New Chain Imputation Methods for Estimating Population Mean in the Presence of Missing Data Using Two Auxiliary Variables

Abstract

This article deals with some new chain imputation methods by using two auxiliary variables under missing completely at random (MCAR) approach. The proposed generalized classes of chain imputation methods are tested from the viewpoint of optimality in terms of MSE. The proposed imputation methods can be considered as an efficient extension to the work of Singh and Horn (Metrika 51:267–276, 2000), Singh and Deo (Stat Pap 44:555–579, 2003), Singh (Stat A J Theor Appl Stat 43(5):499–511, 2009), Kadilar and Cingi (Commun Stat Theory Methods 37:2226–2236, 2008) and Diana and Perri (Commun Stat Theory Methods 39:3245–3251, 2010). The performance of the proposed chain imputation methods is investigated relative to the conventional chain-type imputation methods. The theoretical results are derived and comparative study is conducted and the results are found to be quite encouraging providing the improvement over the discussed work.

Appendix

1.1 Notations and minMSE of the proposed imputation

Let $\epsilon _{0}=\frac{{\bar{y}}_{r}}{{\bar{Y}}}-1$, $\epsilon _{1}=\frac{{\bar{x}}_{r}}{{\bar{X}}}-1$, and $\epsilon _{2}=\frac{{\bar{x}}_{n}}{{\bar{X}}}-1$.

$E\left( \epsilon _{0}\right) =E\left( \epsilon _{1}\right) =E\left( \epsilon _{2}\right) =0$

$E\left( \epsilon _{0}^{2}\right) =f_r C_{y}^{2}$, $E\left( \epsilon _{1}^{2}\right) =f_r C_{x}^{2}$, $E\left( \epsilon _{2}^{2}\right) =f_n C_{x}^{2}$,

$E\left( \epsilon _{0}\epsilon _{1}\right) =f_r \rho _{yx}C_{y}C_{x}$, $E\left( \epsilon _{0}\epsilon _{2}\right) =f_n \rho _{yx}C_{y}C_{x}$, $E\left( \epsilon _{1}\epsilon _{2}\right) =f_n C_{x}^{2}$.

Outline of Derivation of Theorem 3.1. The MSE of $T_{j}$ ($j=2,3$) is given by ${\text {MSE}}(T_{j})={\overline{Y}}^{2} \left[ 1+\gamma _{j}^{2}A_{j} -2\gamma _{j}B_{j} \right] .$

The optimum values of scalars involved are tabulated below for ready reference: $\gamma _{jopt}=\dfrac{B_{j}}{A_{j}}\ ;\left( j=2,3\right)$ substituting the optimum value of $\alpha _{2}$ in ${\text {MSE}}\left( T_{2}\right)$, we get minimum MSE

${\text {MSE}}\left( T_{j}\right) ={\overline{Y}}^{2}\left( 1-\dfrac{B_{j}^{2}}{A_{j}} \right) ,$

where

$$\begin{aligned} A_{2}= & {} 1+f_r C_{y}^{2}+2\theta _{1}^{2}f_{rn} C_{x}^{2} +2\theta _{2}^{2}f_n C_{z}^{2}+\theta _{1}f_{rn} \left( C_{x}^{2}-4\rho _{yx}C_{y}C_{x}\right) \\&+\,\,\theta _{2}f_n \left( C_{z}^{2}-4\rho _{yz}C_{y}C_{z}\right) ,\\ B_{2}= & {} 1+\frac{\theta _{1}^{2}}{2}f_{rn} C_{x}^{2}+\frac{\theta _{2}^{2}}{2}f_n C_{z}^{2}+\frac{\theta _{1}}{2}f_{rn} \left( C_{x}^{2}-2\rho _{yx}C_{y}C_{x}\right) \\&+\,\frac{\theta _{2}}{2} f_n \left( C_{z}^{2}-2\rho _{yz}C_{y}C_{z}\right) ,\\ A_{3}= & {} 1+f_r C_{y}^{2}+3k_{1}^{2}f_{rn} C_{x}^{2}+3k_{2}^{2}f_n C_{z}^{2}-2k_{1}f_{rn}\\&\left( 3C_{x}^{2}-2\rho _{yx}C_{y}C_{x}\right) +f_{rn} \left( 3C_{x}^{2}-4\rho _{yx}C_{y}C_{x}\right) -4k_{2}f_n \rho _{yz}C_{y}C_{z}, \\ B_{3}= & {} 1+k_{1}^{2}f_{rn} C_{x}^{2}+k_{2}^{2}f_n C_{z}^{2}-k_{1}f_{rn} \left( 2C_{x}^{2}-\rho _{yx}C_{y}C_{x}\right) \\&+\,\,f_{rn} \left( C_{x}^{2}-\rho _{yx}C_{y}C_{x}\right) -k_{2}f_n \rho _{yz}C_{y}C_{z}, \end{aligned}$$

where $\theta _{1}=\theta _{1opt}=\rho _{yx}\dfrac{C_{y}}{C_{x}}, \theta _{2}=\theta _{2opt}=\rho _{yz}\dfrac{C_{y}}{C_{z}}, k_{1}=k_{1opt}=\left( 1-\rho _{yx}\dfrac{C_{y}}{C_{x}}\right)$ and $k_{2}=k_{2opt}=\rho _{yz}\dfrac{ C_{y}}{C_{z}}$ are used as optimizing values of the constants in this study, which are the optimum values where $\gamma =1$.

Outline of Derivation of Theorem 4.1. The MSE of $T_{1}$ is given by

${\text {MSE}}\left( T_{1}\right) =\dfrac{{\bar{Y}}^{2}\left[ S_{y}^{2}\left\{ f_r -f_{rn} \rho _{yx}^{2}-f_n \rho _{yz}^{2}\right\} \right] }{\left[ {\bar{Y}}^{2}+S_{y}^{2}\left\{ f_r -f_{rn} \rho _{yx}^{2}-f_n \rho _{yz}^{2}\right\} \right] }=\dfrac{{\bar{Y}}^{2} \left[ {\text {MSE}}\left( t_{1}\right) \right] }{\left[ {\bar{Y}}^{2}+{\text {MSE}}\left( t_{1}\right) \right] }.$

The optimum values of scalars involved are given below:

$\gamma _{1}=\dfrac{1}{\left[ 1+C_{y}^{2}\left\{ f_r -f_{rn} \rho _{yx}^{2}-f_n \rho _{yz}^{2}\right\} \right] }$, $\delta _{1}=-\gamma _{1} \beta _{1}$ and $\delta _{2}=-\gamma _{1} \beta _{2}$,

where $\beta _{1}=\rho _{yx}\dfrac{S_{y}}{S_{x}}$ and $\beta _{2} =\rho _{yz}\dfrac{ S_{y}}{S_{z}}$.