Ratio exponential type estimators with imputation for missing data in sample surveys

Abstract

Twenty exponential type estimators with imputation have been suggested to overcome the problem of missing data for a study variable in sample surveys. It has been shown that the suggested estimators are more efficient than the mean method of imputation, ratio method of imputation, regression method of imputation and the estimators are given by Singh and Horn (2000), Singh and Deo (2003), Singh (2009), Gira (2015) and Singh et al. (2016). The biases and their mean square errors of the suggested estimators are derived. Simulation studies are also demonstrated for comparing the performances of the suggested estimators.

Keywords

Imputation methods missing data bias mean square error (MSE)percent relative efficiency2000 AMS Classification: 62D05

1. Introduction

Non-response is one major problem, which is encountered by practitioners in the field of sample surveys. Repeated surveys are generally more prone to this problem than single-occasion surveys. For examples, in case of milk yield surveys the animal may be sold or may die during the survey period. In agricultural surveys, the crops destroy due to some natural calamities or disease during the course of experiments. Thus, the observations may be missing for some of the time stages. Such type of non-response (missingness) may have different patterns and causes.

It is well recognized fact that inference concerning population parameters can be wrong if the suitable information about the nature of non-response is not known. A natural question arises what one needs to assume to justify ignoring the incomplete mechanism. Rubin (1987) addressed three concepts: missing at random (MAR), observed at random (OAR) and parameter distribution (PD). Heitzan and Basu (1996) have distinguished the meaning of missing at random (MAR) and missing completely at random (MCAR) in a very nice way.

Many methods are used to reduces the negative impact of non-response in sample surveys. Imputation is one which deal with the filling up method of incomplete data for adapting the standared analytic model in statistics. It apparently solves the missing data problem at the beginning of the analysis. To deal with missing values effectively, Sande (1979) and Kalton et al. (1981) intended imputation methods that make incomplete data sets structurally complete and its analysis simply. Imputation method is also be driven out by all of the assist of an auxiliary variable, if one is available. For the MCAR response mechanism, Singh and Horn (2000) indicated a compromised method of imputation. Singh and Deo (2003), Ahmed et al. (2006), Toutenburg et al. (2008), Kadilar and Cingi (2008), Singh (2009), Singh et al. (2010), Diana and Perri (2010), Gira (2015), Singh et al. (2016) and Prasad (2016) have suggested several new imputation based methods with the aid of an auxiliary variable.

Motivated by a recent work of Singh et al. (2016) and using missing completely at random (MCAR) response mechanism, in the present section of this article we have suggested ratio exponential type estimators with imputation to tackle with the problems of non-response in sample surveys. Ratio exponential type estimators have been suggested for estimating the population mean in sample surveys and subsequently their behaviors are studied. Performances of the suggested estimators are compared with the existing estimators and suitable recommendations are made.

The outline of this paper is as follows: in Section 2, we consider several estimators of the finite population mean under non-response that are available in literature. The formulation of suggested estimators are given in Section 3. In Section 4, properties of the suggested estimators are discussed. A simulation study is conducted in Section 5, Analysis of simulation results are given in Section 6 and some concluding remarks are given in Section 7.

Let us consider a finite population of size $N$ with values $Y_{1},Y_{2},\ldots,Y_{N}$ of the study variable and values $X_{1},X_{2},\ldots,X_{N}$ of the auxiliary variable. For the estimation of population mean $\bar{Y}$ , a random sample of size $n$ is drawn according to the procedure of SRSWOR scheme. Assuming the non-response to be random, suppose that there are $r$ response observations $(y_{1},x_{1}),(y_{2},x_{2}),\ldots,(y_{r},x_{r})$ and $(n-r)$ non-response observations. Let $r$ be the size of responding units out of sampled $n$ units, the set of responding units by $R$ and the other of size $(n-r)$ denotes by set of non-responding units $R^{c}$ .

When the non-response observations are discarded, it is customary to estimate population mean $\bar{Y}$ by

$\bar{y}_{r}=\frac{1}{r}\sum_{i=1}^{r}y_{i}$ (1)

When the non-response observations are not discarded and some imputation method is followed, the complete data set is specified by

$y_{.i}=\begin{cases}y_{i}&\mbox{if∼{}}i\in R\\ \tilde{{y}_{i}}&\mbox{if∼{}}i\in R^{c}\end{cases}$ (2)

The general point estimator of population mean $\bar{Y}$ takes the form

$\displaystyle\tau=\frac{1}{n}\sum_{i=1}^{n}y_{.i}=\frac{1}{n}\left[\sum_{i\in R% }y_{.i}+\sum_{i\in R^{c}}y_{.i}\right]=\frac{1}{n}\left[\sum_{i\in R}y_{i}+% \sum_{i\in R^{c}}\tilde{{y}_{i}}\right]$ (3)

Here, the value $\tilde{{y}_{i}}$ denote the imputed value of the study variable corresponding to the $i^{\rm th}$ non-responding units.

2. Related review of some existing estimators

In this section, we consider several estimators for estimating the population mean under non-response.

2.1 Mean method of imputation

Under the mean method of imputation, the point estimator Eq. 3 of population mean $\bar{Y}$ is derived as

$\bar{y}_{m}=\frac{1}{r}\sum_{i=1}^{r}y_{i}=\bar{y}_{r}$ (4)

which is known as the response mean estimator $\bar{y}_{r}$ of population mean $\bar{Y}$ . The variance of the response sample mean $\bar{y}_{r}$ , is given by

$\textit{Var}(\bar{y}_{r})=\left(\frac{1}{r}-\frac{1}{N}\right)\bar{Y}^{2}C_{y}% ^{2}$ (5)

2.2 Ratio method of imputation

Under the ratio method of imputation, the point estimator Eq. 3 of population mean $\bar{Y}$ is derived as

$\bar{y}_{\textit{rat}}=\bar{y}_{r}\frac{\bar{x}_{n}}{\bar{x}_{r}}$ (6)

which is known as the ratio estimator $\bar{y}_{\textit{rat}}$ of population mean $\bar{Y}$ . The MSE of estimator $\bar{y}_{\textit{rat}}$ is obtained under MCAR mechanism upto first order of large approximation, is given by

$\textit{MSE}(\bar{y}_{\textit{rat}})=\textit{Var}(\bar{y}_{r})+\left(\frac{1}{% r}-\frac{1}{n}\right)\bar{Y}^{2}\left(C_{x}^{2}-2\rho_{yx}C_{y}C_{x}\right)$ (7)

2.3 Regression method of imputation

Under the regression method of imputation, the point estimator Eq. 3 of population mean $\bar{Y}$ is given by is given by

$\bar{y}_{\textit{reg}}=\bar{y}_{r}+\hat{b}(\bar{x}_{n}-\bar{x}_{r})$ (8)

which is known as regression estimator $\bar{y}_{\textit{reg}}$ of population mean $\bar{Y}$ . where $\hat{b}=\frac{s_{yx}(r)}{s_{x}^{2}(r)}$ . The MSE of $\bar{y}_{\textit{reg}}$ is obtained under MCAR mechanism upto first order of approximation, is given by

$\textit{MSE}(\bar{y}_{\textit{reg}})=\bar{Y}^{2}C_{y}^{2}\left[\left(\frac{1}{% r}-\frac{1}{N}\right)-\left(\frac{1}{r}-\frac{1}{n}\right)\rho_{yx}^{2}\right]$ (9)

2.4 Singh and Horn (2000) estimator

Singh and Horn (2000) suggested a compromised imputation in survey sampling. Under this method of imputation, the point estimator Eq. 3 of population mean $\bar{Y}$ is derived as

$\bar{y}_{sh}=\alpha\bar{y}_{r}+(1-\alpha)\bar{y}_{r}\frac{\bar{x}_{n}}{\bar{x}% _{r}}$ (10)

where $\alpha$ is a suitable constant. The optimum value of $\alpha$ is $1-\rho_{yx}\frac{C_{y}}{C_{x}}$ .

Now, taking the optimum value of $\alpha$ in Eq. 10, we can obtained the optimum MSE of $\bar{y}_{sh}$ under MCAR mechanism upto the first order of approximation, is given by

$\textit{MSE}(\bar{y}_{sh})_{opt}=\textit{MSE}(\bar{y}_{\textit{rat}})-\left(% \frac{1}{r}-\frac{1}{n}\right)\left(C_{x}-\rho_{yx}C_{y}\right)^{2}\bar{Y}^{2}$ (11)

2.5 Singh and Deo (2003) estimator

Singh and Deo (2003) suggested imputation by power transformation in survey sampling. Under this method of imputation, the point estimator Eq. 3 of population mean $\bar{Y}$ becomes

$\bar{y}_{sd}=\bar{y}_{r}\left(\frac{\bar{x}_{n}}{\bar{x}_{r}}\right)^{\beta}$ (12)

where $\beta$ is a suitable constant. The optimum value of $\beta$ is $\rho_{yx}\frac{C_{y}}{C_{x}}$ .

Now, taking the optimum value of $\beta$ in Eq. 12, we obtained the optimum MSE of $\bar{y}_{sd}$ under MCAR mechanism upto first order of approximation, given by

$\textit{MSE}(\bar{y}_{sd})_{opt}=\textit{MSE}(\bar{y}_{\textit{rat}})-\left(% \frac{1}{r}-\frac{1}{n}\right)S_{x}^{2}\left(\frac{S_{yx}}{S_{x}^{2}}-\frac{% \bar{Y}}{\bar{X}}\right)^{2}$ (13)

2.6 Singh (2009) estimator

Singh (2009) suggested a new method of imputation in survey sampling. Under this method of imputation, The point estimator Eq. 3 of the population mean $\bar{Y}$ becomes

$\bar{y}_{\textit{singh}}=\frac{\bar{y}_{r}\bar{x}_{n}}{\gamma\bar{x}_{r}+(1-% \gamma)\bar{x}_{n}}$ (14)

where $\gamma$ is a suitable constant. The optimum value of $\gamma$ is $\rho_{yx}\frac{C_{y}}{C_{x}}$ .

Now, using the optimum value of $\gamma$ in Eq. 14, we get the optimum MSE of the estimator $\bar{y}_{\textit{singh}}$ under MCAR mechanism upto the first order of large approximation, given by

$\textit{MSE}(\bar{y}_{\textit{singh}})_{opt}=\textit{MSE}(\bar{y}_{\textit{rat% }})-\left(\frac{1}{r}-\frac{1}{n}\right)\left(C_{x}-\rho_{yx}C_{y}\right)^{2}% \bar{Y}^{2}$ (15)

2.7 Gira (2015) estimator

Gira (2015) suggested a new method of ratio type imputation in sample surveys. Under this method of imputation, The point estimator Eq. 3 of the population mean $\bar{Y}$ becomes

$\bar{y}_{\textit{gira}}=\bar{y}_{r}\frac{\phi-\bar{x}_{r}}{\phi-\bar{x}_{n}}$ (16)

where $\phi$ is a suitably chosen constant. The optimum value of $\phi$ is $\bar{X}\left(\frac{C_{x}}{\rho_{yx}C_{y}}-1\right)$ .

Now, using the optimum value of $\phi$ in Eq. 16, we get the optimum MSE of the estimator $\bar{y}_{\textit{gira}}$ under MCAR mechanism upto the first order of large approximation, given by

$\textit{MSE}(\bar{y}_{\textit{gira}})_{\textit{opt}}=\textit{Var}(\bar{y}_{r})% -\left(\frac{1}{r}-\frac{1}{n}\right)\rho_{yx}^{2}\bar{Y}^{2}C_{y}^{2}$ (17)

2.8 Singh et al. (2016) estimator

Singh et al. (2016) proposed single imputation methods and subsequent estimators for estimating the population mean $\bar{Y}$ , are given by

$\displaystyle\bar{y}_{\textit{smkk}_{1}}=\left[\bar{y}_{r}+\hat{b}(\bar{x}_{n}% -\bar{x}_{r})\right]\exp\left(\frac{\bar{X}-\bar{x}_{r}}{\bar{X}+\bar{x}_{r}}\right)$ (18) $\displaystyle\bar{y}_{\textit{smkk}_{2}}=\frac{\bar{y}_{r}}{\bar{x}_{r}}\bar{x% }_{n}\exp\left(\frac{\bar{X}-\bar{x}_{r}}{\bar{X}+\bar{x}_{r}}\right)$ (19) $\displaystyle\bar{y}_{\textit{smkk}_{3}}=\bar{y}_{r}+\hat{b}\left[{\bar{x}_{n}% \exp\left(\frac{\bar{X}-\bar{x}_{r}}{\bar{X}+\bar{x}_{r}}\right)}-\bar{x}_{r}\right]$ (20)

The MSEs of estimators are obtained under MCAR mechanism upto the first order of large sample approximation, are given by

$\displaystyle\textit{MSE}(\bar{y}_{\textit{smkk}_{1}})=\bar{Y}^{2}\left[\left(% \frac{1}{r}-\frac{1}{N}\right)(C_{y}^{2}+\frac{1}{4}C_{x}^{2}-\rho_{yx}C_{y}C_% {x})+\left(\frac{1}{r}-\frac{1}{n}\right)\rho_{yx}C_{y}C_{x}(C_{x}-\rho_{yx}C_% {y})\right]$ (21) $\displaystyle\textit{MSE}(\bar{y}_{\textit{smkk}_{2}})=\bar{Y}^{2}\left[\left(% \frac{1}{r}-\frac{1}{N}\right)(C_{y}^{2}+\frac{9}{4}C_{x}^{2}-3\rho_{yx}C_{y}C% _{x})+2\left(\frac{1}{n}-\frac{1}{N}\right)(\rho_{yx}C_{y}C_{x}-C_{x}^{2})\right]$ (22) $\displaystyle\textit{MSE}(\bar{y}_{\textit{smkk}_{3}})=\bar{Y}^{2}\left(\frac{% 1}{r}-\frac{1}{N}\right)C_{y}^{2}\left(1-\frac{3}{4}\rho_{yx}^{2}\right)$ (23)

3. Formulation of the problem

In this section, ratio exponential type estimators with imputation have been suggested for estimating the population mean $\bar{Y}$ . The study variable after imputation for the suggested method of imputation becomes

$y_{.i}=\begin{cases}y_{i}&\mbox{if∼{}}i\in R\\ \displaystyle\frac{\bar{y}_{r}}{n-r}\left[n\kappa\exp{{\left({\frac{\theta_{i}% \left(1-\bar{x}_{r}/\bar{X}\right)}{1+\left(({p\bar{{x}}_{r}+q})/({p\bar{X}+q}% )\right)}}\right)}}-r\right]&\mbox{if∼{}}i\in R^{c}\end{cases}$ (24)

where $\theta_{i}=\frac{p\bar{X}}{p\bar{X}+q}$ , $i=1,2,\ldots,20$ for different suitable choices of $p$ and $q$ .

Under this suggested method of imputation, the point estimator Eq. 3 of the population mean $\bar{Y}$ , given by

$\tau=\frac{1}{n}\sum_{i=1}^{n}y_{.i}=\frac{1}{n}\left[\sum_{i\in R}y_{.i}+\sum% _{i\in R^{c}}y_{.i}\right]=\kappa\bar{y_{r}}\exp{{\left[{\frac{p\left({\bar{X}% }-\bar{{x}}_{r}\right)}{p\left(\bar{X}+\bar{{x}}_{r}\right)+2q}}\right]}}$ (25)

where $\kappa$ is a suitably chosen constant, such that the MSE of the suggested estimator is minimum and $p\neq 0$ and $q$ are either some real number or the known value of some population parameters of the auxiliary variable $x$ such as standard deviation $(S_{x})$ , coefficient of kurtosis $(\beta_{2}(x))$ , coefficient of skewness $(\beta_{1}(x))$ , coefficient of variation $(C_{x})$ and correlation coefficient $(\rho_{yx})$ .

We would like remark that for different choices of $p$ and $q$ in Eq. 25, we suggest twenty ratio exponential type estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ , as shown in Table 1.

Table 1

Some members of the suggested ratio exponential type estimators with imputation $\tau_{i}$ $(i=1,2,\ldots,20)$

Estimators	$p$	$q$
$\displaystyle\tau_{1}=\kappa\bar{y_{r}}\exp{{\left({\frac{\bar{{X}}-\bar{{x}}_% {r}}{\bar{X}+\bar{{x}}_{r}}}\right)}}$	1	0
$\displaystyle\tau_{2}=\kappa\bar{y_{r}}\exp{\left({\frac{\bar{{X}}-\bar{{x}}_{% r}}{\bar{X}+\bar{{x}}_{r}+2}}\right)}$	1	1
$\displaystyle\tau_{3}=\kappa\bar{y_{r}}\exp{\left({\frac{\bar{X}-\bar{{x}}_{r}% }{\bar{X}+\bar{{x}}_{r}+2\beta_{2}(x)}}\right)}$	1	$\beta_{2}(x)$
$\displaystyle\tau_{4}=\kappa\bar{y_{r}}\exp{\left({\frac{\bar{X}-\bar{{x}}_{r}% }{\bar{X}+\bar{{x}}_{r}+2\beta_{1}(x)}}\right)}$	1	$\beta_{1}(x)$
$\displaystyle\tau_{5}=\kappa\bar{y_{r}}\exp{\left({\frac{\bar{X}-\bar{{x}}_{r}% }{\bar{X}+\bar{{x}}_{r}+2C_{x}}}\right)}$	1	$C_{x}$
$\displaystyle\tau_{6}=\kappa\bar{y_{r}}\exp{\left({\frac{\bar{X}-\bar{{x}}_{r}% }{\bar{X}+\bar{{x}}_{r}+2\rho_{yx}}}\right)}$	1	$\rho_{yx}$
$\displaystyle\tau_{7}=\kappa\bar{y_{r}}\exp{\left[{\frac{S_{x}\left(\bar{X}-% \bar{{x}}_{r}\right)}{S_{x}\left(\bar{X}+\bar{{x}}_{r}\right)+2}}\right]}$	$S_{x}$	1
$\displaystyle\tau_{8}=\kappa\bar{y_{r}}\exp{\left[{\frac{S_{x}\left(\bar{X}-% \bar{{x}}_{r}\right)}{S_{x}\left(\bar{X}+\bar{{x}}_{r}\right)+2\beta_{2}(x)}}% \right]}$	$S_{x}$	$\beta_{2}(x)$
$\displaystyle\tau_{{9}}=\kappa\bar{y_{r}}\exp{\left[{\frac{S_{x}\left(\bar{X}-% \bar{{x}}_{r}\right)}{S_{x}\left(\bar{X}+\bar{{x}}_{r}\right)+2\beta_{1}(x)}}% \right]}$	$S_{x}$	$\beta_{1}(x)$
$\displaystyle\tau_{{10}}=\kappa\bar{y_{r}}\exp{\left[{\frac{S_{x}\left(\bar{X}% -\bar{{x}}_{r}\right)}{S_{x}\left(\bar{X}+\bar{{x}}_{r}\right)+2C_{x}}}\right]}$	$S_{x}$	$C_{x}$
$\displaystyle\tau_{{11}}=\kappa\bar{y_{r}}\exp{\left[{\frac{S_{x}\left(\bar{X}% -\bar{{x}}_{r}\right)}{S_{x}\left(\bar{X}+\bar{{x}}_{r}\right)+2\rho_{yx}}}% \right]}$	$S_{x}$	$\rho_{yx}$
$\displaystyle\tau_{{12}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\beta_{1}(x)\left(% \bar{X}-\bar{{x}}_{r}\right)}{\beta_{1}(x)\left(\bar{X}+\bar{{x}}_{r}\right)+2% }}\right]}$	$\beta_{1}(x)$	1
$\displaystyle\tau_{{13}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\beta_{1}(x)\left(% \bar{X}-\bar{{x}}_{r}\right)}{\beta_{1}(x)\left(\bar{X}+\bar{{x}}_{r}\right)+2% \beta_{2}(x)}}\right]}$	$\displaystyle\beta_{1}(x)$	$\beta_{2}(x)$
$\displaystyle\tau_{{14}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\beta_{1}(x)\left(% \bar{X}-\bar{{x}}_{r}\right)}{\beta_{1}(x)\left(\bar{X}+\bar{{x}}_{r}\right)+2% C_{x}}}\right]}$	$\displaystyle\beta_{1}(x)$	$C_{x}$
$\displaystyle\tau_{{15}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\beta_{1}(x)\left(% \bar{X}-\bar{{x}}_{r}\right)}{\beta_{1}(x)\left(\bar{X}+\bar{{x}}_{r}\right)+2% \rho_{yx}}}\right]}$	$\displaystyle\beta_{1}(x)$	$\rho_{yx}$
$\displaystyle\tau_{{16}}=\kappa\bar{y_{r}}\exp{\left[{\frac{C_{x}\left(\bar{X}% -\bar{{x}}_{r}\right)}{C_{x}\left(\bar{X}+\bar{{x}}_{r}\right)+2\beta_{2}(x)}}% \right]}$	$C_{x}$	$\beta_{2}(x)$
$\displaystyle\tau_{{17}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\rho_{yx}\left(% \bar{X}-\bar{{x}}_{r}\right)}{\rho_{yx}\left(\bar{X}+\bar{{x}}_{r}\right)+2}}% \right]}$	$\displaystyle\rho_{yx}$	1
$\displaystyle\tau_{{18}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\rho_{yx}\left(% \bar{X}-\bar{{x}}_{r}\right)}{\rho_{yx}\left(\bar{X}+\bar{{x}}_{r}\right)+2% \beta_{2}(x)}}\right]}$	$\displaystyle\rho_{yx}$	$\beta_{2}(x)$
$\displaystyle\tau_{{19}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\rho_{yx}\left(% \bar{X}-\bar{{x}}_{r}\right)}{\rho_{yx}\left(\bar{X}+\bar{{x}}_{r}\right)+2% \beta_{1}(x)}}\right]}$	$\displaystyle\rho_{yx}$	$\beta_{1}(x)$
$\displaystyle\tau_{{20}}=\kappa\bar{y_{r}}\exp{\left[{\frac{\rho_{yx}\left(% \bar{X}-\bar{{x}}_{r}\right)}{\rho_{yx}\left(\bar{X}+\bar{{x}}_{r}\right)+2C_{% x}}}\right]}$	$\rho_{yx}$	$C_{x}$

4. Properties of the suggested estimators

\tau_{i}

(i=1,2,\ldots,20)

To obtain the biases and mean square errors (MSEs) of the suggested estimators $\tau_{i}$ (where $i=1,2,\ldots,20$ ) for different suitable choices of $p$ and $q$ upto the first order of large sample approximation are derived under the following transformations:

$\bar{y}_{r}=\bar{Y}(1+e_{1})\text{∼{}and∼{}}\bar{x}_{r}=\bar{X}(1+e_{2})\text{% ∼{}such that∼{}}E(e_{i})=0,|e_{i}|<1∼{}∼{}\forall∼{}i=1,2.$

Under the above large transformations, the estimators $\tau_{i}$ take the following forms:

$\tau_{i}=\kappa\bar{Y}\left(1+e_{1}\right)\exp\left[-\frac{1}{2}\theta e_{2}% \left(1+\frac{1}{2}\theta e_{2}\right)^{-1}\right]$ (26)

where $\theta=\frac{p\bar{X}}{p\bar{X}+q}$ .

For different suitable choices of $p$ and $q$ , the $\theta$ can be changed in $\theta_{i}$ and the Eq. 26 can be written as

$\tau_{i}=\kappa\bar{Y}\left(1+e_{1}\right)\exp\left[-\frac{1}{2}\theta_{i}e_{2% }\left(1+\frac{1}{2}\theta_{i}e_{2}\right)^{-1}\right]$ (27)

where

$\displaystyle\theta_{1}=1,\theta_{2}=\frac{\bar{X}}{\bar{X}+1},\theta_{3}=% \frac{\bar{X}}{\bar{X}+\beta_{2}(x)},\theta_{4}=\frac{\bar{X}}{\bar{X}+\beta_{% 1}(x)},\theta_{5}=\frac{\bar{X}}{\bar{X}+C_{x}},\theta_{6}=\frac{\bar{X}}{\bar% {X}+\rho_{yx}},\theta_{7}=\frac{S_{x}\bar{X}}{S_{x}\bar{X}+1},$ $\displaystyle\theta_{8}=\frac{S_{x}\bar{X}}{S_{x}\bar{X}+\beta_{2}(x)},∼{}% \theta_{9}=\frac{S_{x}\bar{X}}{S_{x}\bar{X}+\beta_{1}(x)},∼{}\theta_{10}=\frac% {S_{x}\bar{X}}{S_{x}\bar{X}+C_{x}},∼{}\theta_{11}=\frac{S_{x}\bar{X}}{S_{x}% \bar{X}+\rho_{yx}},∼{}\theta_{12}=\frac{\beta_{1}(x)\bar{X}}{\beta_{1}(x)\bar{% X}+1},$ $\displaystyle\theta_{13}=\frac{\beta_{1}(x)\bar{X}}{\beta_{1}(x)\bar{X}+\beta_% {2}(x)},∼{}\theta_{14}=\frac{\beta_{1}(x)\bar{X}}{\beta_{1}(x)\bar{X}+C_{x}},∼% {}\theta_{15}=\frac{\beta_{1}(x)\bar{X}}{\beta_{1}(x)\bar{X}+\rho_{yx}},∼{}% \theta_{16}=\frac{C_{x}\bar{X}}{C_{x}\bar{X}+\beta_{2}(x)},$ $\displaystyle\theta_{17}=\frac{\rho_{yx}\bar{X}}{\rho_{yx}\bar{X}+1},∼{}\theta% _{18}=\frac{\rho_{yx}\bar{X}}{\rho_{yx}\bar{X}+\beta_{2}(x)},∼{}\theta_{19}=% \frac{\rho_{yx}\bar{X}}{\rho_{yx}\bar{X}+\beta_{1}(x)}\text{∼{}and∼{}}\theta_{% 20}=\frac{\rho_{yx}\bar{X}}{\rho_{yx}\bar{X}+C_{x}}.$

Table 2

Parameters of data sets

	Data 1	Data 2	Data 3
Parameters	[Source: Cochran (1977)] page 325	[Source: Koyuncu and Kadilar (2009)]	[Source: Lohr (1999)]
$N$	10	923	3059
$n$	4	180	428
$r$	3	144	128
$\bar{Y}$	101.1	436.4345	308582.4
$\bar{X}$	58.8	11440.498	56.5
$C_{y}$	0.1449	1.7183	1.3783
$C_{x}$	0.1281	1.8645	1.2796
$\beta_{2}(x)$	$-$ 0.3814	18.7208	7.5
$\beta_{1}(x)$	0.5764	3.9365	2.4
$\rho_{yx}$	0.6515	0.9543	0.677428

The Eq. 27 can be written neglecting the terms of $e$ ’s having power greater than two, we get

$\tau_{i}-\bar{Y}\cong\bar{Y}\left[(\kappa-1)+\kappa\left(e_{1}-\frac{1}{2}% \theta_{i}e_{2}-\frac{1}{2}\theta_{i}e_{1}e_{2}+\frac{3}{8}\theta_{i}^{2}e_{2}% ^{2}\right)\right]$ (28)

Taking expectation of both sides of Eq. 28, we get the biases of the proposed estimators upto the first order of large approximation as

$\textit{Bias}(\tau_{i})=\bar{Y}\left[(\kappa-1)+\frac{1}{8}\theta_{i}\kappa% \left(\frac{1}{r}-\frac{1}{N}\right)\left(3\theta_{i}C_{x}^{2}-4\rho_{yx}C_{y}% C_{x}\right)\right]$ (29)

Now, after squaring both sides of Eq. 28 and neglecting the terms of $e$ ’s having power greater than two, we have

$\left(\tau_{i}-\bar{Y}\right)^{2}\cong\bar{Y}^{2}\left[(\kappa-1)+\kappa\left(% e_{1}-\frac{1}{2}\theta_{i}e_{2}\right)\right]^{2}$ (30)

Taking expectation of both sides of Eq. 30 respectively, we get the MSEs of the suggested estimators $\tau_{{i}}$ $(i=1,2,\ldots,20)$ for the first order of large approximation as

$\textit{MSE}(\tau_{i})=\bar{Y}^{2}\left[(\kappa-1)^{2}+\left(\frac{1}{r}-\frac% {1}{N}\right)\kappa^{2}\left(C_{y}^{2}+\frac{1}{4}\theta_{i}^{2}C_{x}^{2}-% \theta_{i}\rho_{yx}C_{y}C_{x}\right)\right]$ (31)

Differentiating Eq. 31 with respect to $\kappa$ and equating to zero, we get the optimum value of $\kappa$ is given by

$\kappa_{opt}=\frac{1}{\displaystyle 1+\left(\frac{1}{r}-\frac{1}{N}\right)% \left(C_{y}^{2}+\frac{1}{4}\theta_{i}^{2}C_{x}^{2}-\theta_{i}\rho_{yx}C_{y}C_{% x}\right)}$ (32)

After substituting the optimum value of $\kappa$ i.e., $\kappa_{opt}$ in Eq. 31, we can obtain the optimum MSEs of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ as given as

$\textit{MSE}(\tau_{i})_{opt}=\frac{\displaystyle\left(\frac{1}{r}-\frac{1}{N}% \right)\left(C_{y}^{2}+\frac{1}{4}\theta_{i}^{2}C_{x}^{2}-\theta_{i}\rho_{yx}C% _{y}C_{x}\right)\bar{Y}^{2}}{\displaystyle 1+\left(\frac{1}{r}-\frac{1}{N}% \right)\left(C_{y}^{2}+\frac{1}{4}\theta_{i}^{2}C_{x}^{2}-\theta_{i}\rho_{yx}C% _{y}C_{x}\right)}$ (33)

5. Simulation study

Table 3
Biases, mean square errors and percent relative biases of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ for Data 1

$N=$ 10, $n=$ 4, $r=$ 3, $\rho_{yx}=$ 0.6515
Estimators	Absolute Bias	Mean square error (MSE)	Relative Bias (RB)	Percent relative Bias (PRB)
$\tau_{1}$	0.303351	30.9235	0.0545508	5.45508
$\tau_{2}$	0.307324	31.0804	0.0551257	5.51257
$\tau_{3}$	0.301793	30.8638	0.0543230	5.43230
$\tau_{4}$	0.305661	31.0139	0.0548860	5.48860
$\tau_{5}$	0.303869	30.9436	0.0546262	5.46262
$\tau_{6}$	0.305958	31.0257	0.0549289	5.49289
$\tau_{7}$	0.303888	30.9444	0.0546290	5.46290
$\tau_{8}$	0.303146	30.9156	0.0545208	5.45208
$\tau_{9}$	0.303661	30.9355	0.0545959	5.45959
$\tau_{10}$	0.303420	30.9262	0.0545608	5.45608
$\tau_{11}$	0.303701	30.9371	0.0546018	5.46018
$\tau_{12}$	0.310144	31.1957	0.0555285	5.55285
$\tau_{13}$	0.300631	30.8199	0.0541525	5.41525
$\tau_{14}$	0.304248	30.9584	0.0546813	5.46813
$\tau_{15}$	0.307830	31.1008	0.0551983	5.51983
$\tau_{16}$	0.290497	30.4596	0.0526355	5.26355
$\tau_{17}$	0.309385	31.1643	0.0554205	5.54205
$\tau_{18}$	0.300949	30.8318	0.0541992	5.41992
$\tau_{19}$	0.306875	31.0623	0.0550610	5.50610
$\tau_{20}$	0.304145	30.9544	0.0546664	5.46664

Table 4

Biases, mean square errors and percent relative biases of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ for Data 2

$N=$ 923, $n=$ 180, $r=$ 144, $\rho_{yx}=$ 0.9543
Estimators	Absolute Bias	Mean square error (MSE)	Relative Bias (RB)	Percent relative Bias (PRB)
$\tau_{1}$	2.51937	849.428	0.0864429	8.64429
$\tau_{2}$	2.51991	849.556	0.0864547	8.64547
$\tau_{3}$	2.52932	851.815	0.0866623	8.66623
$\tau_{4}$	2.52147	849.931	0.0864892	8.64892
$\tau_{5}$	2.52037	849.666	0.0864648	8.64648
$\tau_{6}$	2.51988	849.550	0.0864541	8.64541
$\tau_{7}$	2.51937	849.428	0.0864429	8.64429
$\tau_{8}$	2.51938	849.428	0.0864429	8.64429
$\tau_{9}$	2.51938	849.428	0.0864429	8.64429
$\tau_{10}$	2.51937	849.428	0.0864429	8.64429
$\tau_{11}$	2.51937	849.428	0.0864429	8.64429
$\tau_{12}$	2.51951	849.461	0.0864459	8.64459
$\tau_{13}$	2.52190	850.035	0.0864988	8.64988
$\tau_{14}$	2.51963	849.489	0.0864485	8.64485
$\tau_{15}$	2.51950	849.459	0.0864458	8.64458
$\tau_{16}$	2.52471	850.709	0.0865608	8.65608
$\tau_{17}$	2.51993	849.562	0.0864552	8.64552
$\tau_{18}$	2.52979	851.930	0.0866728	8.66728
$\tau_{19}$	2.52157	849.955	0.0864914	8.64914
$\tau_{20}$	2.52041	849.678	0.0864659	8.64659

In this section, the performance of suggested estimators has been evaluated by using statistical data sets previously used in the literature. The percent relative efficiencies of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ with respect to the mean method of imputation, ratio method of imputation, regression method of imputation, Singh and Horn (2000) estimator, Singh and Deo (2003) estimator, Singh (2009) estimator, Gira (2015) estimator and Singh et al. (2016) estimators are computed as

$\displaystyle E_{1}=\textit{PRE}(t_{i},\bar{y}_{r})=\frac{V(\bar{y}_{r})}{% \textit{MSE}(t_{i})_{opt}}\times 100$ (34) $\displaystyle E_{2}=\textit{PRE}(t_{i},\bar{y}_{\textit{rat}})=\frac{\textit{% MSE}(\bar{y}_{\textit{rat}})}{\textit{MSE}(t_{i})_{opt}}\times 100$ (35) $\displaystyle E_{3}=\textit{PRE}(t_{i},\bar{y}_{\textit{reg}})=\frac{\textit{% MSE}(\bar{y}_{\textit{reg}})}{\textit{MSE}(t_{i})_{opt}}\times 100$ (36) $\displaystyle E_{4}=\textit{PRE}(t_{i},\bar{{y}}_{sh})=\frac{\textit{MSE}(\bar% {{y}}_{sh})_{opt}}{\textit{MSE}(t_{i})_{opt}}\times 100$ (37) $\displaystyle E_{5}=\textit{PRE}(t_{i},\bar{{y}}_{sd})=\frac{\textit{MSE}(\bar% {{y}}_{sd})_{opt}}{\textit{MSE}(t_{i})_{opt}}\times 100$ (38) $\displaystyle E_{6}=\textit{PRE}(t_{i},\bar{{y}}_{\textit{singh}})=\frac{% \textit{MSE}(\bar{{y}}_{\textit{singh}})_{opt}}{\textit{MSE}(t_{i})_{opt}}% \times 100$ (39) $\displaystyle E_{7}=\textit{PRE}(t_{i},\bar{{y}}_{\textit{gira}})=\frac{% \textit{MSE}(\bar{{y}}_{\textit{gira}})_{opt}}{\textit{MSE}(t_{i})_{opt}}% \times 100$ (40) $\displaystyle E_{8}=\textit{PRE}(t_{i},\bar{{y}}_{\textit{smkk}_{1}})=\frac{% \textit{MSE}(\bar{{y}}_{\textit{smkk}_{1}})}{\textit{MSE}(t_{i})_{opt}}\times 100$ (41) $\displaystyle E_{9}=\textit{PRE}(t_{i},\bar{{y}}_{\textit{smkk}_{2}})=\frac{% \textit{MSE}(\bar{{y}}_{\textit{smkk}_{2}})}{\textit{MSE}(t_{i})_{opt}}\times 100$ (42) $\displaystyle E_{10}=\textit{PRE}(t_{i},\bar{{y}}_{\textit{smkk}_{3}})=\frac{% \textit{MSE}(\bar{{y}}_{\textit{smkk}_{3}})}{\textit{MSE}(t_{i})_{opt}}\times 100$ (43)

where $t_{i}=\left(\tau_{i}(i=1,2,\ldots,20)\right)$ .

Simulation studies are carried out for the different choices of the parameter $N$ , $n$ , $r$ and $\rho_{yx}$ . Results are presented in the Tables 3–8.

6. Analysis of simulation results

Table 5
Biases, mean square errors and percent relative biases of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ for Data 3

$N=$ 3059, $n=$ 428, $r=$ 128, $\rho_{yx}=$ 0.677428
Estimators	Absolute Bias	Mean square error (MSE)	Relative Bias (RB)	Percent relative Bias (PRB)
$\tau_{1}$	2514.34	7.88E+08	0.0895888	8.95888
$\tau_{2}$	2554.18	7.92E+08	0.0907398	9.07398
$\tau_{3}$	2777.15	8.22E+08	0.0968356	9.68356
$\tau_{4}$	2607.23	7.99E+08	0.0922440	9.22440
$\tau_{5}$	2565.03	7.94E+08	0.0910499	9.10499
$\tau_{6}$	2541.51	7.91E+08	0.0903758	9.03758
$\tau_{7}$	2514.90	7.88E+08	0.0896052	8.96052
$\tau_{8}$	2518.55	7.88E+08	0.0897113	8.97113
$\tau_{9}$	2515.69	7.88E+08	0.0896281	8.96281
$\tau_{10}$	2515.06	7.88E+08	0.0896098	8.96098
$\tau_{11}$	2514.72	7.88E+08	0.0895999	8.95999
$\tau_{12}$	2531.15	7.90E+08	0.0900765	9.00765
$\tau_{13}$	2633.52	8.02E+08	0.0929775	9.29775
$\tau_{14}$	2535.79	7.90E+08	0.0902108	9.02108
$\tau_{15}$	2525.76	7.89E+08	0.0899205	8.99205
$\tau_{16}$	2726.17	8.15E+08	0.0954955	9.54955
$\tau_{17}$	2572.57	7.95E+08	0.0912649	9.12649
$\tau_{18}$	2878.08	8.39E+08	0.0993900	9.93900
$\tau_{19}$	2648.33	8.04E+08	0.0933869	9.33869
$\tau_{20}$	2588.22	7.96E+08	0.0917088	9.17088

Table 6

Percent relative efficiencies of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ over the estimators $\bar{y}_{r}$ , $\bar{y}_{\textit{rat}}$ , $\bar{y}_{\textit{reg}}$ , $\bar{{y}}_{sh}$ , $\bar{{y}}_{sd}$ , $\bar{{y}}_{\textit{singh}}$ , $\bar{{y}}_{\textit{gira}}$ , $\bar{{y}}_{\textit{smkk}_{1}}$ , $\bar{{y}}_{\textit{smkk}_{2}}$ and $\bar{{y}}_{\textit{smkk}_{3}}$ respectively for Data 1

$N=$ 10, $n=$ 4, $r=$ 3, $\rho_{yx}=$ 0.6515
Estimators	$E_{1}$	$E_{2}$	$E_{3}$	$E_{4}$	$E_{5}$	$E_{6}$	$E_{7}$	$E_{8}$	$E_{9}$	$E_{10}$
$\tau_{1}$	161.930	140.511	137.383	137.383	137.383	137.383	137.383	101.426	124.083	110.381
$\tau_{2}$	161.113	139.801	136.689	136.689	136.689	136.689	136.689	100.914	123.457	109.824
$\tau_{3}$	162.243	140.783	137.649	137.649	137.649	137.649	137.649	101.622	124.324	110.595
$\tau_{4}$	161.458	140.101	136.982	136.982	136.982	136.982	136.982	101.130	123.722	110.059
$\tau_{5}$	161.825	140.419	137.294	137.294	137.294	137.294	137.294	101.360	124.003	110.310
$\tau_{6}$	161.396	140.048	136.930	136.930	136.930	136.930	136.930	101.092	123.675	110.018
$\tau_{7}$	161.821	140.416	137.290	137.290	137.290	137.290	137.290	101.358	124.000	110.307
$\tau_{8}$	161.971	140.547	137.418	137.418	137.418	137.418	137.418	101.452	124.115	110.409
$\tau_{{9}}$	161.867	140.456	137.330	137.330	137.330	137.330	137.330	101.387	124.035	110.338
$\tau_{{10}}$	161.916	140.498	137.371	137.371	137.371	137.371	137.371	101.417	124.073	110.372
$\tau_{{11}}$	161.859	140.449	137.323	137.323	137.323	137.323	137.323	101.381	124.029	110.333
$\tau_{{12}}$	160.517	139.285	136.184	136.184	136.184	136.184	136.184	100.541	123.001	109.418
$\tau_{{13}}$	162.474	140.983	137.845	137.845	137.845	137.845	137.845	101.767	124.501	110.752
$\tau_{{14}}$	161.747	140.352	137.228	137.228	137.228	137.228	137.228	101.312	123.944	110.257
$\tau_{{15}}$	161.007	139.710	136.600	136.600	136.600	136.600	136.600	100.848	123.376	109.752
$\tau_{{16}}$	164.396	142.651	139.475	139.475	139.475	139.475	139.475	102.971	125.973	112.062
$\tau_{{17}}$	160.679	139.425	136.321	136.321	136.321	136.321	136.321	100.642	123.125	109.528
$\tau_{{18}}$	162.411	140.928	137.791	137.791	137.791	137.791	137.791	101.728	124.452	110.709
$\tau_{{19}}$	161.206	139.883	136.769	136.769	136.769	136.769	136.769	100.973	123.529	109.888
$\tau_{{20}}$	161.768	140.371	137.246	137.246	137.246	137.246	137.246	101.325	123.960	110.271

Table 7

$N=$ 923, $n=$ 180, $r=$ 144, $\rho_{yx}=$ 0.9543
Estimators	$E_{1}$	$E_{2}$	$E_{3}$	$E_{4}$	$E_{5}$	$E_{6}$	$E_{7}$	$E_{8}$	$E_{9}$	$E_{10}$
$\tau_{1}$	388.045	305.876	304.303	304.303	304.303	304.303	304.303	121.846	126.547	123.004
$\tau_{2}$	387.987	305.830	304.257	304.257	304.257	304.257	304.257	121.828	126.528	122.986
$\tau_{{3}}$	386.958	305.019	303.450	303.450	303.450	303.450	303.450	121.505	126.192	122.659
$\tau_{{4}}$	387.816	305.695	304.123	304.123	304.123	304.123	304.123	121.774	126.472	122.931
$\tau_{{5}}$	387.937	305.790	304.218	304.218	304.218	304.218	304.218	121.812	126.512	122.970
$\tau_{{6}}$	387.990	305.832	304.259	304.259	304.259	304.259	304.259	121.829	126.529	122.986
$\tau_{{7}}$	388.045	305.876	304.303	304.303	304.303	304.303	304.303	121.846	126.547	123.004
$\tau_{{8}}$	388.045	305.876	304.303	304.303	304.303	304.303	304.303	121.846	126.547	123.004
$\tau_{{9}}$	388.045	305.876	304.303	304.303	304.303	304.303	304.303	121.846	126.547	123.004
$\tau_{{10}}$	388.045	305.876	304.303	304.303	304.303	304.303	304.303	121.846	126.547	123.004
$\tau_{{11}}$	388.045	305.876	304.303	304.303	304.303	304.303	304.303	121.846	126.547	123.004
$\tau_{{12}}$	388.031	305.864	304.291	304.291	304.291	304.291	304.291	121.842	126.542	122.999
$\tau_{{13}}$	387.769	305.657	304.086	304.086	304.086	304.086	304.086	121.759	126.457	122.916
$\tau_{{14}}$	388.018	305.854	304.281	304.281	304.281	304.281	304.281	121.838	126.538	122.995
$\tau_{{15}}$	388.031	305.865	304.292	304.292	304.292	304.292	304.292	121.842	126.542	123.000
$\tau_{{16}}$	387.461	305.415	303.845	303.845	303.845	303.845	303.845	121.663	126.357	122.819
$\tau_{{17}}$	387.984	305.828	304.255	304.255	304.255	304.255	304.255	121.827	126.527	122.985
$\tau_{{18}}$	386.906	304.978	303.409	303.409	303.409	303.409	303.409	121.488	126.175	122.643
$\tau_{{19}}$	387.805	305.686	304.114	304.114	304.114	304.114	304.114	121.771	126.469	122.928
$\tau_{{20}}$	387.932	305.786	304.214	304.214	304.214	304.214	304.214	121.810	126.510	122.968

Table 8

$N=$ 3059, $n=$ 428, $r=$ 128, $\rho_{yx}=$ 0.677428
Estimators	$E_{1}$	$E_{2}$	$E_{3}$	$E_{4}$	$E_{5}$	$E_{6}$	$E_{7}$	$E_{8}$	$E_{9}$	$E_{10}$
$\tau_{1}$	171.911	122.122	114.198	114.198	114.198	114.198	114.198	128.199	159.450	112.742
$\tau_{2}$	170.897	121.402	113.525	113.525	113.525	113.525	113.525	127.443	158.510	112.077
$\tau_{3}$	164.632	116.951	109.363	109.363	109.363	109.363	109.363	122.771	152.699	107.969
$\tau_{4}$	169.497	120.407	112.595	112.595	112.595	112.595	112.595	126.398	157.211	111.159
$\tau_{5}$	170.615	121.202	113.338	113.338	113.338	113.338	113.338	127.233	158.249	111.893
$\tau_{6}$	171.223	121.633	113.741	113.741	113.741	113.741	113.741	127.686	158.812	112.291
$\tau_{7}$	171.897	122.112	114.189	114.189	114.189	114.189	114.189	128.188	159.437	112.733
$\tau_{8}$	171.805	122.047	114.128	114.128	114.128	114.128	114.128	128.120	159.352	112.673
$\tau_{9}$	171.877	122.098	114.176	114.176	114.176	114.176	114.176	128.173	159.419	112.720
$\tau_{10}$	171.893	122.109	114.186	114.186	114.186	114.186	114.186	128.185	159.433	112.730
$\tau_{11}$	171.901	122.115	114.192	114.192	114.192	114.192	114.192	128.191	159.441	112.736
$\tau_{12}$	171.487	121.821	113.917	113.917	113.917	113.917	113.917	127.883	159.057	112.464
$\tau_{13}$	168.781	119.899	112.119	112.119	112.119	112.119	112.119	125.865	156.548	110.690
$\tau_{14}$	171.369	121.737	113.838	113.838	113.838	113.838	113.838	127.795	158.948	112.387
$\tau_{15}$	171.624	121.918	114.007	114.007	114.007	114.007	114.007	127.984	159.184	112.554
$\tau_{16}$	166.151	118.030	110.372	110.372	110.372	110.372	110.372	123.903	154.108	108.965
$\tau_{17}$	170.418	121.061	113.207	113.207	113.207	113.207	113.207	127.086	158.066	111.763
$\tau_{18}$	161.481	114.713	107.270	107.270	107.270	107.270	107.270	120.421	149.777	105.902
$\tau_{19}$	168.373	119.608	111.848	111.848	111.848	111.848	111.848	125.560	156.168	110.422
$\tau_{20}$	170.005	120.768	112.932	112.932	112.932	112.932	112.932	126.778	157.683	111.493

The following interpretation can be read out from Tables 3–8.

(1) (1)

From Tables 3–5 it is observed that the percent relative biases of suggested estimators lie between 9% to 10% in Table 5, more than 5% in Table 3 and more than 8% in Table 4.

(2)

From Table 6 it is clear that for a 40% sampled population with response rate 75%, the percent relative efficiencies of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ with respect to the other existing estimators like as the mean method of imputation remains 160.517% to 164.396%; the ratio method of imputation remains 139.285% to 142.651%; the regression method of imputation, Singh and Horn ( $\bar{{y}}_{sh}$ ) estimator, Singh and Deo ( $\bar{{y}}_{sd}$ ) estimator, Singh ( $\bar{{y}}_{\textit{singh}}$ ) estimator and Gira ( $\bar{{y}}_{\textit{gira}}$ ) estimator remain 136.184% to 139.475%; Singh et al. ( $\bar{{y}}_{\textit{smkk}_{1}}$ , $\bar{{y}}_{\textit{smkk}_{2}}$ and $\bar{{y}}_{\textit{smkk}_{3}}$ ) estimators remain 100.541% to 102.971%, 123.001% to 125.973% and 109.418% to 112.062% respectively.

(3)

From Table 7 it is clear that for a 20% sampled population with response rate 80%, the percent relative efficiencies of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ with respect to the other existing estimators like as the mean method of imputation remains 386.906% to 388.045%; the ratio method of imputation remains 304.978% to 305.876%; the regression method of imputation, Singh and Horn ( $\bar{{y}}_{sh}$ ) estimator, Singh and Deo ( $\bar{{y}}_{sd}$ ) estimator, Singh ( $\bar{{y}}_{\textit{singh}}$ ) estimator and Gira ( $\bar{{y}}_{\textit{gira}}$ ) estimator remain 303.409% to 304.303%; Singh et al. ( $\bar{{y}}_{\textit{smkk}_{1}}$ , $\bar{{y}}_{\textit{smkk}_{2}}$ and $\bar{{y}}_{\textit{smkk}_{3}}$ ) estimators remain 121.488% to 121.846%, 126.175% to 126.547% and 122.643% to 123.004% respectively.

(4)

From Table 8 it is seen that for a 14% sampled population with response rate between 20% and 30%, the percent relative efficiencies of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ with respect to the other existing estimators like as the mean method of imputation remains 161.481% to 172.661%; the ratio method of imputation remains 109.288% to 122.122%; the regression method of imputation, Singh and Horn ( $\bar{{y}}_{sh}$ ) estimator, Singh and Deo ( $\bar{{y}}_{sd}$ ) estimator, Singh ( $\bar{{y}}_{\textit{singh}}$ ) estimator and Gira ( $\bar{{y}}_{\textit{gira}}$ ) estimator remain 100.836% to 114.198%; Singh et al. ( $\bar{{y}}_{\textit{smkk}_{1}}$ , $\bar{{y}}_{\textit{smkk}_{2}}$ and $\bar{{y}}_{\textit{smkk}_{3}}$ ) estimators remain 120.421% to 132.243%, 149.777% to 167.612% and 105.902% to 113.235% respectively.

7. Conclusions

In this work, ratio exponential type estimators with imputation $\tau_{i}$ $(i=1,2,\ldots,20)$ have been suggested for the estimating population mean in sample surveys using some known value of population parameters. From Tables 3–5 it is observed that the percent relative biases of the suggested estimators are less than 10% in Tables 3–5. The performance of the suggested estimators are highly justified in simulation studies presented in Tables 6–8. We tried simulation studies for several more populations and found that the percent relative biases in the suggested estimators varying between 11% (minimum) to 50% (maximum). Based on the above analysis (Section 6), it is showed that the suggested estimators are more efficient than the mean method of imputation, ratio method of imputation, regression method of imputation and the estimators are given by Singh and Horn (2000), Singh and Deo (2003)), Singh (2009), Gira (2015) and Singh et al. (2016).

Hence, the use of some known value of auxiliary variables through suggested estimators are highly beneficial and it may be recommended to the survey statisticians for its further use.

Footnotes

Acknowledgments

Author is thankful to the anonymous referee for his valuable suggestions.

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Ratio exponential type estimators with imputation for missing data in sample surveys

Abstract

Keywords

1. Introduction

2.1 Mean method of imputation

Table 3 Biases, mean square errors and percent relative biases of the suggested estimators τ i ( i = 1 , 2 , … , 20 ) for Data 1

Table 5 Biases, mean square errors and percent relative biases of the suggested estimators τ i ( i = 1 , 2 , … , 20 ) for Data 3

Footnotes

Acknowledgments

References

Table 3
Biases, mean square errors and percent relative biases of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ for Data 1

Table 5
Biases, mean square errors and percent relative biases of the suggested estimators $\tau_{i}$ $(i=1,2,\ldots,20)$ for Data 3