Optimal Estimation of Population Mean Using Two Auxiliary Variables in the Presence of Non-response by Simulation Approach

Abstract

In this article, we have proposed a difference-cum-exponential type estimator for estimating population mean by using information on two auxiliary variables under non-response, in which the expressions of bias and mean square error of the proposed estimator have been derived up to the first order of approximation. The suggested estimator has been compared with other existing and adaptive estimators theoretically and empirically. Also, a simulation study has been performed; for this, we have generated two artificial data sets to demonstrate the efficiency of the proposed estimator than the other estimators.

AMS Subject Classification: MSC2020, 62D05

Keywords

Auxiliary variable study variable non-response bias mean square error efficiency

1. Introduction

The problem of non-response in sample surveys is encountered more frequently in mail than in personal interviews.^[1] Hansen and Hurwitz^[2] were the first to address this issue by proposing a sampling plan that involved drawing an initial sample through the first mail attempt, followed by taking a sub-sample from the non-responding population using personal interviews.

It is noted that the presence of auxiliary information can suggest improved estimators of the population mean in typical surveys and plays a vital role in addressing non-response problems as well. Authors such as Cochran,^[3] Rao (1983, 1986, 1987),^[4–6] Khare and Srivastava,^[7–9] Okafor and Lee,^[10] Tabasum and Khan,^{[11, 12]} Singh and Kumar (2008a, 2008b, 2008c, 2009a, 2009b, 2010)^[13–17] and Singh^[18] have utilized auxiliary information to develop improved estimators of the population mean in the presence of non-response.

Consider a finite population of size $N$ , with the values of the study variable $y$ (say) for the units in the population being denoted by $y_{1}, y_{2}, \dots, y_{N}$ . The objective is to estimate the population mean of the $y$ values, denoted as $\bar{Y}$ . To estimate the population mean, a random sample of size $n$ , denoted by $s_{n}$ , is drawn without replacement from the population. In survey sampling involving human populations, it is observed that $n_{1}$ individuals respond on the first attempt, while $n_{2} = n - n_{1}$ individuals do not provide a response, resulting in non-response.

To adjust for non-response at the initial stage, Hansen and Hurwitz^[2] proposed a double sampling plan for estimating the population mean, which is as follows:

A simple random sample of size n is drawn without replacement and a questionnaire is mailed to the sample units.

A sub-sample of size $r = (n_{2} / k), (k \geq 1)$ from $n_{2}$ non-responding units in the initial stage is contacted through personal interviews.

In Hansen and Hurwitz method, the population is supposed to consist of a response stratum of size $N_{1}$ and a non-response stratum of size $N_{2} = (N - N_{1})$ . Let $\bar{Y} = \sum_{i = 1}^{N} y_{i} / N$ and $S_{y}^{2} = \sum_{i = 1}^{N} {(y_{i} - \bar{Y})}^{2} / (N - 1)$ denote the population mean and variance of the study variable y. Let ${\bar{Y}}_{1} = \sum_{i = 1}^{N_{1}} y_{i} / N_{1}$ and $S_{y (1)}^{2} = \sum_{i = 1}^{N_{1}} {(y_{i} - {\bar{Y}}_{1})}^{2} / (N_{1} - 1)$ denote the mean and variance of the respondent group (or strata). Similarly, let ${\bar{Y}}_{2} = \sum_{i = 1}^{N_{2}} y_{i} / N_{2}$ and $S_{y (2)}^{2} = \sum_{i = 1}^{N_{2}} {(y_{i} - {\bar{Y}}_{2})}^{2} / (N_{2} - 1)$ denote the mean and variance of the non-respondent group (or strata). The population mean can be written as:

\bar{Y} = W_{1} {\bar{Y}}_{1} + W_{2} {\bar{Y}}_{2}

where $W_{1} = N_{1} / N$ and $W_{2} = N_{2} / N$ . The sample mean ${\bar{y}}_{1} = \sum_{i = 1}^{n_{1}} y_{i} / n_{1}$ denotes the mean of the $n_{1}$ responding units and ${\bar{y}}_{2} = \sum_{i = 1}^{n_{2}} y_{i} / n_{2}$ denotes the mean of the $n_{2}$ non-responding units.

Let ${\bar{y}}_{2 r} = \sum_{i = 1}^{r} y_{i} / r$ denote the mean of the $r$ sub-sampled units where $r = n_{2} / k$ . Hansen and Hurwitz^[2] suggested an unbiased estimator for the population mean $\bar{Y}$ of the study variable $y$ is given as:

{\bar{y}}^{*} = w_{1} {\bar{y}}_{1} + w_{2} {\bar{y}}_{2 r}

(1.1)

where $w_{1} = n_{1} / n$ and $w_{2} = n_{2} / n$ are the responding proportions and non-responding proportions of the sample. The variance ${\bar{y}}^{*}$ is given as:

V ({\bar{y}}^{*}) = {\bar{Y}}^{2} [(\frac{1 - f}{n}) C_{y}^{2} + \frac{W_{2} (k - 1)}{n} C_{y (2)}^{2}]

(1.2)

where $C_{y}^{2} = S_{y}^{2} / {\bar{Y}}^{2}$ and $C_{y (2)}^{2} = S_{y (2)}^{2} / {\bar{Y}}^{2}$ .

Let $x_{i}$ and $z_{i} (i = 1, 2, \dots N)$ denote the auxiliary variable correlated with the study variable $y_{i} (i = 1, 2, \dots, N)$ , with population mean and variance given as $\bar{X} = \sum_{i = 1}^{N} x_{i} / N$ , $\bar{Z} = \sum_{i = 1}^{N} z_{i} / N$ and $S_{x}^{2} = \sum_{i = 1}^{N} {(x_{i} - \bar{X})}^{2} / (N - 1)$ $S_{z}^{2} = \sum_{i = 1}^{N} {(z_{i} - \bar{Z})}^{2} / (N - 1)$ , respectively. For the respondent group of size $N_{1}$ , let ${\bar{X}}_{1} = \sum_{i = 1}^{N_{1}} x_{i} / N_{1}$ and $S_{x (1)}^{2} = \sum_{i = 1}^{N_{1}} {(x_{i} - {\bar{X}}_{1})}^{2} / (N_{1} - 1)$ , ${\bar{Z}}_{1} = \sum_{i = 1}^{N_{1}} z_{i} / N_{1}$ and $S_{z (1)}^{2} = \sum_{i = 1}^{N_{1}} {(z_{i} - {\bar{Z}}_{1})}^{2} / (N_{1} - 1)$ denote the mean and variance of the respondent group (or strata) for auxiliary variables $x$ and $z$ , respectively. Similarly, let ${\bar{X}}_{2} = \sum_{i = 1}^{N_{2}} x_{i} / N_{2}$ and $S_{x (2)}^{2} = \sum_{i = 1}^{N_{2}} {(x_{i} - {\bar{X}}_{2})}^{2} / (N_{2} - 1)$ and ${\bar{Z}}_{2} = \sum_{i = 1}^{N_{2}} z_{i} / N_{2}$ and $S_{z (2)}^{2} = \sum_{i = 1}^{N_{2}} {(z_{i} - {\bar{Z}}_{2})}^{2} / (N_{2} - 1)$ denote the mean and variance of the non-respondent group (or strata) for auxiliary variables $x$ and $z$ , respectively. Let $\bar{x} = \sum_{i = 1}^{n} x_{i} / n$ , $\bar{z} = \sum_{i = 1}^{n} z_{i} / n$ denote the mean of all the $n$ units, and ${\bar{x}}_{1} = \sum_{i = 1}^{n_{1}} x_{i} / n_{1}$ , ${\bar{z}}_{1} = \sum_{i = 1}^{n_{1}} z_{i} / n_{1}$ denote the mean of the $n_{1}$ responding units and ${\bar{x}}_{2} = \sum_{i = 1}^{n_{2}} x_{i} / n_{2}$ , ${\bar{z}}_{2} = \sum_{i = 1}^{n_{2}} z_{i} / n_{2}$ denote the mean of the $n_{2}$ non-responding units for auxiliary variables $x$ and $z$ , respectively.

Let ${\bar{x}}_{2 r} = \sum_{i = 1}^{r} x_{i} / r$ , ${\bar{z}}_{2 r} = \sum_{i = 1}^{r} z_{i} / r$ denote the mean of the $r$ sub-sampled units from non-respondent strata for auxiliary variables $x$ and $z$ , respectively, where $r = n_{2} / k$ . With this background, an unbiased estimator of the population mean $\bar{X}$ and $\bar{Z}$ is given as ${\bar{x}}^{*} = w_{1} {\bar{x}}_{1} + w_{2} {\bar{x}}_{2 r}$ and ${\bar{z}}^{*} = w_{1} {\bar{z}}_{1} + w_{2} {\bar{z}}_{2 r}$ . The variance of ${\bar{x}}^{*}$ and $z^{*}$ is given as $V ({\bar{x}}^{*}) = {\bar{X}}^{2} [((1 - f) / n) C_{x}^{2} + (W_{2} (k - 1) / n) C_{x (2)}^{2}],$ and $V ({\bar{z}}^{*}) = {\bar{Z}}^{2} [((1 - f) / n) C_{z}^{2} + (W_{2} (k - 1) / n) C_{z (2)}^{2}]$ , where $C_{x}^{2} = S_{x}^{2} / {\bar{X}}^{2}$ and $C_{x (2)}^{2} = S_{x (2)}^{2} / {\bar{X}}^{2}$ and where $C_{z}^{2} = S_{z}^{2} / {\bar{Z}}^{2}$ and $C_{z (2)}^{2} = S_{z (2)}^{2} / {\bar{Z}}^{2}$ .

Let us define the following terms:

{\bar{y}}^{*} = \bar{Y} (1 + ε_{0}), {\bar{x}}^{*} = \bar{X} (1 + ε_{1}), {\bar{z}}^{*} = \bar{Z} (1 + ε_{2})

Then $E (ε_{0}) = E (ε_{1}) = E (ε_{2}) = 0$

E (ε_{0}^{2}) = [ϕ C_{y}^{2} + ψ C_{y (2)}^{2}] = V_{0} (s a y), E (ε_{1}^{2}) = [ϕ C_{x}^{2} + ψ C_{x (2)}^{2}] = V_{1} (s a y), E (ε_{2}^{2}) = [ϕ C_{z}^{2} + ψ C_{z (2)}^{2}] = V_{2} (s a y)

E (ε_{0} ε_{1}) = [ϕ ρ_{y x} C_{y} C_{x} + ψ ρ_{y x (2)} C_{y (2)} C_{x (2)}] = V_{01} (s a y),

E (ε_{0} ε_{2}) = [ϕ ρ_{y z} C_{y} C_{z} + ψ ρ_{y z (2)} C_{y (2)} C_{z (2)}] = V_{02} (s a y)

E (ε_{1} ε_{2}) = [ϕ ρ_{x z} C_{x} C_{z} + ψ ρ_{x z (2)} C_{x (2)} C_{z (2)}] = V_{12} (s a y)

where, $ϕ = (\frac{1 - f}{n})$ , $ψ = \frac{W_{2} (k - 1)}{n}$ and $ρ_{y x} = S_{y x} / S_{x} S_{y}$ , $ρ_{y x (2)} = S_{y x (2)} / S_{x (2)} S_{y (2)}$ ,

ρ_{x z} = \frac{S_{x z}}{S_{z} S_{x}}, ρ_{x z (2)} = \frac{S_{x z (2)}}{S_{z (2)} S_{x (2)}}, R_{1} = \frac{\bar{Y}}{\bar{X}}, R_{2} = \bar{Y} / \bar{Z}

S_{y x} = \frac{1}{N - 1} \sum_{i = 1}^{N} (y_{i} - \bar{Y}) (x_{i} - \bar{X}), S_{y x (2)} = \frac{1}{N_{2} - 1} \sum_{i = 1}^{N_{2}} (y_{i} - {\bar{Y}}_{2}) (x_{i} - {\bar{X}}_{2})

S_{y z} = \frac{1}{N - 1} \sum_{i = 1}^{N} (y_{i} - \bar{Y}) (z_{i} - \bar{Z}), S_{y z (2)} = \frac{1}{N_{2} - 1} \sum_{i = 1}^{N_{2}} (y_{i} - {\bar{Y}}_{2}) (z_{i} - {\bar{Z}}_{(2)})

S_{x z} = \frac{1}{N - 1} \sum_{i = 1}^{N} (x_{i} - \bar{X}) (z_{i} - \bar{Z}), S_{x z (2)} = \frac{1}{N_{2} - 1} \sum_{i = 1}^{N_{2}} (x_{i} - {\bar{X}}_{2}) (z_{i} - {\bar{Z}}_{(2)})

2. Review and Adaptive Estimators

When a few observations are missing in the sample, Hansen and Hurwitz^[2] were the first to propose an estimator for estimating the population mean $\bar{Y}$ . Subsequently, many other researchers have worked on similar situations. Here, we present some of the existing estimators and adaptive estimators using two auxiliary variables for estimating the population mean in cases where some observations are missing.

Rao^[5] proposed a ratio estimator for estimating the population mean $\bar{Y}$ of the study variable $y$ using a single auxiliary variable. Building on this, we adapt the ratio estimator to estimate the population mean $\bar{Y}$ of the study variable $y$ in the presence of non-response utilizing two auxiliary variables $x$ and $y$ . The adapted estimator is given as:

S_{1} = {\bar{y}}^{*} (\frac{\bar{X}}{{\bar{x}}^{*}}) (\frac{\bar{Z}}{{\bar{z}}^{*}})

(2.1)

The bias and mean square error (MSE) of S₁ are given as:

B (S_{1}) = \bar{Y} [ϕ (C_{x}^{2} + C_{z}^{2} - ρ_{y x} C_{y} C_{x} - ρ_{y z} C_{y} C_{z} + ρ_{x z} C_{x} C_{z}) + \frac{W_{2} (k - 1)}{n} (C_{x (2)}^{2} + C_{z (2)}^{2} - ρ_{y x (2)} C_{y (2)} C_{x (2)} - ρ_{y z (2)} C_{y (2)} C_{z (2)} + ρ_{x z (2)} C_{x (2)} C_{z (2)})] = \bar{Y} (V_{1} + V_{2} - V_{01} - V_{02} + V_{12})

M S E (S_{1}) = {\bar{Y}}^{2} [ϕ (C_{y}^{2} + C_{x}^{2} + C_{z}^{2} - 2 ρ_{y x} C_{y} C_{x} - 2 ρ_{y z} C_{y} C_{z} + 2 ρ_{x z} C_{x} C_{z}) + ψ (C_{y (2)}^{2} + C_{x (2)}^{2} + C_{z (2)}^{2} - 2 ρ_{y x (2)} C_{y (2)} C_{x (2)} - 2 ρ_{y z (2)} C_{y (2)} C_{z (2)} + 2 ρ_{x z (2)} C_{x (2)} C_{z (2)})] = {\bar{Y}}^{2} (V_{0} + V_{1} + V_{2} - 2 V_{01} - 2 V_{02} + 2 V_{12})

(2.2)

Khare and Srivastava^[4] suggested a product estimator for estimating the population mean $\bar{Y}$ of the study variable $y$ for a single auxiliary variable. Adapting this, the product estimator for estimating the population mean $\bar{Y}$ of the study variable $y$ in the presence of non-response using two auxiliary variables $x$ and $y .$

S_{2} = {\bar{y}}^{*} (\frac{{\bar{x}}^{*}}{\bar{X}}) (\frac{{\bar{z}}^{*}}{\bar{Z}})

(2.3)

The bias and MSE of $S_{2}$ are given as:

B i a s (S_{2}) = \bar{Y} [ϕ (ρ_{y x} C_{y} C_{x} + ρ_{y z} C_{y} C_{z} + ρ_{x z} C_{x} C_{z}) + ψ (ρ_{y x (2)} C_{y (2)} C_{x (2)} + ρ_{y z (2)} C_{y (2)} C_{z (2)} + ρ_{x z (2)} C_{x (2)} C_{z (2)})] = \bar{Y} (V_{01} + V_{02} + V_{12})

M S E (S_{2}) = {\bar{Y}}^{2} [ϕ (C_{y}^{2} + C_{x}^{2} + C_{z}^{2} + 2 ρ_{y x} C_{y} C_{x} + 2 ρ_{y z} C_{y} C_{z} + 2 ρ_{x z} C_{x} C_{z}) + ψ (C_{y (2)}^{2} + C_{x (2)}^{2} + C_{z (2)}^{2} + 2 ρ_{y x (2)} C_{y (2)} C_{x (2)} + 2 ρ_{y z (2)} C_{y (2)} C_{z (2)} + 2 ρ_{x z (2)} C_{x (2)} C_{z (2)})] = {\bar{Y}}^{2} (V_{0} + V_{1} + V_{2} + 2 V_{01} + 2 V_{02} + 2 V_{12})

(2.4)

Singh et al. (2008a, 2008b), for estimating the finite population mean of a study variable having one auxiliary variable is known. Singh et al. (2008a, 2008b) suggested exponential ratio-type estimators for the population mean $\bar{Y}$ of the study variable $y$ using one auxiliary variable $x$ . The adapted exponential ratio-type estimator for estimating the population mean $\bar{Y}$ of the study variable $y$ in the presence of non-response using two auxiliary variables $x$ and $y$ is given as:

S_{3} = {\bar{y}}^{*} exp (\frac{\bar{X} - {\bar{x}}^{*}}{\bar{X} + {\bar{x}}^{*}}) exp (\frac{\bar{Z} - {\bar{z}}^{*}}{\bar{Z} + {\bar{z}}^{*}})

(2.5)

The bias and MSE of the $S_{3}$ are given as:

B i a s (S_{3}) = \bar{Y} [ϕ (\frac{3}{8} C_{x}^{2} + \frac{3}{8} C_{z}^{2} - \frac{1}{2} ρ_{y x} C_{y} C_{x} - \frac{1}{2} ρ_{y z} C_{y} C_{z} + \frac{1}{2} ρ_{x z} C_{x} C_{z}) + ψ (\frac{3}{8} C_{x (2)}^{2} + \frac{3}{8} C_{z (2)}^{2} - \frac{1}{2} ρ_{y x (2)} C_{y (2)} C_{x (2)} - \frac{1}{2} ρ_{y z (2)} C_{y (2)} C_{z (2)} + \frac{1}{2} ρ_{x z (2)} C_{x (2)} C_{z (2)})] = \bar{Y} [\frac{3}{8} V_{1} + \frac{3}{8} V_{2} - \frac{1}{2} V_{01} - \frac{1}{2} V_{02} + \frac{1}{2} V_{12}]

M S E (S_{3}) = {\bar{Y}}^{2} [ϕ (C_{y}^{2} + \frac{1}{4} C_{x}^{2} + \frac{1}{4} C_{z}^{2} - ρ_{y x} C_{y} C_{x} - ρ_{y z} C_{y} C_{z} + \frac{1}{2} ρ_{x z} C_{x} C_{z}) + ψ (C_{y (2)}^{2} + \frac{1}{4} C_{x (2)}^{2} + \frac{1}{4} C_{z (2)}^{2} - ρ_{y x (2)} C_{y (2)} C_{x (2)} - ρ_{y z (2)} C_{y (2)} C_{z (2)} + \frac{1}{2} ρ_{x z (2)} C_{x (2)} C_{z (2)})] = {\bar{Y}}^{2} [V_{0} + \frac{1}{4} V_{1} + \frac{1}{4} V_{2} - V_{01} - V_{02} + \frac{1}{2} V_{12}]

(2.6)

Singh et al. (2008a, 2008b) suggested exponential product-type estimators for the population mean $\bar{Y}$ of the study variable $y$ having one auxiliary variable $x$ . Building on this, we adapt the exponential product-type estimators for estimating the population mean $\bar{Y}$ of the study variable $y$ in the presence of non-response using two auxiliary variables $x$ and $y$ .

S_{4} = {\bar{y}}^{*} \exp (\frac{{\bar{x}}^{*} - \bar{X}}{{\bar{x}}^{*} + \bar{X}}) \exp (\frac{{\bar{z}}^{*} - \bar{Z}}{{\bar{z}}^{*} + \bar{Z}})

(2.7)

The bias and MSE of $S_{4}$ are given as:

B i a s (S_{4}) = \bar{Y} [ϕ (- \frac{1}{8} C_{x}^{2} - \frac{1}{8} C_{z}^{2} + \frac{1}{2} ρ_{y x} C_{y} C_{x} + \frac{1}{2} ρ_{y z} C_{y} C_{z} + \frac{1}{4} ρ_{x z} C_{x} C_{z}) + ψ (- \frac{1}{8} C_{x (2)}^{2} - \frac{1}{8} C_{z (2)}^{2} + \frac{1}{2} ρ_{y x (2)} C_{y (2)} C_{x (2)} + \frac{1}{2} ρ_{y z (2)} C_{y (2)} C_{z (2)} + \frac{1}{4} ρ_{x z (2)} C_{x (2)} C_{z (2)})] = \bar{Y} [- \frac{1}{8} V_{1} - \frac{1}{8} V_{2} + \frac{1}{2} V_{01} + \frac{1}{2} V_{02} + \frac{1}{4} V_{12}]

M S E (S_{4}) = {\bar{Y}}^{2} [ϕ (C_{y}^{2} + \frac{1}{4} C_{x}^{2} + \frac{1}{4} C_{z}^{2} + ρ_{y x} C_{y} C_{x} + ρ_{y z} C_{y} C_{z} + \frac{1}{2} ρ_{x z} C_{x} C_{z}) + ψ (C_{y (2)}^{2} + \frac{1}{4} C_{x (2)}^{2} + \frac{1}{4} C_{z (2)}^{2} + ρ_{y x (2)} C_{y (2)} C_{x (2)} + ρ_{y z (2)} C_{y (2)} C_{z (2)} + \frac{1}{2} ρ_{x z (2)} C_{x (2)} C_{z (2)})] = {\bar{Y}}^{2} [V_{0} + \frac{1}{4} V_{1} + \frac{1}{4} V_{2} + V_{01} + V_{02} + \frac{1}{2} V_{12}]

(2.8)

Kumar and Bhougal^[19] proposed a ratio-product type exponential estimator following Singh et al. (2008a, 2008b), for estimating the finite population mean $\bar{Y}$ of a study variable $y$ having one auxiliary variable $x$ . Following Kumar and Bhougal,^[7] we adapt the exponential ratio-type estimators for estimating the population mean $\bar{Y}$ of the study variable $y$ in the presence of non-response, utilizing two auxiliary variables $x$ and $z$ , giving as:

S_{5} = {\bar{y}}^{*} [λ exp (\frac{\bar{X} - {\bar{x}}^{*}}{\bar{X} + {\bar{x}}^{*}}) exp (\frac{\bar{Z} - {\bar{z}}^{*}}{\bar{Z} + {\bar{z}}^{*}}) + (1 - λ) exp (\frac{{\bar{x}}^{*} - \bar{X}}{{\bar{x}}^{*} + \bar{X}}) exp (\frac{{\bar{z}}^{*} - \bar{Z}}{{\bar{z}}^{*} + \bar{Z}})]

(2.9)

The bias and MSE of $S_{5}$ are given as:

B (S_{5}) = \bar{Y} [ϕ (\frac{1}{4} ρ_{x z} C_{x} C_{z} - \frac{1}{8} C_{x}^{2} - \frac{1}{8} C_{z}^{2} + \frac{λ}{2} C_{x}^{2} + \frac{λ}{2} C_{z}^{2} + \frac{1}{2} ρ_{y x} C_{y} C_{x} + \frac{1}{2} ρ_{y z} C_{y} C_{z} - λ ρ_{y x} C_{y} C_{x} - λ ρ_{y z} C_{y} C_{z}) + ψ (\frac{1}{4} ρ_{x z (2)} C_{x (2)} C_{z (2)} - \frac{1}{8} C_{_{x (2)}}^{2} - \frac{1}{8} C_{_{z (2)}}^{2} + \frac{λ}{2} C_{_{x (2)}}^{2} + \frac{λ}{2} C_{_{z (2)}}^{2} + \frac{1}{2} ρ_{y x (2)} C_{y (2)} C_{x (2)} + \frac{1}{2} ρ_{y z (2)} C_{y (2)} C_{z (2)} - λ ρ_{y x (2)} C_{y (2)} C_{x (2)} - λ ρ_{y z (2)} C_{y (2)} C_{z (2)})] = \bar{Y} [\frac{1}{4} V_{12} - \frac{1}{8} V_{1} - \frac{1}{8} V_{2} + \frac{λ}{2} V_{1} + \frac{λ}{2} V_{2} + \frac{1}{2} V_{01} + \frac{1}{2} V_{02} - λ V_{01} - λ V_{02}]

M S E (S_{5}) = Y ¯^{2} [ϕ (1 / 4 C_{x}^{2} + C_{y}^{2} + ρ_{y} x C_{y} C_{x} + 1 / 4 C_{z}^{2} + ρ_{y} z C_{y} C_{z} + 1 / 2 ρ_{x} z C_{x} C_{z} - λ C_{x}^{2} - 2 λ ρ_{y} x C_{y} C_{x} - λ C_{z}^{2} - 2 λ ρ_{y} z C_{y} C_{z} - 2 λ ρ_{x} z C_{x} C_{z} + λ^{2} C_{x}^{2} + λ^{2} C_{z}^{2} + 2 λ^{2} ρ_{x} z C_{x} C_{z}) + ψ (1 / 4 C_{(} x (2))^{2} + C_{(} y (2))^{2} + ρ_{(} y x (2)) C_{(} y (2)) C_{(} x (2)) + 1 / 4 C_{(} z (2))^{2} + ρ_{(} y z (2)) C_{(} y (2)) C_{(} z (2)) + 1 / 2 ρ_{(} x z (2)) C_{(} x (2)) C_{(} z (2)) - λ C_{(} x (2))^{2} - 2 λ ρ_{(} y x (2)) C_{(} y (2)) C_{(} x (2)) - λ C_{(} z (2))^{2} - 2 λ ρ_{(} y z (2)) C_{(} y (2)) C_{(} z (2)) - 2 λ ρ_{(} x z (2)) C_{(} x (2)) C_{(} z (2)) + λ^{2} C_{(} x (2))^{2} + λ^{2} C_{(} z (2))^{2} + 2 λ^{2} ρ_{(} x z (2)) C_{(} x (2)) C_{(} z (2)))] = Y ¯^{2} [1 / 4 V_{1} + V_{0} + V_{0} 1 + 1 / 4 V_{2} + V_{0} 2 + 1 / 2 V_{1} 2 - λ V_{1} - 2 λ V_{0} 1 - λ V_{2} - 2 λ V_{0} 2 - 2 λ V_{1} 2 + λ^{2} V_{1} + λ^{2} V_{2} + 2 λ^{2} V_{1} 2]

where $λ = (V_{1} + V_{2} + 2 V_{01} + 2 V_{02} + 2 V_{12}) / (2 (V_{1} + V_{2} + 2 V_{12}))$ .

3. Proposed Estimator

Following Hansen and Hurwitz^[2] technique of dealing with non-response, and motivated by the estimators in the review section, in this article, we propose a difference-cum-exponential type estimator using two auxiliary variables $x$ and $y$ to estimate the population mean $\bar{Y}$ in the presence of few missing observations when the non-response occurs on both study variable $y$ as well as auxiliary variables $(x, z$ ), and the population means $\bar{X} and \bar{Z}$ of the auxiliary variable is known. The proposed estimator is given as:

S_{0} = [α {\bar{y}}^{*} + β (\bar{X} - {\bar{x}}^{*}) + γ (\bar{Z} - {\bar{z}}^{*})] exp [\frac{\bar{Z} - {\bar{z}}^{*}}{\bar{Z} + {\bar{z}}^{*}}]

(3.1)

where $α, β and γ$ are real constant to be determined such that the MSE of $S_{0}$ is minimum.

3.1 Bias and MSE of the Proposed Estimator

The bias and MSEs of the proposed estimator $S_{0}$ to the first order of approximations are derived under large sample approximations.

Expressing $S_{0}$ in terms of $ε$ s, we can write:

\begin{matrix} S_{0} = [α \bar{Y} (1 + ε_{0}) + β (\bar{X} - \bar{X} (1 + ε_{1})) + γ (\bar{Z} - \bar{Z} (1 + ε_{2}))] \exp [\frac{\bar{Z} - \bar{Z} (1 + ε_{2})}{\bar{Z} + \bar{Z} (1 + ε_{2})}] \\ = [α \bar{Y} (1 + ε_{0}) - β \bar{X} ε_{1} - γ \bar{Z} ε_{2}] \exp [\frac{- ε_{2}}{2 + ε_{2}}] \\ = [α \bar{Y} (1 + ε_{0}) - β \bar{X} ε_{1} - γ \bar{Z} ε_{2}] \exp [- \frac{ε_{2}}{2} {(1 + \frac{ε_{2}}{2})}^{- 1}] \end{matrix}

(3.2)

Expanding the right-hand side of Equation (3.2):

S_{0} = α \bar{Y} (1 + ε_{0} - \frac{ε_{2}}{2} - \frac{ε_{0} ε_{2}}{2} + \frac{3}{8} ε_{2}^{2} \dots) - β \bar{X} (ε_{1} - \frac{ε_{1} ε_{2}}{2} \dots) - γ \bar{Z} (ε_{2} - \frac{ε_{2}^{2}}{2} \dots)

and neglecting the terms involving powers of $ε$ s greater than two, we have

S_{0} - \bar{Y} = (α - 1) \bar{Y} + α \bar{Y} (ε_{0} - \frac{ε_{2}}{2} - \frac{ε_{0} ε_{2}}{2} + \frac{3}{8} ε_{2}^{2}) - β \bar{X} (ε_{1} - \frac{ε_{1} ε_{2}}{2}) - γ \bar{Z} (ε_{2} - \frac{ε_{2}^{2}}{2})

(3.3)

Taking expectation on both sides of Equation (3.3), we get the approximate expression for the bias of the estimator $S_{0}$ as:

B (S_{0}) = \bar{Y} [(α - 1) + ϕ (\frac{3}{8} α C_{z}^{2} - \frac{1}{2} α ρ_{y z} C_{y} C_{z} + \frac{1}{2} \frac{β}{R_{1}} ρ_{x z} C_{x} C_{z} + \frac{1}{2} \frac{γ}{R_{2}} C_{z}^{2}) + ψ (\frac{3}{8} α C_{z (2)}^{2} - \frac{1}{2} α ρ_{y z (2)} C_{y (2)} C_{z (2)} + \frac{1}{2} \frac{β}{R_{1}} ρ_{x z (2)} C_{x (2)} C_{z (2)} + \frac{1}{2} \frac{γ}{R_{2}} C_{z (2)}^{2})]

= \bar{Y} [(α - 1) + α \frac{1}{8} (3 V_{2} - 4 V_{02}) + \frac{1}{2} \frac{β}{R_{1}} V_{12} + \frac{1}{2} \frac{γ}{R_{2}} V_{2}]

(3.4)

Squaring of Equation (3.3) on both sides and neglecting terms of $ε$ s involving powers greater than two, we have:

{(S_{0} - \bar{Y})}^{2} ≅ {[(α - 1) \bar{Y} + α \bar{Y} (ε_{0} - \frac{ε_{2}}{2} - \frac{ε_{0} ε_{2}}{2} + \frac{3}{8} ε_{2}^{2}) - β \bar{X} (ε_{1} - \frac{ε_{1} ε_{2}}{2}) - γ \bar{Z} (ε_{2} - \frac{ε_{2}^{2}}{2})]}^{2}

= {(α - 1)}^{2} {\bar{Y}}^{2} + α^{2} {\bar{Y}}^{2} (ε_{0}^{2} + \frac{ε_{2}^{2}}{4} - ε_{0} ε_{2}) + β^{2} {\bar{X}}^{2} ε_{1}^{2} + γ^{2} {\bar{Z}}^{2} ε_{2}^{2} + 2 α (α - 1) {\bar{Y}}^{2} (ε_{0} - \frac{ε_{2}}{2} - \frac{ε_{0} ε_{2}}{2} + \frac{3}{8} ε_{2}^{2}) - 2 β (α - 1) \bar{X} \bar{Y} (ε_{1} - \frac{ε_{1} ε_{2}}{2}) - 2 γ (α - 1) \bar{Y} \bar{Z} (ε_{2} - \frac{ε_{2}^{2}}{2}) - 2 α β \bar{X} \bar{Y} (ε_{0} ε_{1} - \frac{ε_{1} ε_{2}}{2}) - 2 α γ \bar{Y} \bar{Z} (ε_{0} ε_{2} - \frac{ε_{2}^{2}}{2}) + 2 β γ \bar{X} \bar{Z} ε_{1} ε_{2}

(3.5)

Taking expectation on both sides of Equation (3.5), we get the first order approximation of the MSE of as:

Y ¯^{2} [(α^{2} + 1) + ((1 - f) / n) (α^{2} C_{y}^{2} + α^{2} C_{z}^{2} - 2 α^{2} ρ_{y} z C_{y} C_{z} + β^{2} / (R_{1}^{2}) C_{x}^{2} + γ^{2} / (R_{2}^{2}) C_{z}^{2} + 2 α β / R_{1} ρ_{x} z C_{x} C_{z} - 2 α β / R_{1} ρ_{y} x C_{y} C_{x} + 2 α γ / R_{2} C_{z}^{2} - 2 α γ / R_{2} ρ_{y} z C_{y} C_{z} + 2 β γ / (R_{1} R_{2}) ρ_{x} z C_{x} C_{z} - 3 / 4 α C_{z}^{2} + 3 / 4 α ρ_{y} z C_{y} C_{z} - β / R_{1} ρ_{x} z C_{x} C_{z} - γ / R_{2} C_{z}^{2}) + (W_{2} (k - 1)) / n (α^{2} C_{(} y (2))^{2} + α^{2} C_{(} z (2))^{2} - 2 α^{2} ρ_{(} y z (2)) C_{(} y (2)) C_{(} z (2)) + β^{2} / (R_{1}^{2}) C_{(} x (2))^{2} + γ^{2} / (R_{2}^{2}) C_{(} z (2))^{2} + 2 α β / R_{1} ρ_{(} x z (2)) C_{(} x (2)) C_{(} z (2)) - 2 α β / R_{1} ρ_{(} y x (2)) C_{(} y (2)) C_{(} x (2)) + 2 α γ / R_{2} C_{(} z (2))^{2} - 2 α γ / R_{2} ρ_{(} y z (2)) C_{(} y (2)) C_{(} z (2)) + 2 β γ / (R_{1} R_{2}) ρ_{(} x z (2)) C_{(} x (2)) C_{(} z (2)) - 3 / 4 α C_{(} z (2))^{2} + 3 / 4 α ρ_{(} y z (2)) C_{(} y (2)) C_{(} z (2)) - β / R_{1} ρ_{(} x z (2)) C_{(} x (2)) C_{(} z (2)) - γ / R_{2} C_{(} z (2))^{2})]

= {\bar{Y}}^{2} α^{2} (1 + V_{0} + V_{2} - 2 V_{02}) + β^{2} {\bar{X}}^{2} V_{1} + γ^{2} {\bar{Z}}^{2} V_{2} + 2 α β \bar{X} \bar{Y} (V_{12} - V_{01}) + 2 α γ \bar{Y} \bar{Z} (V_{2} - V_{02}) + 2 β γ \bar{X} \bar{Z} V_{12} - α {\bar{Y}}^{2} (\frac{3}{4} V_{2} - V_{02}) - β \bar{X} \bar{Y} V_{12} - γ \bar{Y} \bar{Z} V_{2} + {\bar{Y}}^{2}

M S E (S_{0}) = {\bar{Y}}^{2} [α^{2} (1 + V_{0} + V_{2} - 2 V_{02}) + \frac{β^{2}}{R_{1}^{2}} V_{1} + \frac{γ^{2}}{R_{2}^{2}} V_{2} - 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) + 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) + 2 \frac{β γ}{R_{1} R_{2}} V_{12} - α (\frac{3}{4} V_{2} - V_{02}) - \frac{β}{R_{1}} V_{12} - \frac{γ}{R_{2}} V_{2} + 1]

(3.6)

= {\bar{Y}}^{2} [1 + α^{2} A + β^{2} B + γ^{2} C - 2 α β D + 2 α γ E + 2 β γ F - α G - β H - γ I]

(3.7)

where

A = (1 + V_{0} + V_{2} - 2 V_{02}), B = V_{1} / R_{1}^{2}, C = \frac{V_{2}}{R_{2}^{2}}, D = (V_{12} - V_{01}) / R_{1}, E = (V_{2} - V_{02}) / R_{2}, F = V_{12} / R_{1} R_{2},

G = (3 / 4 V_{2} - V_{02}), H = \frac{V_{12}}{R_{1}}, I = V_{2} / R_{2} .

3.2 Optimum Choice of

α, β a n d γ

and the Minimum MSE of Proposed Estimator

S_{0}

To obtain the optimum values of $α, β$ and $γ$ , corresponding to minimum variance, differentiate the expression of MSE in Equation (3.7) with respect to $α, β$ and $γ$ and equate them to zero. The optimum value of $α, β$ and $γ$ accordingly are obtained by solving the following set of simultaneous equations:

\begin{matrix} 2 α A - 2 β D + 2 γ E = G \\ 2 β B - 2 α D + 2 γ E = H \\ 2 γ C + 2 α E + 2 β F = I \end{matrix}\}

(3.8)

This system can be written in matrix form as:

Q l^{'} = b

(3.9)

where, $Q = [\begin{matrix} A & - D & E \\ - D & B & F \\ E & F & C \end{matrix}]$ , $l^{'} = (α, β, γ)$ , $b = (G, H, I)$

Solving Equation (3.9), we get the optimum values of ( $α, β, γ)$ as:

α_{o p t} = \frac{1}{2} \frac{Q_{1}}{D e t (Q)}, β_{o p t} = \frac{1}{2} \frac{Q_{2}}{D e t (Q)}, γ_{o p t} = \frac{1}{2} \frac{Q_{3}}{D e t (Q)}

where

Q_{1} = |\begin{matrix} G & - D & E \\ H & B & F \\ I & F & C \end{matrix}| = G (B C - F^{2}) + D (H C - F I) + E (F H - F I)

= B C G - G F^{2} + C D H + E F H - D F I - B E I

Q_{2} = |\begin{matrix} A & G & E \\ - D & H & F \\ E & I & C \end{matrix}| = A (H C - F I) + G (D C + E F) - E (D I + E H)

= A H C - A F I + C D G + E F G - E^{2} H - D E I

Q_{3} = |\begin{matrix} A & - D & G \\ - D & B & H \\ E & F & I \end{matrix}| = A (B I - F H) - D (D I + E H) - G (D F + E B)

= A B I - D^{2} I - D F G - B E G - A F H - D E H

D e t (Q) = |\begin{matrix} A & - D & E \\ - D & B & F \\ E & F & C \end{matrix}| = (A B C - A F^{2} - C D^{2} - 2 D E F - B E^{2})

Substituting the optimum values of $α$ , $β$ and $γ$ in Equation (3.7), the resulting minimum MSE of the proposed estimator $S_{0}$ is given by:

min M S E (S_{0}) = {\bar{Y}}^{2} [1 - \frac{1}{4} \frac{G Q_{1} + + H Q_{2} + T Q_{3}}{det (Q)}]

(3.10)

If $0 < \frac{G Q_{1} + + H Q_{2} + T Q_{3}}{det (Q)} < 4$ and $D e t (Q) > 0$ are satisfied.

4. Theoretical Efficiency Comparison

The proposed estimator is better in terms of efficiency with ${\bar{y}}^{*}$ and the other estimators $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ , utilizing two auxiliary variates, if the following conditions are satisfied:

$V a r (y^{*}) - M S E (S_{0}) \geq 0 i f$

V_{0} (1 - α^{2}) - α^{2} V_{2} - 2 α^{2} V_{02} - \frac{β^{2}}{R_{1}^{2}} V_{1} - \frac{γ^{2}}{R_{2}^{2}} V_{2} + α (\frac{3}{4} V_{2} - V_{02}) + \frac{β}{R_{1}} V_{12} + 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) - 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) - 2 \frac{β γ}{R_{1} R_{2}} V_{12} + \frac{γ}{R_{2}} V_{2} - (α^{2} + 1) \geq 0

(4.1)

$M S E (S_{1}) - M S E (S_{0}) \geq 0 i f$

V_{0} + V_{1} + V_{2} - 2 V_{01} - 2 V_{02} + 2 V_{12} - α^{2} (V_{0} + V_{2} - 2 V_{02}) - \frac{β^{2}}{R_{1}^{2}} V_{1} - \frac{γ^{2}}{R_{2}^{2}} V_{2} + 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) - 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) - 2 \frac{β γ}{R_{1} R_{2}} V_{12} + α (\frac{3}{4} V_{2} - V_{02}) + \frac{β}{R_{1}} V_{12} + \frac{γ}{R_{2}} V_{2} - (α^{2} + 1) \geq 0

(4.2)

$M S E (S_{2}) - M S E (S_{0}) \geq 0 i f$

V_{0} + V_{1} + V_{2} + 2 V_{01} + 2 V_{02} + 2 V_{12} - α^{2} (V_{0} + V_{2} - 2 V_{02}) - \frac{β^{2}}{R_{1}^{2}} V_{1} - \frac{γ^{2}}{R_{2}^{2}} V_{2} + 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) - 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) - 2 \frac{β γ}{R_{1} R_{2}} V_{12} + α (\frac{3}{4} V_{2} - V_{02}) + \frac{β}{R_{1}} V_{12} + \frac{γ}{R_{2}} V_{2} - (α^{2} + 1) \geq 0

(4.3)

$M S E (S_{3}) - M S E (S_{0}) \geq 0 i f$

V_{0} + \frac{1}{4} V_{1} + \frac{1}{4} V_{2} - V_{01} - V_{02} + \frac{1}{2} V_{12} - α^{2} (V_{0} + V_{2} - 2 V_{02}) - \frac{β^{2}}{R_{1}^{2}} V_{1} - \frac{γ^{2}}{R_{2}^{2}} V_{2} + 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) - 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) - 2 \frac{β γ}{R_{1} R_{2}} V_{12} + α (\frac{3}{4} V_{2} - V_{02}) + \frac{β}{R_{1}} V_{12} + \frac{γ}{R_{2}} V_{2} - (α^{2} + 1) \geq 0

(4.4)

$M S E (S_{4}) - M S E (S_{0}) \geq 0 i f$

V_{0} + \frac{1}{4} V_{1} + \frac{1}{4} V_{2} + V_{01} + V_{02} + \frac{1}{2} V_{12} - α^{2} (V_{0} + V_{2} - 2 V_{02}) - \frac{β^{2}}{R_{1}^{2}} V_{1} - \frac{γ^{2}}{R_{2}^{2}} V_{2} + 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) - 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) - 2 \frac{β γ}{R_{1} R_{2}} V_{12} + α (\frac{3}{4} V_{2} - V_{02}) + \frac{β}{R_{1}} V_{12} + \frac{γ}{R_{2}} V_{2} - (α^{2} + 1) \geq 0

(4.5)

$M S E (S_{5}) - M S E (S_{0}) \geq 0 i f$

\frac{1}{4} V_{1} + V_{0} + V_{01} + \frac{1}{4} V_{2} + V_{02} + \frac{1}{2} V_{12} - λ V_{1} - 2 λ V_{01} - λ V_{2} - 2 λ V_{02} - 2 λ V_{12} + λ^{2} V_{1} + λ^{2} V_{2} + 2 λ^{2} V_{12} - α^{2} (V_{0} + V_{2} - 2 V_{02}) - \frac{β^{2}}{R_{1}^{2}} V_{1} - \frac{γ^{2}}{R_{2}^{2}} V_{2} + 2 \frac{α β}{R_{1}} (V_{12} - V_{01}) - 2 \frac{α γ}{R_{2}} (V_{2} - V_{02}) - 2 \frac{β γ}{R_{1} R_{2}} V_{12} + α (\frac{3}{4} V_{2} - V_{02}) + \frac{β}{R_{1}} V_{12} + \frac{γ}{R_{2}} V_{2} - (α^{2} + 1) \geq 0

(4.6)

5. Numerical Illustration

To illustrate the efficiency comparison of the theoretical results, we consider a real data set used by Khare and Sinha.^[20] The description of the dataset is given below:

Population 1: The data on physical growth of the upper socio-economic group of 95 schoolchildren of Varanasi under an Indian Council of Medical Research (ICMR) study, Department of Paediatrics, Banaras Hindu University, during 1983–1984 have been taken under study. The first 25 per cent (i.e., 24 children) units have been considered as non-responding units. Here, we have taken the study characters and the auxiliary characters as follows:

$y$ : weight (in kg) of the children

$x$ : skull circumference (in cm) of the children

$z$ : chest circumference (in cm) of the children

\bar{Y} = 19.4968, \bar{X} = 51.1726, \bar{Z} = 55.8611, C_{y} = 0.15613, C_{x} = 0.03006,

C_{z} = 0.05860, C_{y (2)} = 0.12075, C_{x (2)} = 0.02478, C_{z (2)} = 0.05402, ρ_{y x} = 0.328,

ρ_{y x (2)} = 0.477, ρ_{y z} = 0.846, ρ_{y z (2)} = 0.729, ρ_{x z} = 0.297, ρ_{x z (2)} = 0.570,

N = 95, n = 35, W_{2} = 0.25

The MSEs and percent-relative efficiencies (PREs) of the proposed various existing estimators with respect to the usual unbiased estimator ${\bar{y}}^{*}$ for different values of k are calculated by using the formulae:

P R E (S_{j}, {\bar{y}}^{*}) = \frac{V a r ({\bar{y}}^{*})}{M S E (S_{j})} \times 100

where $S_{j} = S_{0}, S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ . The results are shown in Table 1.

Table 1.

Mean Square Error (MSE) and Percent-relative Efficiency (PRE) of the Various Estimators of Population 1 (Real Data Set).

Estimators	(1/k)
	k = 2		k = 3		k = 4		k = 5		k = 6
	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$S_{1}$	0.0970	98.14	0.1167	102.87	0.1365	106.23	0.1562	108.74	0.1760	110.68
$S_{2}$	0.4113	23.15	0.4982	24.11	0.5851	24.78	0.6720	25.28	0.7589	25.67
$S_{3}$	0.1400	68.00	0.1663	72.24	0.1925	75.33	0.2187	77.68	0.2450	79.53
$S_{4}$	0.29723	32.04	0.35704	33.65	0.4168	34.79	0.4766	35.65	0.5364	36.32
$S_{5}$	0.0894	124.40	0.1119	123.04	0.1344	122.93	0.1568	122.20	0.1793	122.04
$S_{0}$	0.0675	140.96	0.0875	137.29	0.1069	135.68	0.1259	134.89	0.1448	134.49

6. Simulation Study

For the purpose of comparison of the proposed estimator, we conducted the simulation study and computed the PRE of the estimator for artificially generated data. We computed the PREs of various existing estimators with respect to the usual unbiased estimator ${\bar{y}}^{*}$ for different values of k by using the formulae:

P R E (S_{j}, {\bar{y}}^{*}) = \frac{V a r ({\bar{y}}^{*})}{M S E (S_{j})} \times 100, S_{j} = S_{0}, S_{1}, S_{2}, S_{3}, S_{4}, S_{5}

The simulation study has been carried out to illustrate and compare the performance of the proposed estimator $S_{0}$ with the existing and adaptive estimators $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ . In this simulation study, we consider two artificial populations described as follows:

Population 2: A population of size $N = 1, 000$ , with one study variable $y$ and two auxiliary variables $x$ and $z$ , is generated from the normal distribution, where the study variable $y$ is correlated with auxiliary variables $x$ and $z$ . Non-response occurs in both study and auxiliary variables. The variables $(y, x, z)$ are generated using the MVNORM package in the R software. From this population, we draw a sample of size $n = 250$ , and then consider $8 %$ cases as missing randomly, that is, there are $n_{2} = 20$ non-respondent units in the sample. The values of MSE for all the considered estimators and PRE with respect to ${\bar{y}}^{*}$ are calculated for the adapted and proposed estimators for different values of $k (= 2, 3, 4, 5, 6)$ . The result of these calculations is shown in Table 2.

Population 3: An artificial population is generated of size $N = 2, 000$ , which involves one study variable $y$ and two auxiliary variables $x and z$ , where both the variables are correlated. Non-response occurs in both the study variable $y$ and auxiliary variables $x and z$ . The variables $(y, x, z)$ are generated using the MVNORM package in the R software. From this population, we draw a sample of size $n = 256$ and consider $12 %$ as non-response $(n_{2} = 30)$ from the sample and calculate PRE for different values of $k (= 2, 3, 4, 5, 6)$ (shown in Table 3).

The PREs of the proposed estimators are computed through 10,000 repeated samples of size $n$ , following the non-response technique. The process involves the following steps:

Draw a random sample of size $n$ from the population of size $N$ .

From each selected sample $n_{2}$ units are treated as non-responses.

Calculate the estimators and their MSEs for each sample, and then compute the mean over all 10,000 samples.

The MSE and PREs are given by:

M S E (S_{j}) = \frac{1}{10000} \sum_{i = 1}^{10000} {(S_{j} - \bar{Y})}^{2}, S_{j} = S_{0}, S_{1}, S_{2}, S_{3}, S_{4}, S_{5}

P R E (S_{j}, {\bar{y}}^{*}) = \frac{V a r ({\bar{y}}^{*})}{M S E (S_{j})} \times 100, S_{j} = S_{0}, S_{1}, S_{2}, S_{3}, S_{4}, S_{5}

Based on 10,000 repeated samples, the values of PREs are recorded and reported in Tables 2 and 3.

Table 2.

Percent-relative Efficiency Based on Population 2 (Artificially Generated Normal Population).

$P R E (S_{j}, {\bar{y}}^{*})$	$(1 / k)$
$P R E (S_{j}, {\bar{y}}^{*})$	k = 2	k = 3	k = 4	k = 5	k = 6
$P R E ({\bar{y}}^{*})$	100.00	100.00	100.00	100.00	100.00
$P R E (S_{1}, {\bar{y}}^{*})$	45.52	53.97	62.43	70.88	79.33
$P R E (S_{2}, {\bar{y}}^{*})$	11.38	13.49	15.61	17.72	19.83
$P R E (S_{3}, {\bar{y}}^{*})$	194.05	230.09	266.13	302.17	338.21
$P R E (S_{4}, {\bar{y}}^{*})$	26.41	31.31	36.22	41.12	46.026
$P R E (S_{5}, {\bar{y}}^{*})$	220.47	261.19	301.87	342.51	383.10
$P R E (S_{0}, {\bar{y}}^{*})$	243.98	288.59	332.79	376.48	419.61

Table 3.

Percent-relative Efficiency Based on Population 3 (Artificially Generated Normal Population).

$P R E (S_{j}, {\bar{y}}^{*})$	$(1 / k)$
$P R E (S_{j}, {\bar{y}}^{*})$	k = 2	k = 3	k = 4	k = 5	k = 6
$P R E ({\bar{y}}^{*})$	100.00	100.00	100.00	100.00	100.00
$P R E (S_{1}, {\bar{y}}^{*})$	43.5905	48.1178	52.6452	57.1725	61.6998
$P R E (S_{2}, {\bar{y}}^{*})$	8.1391	8.9845	9.8298	10.6752	11.5205
$P R E (S_{3}, {\bar{y}}^{*})$	189.3557	209.0224	228.6890	248.3556	268.0222
$P R E (S_{4}, {\bar{y}}^{*})$	20.6617	22.8076	24.9536	27.0995	29.2454
$P R E (S_{5}, {\bar{y}}^{*})$	219.0583	241.5701	264.0697	286.5604	309.0444
$P R E (S_{0}, {\bar{y}}^{*})$	237.6378	262.7540	287.8705	313.0039	338.1565

7. Interpretation of the Computational Results

The following interpretation can be drawn from Tables 1 –3:

For the real population, the results are shown in Table 1. It is evident that the proposed estimator outperforms the other existing estimators, including adaptive estimators, in terms of MSE and PRE. Table 1 indicates that the PREs of $S_{1}, S_{2}, S_{3} and S_{4}$ increase, while the PREs of the $S_{0}$ and $S_{5}$ decrease as the value of $k$ increases.

The PRE values calculated in Tables 2 and 3 for artificially generated populations from a normal distribution, for various values of $k$ and different non-response $(n_{2} (in %))$ , show that the proposed estimator consistently yields better results than the existing and adaptive estimators. From Tables 2 and 3, it can be observed that the PREs of the $S_{0}, S_{1}, S_{2}, S_{3}, S_{4}$ and $S_{5}$ increase as the value of $k$ increases. The results of this simulation study clearly demonstrate that the proposed estimator $S_{0}$ performs better in comparison to the conventional estimators.

8. Conclusion

In this article, we have proposed an estimator $S_{0}$ of the finite population mean of the study variable using two auxiliary variables, considering non-response in both the study variable and the auxiliary variables. The proposed estimator is compared with the usual mean estimator, as well as other existing and adaptive estimators. Computational results based on one real and two artificial populations demonstrate the superiority of the proposed method.

The simulation results confirm the finding derived from the theoretical formula of the MSE. The simulation results also show that the proposed estimator $S_{0}$ under different populations and varying non-response rates performs better than the other estimators under optimality conditions, as shown in Tables 2 and 3. Therefore, the suggested estimator $S_{0}$ is to be recommended for practical use.

Footnotes

Acknowledgement

The authors are grateful to the experienced referees for their useful comments and suggestions.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

ORCID iDs

Udita Gupta

Anjali Bhardwaj

References

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13.

Singh

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14.

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15.

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16.

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A general class of estimators of the population mean in survey sampling using auxiliary information with sub sampling the non-respondents. Korean J Appl Stat 2009b; 22(2): 387–402.

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18.

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19.

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20.

Khare

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Estimation of the ratio of the two population means using multi auxiliary characters in the presence of non-response. In: BN Pandey, editor. Statistical Techniques in Life Testing, Reliability, Sampling Theory and Quality Control . New Delhi: Narosa Publishing House; 2007. pp. 163–171.