Two new estimators of finite population variance under random non-response: An application

Abstract

In this paper, an attempt has been made to study the estimation of finite population variance in simple random sampling without replacement (SRSWOR) in presence of non-response by proposing two estimators. The two estimators (dual to ratio and ratio cum dual to ratio type) have been proposed on the basis of availability and non-availability of auxiliary information. The properties such as bias and mean square error of the proposed estimators have also been studied up to first order of approximation. The proposed estimators are compared with some existing estimators and are also mutually compared. An empirical study, based on both vegetable crop data and simulated data, has been performed to find out the best estimator. It has been seen that out of the two proposed estimators, the ratio cum dual to ratio type estimator performed better than dual to ratio estimator.

Keywords

Auxiliary information bias mean square error non-response percentage relative efficiency

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1. Introduction

The auxiliary information includes a known variable (auxiliary variable) to which the variable of interest (study variable) is related. For example, production of a crop depends on area under cultivation. It is well known that the proper use of auxiliary information results in substantial increase in the precision of estimators. By applying auxiliary information in different forms, estimators for population parameters generally population mean and variance are made by various statisticians, namely, Cochran (1940), Olkin (1958), Srivastava and Jhajj (1981), Bahl and Tuteja (1991) and Singh et al. (2016).

Sometimes, there also arises a situation where information about some observations which are included in the sample is not available leading to incompleteness in the data. This type of problem is termed as non-response. Hansen and Hurwitz (1946) were the first to deal with the problem of incomplete samples in mail surveys. They proposed a model for mail survey design to provide unbiased estimators of population mean or total. Okafor and Lee (2000) applied Hansen and Hurwitz (1946) technique for ratio and regression estimator. Tracy and Osahan (1994) studied the effect of random non-response on the usual ratio estimator of population mean in two situations: (i) non-response in study as well as the auxiliary variable and (ii) non-response in the study variable only.

Singh and Joarder (1998) considered the problem of estimation of finite population variance in the presence of non-response in survey sampling. Singh (2003) compared the efficiency of estimators proposed by Singh and Joarder (1998) by using the data set on real estate farm loans and non real estate farm loans. After that, various researchers including Ahmed et al. (2005), Dubey and Uprety (2008), Kumar and Bhougal (2011), Kumar (2014), Shahzad et al. (2017), Bhat et al. (2018), Kumar et al. (2018), Irfan et al. (2018), Mittal and Kumar (2021) and Priyanka (2021) have contributed for the estimation of population mean and population variance under random non-response.

2. Distribution of random non response and notations

Consider a finite population $\Omega=(v_{1},v_{2},\dots,v_{N})$ of $N$ identifiable units from which an SRSWOR sample of size $n$ is drawn. If $r(r=0,1,\ldots,(n-2))$ denotes the number of sampling units on which information could not be obtained due to random non-response, then the remaining $(n-r)$ units in the sample can be treated as an SRSWOR sample from $\Omega$ . We assume that $r$ should be less than $(n-1)$ , that is, $0\leqslant r\leqslant(n-2)$ , as we are interested in unbiased estimation of finite population variance. If $p$ denotes the probability of non-response among the $(n-2)$ possible values of responses, then Singh and Joarder (1998) assumed that $r$ has following discrete distribution

$\displaystyle P(r)=\frac{n-r}{nq+2p}\left(\genfrac{}{}{0.0pt}{}{n-2}{r}\right)% p^{r}q^{n-2-r}$ (1)

where $q=1-p$ and $(\genfrac{}{}{0.0pt}{}{n-2}{r})$ denotes the number of ways to obtain $r$ non-responses out of possible $(n-2)$ .

We write, $\varepsilon=(s^{*2}_{y}/S^{2}_{y})-1$ , $\delta=(s^{*2}_{x}/S^{2}_{x})-1$ , and $\eta=(s^{2}_{x}/S^{2}_{x})-1$ , where,

$\displaystyle s^{*2}_{y}={(n-r-1)}^{-1}\sum\nolimits^{n-r}_{i=1}{{(y_{i}-{% \overline{y}}^{*})}^{2}}$

and

$\displaystyle s^{*2}_{x}={(n-r-1)}^{-1}\sum\nolimits^{n-r}_{i=1}{{(x_{i}-{% \overline{x}}^{*})}^{2}}$

are conditionally unbiased estimators of finite population variance of study variable $y$ , i.e.

$\displaystyle S^{2}_{y}={(N-1)}^{-1}\sum\nolimits^{N}_{i=1}{{(Y_{i}-\overline{% Y})}^{2}}$

and auxiliary variable $x$ , i.e.

$\displaystyle S^{2}_{x}={(N-1)}^{-1}\sum\nolimits^{N}_{i=1}{{(X_{i}-\overline{% X})}^{2}}$

respectively, and

$\displaystyle s^{2}_{x}={(n-1)}^{-1}\sum\nolimits^{n}_{i=1}{{(x_{i}-\overline{% x})}^{2}},{\overline{y}}^{*}={(n-r)}^{-1}\sum\nolimits^{n-r}_{i=1}{{y}_{i}},$ $\displaystyle{\overline{x}}^{*}={(n-r)}^{-1}\sum\nolimits^{n-r}_{i=1}{{x}_{i}}% ,\overline{x}=n^{-1}\sum\nolimits^{n}_{i=1}{x_{i}}\overline{Y}=N^{-1}\sum% \nolimits^{N}_{i=1}{{Y}_{i}}$

and

$\displaystyle\overline{X}=N^{-1}\sum\nolimits^{N}_{i=1}{X_{i}}.$

Under (1), the following results are obtained,

$\displaystyle E(\varepsilon)=E(\delta)=E(\eta)=0,E({\varepsilon}^{2})=\left(% \frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1),$ $\displaystyle E({\delta}^{2})=\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({% \lambda}_{04}-1),E({\eta}^{2})=\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_% {04}-1),$ $\displaystyle E(\varepsilon\delta)=\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)(% {\lambda}_{22}-1),E(\varepsilon\eta)=\left(\frac{1}{n}-\frac{1}{N}\right)({% \lambda}_{22}-1),$

and

$\displaystyle E(\delta\eta)=\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{04% }-1)$

where

$\displaystyle{\lambda}_{ls}=\frac{{\mu}_{ls}}{{\mu}^{{l}/{2}}_{20}{\mu}^{{s}/{% 2}}_{02}}$

and

$\displaystyle{\mu}_{ls}=\frac{1}{N-1}\sum\nolimits^{N}_{i=1}{{(Y_{i}-\overline% {Y})}^{l}}{(X_{i}-\overline{X})}^{s}.$

For estimating the population variance $S^{2}_{y}$ of study variable $y$ , Singh and Joarder (1998) proposed two situations in presence of non-response viz.

Situation I: When random non-response exists on both the study variable and the auxiliary variable which are denoted by $y$ and $x$ respectively and the population variance $S^{2}_{x}$ of the auxiliary variable is known.

Situation II: When information on study variable $y$ could not be obtained for $r$ units while information on the auxiliary variable $x$ is available and population variance $S^{2}_{x}$ of the auxiliary variable is known.

3. Existing estimators

The usual unbiased estimator for the population variance $S^{2}_{y}$ of study variable $y$ under non-response is given by

$\displaystyle{\hat{T}}_{0}=s^{*2}_{y}$ (2)

The variance of ${\hat{T}}_{0}$ to the first degree of approximation under non response is given by

$\displaystyle\textit{Var}({\hat{T}}_{0})=\textit{Var}(s^{*2}_{y})=\textit{Var}% (S^{2}_{y}(1+\varepsilon))=S^{4}_{y}[E({\varepsilon}^{2})-E(\varepsilon)]=S^{4% }_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1)$ (3)

Singh and Joarder (1998) proposed following estimators under situation I and II and studied the properties of estimators under non-response.

$\displaystyle{\hat{T}}_{RI}=s^{*2}_{y}\frac{S^{2}_{x}}{s^{*2}_{x}}$ (4) $\displaystyle{\hat{T}}_{\textit{RII}}=s^{*2}_{y}\frac{S^{2}_{x}}{s^{2}_{x}}$ (5)

The bias and mean square error (MSE) of ${\hat{T}}_{RI}$ and ${\hat{T}}_{\textit{RII}}$ to the first degree of approximation are given by

$\displaystyle\textit{Bias}({\hat{T}}_{RI})=S^{2}_{y}\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)({\lambda}_{04}-{\lambda}_{22})$ (6) $\displaystyle\textit{MSE}({\hat{T}}_{RI})=S^{4}_{y}\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)({\lambda}_{40}+{\lambda}_{04}-2{\lambda}_{22})$ (7) $\displaystyle\textit{Bias}({\hat{T}}_{\textit{RII}})=S^{2}_{y}\left(\frac{1}{n% }-\frac{1}{N}\right)({\lambda}_{04}-{\lambda}_{22})$ (8) $\displaystyle\textit{MSE}({\hat{T}}_{\textit{RII}})=S^{4}_{y}\left(\left(\frac% {1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1)+\left(\frac{1}{n}-\frac{1}{N% }\right)({\lambda}_{04}-1)-2\left(\frac{1}{n}-\frac{1}{N}\right)(\lambda_{22}-% 1)\right)$ (9)

4. Proposed estimators

In this section, two estimators viz. dual to ratio and ratio cum dual to ratio type estimators of population variance under two different situations of random non-response have been proposed by modifying the population variance estimator of Yadav and Kadilar (2013).

4.1 Dual to ratio estimator under different situations of random non-response

Under Situation I

$\displaystyle{{\hat{T}}_{dI}=s}^{*2}_{y}\left(\frac{s^{{}^{\prime}2}_{x}}{S^{2% }_{x}}\right)$ (10)

where,

$\displaystyle s^{{}^{\prime}2}_{x}=\frac{NS^{2}_{x}-ns^{*2}_{x}}{N-n}{=(1+k)S}% ^{2}_{x}-ks^{*2}_{x},k=\frac{n}{N-n}.$

Under Situation II

$\displaystyle{\hat{T}}_{\textit{dII}}=s^{*2}_{y}\left(\frac{s^{{}^{\prime% \prime}2}_{x}}{S^{2}_{x}}\right)$ (11)

where,

$\displaystyle s^{{}^{\prime\prime}2}_{x}=\frac{NS^{2}_{x}-ns^{2}_{x}}{N-n}{=(1% +k)S}^{2}_{x}-ks^{2}_{x},k=\frac{n}{N-n}.$

Theorem 4.1.1: The bias and MSE of the proposed estimator ${\hat{T}}_{dI}$ are given by

$\displaystyle\textit{Bias}({\hat{T}}_{dI})=-kS^{2}_{y}\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)({\lambda}_{22}-1)$ (12) $\displaystyle\textit{MSE}({\hat{T}}_{dI})={S}^{4}_{y}\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)[({\lambda}_{40}-1){+k}^{2}({\lambda}_{04}-1)-2k({\lambda}_{% 22}-1)]$ (13)

Proof: The estimator ${\hat{T}}_{dI}$ in terms of $\varepsilon$ and $\delta$ can be written as

$\displaystyle{\hat{T}}_{dI}=s^{*2}_{y}\left(\frac{(1+k)S^{2}_{x}-ks^{*2}_{x}}{% S^{2}_{x}}\right)=S^{2}_{y}(1+\varepsilon)[(1+k)-k(1+\delta)]=S^{2}_{y}[1+% \varepsilon-k(\delta+\varepsilon\delta)]$ (14)

Taking expected value on both sides of Eq. (14) and using the results on expectations from Section 2, we get

$\displaystyle E({\hat{T}}_{dI})=S^{2}_{y}[1+E(\varepsilon)-k(E(\delta)+E(% \varepsilon\delta))]=S^{2}_{y}\left[1-k\left(\frac{1}{(nq+2p)}-\frac{1}{N}% \right)({\lambda}_{22}-1)\right]$ $\displaystyle{E({\hat{T}}_{dI})-S}^{2}_{y}=-kS^{2}_{y}\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)({\lambda}_{22}-1)$

The bias of estimator ${\hat{T}}_{dI}$ , up to terms of order $n^{-1}$ , is $\textit{Bias}({\hat{T}}_{dI})=-kS^{2}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}% \right)({\lambda}_{22}-1)$ .

From Eq. (14), we have

$\displaystyle{\hat{T}}_{dI}-S^{2}_{y}=S^{2}_{y}[\varepsilon-k(\delta+% \varepsilon\delta)]$ (15)

Squaring on both sides of Eq. (15) and neglecting the higher order terms, we get

$\displaystyle{({\hat{T}}_{dI}-S^{2}_{y})}^{2}=S^{4}_{y}(\varepsilon-k\delta)2=% {({\hat{T}}_{dI}-S^{2}_{y})}^{2}=S^{4}_{y}({\varepsilon}^{2}+k^{2}{\delta}^{2}% -2k\varepsilon\delta)$ (16)

Taking expected value on both sides of Eq. (16) and using the results on expectations from Section 2, we get

$\displaystyle E{({\hat{T}}_{dI}-S^{2}_{y})}^{2}=S^{4}_{y}E({\varepsilon}^{2}+k% ^{2}{\delta}^{2}-2k\varepsilon\delta)=S^{4}_{y}(E({\varepsilon}^{2})+k^{2}E({% \delta}^{2})-2kE(\varepsilon\delta))=S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)({\lambda}_{40}-1)+k^{2}\left(\frac{1}{(nq+2p)}-\frac{1}{N}% \right)({\lambda}_{04}-1)\right.{}\left.-{2}k\left(\frac{1}{(nq+2p)}-\frac{1}{% N}\right)({\lambda}_{22}-1)\right)$

The mean square error of estimator ${\hat{T}}_{dI}$ , up to terms of order $n^{-1}$ , is

$\displaystyle\textit{MSE}({\hat{T}}_{dI})=S^{4}_{y}\left(\frac{1}{(nq+2p)}-% \frac{1}{N}\right)[({\lambda}_{40}-1){+k}^{2}({\lambda}_{04}-1)-2k({\lambda}_{% 22}-1)]$

Hence the theorem.

Theorem 4.1.2: The bias and MSE of the proposed estimator ${\hat{T}}_{\textit{dII}}$ are given by

$\displaystyle\textit{Bias}({\hat{T}}_{\textit{dII}})=-kS^{2}_{y}\left(\frac{1}% {n}-\frac{1}{N}\right)({\lambda}_{22}-1)$ (17) $\displaystyle\textit{MSE}({\hat{T}}_{\textit{dII}})=S^{4}_{y}\left(\left(\frac% {1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1)+\left(\frac{1}{n}-\frac{1}{N% }\right)(k^{2}({\lambda}_{04}-1)-2k({\lambda}_{22}-1))\right)$ (18)

Proof: The estimator ${\hat{T}}_{\textit{dII}}$ in terms of $\varepsilon$ and $\eta$ can be written as

$\displaystyle{\hat{T}}_{\textit{dII}}=s^{*2}_{y}\left(\frac{(1+k)S^{2}_{x}-ks^% {2}_{x}}{S^{2}_{x}}\right)=S^{2}_{y}(1+\varepsilon)[(1+k)-k(1+\eta)]=S^{2}_{y}% [1+\varepsilon-k(\eta+\varepsilon\eta)]$ (19)

Taking expected value on both sides of Eq. (19) and using the results on expectations from Section 2, we get

$\displaystyle E({\hat{T}}_{\textit{dII}})=S^{2}_{y}[1+E(\varepsilon)-k(E(\eta)% +E(\varepsilon\eta))]=S^{2}_{y}\left[1-k\left(\frac{1}{n}-\frac{1}{N}\right)({% \lambda}_{22}-1)\right]$ $\displaystyle E({\hat{T}}_{\textit{dII}}){-S}^{2}_{y}=-kS^{2}_{y}\left(\frac{1% }{n}-\frac{1}{N}\right)({\lambda}_{22}-1)$

The bias of estimator ${\hat{T}}_{\textit{dII}}$ , up to terms of order $n^{-1}$ , is $\textit{Bias}({\hat{T}}_{\textit{dII}})=-kS^{2}_{y}\left(\frac{1}{n}-\frac{1}{% N}\right)({\lambda}_{22}-1)$ .

From Eq. (19), we have

$\displaystyle{\hat{T}}_{\textit{dII}}-S^{2}_{y}=S^{2}_{y}[\varepsilon-k(\eta+% \varepsilon\eta)]$ (20)

Squaring on both sides of Eq. (20) and neglecting the higher order terms, we get

$\displaystyle{({\hat{T}}_{\textit{dII}}-S^{2}_{y})}^{2}=S^{4}_{y}(\varepsilon-% k\eta)2=S^{4}_{y}({\varepsilon}^{2}+k^{2}{\eta}^{2}-2k\varepsilon\eta)$ (21)

Taking expected value on both sides of Eq. (21) and using the results on expectations from Section 2, we get

$\displaystyle E({\hat{T}}_{\textit{dII}}-S^{2}_{y})^{2}=S^{4}_{y}E({% \varepsilon}^{2}+k^{2}{\eta}^{2}-2k\varepsilon=S^{4}_{y}(E({\varepsilon}^{2})+% k^{2}E({\eta}^{2})-2kE(\varepsilon\eta))=S^{4}_{y}\left(\left(\frac{1}{(nq+2p)% }-\frac{1}{N}\right)({\lambda}_{40}-1)+k^{2}\left(\frac{1}{n}-\frac{1}{N}% \right)({\lambda}_{04}-1)\right.\left.{}-2k\left(\frac{1}{n}-\frac{1}{N}\right% )({\lambda}_{22}-1)\right)$

The mean square error of estimator ${\hat{T}}_{\textit{dII}}$ , up to terms of order $n^{-1}$ , is

$\displaystyle\textit{MSE}({\hat{T}}_{\textit{dII}})=S^{4}_{y}\left(\left(\frac% {1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1)+\left(\frac{1}{n}-\frac{1}{N% }\right)(k^{2}({\lambda}_{04}-1)-2k({\lambda}_{22}-1))\right)$

Hence the theorem.

4.2 Ratio cum dual to ratio type estimator under different situations of random non-response

Under Situation I

$\displaystyle{\hat{T}}_{\textit{dRI}}=s^{*2}_{y}\left[\alpha\left(\frac{S^{2}_% {x}}{s^{*2}_{x}}\right)+(1-\alpha)\left(\frac{s^{{}^{\prime}2}_{x}}{S^{2}_{x}}% \right)\right]$ (22)

where $\alpha$ is a constant to be determined such that mean square error of estimator in Eq. (22) is minimum.

Under Situation II

$\displaystyle{\hat{T}}_{\textit{dRII}}=s^{*2}_{y}\left[\alpha\left(\frac{S^{2}% _{x}}{s^{2}_{x}}\right)+(1-\alpha)\left(\frac{s^{{}^{\prime\prime}2}_{x}}{S^{2% }_{x}}\right)\right]$ (23)

where $\alpha$ is a constant to be determined such that mean square error of estimators in Eq. (23) is minimum.

Theorem 4.2.1: The minimum bias and minimum MSE of the proposed estimator ${\hat{T}}_{\textit{dRI}}$ are given by

$\displaystyle\textit{nin.Bias}({\hat{T}}_{\textit{dRI}})=S^{2}_{y}\left(\frac{% 1}{(nq+2p)}-\frac{1}{N}\right)({\alpha}_{1(\textit{opt})}({\lambda}_{04}-1)-g(% {\lambda}_{22}-1))$ (24) $\displaystyle\textit{min.MSE}({\hat{T}}_{\textit{dRI}})=S^{4}_{y}\left(\frac{1% }{(nq+2p)}-\frac{1}{N}\right)(({\lambda}_{40}-1)-g({\lambda}_{22}-1))$ (25)

Proof: ${\hat{T}}_{\textit{dRI}}=s^{*2}_{y}\left[\alpha\left(\frac{S^{2}_{x}}{s^{*2}_{% x}}\right)+(1-\alpha)\left(\frac{{(1+k)S}^{2}_{x}-ks^{*2}_{x}}{S^{2}_{x}}% \right)\right]=s^{*2}_{y}\left[\alpha\left(\frac{S^{2}_{x}}{s^{*2}_{x}}\right)% +(1-\alpha)\left((1+k)-\frac{ks^{*2}_{x}}{S^{2}_{x}}\right)\right]$

The estimator ${\hat{T}}_{\textit{dRI}}$ in terms of $\varepsilon$ and $\delta$ can be written as

$\displaystyle{\hat{T}}_{\textit{dRI}}=S^{2}_{y}(1+\varepsilon)\left(\frac{% \alpha}{1+\delta}+(1-\alpha)(1+k-k(1+\delta))\right)=S^{2}_{y}(1+\varepsilon)(% \alpha{(1+\delta)}^{-1}+(1-\alpha)(1-k\delta))$

We assume that $|\delta|<1$ , so that ${(1+\delta)}^{-1}$ is expandable in terms of power series. Simplifying and ignoring terms of order greater than 2, we get

$\displaystyle{\hat{T}}_{\textit{dRI}}=S^{2}_{y}(1+\varepsilon)(\alpha(1-\delta% +{\delta}^{2}+\ldots)+(1-\alpha-k\delta+\alpha k\delta))=S^{2}_{y}(1+% \varepsilon-(\alpha+k-\alpha k)\delta-(\alpha+k-\alpha k)\varepsilon\delta+{% \alpha\delta}^{2})$ (26)

Taking expected value on both sides of Eq. (26) and using the results on expectations from Section 2, we get

$\displaystyle E({\hat{T}}_{\textit{dRI}})=S^{2}_{y}(1+E(\varepsilon)-(\alpha+k% -\alpha k)E(\delta)-(\alpha+k-\alpha k)E(\varepsilon\delta)+{\alpha E(\delta}^% {2}))$ $\displaystyle\quad=S^{2}_{y}\left(1-(\alpha+k-\alpha k)\left(\frac{1}{(nq+2p)}% -\frac{1}{N}\right)({\lambda}_{22}-1)+\alpha\left(\frac{1}{(nq+2p)}-\frac{1}{N% }\right)({\lambda}_{04}-1)\right)$ $\displaystyle{E({\hat{T}}_{\textit{dRI}})-S}^{2}_{y}=S^{2}_{y}\left(\alpha% \left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{04}-1)-(\alpha+k-\alpha k% )\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{22}-1)\right)$

The bias of estimator ${\hat{T}}_{\textit{dRI}}$ , up to terms of order $n^{-1}$ , is

$\displaystyle\textit{Bias}({\hat{T}}_{\textit{dRI}})=S^{2}_{y}\left(\frac{1}{(% nq+2p)}-\frac{1}{N}\right)(\alpha({\lambda}_{04}-1)-(\alpha+k-\alpha k)({% \lambda}_{22}-1))$ (27)

The bias in estimator ${\hat{T}}_{\textit{dRI}}$ is zero, when $\alpha=\frac{{k(\lambda}_{22}-1)}{({\lambda}_{04}-1)-(1-k){(\lambda}_{22}-1)}$ .

Also,

$\displaystyle{\hat{T}}_{\textit{dRI}}-S^{2}_{y}=S^{2}_{y}(\varepsilon-(\alpha+% k-\alpha k)\delta-(\alpha+k-\alpha k)\varepsilon\delta+{\alpha\delta}^{2})$ (28)

Squaring on both sides of Eq. (28) and neglecting the higher order terms, we get

$\displaystyle({\hat{T}}_{\textit{dRI}}-S^{2}_{y})^{2}=S^{4}_{y}(\varepsilon-(% \alpha+k-\alpha k)\delta)2=S^{4}_{y}({\varepsilon}^{2}+{(\alpha+k-\alpha k)}^{% 2}{\delta}^{2}-2(\alpha+k-\alpha k)\varepsilon\delta)$ (29)

Taking expected value on both sides of Eq. (29) and using the results on expectations from Section 2, we get

$\displaystyle E({\hat{T}}_{\textit{dRI}}-S^{2}_{y})^{2}=S^{4}_{y}(E({% \varepsilon}^{2})+{(\alpha+k-\alpha k)}^{2}E({\delta}^{2})-2(\alpha+k-\alpha k% )E(\varepsilon\delta)=S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right% )({\lambda}_{40}-1)+{(\alpha+k-\alpha k)}^{2}\left(\frac{1}{(nq+2p)}-\frac{1}{% N}\right)({\lambda}_{04}-1)\right.\left.{}-2(\alpha+k-\alpha k)\left(\frac{1}{% (nq+2p)}-\frac{1}{N}\right)({\lambda}_{22}-1)\right)$

The mean square error of estimator ${\hat{T}}_{\textit{dRI}}$ , up to terms of order $n^{-1}$ , is

$\displaystyle\textit{MSE}({\hat{T}}_{\textit{dRI}})={S}^{4}_{y}\left(\frac{1}{% (nq+2p)}-\frac{1}{N}\right)$ (30) $\displaystyle\quad(({\lambda}_{40}-1)+{(\alpha+k-\alpha k)}^{2}({\lambda}_{04}% -1)-2(\alpha+k-\alpha k)({\lambda}_{22}-1))$

Differentiating Eq. (38) with respect to $(\alpha+k-\alpha k)$ and equating to zero, $\frac{\partial\{\textit{MSE}({\hat{T}}_{\textit{dRI}})\}}{\partial(\alpha+k-% \alpha k)}=0$

$\displaystyle{S}^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)\frac{% \partial}{\partial k}[({\lambda}_{40}-1)+{(\alpha+k-\alpha k)}^{2}({\lambda}_{% 04}-1)-2(\alpha+k-\alpha k)({\lambda}_{22}-1)]=0$ $\displaystyle{S}^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)[2(\alpha+k-% \alpha k)({\lambda}_{04}-1)-2({\lambda}_{22}-1)]=0$ $\displaystyle 2(\alpha+k-\alpha k)({\lambda}_{04}-1)-2({\lambda}_{22}-1)=0$ $\displaystyle(\alpha+k-\alpha k)=(\alpha(1-k)+k)=\frac{{\lambda}_{22}-1}{{% \lambda}_{04}-1}=g(\text{say})$ (31) $\displaystyle\alpha=\frac{1}{(1-k)}\left(\frac{{\lambda}_{22}-1}{{\lambda}_{04% }-1}-k\right)=\frac{g-k}{1-k}$

Optimum value of $\alpha$ is given by

$\displaystyle\alpha=\frac{g-k}{1-k}={\alpha}_{1(\textit{opt})}(\text{say})$ (32)

Putting Eqs (4.2) and (32) in Eqs (27) and (4.2), we get Eqs (24) and (25). Hence the theorem.

Theorem 4.2.2: The minimum bias and minimum MSE of the proposed estimator ${\hat{T}}_{\textit{dRII}}$ are given by

$\displaystyle\textit{min.Bias}({\hat{T}}_{\textit{dRII}})=S^{2}_{y}\left(\frac% {1}{n}-\frac{1}{N}\right)({\alpha}_{1(\textit{opt})}({\lambda}_{04}-1)-g({% \lambda}_{22}-1))$ (33) $\displaystyle\textit{min.MSE}({\hat{T}}_{\textit{dRII}})=S^{4}_{y}\left(\left(% \frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1)-g\left(\frac{1}{n}-% \frac{1}{N}\right)({\lambda}_{22}-1)\right)$ (34)

where ${\alpha}_{1(\textit{opt})}=\frac{g-k}{1-k}$ and $g=\frac{{\lambda}_{22}-1}{{\lambda}_{04}-1}$ .

Proof: ${\hat{T}}_{\textit{dRII}}=s^{*2}_{y}\left[\alpha\left(\frac{S^{2}_{x}}{s^{2}_{% x}}\right)+(1-\alpha)\left(\frac{{(1+k)S}^{2}_{x}-ks^{2}_{x}}{S^{2}_{x}}\right% )\right]=s^{*2}_{y}[\alpha\left(\frac{S^{2}_{x}}{s^{2}_{x}}\right)+(1-\alpha)% \left(1+k-\frac{ks^{2}_{x}}{S^{2}_{x}}\right)$

The estimator ${\hat{T}}_{\textit{dRII}}$ in terms of $\varepsilon$ and $\eta$ can be written as

$\displaystyle{\hat{T}}_{\textit{dRII}}=S^{2}_{y}(1+\varepsilon)\left(\frac{% \alpha}{1+\eta}+(1-\alpha)(1+k-k(1+\eta))\right)=S^{2}_{y}(1+\varepsilon)(% \alpha{(1+\eta)}^{-1}+(1-\alpha)(1-k\eta))$

We assume that $|\eta|<1$ , so that ${(1+\eta)}^{-1}$ is expandable in terms of power series. Simplifying and ignoring terms of order greater than 2, we get

$\displaystyle{\hat{T}}_{\textit{dRII}}=S^{2}_{y}(1+\varepsilon)(\alpha(1-\eta+% {\eta}^{2}+\ldots)+(1-\alpha-k\eta+\alpha k\eta))=S^{2}_{y}(1+\varepsilon-(% \alpha+k-\alpha k)\eta-(\alpha+k-\alpha k)\varepsilon\eta+{\alpha\eta}^{2})$ (35)

Taking expected value on both sides of Eq. (35) and using the results on expectations from Section 2, we get

$\displaystyle\mathrm{E}({\hat{T}}_{\textit{dRII}})=S^{2}_{y}(1+E(\varepsilon)-% (\alpha+k-\alpha k)E(\eta)-(\alpha+k-\alpha k)E(\varepsilon\eta)+{\alpha E(% \eta}^{2}))$ $\displaystyle\quad=S^{2}_{y}\left(1-(\alpha+k-\alpha k)\left(\frac{1}{n}-\frac% {1}{N}\right)({\lambda}_{22}-1)+\alpha\left(\frac{1}{n}-\frac{1}{N}\right)({% \lambda}_{04}-1)\right)$ $\displaystyle\mathrm{E}({\hat{T}}_{\textit{dRII}})-S^{2}_{y}=S^{2}_{y}\left(% \alpha\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{04}-1)-(\alpha+k-\alpha k% )\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{22}-1)\right)$

The bias of estimator ${\hat{T}}_{\textit{dRII}}$ , up to terms of order $n^{-1}$ , is

$\displaystyle\textit{Bias}({\hat{T}}_{\textit{dRII}})=S^{2}_{y}\left(\frac{1}{% n}-\frac{1}{N}\right)(\alpha({\lambda}_{04}-1)-(\alpha+k-\alpha k)({\lambda}_{% 22}-1))$ (36)

The bias in Eq. (36) is zero, when $\alpha=\frac{{k(\lambda}_{22}-1)}{({\lambda}_{04}-1)-(1-k){(\lambda}_{22}-1)}$

Also,

$\displaystyle{\hat{T}}_{\textit{dRII}}-S^{2}_{y}=S^{2}_{y}(\varepsilon-(\alpha% +k-\alpha k)\eta-(\alpha+k-\alpha k)\varepsilon\eta+{\alpha\eta}^{2})$ (37)

Squaring on both sides of Eq. (37) and neglecting the higher order terms, we get

$\displaystyle({\hat{T}}_{\textit{dRII}}-S^{2}_{y})^{2}=S^{4}_{y}(\varepsilon-(% \alpha+k-\alpha k)\eta)^{2}=S^{4}_{y}({\varepsilon}^{2}+(\alpha+k-\alpha k)^{2% }{\eta}^{2}-2(\alpha+k-\alpha k)\varepsilon\eta)$ (38)

Taking expected value on both sides of Eq. (38) and using the results on expectations from Section 2, we get

$\displaystyle E({\hat{T}}_{\textit{dRII}}-S^{2}_{y})^{2}=S^{4}_{y}(E({% \varepsilon}^{2})+{(\alpha+k-\alpha k)}^{2}E({\eta}^{2})-2(\alpha+k-\alpha k)E% (\varepsilon\eta))=S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({% \lambda}_{40}-1)+{(\alpha+k-\alpha k)}^{2}\left(\frac{1}{n}-\frac{1}{N}\right)% ({\lambda}_{04}-1)\right.\left.{}-2(\alpha+k-\alpha k)\left(\frac{1}{n}-\frac{% 1}{N}\right)({\lambda}_{22}-1)\right)$

The mean square error of estimator ${\hat{T}}_{\textit{dRII}}$ , up to terms of order $n^{-1}$ , is

$\displaystyle\textit{MSE}({\hat{T}}_{\textit{dRII}})=S^{4}_{y}\left(\left(% \frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{40}-1)+\left(\frac{1}{n}-\frac% {1}{N}\right)\right.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!% \left.\phantom{\frac{1}{(nq+2p)}}((\alpha+k-\alpha k)^{2}({\lambda}_{04}-1)-2(% \alpha+k-\alpha k)({\lambda}_{22}-1))\right)$ (39)

Differentiating Eq. (39) with respect to $(\alpha+k-\alpha k)$ and equating to zero, we get same optimum values as in situation I shown by Eqs (4.2) and (32).

Putting Eqs (4.2) and (32) in Eqs (36) and (39), we get Eqs (33) and (34). Hence the theorem.

5. Efficiency comparisons

For efficiency comparisons following conditions have been obtained by comparing the proposed estimators ${\hat{T}}_{\textit{dRI}}$ and ${\hat{T}}_{\textit{dRII}}$ with respect to usual unbiased estimator ${\hat{T}}_{0}$ , ratio estimator ${\hat{T}}_{RI}$ and ${\hat{T}}_{\textit{RII}}$ and dual to ratio estimator ${\hat{T}}_{dI}$ and ${\hat{T}}_{\textit{dII}}$ under situation I and situation II.

Under situation I

(i)
From Eqs (3) and (25), we get $\textit{Var}({\hat{T}}_{0})-\textit{min.MSE}({\hat{T}}_{\textit{dRI}})>0$

$\displaystyle S^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{4% 0}-1)-S^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)(({\lambda}_{40}-1)-g% ({\lambda}_{22}-1))>0$ (40)

After some simplification, we get ${\lambda}_{04}>1$ .
(ii)
From Eqs (7) and (25), we get $\textit{MSE}({\hat{T}}_{RI})-\textit{min.MSE}({\hat{T}}_{\textit{dRI}})>0$

$\displaystyle S^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{4% 0}+{\lambda}_{04}-2{\lambda}_{22})-S^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N% }\right)$ (41) $\displaystyle(({\lambda}_{40}-1)-g({\lambda}_{22}-1))>0$

After some simplification, we get ${(1-g)}^{2}>0$ .
(iii)
From Eqs (13) and (25), we get $\textit{MSE}({\hat{T}}_{dI})-\textit{min.MSE}({\hat{T}}_{\textit{dRI}})>0$

$\displaystyle{S}^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)[({\lambda}_% {40}-1){+k}^{2}({\lambda}_{04}-1)-2k({\lambda}_{22}-1)]$ (42) $\displaystyle-S^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)(({\lambda}_{% 40}-1)-g({\lambda}_{22}-1))>0$

After some simplification, we get ${(k-g)}^{2}>0$ .

Under Situation II

(i)
From Eqs (3) and (34), we get $\textit{Var}({\hat{T}}_{0})-\textit{min.MSE}({\hat{T}}_{\textit{dRII}})>0$

$\displaystyle S^{4}_{y}\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({\lambda}_{4% 0}-1)$ (43) $\displaystyle\quad{}-S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)% ({\lambda}_{40}-1)-g\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{22}-1)% \right)>0$

After some simplification, we get ${\lambda}_{04}>1$
(ii)
From Eqs (9) and (34), we get $\textit{MSE}({\hat{T}}_{\textit{RII}})-\textit{min.MSE}({\hat{T}}_{\textit{% dRII}})>0$

$\displaystyle S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({% \lambda}_{40}-1)+\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{04}-1)-2\left% (\frac{1}{n}-\frac{1}{N}\right)(\lambda_{22}-1)\right)$ (44) $\displaystyle\quad{}-S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)% ({\lambda}_{40}-1)-g\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{22}-1)% \right)>0$

After some simplification, we get ${(1-g)}^{2}>0$
(iii)
From Eqs (18) and (34), we get $\textit{MSE}({\hat{T}}_{\textit{dII}})-\textit{min.MSE}({\hat{T}}_{\textit{% dRII}})>0$

$\displaystyle S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)({% \lambda}_{40}-1)+\left(\frac{1}{n}-\frac{1}{N}\right)(k^{2}({\lambda}_{04}-1)-% 2k({\lambda}_{22}-1))\right)$ (45) $\displaystyle\quad{}-S^{4}_{y}\left(\left(\frac{1}{(nq+2p)}-\frac{1}{N}\right)% ({\lambda}_{40}-1)-g\left(\frac{1}{n}-\frac{1}{N}\right)({\lambda}_{22}-1)% \right)>0$

After some simplification, we get ${(k-g)}^{2}>0$

From Eqs (40) to ((iii)) it is observed that proposed ratio cum dual to ratio type estimator performed better than the usual unbiased, usual ratio and dual to ratio estimator under both situations of non-response.
6. Empirical results

In this section, we report the results of an empirical study that investigates the performance of proposed estimators. The empirical study includes application of both real data and simulation.

6.1 Real data application

The area and production of vegetable crop for Haryana (Source: Horticulture Department, Government of Haryana) from year 1970-71 to 2019-20 are considered for empirical study of the proposed estimators. The production under vegetable crop is considered as study variable $y$ whereas area under vegetable crop is considered as auxiliary variable $x$ . The following parameters are obtained from the studied data

$N=$ 50, $n=$ 20, $r=$ 4, $p=$ 0.23355, $q=$ 0.76645, ${\mu}^{*}_{40}=$ 72.74838, ${\mu}^{*}_{04}=$ 90442.12, ${\mu}^{*}_{22}=$ 1777.548, ${\overline{x}}^{*}=$ 13.22687, ${\overline{y}}^{*}=$ 2.182292, $s^{*2}_{y}=$ 5.35533, $s^{*2}_{x}=$ 182.8243, ${\lambda}^{*}_{40}=$ 2.536401, ${\lambda}_{04}=$ 2.283459 and ${\lambda}^{*}_{22}=$ 1.815452

Percentage relative efficiency (PRE) of the proposed estimators is defined by $\textit{PRE}=100\frac{\textit{Var}({\hat{T}}_{0})}{\textit{MSE}(\mathrm{% \blacksquare})}$ where, $\mathrm{\blacksquare}={\hat{T}}_{Ri}$ , ${\hat{T}}_{di}$ and ${\hat{T}}_{\textit{dRi}},i(=I,II)$ .

Table 1 shows the numerical results of the MSE and PRE of estimators obtained through real data. It can be easily observed that out of the two proposed estimators ${\hat{T}}_{di}$ and ${\hat{T}}_{\textit{dRi}},i(=I,II)$ , only estimator ${\hat{T}}_{\textit{dRi}},i(=I,II)$ is always more efficient than the usual unbiased estimator ${\hat{T}}_{0}$ and usual ratio estimator ${\hat{T}}_{Ri},i(=I,II)$ respectively. Due to low value of PREs, the proposed estimator ${\hat{T}}_{di},i(=I,II)$ is not performing better than the usual unbiased estimator ${\hat{T}}_{0}$ and usual ratio estimator ${\hat{T}}_{Ri},i(=I,II)$ .

Table 1
Efficiency comparisons of the existing and proposed estimators through real data

Situation I			Situation II
Estimators	MSE	PRE	Estimators	MSE	PRE
${\hat{T}}_{0}$	0.970433	100	${\hat{T}}_{0}$	0.970433	100
${\hat{T}}_{RI}$	0.479038	202.5794	${\hat{T}}_{\textit{RII}}$	0.592149	163.8831
${\hat{T}}_{dI}$	1.849531	52.4691	${\hat{T}}_{\textit{dII}}$	1.139244	85.1821
${\hat{T}}_{\textit{dRI}}$	0.424216	288.7591	${\hat{T}}_{\textit{dRII}}$	0.574165	169.0165

Bold values represent the maximum percentage relative efficiency among the proposed and existing estimators.

Table 2

Efficiency comparisons of the existing and proposed estimators under situation I through simulation study

Estimators	Percent relative efficiency
	$r=$ 2 (10%)	$r=$ 4 (20%)	$r=$ 6 (30%)	$r=$ 8 (40%)	$r=$ 10 (50%)
${\hat{T}}_{0}$	100	100	100	100	100
${\hat{T}}_{RI}$	628.816	619.928	566.948	578.819	527.981
${\hat{T}}_{dI}$	316.519	314.536	292.854	274.201	279.807
${\hat{T}}_{\textit{dRI}}$	1324.099	1345.249	1168.160	946.551	831.835

Bold values represent the maximum percentage relative efficiency among the proposed and existing estimators.

Table 3

Efficiency comparisons of the existing and proposed estimators under situation II through simulation study

Estimators	Percent relative efficiency
	$r=$ 2 (10%)	$r=$ 4 (20%)	$r=$ 6 (30%)	$r=$ 8 (40%)	$r=$ 10 (50%)
${\hat{T}}_{0}$	100	100	100	100	100
${\hat{T}}_{\textit{RII}}$	330.780	215.233	166.356	137.585	121.020
${\hat{T}}_{\textit{dII}}$	240.089	196.933	162.986	124.322	113.243
${\hat{T}}_{\textit{dRII}}$	404.596	250.564	191.785	155.732	136.967

Bold values represent the maximum percentage relative efficiency among the proposed and existing estimators.

Figure 1.

Percent relative efficiency of existing and proposed class of estimators under situation I and II for different values of r (or p).

6.2 Simulation results

Simulation is performed to calculate the MSE of the estimators of ${\hat{S}}^{2}_{y}$ . Consider a real population of size $N=$ 50 for the simulation study. The following steps involved in simulations are

Step 1.
Select a (SRSWOR) of size 20 from the population of size 50.
Step 2.
Select value of $r$ , say $r$ .
Step 3.
Drop $r$ units randomly from each sample in step 1.
Step 4.
Find the value of the estimator based on the $(n-r)$ observation.
Step 5.
Repeat steps (1) to (4) 50,000 times. Thus, we obtain 50,000 values for ${\hat{S}}^{2}_{y}$ say ${\hat{S}}^{2}_{y1},\ldots,{\hat{S}}^{2}_{y50,000}$ .
Step 6.
The MSE of proposed estimator is obtained by $\textit{MSE}=\frac{1}{50,000}\sum\nolimits^{50,000}_{i=1}{{(\mathrm{% \blacksquare}-{\hat{S}}^{2}_{y})}^{2}}$ where, $\mathrm{\blacksquare}={\hat{T}}_{Ri}$ , ${\hat{T}}_{di}$ and ${\hat{T}}_{\textit{dRi}},i(=I,II)$ .

Simulation results from Tables 2 and 3 shows that both the proposed estimators are more efficient than the usual unbiased estimator for different values of r (or p) (see Fig. 1). The critical findings of the simulation are listed below:

1.
For different values of r (or p), the proposed estimator ${\hat{T}}_{dI}$ and ${\hat{T}}_{\textit{dII}}$ under two different situations perform better than the usual unbiased estimator ${\hat{T}}_{0}$ only.
2.
For different values of r (or p), the proposed estimator ${\hat{T}}_{\textit{dRI}}$ and ${\hat{T}}_{\textit{dRII}}$ under two different situations perform better than both the existing estimators, the usual unbiased estimator ${\hat{T}}_{0}$ and ratio estimator, ${\hat{T}}_{RI}$ and ${\hat{T}}_{\textit{RII}}$ , respectively.
3.
Both the proposed estimators ${\hat{T}}_{dI}$ and ${\hat{T}}_{\textit{dRI}}$ have maximum PRE in situation I as compared to the estimators ${\hat{T}}_{\textit{dII}}$ and ${\hat{T}}_{\textit{dRII}}$ in situation II.

7. Conclusion

The dual to ratio and ratio cum dual to ratio type estimators are proposed for estimation of population variance under non-response considering two situations: i) when random non-response occurs on both $y$ and $x$ and $S^{2}_{x}$ is known. ii) when random non-response occurs only on $y$ and $S^{2}_{x}$ is known. It is observed from the results obtained from vegetable crop data that estimators ${\hat{T}}_{\textit{dRi}},i(=I,II)$ are found to be the best with maximum PRE. However, the results obtained from simulation proved that both the proposed estimators are better than the usual unbiased estimator only. So, the estimators can be arranged in decreasing order of their efficiencies on the basis of simulation study as, ${\hat{T}}_{\textit{dRi}}>{\hat{T}}_{Ri}>{\hat{T}}_{di}>{\hat{T}}_{0},i(=I,II)$ . Thus, the proposed estimators proved their practicability in case of non-response.

Footnotes

Acknowledgments

The authors are thankful to the reviewers for their valuable suggestions, which helped in improving the quality of this paper.

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Two new estimators of finite population variance under random non-response: An application

Abstract

Keywords

1. Introduction

2. Distribution of random non response and notations

4.1 Dual to ratio estimator under different situations of random non-response

6.1 Real data application

Table 1 Efficiency comparisons of the existing and proposed estimators through real data

Footnotes

Acknowledgments

References

Table 1
Efficiency comparisons of the existing and proposed estimators through real data