New estimators for estimating population total: an application to water demand in Thailand under unequal probability sampling without replacement for missing data

Basic setup

In this section, we introduce notations and basic notions about the population total estimator and their variance estimators under the reverse framework. Let $y$ be a study variable and the population total of $y$ is $Y=\sum _{i\in U}{y}_i$ where $U=\{1,2,...,N\}$ and $N$ is the population size. Suppose the auxiliary variables $x$, $w$ and the size variable $k$ are available and highly positively correlated with the study variable. The calibration variables $u_1,u_2,...,u_q$ where $q\ge 1$ are also available and they are correlated with the auxiliary variable $x$. Let, ${\mathrm{u}}_i=(\begin{array}{cccc}1& u_{i1}& L& u_{iq}\end{array}{)}^{\mathrm{\prime }}$ and ${\mathrm{U}}_N=(\begin{array}{cccc}{\mathrm{u}}_1& {\mathrm{u}}_2& L& {\mathrm{u}}_N\end{array}{)}^{\mathrm{\prime }}$ be the $N\times (q+1)$ matrix values of ${\mathrm{u}}_i$. We are using the GREG estimator model from Särndal, Swensson & Wretman (1992) and Särndal (2007) in which the linear assisting model $\xi$, $E_{\xi }(x_i)={\beta }^{\mathrm{\prime }}{\mathrm{u}}_i$ and $V_{\xi }(x_i)={\sigma }_i^2$. The linear assisting model $\xi$ is a model describing the relationship between the study variable and auxiliary variable. Let $q_i$ be determined by the linear assisting model $\xi$ that is $q_i=1/{\sigma }_i^2$. Usually, the standard choice of $q_i$ is $q_i=1$ and it is determined by the linear assisting model $\xi$: $E_{\xi }(x_i)={\beta }^{\mathrm{\prime }}{\mathrm{u}}_i$ and $V_{\xi }(x_i)={\sigma }^2$.

Let, $\mathrm{F}$ be the set of all possible subsets of $U$ and the sample $s$ of size $n$ was selected from the population $U$ under UPWOR. A sampling design $p(.)$ is a probability distribution on $\mathrm{F}$, i.e., $P(s)\ge 0$ for all $s\in \mathrm{F}$ and $\sum _{s\in \mathrm{F}}p(s)=1$. Let, ${\pi }_i=P(i\in s)=\sum _{s∍i}P(s)$ be the first order inclusion probability and ${\pi }_{ij}=P(i\wedge j\in s)=\sum _{s\supset \{i,j\}}P(s)$ be the second order inclusion probability. We also define $E_S(\bullet )$ and $V_S(\bullet )$ as the expectation and variance operators with respect to the UPWOR sampling design.

In the presence of nonresponse, let subscript $R$ and $r_i$ be the nonresponse mechanism and nonresponse indicator variable of $y_i$ which $r_i=1$ if unit $i$ responds to item $y$ otherwise $r_i=0$. Let, $\mathrm{R}=(\begin{array}{cccc}{r}_1& r_2& \cdots & r_N\end{array}{)}^{\mathrm{\prime }}$ be the vector of the response indicator and $p_i=P(r_i=1)$ be the response probability under MAR nonresponse. Let, $E_R(\bullet )$ and $V_R(\bullet )$ be the expectation and variance operators with respect to the nonresponse mechanism. Three assumptions are defined; $(A_1)$ the response mechanism is uniform response. $(A_2){\hat{\beta }}_r-\beta =O_p(n_r^{-{\displaystyle \frac{1}{2}}})$ and $(A_3)V\left(\sum _{i\in s}{\displaystyle \frac{b_i}{{\pi }_i}}\right)\to 0$ as $n\to \mathrm{\infty }$ where $b_i=w_i$ or $r_i$. We also consider three more conditions for investigating the estimator of $Y=\sum _{i\in U}{y}_i$ as follows. $(B_1)$ nonresponse occurs only on $y$, the information on $x_i$ is available for all $i\in s$ and $\overline{X}$ is known. $(B_2)$ nonresponse occurs on both $y$ and $x$ and $\overline{X}$ is known and $(B_3)$ nonresponse occurs both with $y$ and $x$ and $\overline{X}$ is unknown but information on $u_1,u_2,...,u_q$ are available for all $i\in s$ and ${\overline{U}}_j={\displaystyle \frac{1}{N}\sum _{i\in U}{u}_{ij}}$, $j=1,2,...,q$ are known.

Throughout this article, we consider variance estimation of the population total estimator in the presence of nonresponse under the reverse framework. Therefore, we discuss three steps to investigate the variance and its nonlinear estimator such as the ratio estimator when nonresponse occurs in the study variable. Assume that we have $K$ variables consisting of $t_1$, the study variable and $t_2,t_3,...,t_K$, auxiliary variables. Let ${\hat{Y}}_s$ be a nonlinear estimator and be defined by,

(1) $${\hat{Y}}_s=\psi (\hat{\mathrm{T}}),$$

where $\psi$ is a known smooth function, $\hat{\mathrm{T}}=\left[\begin{array}{cccc}{\hat{t}}_1& {\hat{t}}_2& \cdots & {\hat{t}}_K\end{array}\right]$, $K\ge 2$ ${\hat{t}}_k=\sum _{i\in s}{\displaystyle \frac{r_i{t}_{ki}}{{\pi }_i{p}_i}}$ if the variable $t_k$ exhibits nonresponse otherwise it can be obtained by ${\hat{Y}}_s.$ Under the reverse framework, variance of ${\hat{t}}_k=\sum _{i\in s}{\displaystyle \frac{t_{ki}}{{\pi }_i}}$

(2) $$V({\hat{Y}}_s)=E_R{V}_S\left({\hat{Y}}_s|\mathrm{R}\right)+V_R{E}_S\left({\hat{Y}}_s|\mathrm{R}\right)=V_1+V_2,$$

where $V_1=E_R{V}_S\left({\hat{Y}}_s|\mathrm{R}\right)$ and $V_2=V_R{E}_S\left({\hat{Y}}_s|\mathrm{R}\right)$. The formula of $V({\hat{Y}}_s)$ consists of three steps as below.

Step 1: Investigate a formula of $V_1=E_R{V}_S\left({\hat{Y}}_s|\mathrm{R}\right)$. Since ${\hat{Y}}_s$ is in a form of a nonlinear estimator then $V_1=E_R{V}_S\left({\hat{Y}}_s|\mathrm{R}\right)$ can be approximated by,

(3) $$V_1^{\mathrm{\prime }}\cong E_R{V}_S\left({\hat{Y}}_{s(1)}|\mathrm{R}\right),$$

where ${\hat{Y}}_{s(1)}$ is a linear estimator of ${\hat{Y}}_s$ under the Taylor linearization approach.

Step 2: Investigate the formula of $V_2=V_R{E}_S\left({\hat{Y}}_s|\mathrm{R}\right)$. The formula of $V_2=V_R{E}_S\left({\hat{Y}}_s|\mathrm{R}\right)$ can be approximated by,

(4) $$V_2^{\mathrm{\prime }}\cong V_R\left({\tilde{Y}}_s|\mathrm{R}\right),$$

where ${\tilde{Y}}_s=E_S\left({\hat{Y}}_s|\mathrm{R}\right)$.

Step 3: Approximate the value of $V({\hat{Y}}_s)$ and its estimator. The value of $V({\hat{Y}}_s)$ can be obtained by,

(5) $$V({\hat{Y}}_s)=V_1^{\mathrm{\prime }}+V_2^{\mathrm{\prime }}.$$

The estimator of $V({\hat{Y}}_s)$ can be obtained by substituting estimators for the unknown parameter in (5). Then, the estimator of $V({\hat{Y}}_s)$ is defined by,

(6) $$\hat{V}({\hat{Y}}_s)={\hat{V}}_1^{\mathrm{\prime }}+{\hat{V}}_2^{\mathrm{\prime }},$$

where ${\hat{V}}_1^{\mathrm{\prime }}$, ${\hat{V}}_2^{\mathrm{\prime }}$ are the estimators of $V_1^{\mathrm{\prime }}$, $V_2^{\mathrm{\prime }}$ respectively.

Existing estimators under uniform nonresponse

Uniform nonresponse or missing completely at random (MCAR) is a nonresponse mechanism in which the probability of response of the study variable $y$ neither depends on itself nor another variable such as $x,k$ or $w.$ In this section, we discuss two estimators for estimating population total in the presence of uniform nonresponse namely ratio and GREG estimators proposed by Ponkaew & Lawson (2018) and Ponkaew (2018), respectively. The variance estimation of both ratio and GREG estimators are considered under the reverse framework and the sampling fraction is negligible with the UPWOR sampling design.

The ratio estimator

When nonresponse occurs only with $y$ but the population mean and its estimator of $x$ are available, Ponkaew & Lawson (2018) proposed ratio estimators to estimate population mean and the total of $y$ under unequal probability sampling without replacement and the nonresponse mechanism is MCAR. The Ponkaew & Lawson (2018) estimator for population mean is

(7) $$\widehat{{\overline{Y}}_R}={\displaystyle \frac{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_{i}p}}}}{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}}}\overline{X}={\displaystyle \frac{{\hat{\overline{Y}}}_r}{{\hat{\overline{X}}}_{HT}}\overline{X}={\hat{R}}_r\overline{X},}}$$

where $\widehat{{\overline{Y}}_r}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_{i}p}}}$, $\widehat{{\overline{X}}_{HT}}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}}$ and ${\hat{R}}_r=\widehat{{\overline{Y}}_r}(\widehat{{\overline{X}}_{HT}}{)}^{-1}$. Ponkaew & Lawson’s (2018) estimator for population total is

(8) $${\hat{Y}}_R=N\widehat{{\overline{Y}}_R}=N\overline{X}{\hat{R}}_r.$$

We note that, if $p$ is unknown the estimator of $p$ is equal to $\hat{p}=\left(\sum _{i\in s}{\displaystyle \frac{r_i}{{\pi }_i}}\right){\left(\sum _{i\in s}{\displaystyle \frac{1}{{\pi }_i}}\right)}^{-1}$. The variance and associated estimators of the estimator in (8) is defined in (9),

(9) $$V\left({\hat{Y}}_R\right)=\sum _{i\in U}{D}_i(y_i-R{x}_i{)}^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}(y_i-R{x}_i)(y_j-R{x}_j)+\sum _{i\in U}{\displaystyle \frac{1-p_i}{p_i}{y}_i^2},$$

where $R=\overline{Y}{\overline{X}}^{-1}$. The estimator of $V\left({\hat{Y}}_R\right)$ is given in (10),

(10) $$\hat{V}\left({\hat{Y}}_R\right)=\sum _{i\in s}{\hat{D}}_i(y_i-{\hat{R}}_r{x}_i{)}^2+\sum _{i\in s}\sum _{i\mathrm{\setminus }\{j\}\in s}{\hat{D}}_{ij}(y_i-{\hat{R}}_r{x}_i)(y_j-{\hat{R}}_r{x}_j)+\sum _{i\in s}{\hat{E}}_i{y}_i^2,$$

where ${\hat{R}}_r^{}=\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_{i}p}}{\left(\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}\right)}^{-1}$, ${\hat{E}}_i={\displaystyle \frac{r_i{E}_i}{{\pi }_i}}$, ${\hat{D}}_i={\displaystyle \frac{r_i{D}_i}{p}}$ and ${\hat{D}}_{ij}={\displaystyle \frac{r_i{r}_j{D}_{ij}}{p^2}}$.

The GREG estimator

The GREG estimators for estimating population mean or population total of the study variable is a powerful method when the calibration variables $u_1,u_2,...,u_q$ are present where $q\ge 1$ are also available. In full response, Särndal, Swensson & Wretman (1992) and Särndal (2007) proposed a GREG estimator under the linear assisting model $\xi$,

(11) $$E_{\xi }(x_i)={\beta }^{\mathrm{\prime }}{\mathrm{u}}_i\mathrm{a}\mathrm{n}\mathrm{d}{V}_{\xi }(x_i)={\sigma }_i^2.$$

Let $Q_s=diag(q_i{)}_{s\times s}$ and $q_i$ be determined by the linear assisting model $\xi$ in (5.1) i.e., $q_i={\sigma }_i^{-2}$. In the presence of nonresponse, Särndal & Lundström (2005) proposed a linear GREG estimator to estimate population total. They investigated variance and associated estimators under the two-phase framework. Ponkaew (2018) proposed linear GREG estimators for estimating the population mean of $x$ under the MCAR mechanism which is defined by,

(12) $$\widehat{{\overline{X}}_{GREG}^{(1)}}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_{i}p}}+{\left[\overline{\mathrm{U}}-{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{\mathrm{u}}_i}{{\pi }_{i}p}}}\right]}^{\mathrm{\prime }}{\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{{\mathrm{u}}^{\mathrm{\prime }}}_i}{{\pi }_{i}p}}\right)}^{-1}\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{x}_i}{{\pi }_{i}p}}\right)=\widehat{{\overline{X}}_r^{(1)}}+{\left[\overline{\mathrm{U}}-{\widehat{\overline{\mathrm{U}}}}_r^{(1)}\right]}^{\mathrm{\prime }}{\hat{\beta }}_r,}$$

where $\widehat{{\overline{X}}_r^{(1)}}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_{i}p}},}$ ${\widehat{\overline{\mathrm{U}}}}_r^{(1)}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{\mathrm{u}}_i}{{\pi }_{i}p}}}$, and ${\hat{\beta }}_r={\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{{\mathrm{u}}^{\mathrm{\prime }}}_i}{{\pi }_{i}p}}\right)}^{-1}\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{x}_i}{{\pi }_{i}p}}\right)={\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{{\mathrm{u}}^{\mathrm{\prime }}}_i}{{\pi }_i}}\right)}^{-1}\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{x}_i}{{\pi }_i}}\right)$.

Then, the GREG estimator to estimate the population total of $x$ is

(13) $${\hat{X}}_{GREG}^{(1)}=N\widehat{{\overline{X}}_{GREG}^{(1)}}=\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_{i}p}}+{\left[\mathrm{U}-\sum _{i\in s}{\displaystyle \frac{r_i{\mathrm{u}}_i}{{\pi }_{i}p}}\right]}^{\mathrm{\prime }}{\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{{\mathrm{u}}^{\mathrm{\prime }}}_i}{{\pi }_{i}p}}\right)}^{-1}\left(\sum _{i\in s}{\displaystyle \frac{r_i{q}_i{\mathrm{u}}_i{x}_i}{{\pi }_{i}p}}\right),={\hat{X}}_r^{(1)}+{\left[\mathrm{U}-{\hat{\mathrm{U}}}_r^{(1)}\right]}^{\mathrm{\prime }}{\hat{\beta }}_r,$$

where ${\hat{X}}_r^{(1)}=N\widehat{{\overline{X}}_r^{(1)}}=\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_{i}p}},$ ${\hat{\mathrm{U}}}_r^{(1)}=N{\widehat{\overline{\mathrm{U}}}}_r^{(1)}=\sum _{i\in s}{\displaystyle \frac{r_i{\mathrm{u}}_i}{{\pi }_{i}p}}$.

Under the reverse framework and when sampling fraction is negligible the variance of ${\hat{W}}_{GREG}^{(1)}$ is

(14) $$V({\hat{X}}_{GREG}^{(1)})\cong \sum _{i\in U}{D}_{1i}{e}_i^2+\sum _{i\in U}\sum _{j\in U\mathrm{\setminus }\{i\}}{D}_{ij}{e}_i{e}_j,$$

where $D_{1i}={\displaystyle \frac{(1-{\pi }_i)}{{\pi }_{i}p}}$, $D_{ij}={\displaystyle \frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}}$ and $e_i=(x_i-{{\mathrm{u}}^{\mathrm{\prime }}}_i\beta )$.

The estimator of $V({\hat{W}}_{GREG}^{(1)})$ is equal to.

(15) $$\hat{V}({\hat{X}}_{GREG}^{(1)})\cong {\displaystyle \frac{1}{{p^{\mathrm{\prime }}}^2}\left[\sum _{i\in s}{\hat{D}}_i{r}_i{\hat{e}}_i^2+\sum _{i\in s}\sum _{j\in s\mathrm{\setminus }\{i\}}{\hat{D}}_{ij}{r}_i{r}_j{\hat{e}}_i{\hat{e}}_j\right],}$$

where ${\hat{e}}_i=(x_i-{{\mathrm{u}}^{\mathrm{\prime }}}_i{\hat{\beta }}_r)$, ${\hat{D}}_i={\displaystyle \frac{1-{\pi }_i}{{\pi }_i^2}}$, ${\hat{D}}_{ij}={\displaystyle \frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_{ij}{\pi }_i{\pi }_j}}$, $p^{\mathrm{\prime }}=\left(\sum _{i\in s}{\displaystyle \frac{1}{{\pi }_i}}\right){\left(\sum _{i\in s}{\displaystyle \frac{r_i}{{\pi }_i}}\right)}^{-1}$ if $p$ is unknown otherwise $p^{\mathrm{\prime }}=p$.

The proposed new ratio estimators

In the previous section, we introduced two estimators of the population total: ratio and GREG estimators in the presence of uniform nonresponse. The variance estimation for both ratio and GREG estimators are considered under the UPWOR sampling design and when the sampling fraction is negligible. However, the ratio estimators in (7) and (8) are considered under a situation where nonresponse occurs in $y$ only and they require the value of the population mean of $x$. Then, in this section we aim to propose new ratio estimators when nonresponse occurs in both variables $y$ and $x$. We also consider two distinct situations of $\overline{X}$ that are known or unknown. In the situation where $\overline{X}$ is unknown we estimate it from the calibration variables $u_1,u_2,...,u_q$ using the GREG estimator. In the context of nonresponse, we investigate the proposed ratio estimator under the MAR mechanism because it has weak assumptions and tends to occur in real life more often than the MCAR mechanism. However, we still consider new ratio estimators under the MCAR mechanism for comparing the efficiency of the proposed estimators. First of all, we extended the Ponkaew & Lawson (2018) estimators to the MAR mechanism. The ratio estimator of Ponkaew & Lawson (2018) for estimating population mean under the MAR mechanism is equal to,

(16) $$\widehat{{\overline{Y}}_R^{(1)}}={\displaystyle \frac{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p}_i}}}}{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}}}\overline{X}={\displaystyle \frac{\widehat{{\overline{Y}}_r^{(1)}}}{{\hat{\overline{X}}}_{HT}}\overline{X}={\hat{R}}_r^{(1)}\overline{X},}}$$

where $\widehat{{\overline{Y}}_r^{(1)}}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p}_i}}}$, $\widehat{{\overline{X}}_{HT}}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}}$ and ${\hat{R}}_r^{(1)}=\widehat{{\overline{Y}}_r^{(1)}}(\widehat{{\overline{X}}_{HT}}{)}^{-1}$.

Then, the ratio estimator for estimating population total under the MAR mechanism is

(17) $${\hat{Y}}_R^{(1)}=N\widehat{{\overline{Y}}_R^{(1)}}=N\overline{X}{\hat{R}}_r^{(1)},$$

Under the MAR mechanism if $p_i$ is unknown then it is estimated using the probit or logistic regression models. The variance and associated estimators of ${\hat{Y}}_R^{(1)}$ are discussed in Theorem 4.1.

Theorem 1. Under condition $(B_1)$ with the reverse framework and the nonresponse mechanism is MAR.

(1) The variance of ${\hat{Y}}_R^{\mathrm{\prime }(1)}$ is

$$V\left({\hat{Y}}_R^{\mathrm{\prime }(1)}\right)\cong \sum _{i\in U}{D}_i{A}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{A}_i{A}_j+\sum _{i\in U}{E}_i{y}_i^2,$$

where $A_i=y_i-R{x}_i$, $R=\overline{Y}{\overline{X}}^{-1}$ and $E_i={\displaystyle \frac{1-p_i}{p_i}}$.

(2) The estimator of $V\left({\hat{Y}}_R^{\mathrm{\prime }(1)}\right)$ is

$$\hat{V}\left({\hat{Y}}_R^{\mathrm{\prime }(1)}\right)\cong \sum _{i\in s}{\hat{D}}_i{\hat{A}}_i^{(1)2}+\sum _{i\in s}\sum _{i\mathrm{\setminus }\{j\}\in s}{\hat{D}}_{ij}{\hat{A}}_i^{(1)}{\hat{A}}_j^{(1)}+\sum _{i\in s}{\hat{E}}_i{y}_i^2,$$

where ${\hat{A}}_i^{(1)}=y_i-{\hat{R}}_r^{(1)}{x}_i$, ${\hat{R}}_r^{(1)}=\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p}_i}}{\left(\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}\right)}^{-1}$, ${\hat{E}}_i={\displaystyle \frac{r_i{E}_i}{{\pi }_i}}$, ${\hat{D}}_i={\displaystyle \frac{r_i{D}_i}{p_i}}$ and ${\hat{D}}_{ij}={\displaystyle \frac{r_i{r}_j{D}_{ij}}{p_i{p}_j}}$.

Proof. Let ${\hat{Y}}_R^{(1)}$ be defined in (17). Therefore, variance of ${\hat{Y}}_R^{(1)}$ is

(18) $$V({\hat{Y}}_R^{(1)})=V(N\overline{X}{\hat{R}}_r^{(1)})=N^2{\overline{X}}^{2}V({\hat{R}}_r^{(1)}).$$

Furthermore, the estimator of $V({\hat{Y}}_R^{(1)})$ can be obtained by,

(19) $$\hat{V}({\hat{Y}}_R^{(1)})=N^2{\overline{X}}^2\hat{V}({\hat{R}}_r^{(1)}).$$

Since ${\hat{R}}_r^{(1)}$ is a nonlinear estimator then the variance of this estimator is equal to,

(20) $$V({\hat{R}}_r^{(1)})=E_R{V}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)+V_R{E}_S\left({\hat{R}}_r^{(1)}\mathrm{R}\right)=V_1+V_2,$$

where $V_1=E_R{V}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)$, $V_2=V_R{E}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)$.

Step 1: Investigate the formula of $V_1=E_R{V}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)$.

By using the Taylor linearization approach the linear estimator of ${\hat{R}}_r^{(1)}$ is

(21) $${\hat{R}}_r^{(1)})\equiv Constant+{\displaystyle \frac{1}{N\overline{X}}\sum _{i\in s}{\displaystyle \frac{{\tilde{A}}_i}{{\pi }_i}},}$$

where ${\tilde{A}}_i=\left({\displaystyle \frac{r_i{y}_i}{p_i}-{\tilde{R}}_r^{(1)}{x}_i}\right)$. Then $V_1=E_R{V}_S\left({\hat{R}}_r^{(1)})|\mathrm{R}\right)$ can be approximated by,

$$\begin{gathered} V_1^{\mathrm{\prime }}\cong E_R{V}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)=E_R{V}_S\left(Constant+{\displaystyle \frac{1}{N\overline{X}}\sum _{i\in s}{\displaystyle \frac{{\tilde{A}}_i}{{\pi }_i}}}|\mathrm{R}\right) \\ ={\displaystyle \frac{1}{N^2{\overline{X}}^2}{E}_R\left(\sum _{i\in U}{D}_i{\tilde{A}}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{\tilde{A}}_i{\tilde{A}}_j|\mathrm{R}\right)} \\ ={\displaystyle \frac{1}{N^2{\overline{X}}^2}\left(\sum _{i\in U}{D}_i{A}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{A}_i{A}_j\right),} \end{gathered}$$

where $A_i=E_R{V}_S\left({\tilde{A}}_i|\mathrm{R}\right)=y_i-R{x}_i$ and $R=\overline{Y}{\overline{X}}^{-1}$.

Therefore,

(22) $$V_1^{\mathrm{\prime }}\cong {\displaystyle \frac{1}{N^2{\overline{X}}^2}\left(\sum _{i\in U}{D}_i{A}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{A}_i{A}_j\right).}$$

Step 2: Investigate the formula of $V_2=V_R{E}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)$.

The formula of $V_2=V_R{E}_S\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)$ can be approximated by,

$$\begin{gathered} V_2^{\mathrm{\prime }}\cong V_R{E}_R\left({\hat{R}}_r^{(1)}|\mathrm{R}\right)=V_R{E}_R\left({\displaystyle \frac{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p}_i}}}}{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{x_i}{{\pi }_i}}}}}|\mathrm{R}\right)=V_R\left({\displaystyle \frac{{\displaystyle \frac{1}{N}\sum _{i\in U}{\displaystyle \frac{r_i{y}_i}{p_i}}}}{\overline{X}}}|\mathrm{R}\right) \\ ={\displaystyle \frac{1}{N^2{\overline{X}}^2}\sum _{i\in U}{\displaystyle \frac{(1-p_i)y_i^2}{p_i}}={\displaystyle \frac{1}{N^2{\overline{X}}^2}\sum _{i\in U}{E}_i{y}_i^2}} \end{gathered}$$

where $E_i={\displaystyle \frac{(1-p_i)}{p_i}}$.

Then,

(23) $$V_2^{\mathrm{\prime }}\cong {\displaystyle \frac{1}{N^2{\overline{X}}^2}\sum _{i\in U}{E}_i{y}_i^2.}$$

Step 3: Approximate the value of $V({\hat{R}}_r^{(1)})$ and its estimators.

The value of $V({\hat{R}}_r^{(1)})$ can be approximated by,

(24) $$\begin{gathered} V({\hat{R}}_r^{(1)})\cong V_1^{\mathrm{\prime }}+V_2^{\mathrm{\prime }} \\ ={\displaystyle \frac{1}{N^2{\overline{X}}^2}\left(\sum _{i\in U}{D}_i{A}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{A}_i{A}_j+\sum _{i\in U}{E}_i{y}_i^2\right).} \end{gathered}$$

The estimator of $V({\hat{R}}_r^{(1)})$ is

(25) $$\hat{V}\left({\hat{R}}_r^{(1)}\right)={\displaystyle \frac{1}{N^2{\overline{X}}^2}\left(\sum _{i\in s}{\hat{D}}_i{\hat{A}}_i^{(1)2}+\sum _{i\in s}\sum _{i\mathrm{\setminus }\{j\}\in s}{\hat{D}}_{ij}{\hat{A}}_i^{(1)}{\hat{A}}_j^{(1)}+\sum _{i\in s}{\hat{E}}_i{y}_i^2\right).}$$

Replace (25) into (18) then the variance of ${\hat{Y}}_R^{(1)}$ is

(26) $$V({\hat{Y}}_R^{(1)})\cong \sum _{i\in U}{D}_i{A}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{A}_i{A}_j+\sum _{i\in U}{E}_i{y}_i^2.$$

Furthermore, the estimator of $V({\hat{Y}}_R^{(1)})$ can be obtained by substituting (26) in (19) then,

(27) $$\hat{V}({\hat{Y}}_R^{(1)})\cong \sum _{i\in s}{\hat{D}}_i{\hat{A}}_i^{(1)2}+\sum _{i\in s}\sum _{i\mathrm{\setminus }\{j\}\in s}{\hat{D}}_{ij}{\hat{A}}_i^{(1)}{\hat{A}}_j^{(1)}+\sum _{i\in s}{\hat{E}}_i{y}_i^2.$$

In (16) and (17), we extend the ratio estimators of Ponkaew & Lawson (2018) to the MAR mechanism and discussed the variance and its estimators in Theorem 1. However, the ratio estimator for population mean in (16) and for population total in (17) can be used under the condition $(B_1)$ that is, when nonresponse occurs only with the $y$ variable but information on $x_i$ for all $i\in s$ and $\overline{X}$ needs to be known. Next, we proposed new ratio estimators under condition $(B_2)$ where nonresponse occurs on both $y$ and $x$ but $\overline{X}$ is known and condition $(B_3)$ nonresponse occurs both $y$ and $x$ and $\overline{X}$ is unknown but information of $u_1,u_2,...,u_q$ are available for all $i\in s$ and the population mean of $u_1,u_2,...,u_q$ are also known.

The new ratio estimator when $\overline{X}$ is known

Assume that the condition $(B_2)$ is satisfied when nonresponse occurs with both variables $y$ and $x$ but $\overline{X}$ is known. The new ratio estimator for estimating population mean is given below,

(28) $${\hat{\overline{Y}}}_R^{(2)}=\left({\displaystyle \frac{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p}_i}}}}{{\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_i{p}_i}}}}}\right){\displaystyle \frac{1}{N}\sum _{i\in U}{x}_i={\displaystyle \frac{{\hat{\overline{Y}}}_r}{{\hat{\overline{X}}}_r}\overline{X}={\hat{R}}_r^{(2)}\overline{X},}}$$

where $\widehat{{\overline{Y}}_r}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p}_i}}}$, $\widehat{{\overline{X}}_r}={\displaystyle \frac{1}{N}\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_i{p}_i}}}$, ${\hat{R}}_r^{(2)}=\widehat{{\overline{Y}}_r}(\widehat{{\overline{X}}_r}{)}^{-1}$. Furthermore, the estimator of $p_i$ can be obtained by using the probit or logistic regression models under the MAR mechanism. Then, the new ratio estimator for the population total is

(29) $${\hat{Y}}_R^{\mathrm{\prime }(2)}=N\widehat{{\overline{Y}}_R^{(2)}}=N\overline{X}{\hat{R}}_r^{(2)}.$$

The variance and associated estimators of ${\hat{Y}}_R^{\mathrm{\prime }(2)}$ are discussed in Theorem 2.

Theorem 2. Under condition $(B_2)$ with reverse framework and where the nonresponse mechanism is MAR.

(1) The variance of ${\hat{Y}}_R^{\mathrm{\prime }(2)}$ is

(30) $$V\left({\hat{Y}}_R^{\mathrm{\prime }(2)}\right)=\sum _{i\in U}{D}_i{A}_i^2+\sum _{i\in U}\sum _{i\mathrm{\setminus }\{j\}\in U}{D}_{ij}{A}_i{A}_j+\sum _{i\in U}{E}_i{y}_i^2,$$

where $A_i=y_i-R{x}_i$, $R=\overline{Y}{\overline{X}}^{-1}$ and $E_i={\displaystyle \frac{1-p_i}{p_i}}$.

(2) The estimator of $V\left({\hat{Y}}_R^{\mathrm{\prime }(2)}\right)$ is

(31) $$\hat{V}\left({\hat{Y}}_R^{\mathrm{\prime }(2)}\right)=\sum _{i\in s}{\hat{D}}_i{\hat{A}}_i^{(2)2}+\sum _{i\in s}\sum _{i\mathrm{\setminus }\{j\}\in s}{\hat{D}}_{ij}{\hat{A}}_i^{(2)}{\hat{A}}_j^{(2)}+\sum _{i\in s}{\hat{E}}_i{y}_i^2,$$

where ${\hat{A}}_i^{(2)}=y_i-{\hat{R}}_r^{(2)}{x}_i$, ${\hat{R}}_r^{(2)}=\sum _{i\in s}{\displaystyle \frac{r_i{y}_i}{{\pi }_i{p^{\mathrm{\prime }}}_i}}{\left(\sum _{i\in s}{\displaystyle \frac{r_i{x}_i}{{\pi }_i{p}^{\mathrm{\prime }}}}\right)}^{-1}$, ${\hat{D}}_i={\displaystyle \frac{r_i{D}_i}{{\pi }_i{p^{\mathrm{\prime }}}_i}}$, ${\hat{D}}_{ij}={\displaystyle \frac{r_i{D}_{ij}}{{\pi }_i{p^{\mathrm{\prime }}}_i}}$, ${\hat{E}}_i={\displaystyle \frac{r_i{E}_i}{{\pi }_i{p^{\mathrm{\prime }}}_i}}$. The value of $p_i^{\mathrm{\prime }}$, $p_i^{\mathrm{\prime }}=p_i$ if $p_i$ is known otherwise $p_i^{\mathrm{\prime }}={\hat{p}}_i$. ${\hat{p}}_i$ is the estimator of $p_i$ from the probit or logistic regression models.