$	$\cong {\mu }_y^2{\theta }_2{C}_y^2+\{B_1{\mu }_x+{\mu }_y(1-{\gamma }_l)\}({\theta }_2-{\theta }_1)C_x^2$
	$\mathrm{\hspace{1em}}\{B_1{\mu }_x+{\mu }_y(1-{\gamma }_l)-2{\mu }_y{H}_{yx}\}$	(8)

where

$${\gamma }_1=0,{\gamma }_2=\frac{C_x}{{\mu }_x},{\gamma }_3=\frac{{\beta }_2(x)}{{\mu }_x},$$
$${\gamma }_4=\frac{C_x}{{\mu }_x\cdot {\beta }_2(x)}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\gamma }_5=\frac{{\beta }_2(x)}{{\mu }_x\cdot C_x}$$	(9)

In the presence of outlier, in Equation (6) failed to produced reliable results. So Noor-ul-Amin et al. (2016) proposed the following robust regression-in-ratio estimators for population mean in two phase sapling using Huber (1964) M-estimator

$${\overline{y}}_{\mathit{robl}}=\frac{{\overline{y}}_2+b_2({\overline{x}}_1-{\overline{x}}_2)}{({\lambda }_l{\overline{x}}_2+{\phi }_l)}({\lambda }_l{\overline{x}}_1+{\phi }_l)$$

(10)

Where $b_2$ is obtained by minimizing the $\sum {\rho }_2(y-a-bx)$ by using the objective function defined in Equation (1). Unfortunately, there are some typing errors in the expression of MSE given by Noor-ul-Amin et al. (2016), where the correct expression of MSE for the estimator given in Equation (10) is given as

$\mathit{MSE}({\widehat{\overline{y}}}_{\mathit{robl}})$	$\cong {\mu }_y^2{\theta }_2{C}_y^2+\{B_2{\mu }_x+{\mu }_y(1-{\gamma }_l)\}({\theta }_2-{\theta }_1)C_x^2$
	$\mathrm{\hspace{1em}}\{B_2{\mu }_x+{\mu }_y(1-{\gamma }_l)-2{\mu }_y{H}_{yx}\}$	(11)

The values of ${\gamma }_l$ are similar as given in Equation (9).

3 Proposed Regression-in-Ratio Estimators in Two Phase Sampling

As mentioned earlier, Huber’s (1964) M-estimator does not perform well for larger values of residuals. To overcome this deficiency, we have proposed the following regression-in-ratio estimators for population mean in two phase sampling using the objective function developed by Raza et al. (2019)

$${\overline{y}}_{\mathit{prol}}=\frac{{\overline{y}}_2+b_3({\overline{x}}_1-{\overline{x}}_2)}{({\lambda }_l{\overline{x}}_2+{\phi }_l)}({\lambda }_l{\overline{x}}_1+{\phi }_l)$$

(12)

where $b_3$ is obtained by minimizing the $\sum {\rho }_3(y-a+bx)$ by using objective function defined in Equation (2). The expression of MSE of proposed regression-in-ratio estimators is obtained as under

$\mathit{MSE}({\overline{y}}_{\mathit{prol}})$	$={\mu }_y^2{\theta }_2{C}_y^2+\{B_3{\mu }_x+{\mu }_y(1-{\gamma }_l)\}({\theta }_2-{\theta }_1)C_x^2$
	$\mathrm{\hspace{1em}}\{B_3{\mu }_x+{\mu }_y(1-{\gamma }_l)-2{\mu }_y{H}_{yx}\}$	(13)

4 Efficiency Comparison

In this session, we compared the efficiencies of proposed estimators with existing estimators of the basis of MSE. The proposed estimators performed better if we have,

$$\mathit{MSE}({\overline{y}}_{\mathit{prol}})<\mathit{MSE}({\overline{y}}_{\mathit{robl}}),l=1,2,3,4,5$$
$$\mathrm{\{}\{{\mu }_y(1-{\gamma }_l)+B_3{\mu }_x\}-\{{\mu }_y(1-{\gamma }_l)+B_2{\mu }_x\}-2{\mu }_y{H}_{yx}\}<0$$
$$B_3-B_2<0\mathit{and}{B}_3-B_2>\frac{2H_{yx}}{1-{\gamma }_1}$$
$$or$$
$$B_3-B_2>0\mathit{and}{B}_3-B_2<\frac{2H_{yx}}{1-{\gamma }_1}$$
$$\min (2B_2,\frac{2H_{yx}}{1-{\gamma }_1})<\left(B_3-B_2\right)<\max (2B_2,\frac{2H_{yx}}{1-{\gamma }_1})$$	(14)

5 Applications

In this section, performance of proposed robust regression-in-ratio estimators of population mean is compared with and considered estimators using MSEs. Superiority of proposed regression-in-ratio estimators is supported by using a real life data example and using extensive simulation study.

5.1 Real Life Data Example

The data used in this example is about the level of apple production ‘y’ in tons as variable of interest and numbers of apple trees (X, 1 unit $=$ 100 trees) as auxiliary variable in 94 villages of the Akdeniz Region in Turkey, 1999. (Source: Institute of Statistics, Republic of Turkey). Statistics regarding the population of Example are given in Table 1 and Graph of the used data is presented in Figure 1. Presence of outliers is verified from this Figure 1.

Table 1 Statistics of data used in Example 4.1

$N=94$	${\sigma }_{yx}=43324466$
$\rho =0.9011$	${\mu }_x=724.099$
$B_2=12.287$	${\mu }_y=9384.309$
$B_1=16.764$	${\sigma }_x=1607.573$
$B_3=7.0560$	${\sigma }_y=29907.48$
$C_x=2.220$	$B_2\left(x\right)=27.703$

Figure 1 Apple production and number of trees.

A sample of size $n_1=50$ is taken in first phase from the population using SRS without replacement to obtain the auxiliary information. Three different samples of sizes 5, 10 and 30 at phase II are considered to check the efficiencies of proposed estimators.

By incorporating the data in R programming, the MSE of proposed and considered estimators are calculated using Equations (7), (11) and (13). The R.Es of proposed estimators with considered estimators are obtained by using Equation (14) and presented in Table 2.

$$R.E({\overline{y}}_{\mathit{prol}})=\frac{\mathit{MSE}({\overline{y}}_{ul})}{\mathit{MSE}({\overline{y}}_{\mathit{prol}})},l=1,2,\mathrm{\dots }5\text{and}u=1,2$$

(15)

where ${\overline{y}}_{1l}={\overline{y}}_{\mathit{OLSl}}$ and ${\overline{y}}_{2l}={\overline{y}}_{\mathit{robl}}$.

Table 2 R.E of proposed estimators w.r.t considered estimators in two phase sampling at $n_1=50$

$n_2$	R.E	${\overline{y}}_{\mathit{OLS1}}$	${\overline{y}}_{\mathit{OLS2}}$	${\overline{y}}_{\mathit{OLS3}}$	${\overline{y}}_{\mathit{OLS4}}$	${\overline{y}}_{\mathit{OLS5}}$	${\overline{y}}_{\mathit{rob1}}$	${\overline{y}}_{\mathit{rob2}}$	${\overline{y}}_{\mathit{rob3}}$	${\overline{y}}_{\mathit{rob4}}$	${\overline{y}}_{\mathit{rob5}}$
5	${\overline{y}}_{\mathit{pro1}}$	2.68	2.66	2.54	2.68	2.68	1.65	1.65	1.70	1.65	1.60
	${\overline{y}}_{\mathit{pro2}}$	2.68	2.67	2.55	2.68	2.69	1.64	1.65	1.70	1.64	1.59
	${\overline{y}}_{\mathit{pro3}}$	2.76	2.75	2.62	2.76	2.77	1.56	1.57	1.61	1.56	1.51
	${\overline{y}}_{\mathit{pro4}}$	2.68	2.67	2.54	2.68	2.68	1.65	1.65	1.70	1.65	1.60
	${\overline{y}}_{\mathit{pro5}}$	2.59	2.58	2.46	2.59	2.60	1.67	1.67	1.72	1.67	1.62
10	${\overline{y}}_{\mathit{pro1}}$	2.35	2.34	2.24	2.35	2.36	1.52	1.53	1.56	1.52	1.49
	${\overline{y}}_{\mathit{pro2}}$	2.36	2.35	2.25	2.36	2.36	1.52	1.52	1.56	1.52	1.48
	${\overline{y}}_{\mathit{pro3}}$	2.41	2.40	2.30	2.41	2.42	1.45	1.46	1.49	1.45	1.42
	${\overline{y}}_{\mathit{pro4}}$	2.35	2.34	2.24	2.35	2.36	1.52	1.53	1.56	1.52	1.48
	${\overline{y}}_{\mathit{pro5}}$	2.29	2.28	2.19	2.29	2.30	1.54	1.54	1.58	1.54	1.50
30	${\overline{y}}_{\mathit{pro1}}$	1.66	1.65	1.60	1.66	1.66	1.25	1.25	1.27	1.25	1.24
	${\overline{y}}_{\mathit{pro2}}$	1.66	1.65	1.61	1.66	1.66	1.25	1.25	1.27	1.25	1.23
	${\overline{y}}_{\mathit{pro3}}$	1.68	1.67	1.62	1.68	1.68	1.22	1.22	1.23	1.22	1.20
	${\overline{y}}_{\mathit{pro4}}$	1.66	1.65	1.60	1.66	1.66	1.25	1.25	1.27	1.25	1.24
	${\overline{y}}_{\mathit{pro5}}$	1.64	1.63	1.58	1.64	1.64	1.26	1.26	1.28	1.26	1.25

The Table 2 indicates that proposed regression-in-ratio estimators worked efficiently in two phase sampling to estimate the population mean as compare to the considered estimators, for all samples sizes. Performance of estimators given by Noor-ul-Amin et al. (2016) is better than OLS estimators but significantly less than proposed estimators for all conditions. At small sample of size $n_2=5$, proposed estimator ${\overline{y}}_{\mathit{pro1}}$ is 268% efficient than OLS estimators ${\overline{y}}_{\mathit{OLS1}}$ and 165% efficient than estimator ${\overline{y}}_{\mathit{rob1}}$. As the sample size of phase II increases i.e. $n_2=30$, efficiency of estimator ${\overline{y}}_{\mathit{pro1}}$ is 166% as compare to estimator ${\overline{y}}_{\mathit{OLS1}}$ and 125%, relative to estimators ${\overline{y}}_{\mathit{rob1}}$. So increase of phase II sample size decreases the R.E of the proposed estimator but it remains higher than considered estimators.

5.2 Simulation Study

The usefulness of suggested regression-in-ratio estimators to estimate the population mean in two phase sampling is evaluated using extensive simulation study. The R-programing is used to generate a bivariate normal population of size $N=1000$ with mean vector $\mu =[{\mu }_x,{\mu }_y]=[10,2]$ and variance covariance matrix $\sum =\left[\begin{array}{cc}1\hfill & 0.85\hfill \\ 0.85\hfill & 1\hfill \end{array}\right]$. To examine the robustness of proposed estimators, two different levels of contaminations of outliers are used in the population i.e. 2%, and 6% which are termed as low level and high level of containments respectively using a bivariate normal distribution with mean vector $\mu =[30,15]$ and variance covariance vector is similar as discussed above. A sample of 100 values is chosen using SRS without replacement from the considered population in Phase I to obtain auxiliary information and sample of sizes 10, 20 and 30 are drawn using SRS without replacement in phase II to obtain the information regarding the variable of interest and auxiliary variable.

Figure 2 Generated population with 2% outliers.

For each sample size, 50000 iterations are carried out to obtained the MSE of ${\overline{y}}_{ul}$ using following formula

$\mathit{MSE}({\overline{y}}_{ul})={\displaystyle \frac{1}{50000}}{\displaystyle \sum _{i=1}^{50000}}{({\overline{y}}_{iul}-{\mu }_y)}^2,l=1,2,3,4,5\text{ and}u=1,2,3$

(16)

where ${\overline{y}}_{1l}={\overline{y}}_{\mathit{OLSl}}$, ${\overline{y}}_{2l}={\overline{y}}_{\mathit{robl}}$, ${\overline{y}}_{3l}={\overline{y}}_{\mathit{prol}}$ and ${\mu }_y$ is the mean of ${\overline{y}}_{\mathit{iul}}$ from 50000 samples. The graph of generated population with 2% contamination of outliers is presented in Figure 2.

The MSE of proposed and considered estimators are calculated by using Equation (16). The R.Es of considered regression-in-ratio estimators of population mean are calculated by using Equation (15) and presented in Tables 3 and 4.

Table 3 R.E of proposed estimators w.r.t. considered estimators in two phase sampling at $n_1=100$ and 2% outliers

$n_2$	R.E	${\overline{y}}_{\mathit{OLS1}}$	${\overline{y}}_{\mathit{OLS2}}$	${\overline{y}}_{\mathit{OLS3}}$	${\overline{y}}_{\mathit{OLS4}}$	${\overline{y}}_{\mathit{OLS5}}$	${\overline{y}}_{\mathit{rob1}}$	${\overline{y}}_{\mathit{rob2}}$	${\overline{y}}_{\mathit{rob3}}$	${\overline{y}}_{\mathit{rob4}}$	${\overline{y}}_{\mathit{rob5}}$
10	${\overline{y}}_{\mathit{pro1}}$	3.34	3.26	2.62	3.34	2.60	1.85	1.78	28	1.85	1.27
	${\overline{y}}_{\mathit{pro2}}$	3.51	3.42	2.75	3.50	2.73	1.94	1.87	1.35	1.94	1.34
	${\overline{y}}_{\mathit{pro3}}$	5.40	5.26	4.22	5.39	4.20	2.99	2.88	2.07	2.98	2.05
	${\overline{y}}_{\mathit{pro4}}$	3.35	3.26	2.62	3.34	2.61	1.85	1.79	1.29	1.85	1.27
	${\overline{y}}_{\mathit{pro5}}$	5.46	5.32	4.27	5.45	4.25	3.02	2.91	2.10	3.02	2.08
20	${\overline{y}}_{\mathit{pro1}}$	3.01	2.92	2.24	3.01	2.23	1.88	1.81	1.30	1.88	1.29
	${\overline{y}}_{\mathit{pro2}}$	3.15	3.05	2.35	3.15	2.33	1.97	1.89	1.36	1.96	1.35
	${\overline{y}}_{\mathit{pro3}}$	4.56	4.43	3.40	4.56	3.38	2.85	2.74	1.98	2.84	1.96
	${\overline{y}}_{\mathit{pro4}}$	3.02	2.92	2.25	3.01	2.23	1.88	1.81	1.31	1.88	1.30
	${\overline{y}}_{\mathit{pro5}}$	4.60	4.46	3.43	4.59	3.41	2.87	2.76	1.99	2.87	1.98
30	${\overline{y}}_{\mathit{pro1}}$	2.98	2.89	2.23	2.98	2.21	1.74	1.68	1.23	1.74	1.22
	${\overline{y}}_{\mathit{pro2}}$	3.10	3.00	2.32	3.09	2.30	1.81	1.74	1.28	1.81	1.27
	${\overline{y}}_{\mathit{pro3}}$	4.15	4.03	3.10	4.14	3.08	2.42	2.34	1.72	2.42	1.70
	${\overline{y}}_{\mathit{pro4}}$	2.99	2.90	2.23	2.98	2.21	1.74	1.68	1.23	1.74	1.22
	${\overline{y}}_{\mathit{pro5}}$	4.18	4.06	3.13	4.18	3.10	2.44	2.36	1.73	2.44	1.71

Table 3 depicted that performance of proposed estimators for population mean in two phase sampling is better than OLS and considered robust estimators for all sample sizes at low level contamination of outliers i.e. 2%. The R.E of proposed estimator ${\overline{y}}_{\mathit{pro1}}$ at sample of size 10 is 3.34 i.e. 334 % as compare to estimator ${\overline{y}}_{\mathit{OLS1}}$ and 1.85 i.e. 185% as compare to estimator ${\overline{y}}_{\mathit{rob1}}$. These R.Es are 298% and 174% respectively at phase II sample of size 30 with reference to estimator ${\overline{y}}_{\mathit{pro1}}$. It is also concluded from Table 3 that estimators given by Noor-ul-Amin et al. (2016) have overall high R.Es than OLS estimators. It is due to low level of contamination of outliers. As the sample size of Phase II increases performance of considered estimators of population mean improves significantly but still remains less than proposed regression-in-ratio estimators for population mean.

Table 4 R.E of proposed estimators w.r.t. considered estimators in two phase sampling at $n_1=100$ and 6% outliers

$n_2$	R.E	${\overline{y}}_{\mathit{OLS1}}$	${\overline{y}}_{\mathit{OLS2}}$	${\overline{y}}_{\mathit{OLS3}}$	${\overline{y}}_{\mathit{OLS4}}$	${\overline{y}}_{\mathit{OLS5}}$	${\overline{y}}_{\mathit{rob1}}$	${\overline{y}}_{\mathit{rob2}}$	${\overline{y}}_{\mathit{rob3}}$	${\overline{y}}_{\mathit{rob4}}$	${\overline{y}}_{\mathit{rob5}}$
10	${\overline{y}}_{\mathit{pro1}}$	5.18	4.97	4.15	5.16	4.22	5.14	4.93	4.11	5.12	4.18
	${\overline{y}}_{\mathit{pro2}}$	5.64	5.42	4.52	5.62	4.60	5.59	5.37	4.47	5.57	4.55
	${\overline{y}}_{\mathit{pro3}}$	8.46	8.12	6.78	8.43	6.89	8.38	8.05	6.71	8.35	6.83
	${\overline{y}}_{\mathit{pro4}}$	5.22	5.01	4.18	5.20	4.25	5.17	4.96	4.14	5.15	4.21
	${\overline{y}}_{\mathit{pro5}}$	8.12	7.80	6.50	8.09	6.62	8.05	7.72	6.44	8.02	6.55
20	${\overline{y}}_{\mathit{pro1}}$	3.80	3.64	2.98	3.78	3.03	3.76	3.61	2.95	3.75	3.00
	${\overline{y}}_{\mathit{pro2}}$	4.06	3.90	3.19	4.05	3.24	4.03	3.86	3.16	4.01	3.21
	${\overline{y}}_{\mathit{pro3}}$	5.63	5.39	4.42	5.61	4.49	5.58	5.35	4.37	5.56	4.45
	${\overline{y}}_{\mathit{pro4}}$	3.82	3.66	3.00	3.80	3.05	3.78	3.63	2.97	3.77	3.02
	${\overline{y}}_{\mathit{pro5}}$	5.48	5.25	4.30	5.46	4.37	5.43	5.21	4.26	5.41	4.33
30	${\overline{y}}_{\mathit{pro1}}$	3.05	2.93	2.42	3.04	2.46	3.04	2.91	2.41	3.03	2.45
	${\overline{y}}_{\mathit{pro2}}$	3.23	3.10	2.56	3.22	2.61	3.22	3.09	2.55	3.20	2.60
	${\overline{y}}_{\mathit{pro3}}$	4.20	4.03	3.33	4.18	3.39	4.18	4.01	3.32	4.16	3.38
	${\overline{y}}_{\mathit{pro4}}$	3.06	2.94	2.43	3.05	2.48	3.05	2.93	2.42	3.04	2.47
	${\overline{y}}_{\mathit{pro5}}$	4.10	3.93	3.26	4.08	3.31	4.08	3.92	3.24	4.07	3.30

From Table 4, it is concluded that as the percentage of outliers increases in the population, efficiencies of considered estimators given in Equation (12) for population mean decrease like the estimators given in Equation (6) and the level of robustness decayed significantly due to high level of contaminations of outliers. For example, at phase II sample of size 10, the R.E of ${\overline{y}}_{\mathit{pro1}}$ is 518% and 514% as compare to OLS estimator ${\overline{y}}_{\mathit{OLS1}}$ and robust estimator ${\overline{y}}_{\mathit{rob1}}$ respectively. These R.Es become 305% and 304% respectively at $n_2=30$. It is practically proved that estimators for population mean based on redescending estimator proposed by Huber (1964) failed to produce robust estimates at high level of contamination of outliers as their performance is similar to the estimators based on OLS technique at this level of contamination.

6 Conclusion

Results obtained in above sections concluded that proposed regression-in-ratio estimators performed significantly better than considered estimators of population mean for all condition and for all sample sizes in two phase sampling design. It is suggested that devised estimators should be utilized for estimation of population mean in two phase sampling design as they outfit the considered estimators for population mean. The proposed estimators can be used in other sampling designs and in quality control to obtain efficient estimator of mean when data have outliers.

Acknowledgement

The authors are thankful to Prof. Cem Kadilar for providing the data of example 1.

Appendix

Using the notations given in Section 1.1, following expression is obtained for the estimator given in (14).

${\overline{y}}_{\mathit{prol}}$	$\cong {\displaystyle \frac{[{\mu }_y(1+{\overline{e}}_{y_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})]({\lambda }_l{\mu }_x(1+{\overline{e}}_{x_1})+{\phi }_l)}{({\lambda }_l{\mu }_x(1+{\overline{e}}_{x_2})+{\phi }_l)}}$
${\overline{y}}_{\mathit{prol}}$	$\cong {\displaystyle \frac{[{\mu }_y(1+{\overline{e}}_{y_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})]((1+{\overline{e}}_{x_1})+\frac{{\phi }_l}{{\lambda }_l{\mu }_x})}{((1+{\overline{e}}_{x_2})+\frac{{\phi }_l}{{\lambda }_l{\mu }_x})}}$
${\overline{y}}_{\mathit{prol}}$	$\cong {\displaystyle \frac{[{\mu }_y(1+{\overline{e}}_{y_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})]((1+{\overline{e}}_{x_1})+{\gamma }_l)}{((1+{\overline{e}}_{x_2})+{\gamma }_l)}}$

where

${\gamma }_l$	$={\displaystyle \frac{{\phi }_l}{{\lambda }_l{\mu }_x}}$
${\overline{y}}_{\mathit{prol}}$	$\cong [{\mu }_y(1+{\overline{e}}_{y_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})]((1+{\overline{e}}_{x_1})+{\gamma }_l){((1+{\overline{e}}_{x_2})+{\gamma }_l)}^{-1}$

Expanding last term up to order one

${\overline{y}}_{\mathit{prol}}$	$\cong [{\mu }_y(1+{\overline{e}}_{y_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})]((1+{\overline{e}}_{x_1})+{\gamma }_l)$
	$\mathrm{\hspace{1em}}(1-{\overline{e}}_{x_2}-{\gamma }_l+2{\gamma }_l{\overline{e}}_{x_2})$
	$\cong [{\mu }_y(1+{\overline{e}}_{y_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})](1+(1-{\gamma }_l))({\overline{e}}_{x_1}-{\overline{e}}_{x_2})$
${\overline{y}}_{\mathit{prol}}-{\mu }_y$	$\cong [{\mu }_y(1+{\overline{e}}_{y_2})(1+(1-{\gamma }_l))({\overline{e}}_{x_1}-{\overline{e}}_{x_2})$
	$\mathrm{\hspace{1em}}+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})(1+(1-{\gamma }_l))({\overline{e}}_{x_1}-{\overline{e}}_{x_2})]$

Ignoring square and higher order terms

${\overline{y}}_t-{\mu }_y$	$\cong {\mu }_y{\overline{e}}_{y_2}+{\mu }_y(1-{\gamma }_l)({\overline{e}}_{x_1}-{\overline{e}}_{x_2})+b_3{\mu }_x({\overline{e}}_{x_1}-{\overline{e}}_{x_2})$
${\overline{y}}_t-{\mu }_y$	$\cong {\mu }_y{\overline{e}}_{y_2}+\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}({\overline{e}}_{x_1}-{\overline{e}}_{x_2})$

Squaring and applying expectation on both sides

$\mathit{MSE}(\overline{y})$	$\cong {\mu }_y^2{\theta }_2{C}_y^2+C_x^2{\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}}^2({\theta }_2-{\theta }_1)$
	$\mathrm{\hspace{1em}}+2{\mu }_y\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}({\theta }_1{\rho }_{xy}{C}_x{C}_y-{\theta }_2{\rho }_{xy}{C}_x{C}_y)$
$\mathit{MSE}(\overline{y})$	$={\mu }_y^2{\theta }_2{C}_y^2+C_x^2{\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}}^2({\theta }_2-{\theta }_1)$
	$\mathrm{\hspace{1em}}+2{\mu }_y\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}({\theta }_1-{\theta }_2)H_{yx}{C}_x^2$
$\mathit{MSE}(\overline{y})$	$={\mu }_y^2{\theta }_2{C}_y^2+C_x^2{\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}}^2({\theta }_2-{\theta }_1)$
	$\mathrm{\hspace{1em}}+2{\mu }_y\{{\mu }_y(1-{\gamma }_l)+b_3{\mu }_x\}({\theta }_1-{\theta }_2)H_{yx}{C}_x^2$

The MSE of proposed regression-in-ratio estimators is obtained as under

$\mathit{MSE}({\overline{y}}_{\mathit{prol}})$	$={\mu }_y^2{\theta }_2{C}_y^2+\{B_3{\mu }_x+{\mu }_y(1-{\gamma }_l)\}({\theta }_2-{\theta }_1)C_x^2$
	$\mathrm{\hspace{1em}}\{B_3{\mu }_x+{\mu }_y(1-{\gamma }_l)-2{\mu }_y{H}_{yx}\}$

References

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Biographies

Muhammad Noor-ul-Amin received his Ph.D. degree from NCBA&E, Lahore, Pakistan. He has working experience in various universities for teaching and research that includes the Virtual University of Pakistan, University of Sargodha, Pakistan, and the University of Burgundy, France. He is currently working as an Assistant Professor at COMSATS University Islamabad-Lahore Campus. His research interests include sampling techniques and control charting techniques. He is an HEC approved supervisor.

Aamir Raza received his Ph.D. degree from National College of Business Administration & Economics (NCBA&E), Lahore, Pakistan. He did his M. Phil & M.Sc. Statistics from University of the Punjab, Lahore. He is currently working as Lecturer in Statistics at Govt. Graduate College, Mandi Bahauddin. His research interests include sampling techniques and Robust Regression.

Journal of Reliability and Statistical Studies, Vol. 14, Issue 2 (2021), 527–540.
doi: 10.13052/jrss0974-8024.1427
© 2021 River Publishers

Abstract

1 Introduction

1.1 Notations

2 Regression-in-Ratio Estimators in Two Phase Sampling

3 Proposed Regression-in-Ratio Estimators in Two Phase Sampling

4 Efficiency Comparison

5 Applications

5.1 Real Life Data Example

5.2 Simulation Study

6 Conclusion

Acknowledgement

Appendix

References

Biographies