Geographically Weighted Regression-Based Model Calibration Estimation of Finite Population Total Under Geo-referenced Complex Surveys

Abstract

In sample surveys, the model calibration approach is an improvement over the usual calibration approach, where the concept of the calibration approach is generalized to obtain a model-assisted estimator using more complex models based on complete auxiliary information. In many surveys, the study and auxiliary variables vary across locations and the observations tend to be similar for the nearby units than those located further apart. In such situations, a simple global model cannot explain the relationships between some sets of variables. This phenomenon is known as spatial non-stationarity which is considered by the geographically weighted regression (GWR) model. It can capture the spatially varying relationship between different variables. In the present study, GWR-based model calibration estimators of population total of the study variable were developed in the context of geo-referenced complex survey designs when complete auxiliary information along with their spatial locations is available at population level. The asymptotic properties of the developed GWR-based model calibration estimators were evaluated under a set of assumptions. Under the same set of assumptions, the variances and estimators of variances of the developed estimators were given. Through a spatial simulation study, the performance of the developed estimators was compared to that of existing estimators and found to be more efficient than the existing ones. Supplementary materials accompanying this paper appear online

Appendix

Proof of Theorem 1

By applying Taylor series approximation to $\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right)$ at $\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right)$, we get

$$\begin{aligned} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{+}\left\{ \left. \frac{\partial }{\partial \varvec{t}_{\varvec{i}}}\varvec{x}_{i}^{T}\varvec{t}_{\varvec{i}} \right| _{\varvec{t}_{\varvec{i}}\varvec{=}{\varvec{\beta }\left( u_{i} \right) }^{{*}}} \right\} ^{\varvec{T}}\left[ \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] \end{aligned}$$

(A1)

where ${\varvec{\beta }\left( u_{i} \right) }^{{*}}\in \{\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) ,\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \}$ or $\{\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) ,\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \}$.

Using assumptions (ii) and (iii) and summing both side of Eq. (A1) over whole population, we get

$$\begin{aligned} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) } =\sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } \varvec{+}O_{p}\mathbf {(}n^{\mathbf {-}\frac{1}{2}}\mathbf {)}. \end{aligned}$$

(A2)

Using assumptions (ii) and (iii) and multiplying both side of Eq. (A1) by survey weights and summing over whole sample, we get

$$\begin{aligned} \sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{+}O_{p}\mathbf {(}n^{\mathbf {-}\frac{1}{2}}\mathbf {)}. \end{aligned}$$

(A3)

Using assumption (i), we get

$$\begin{aligned} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}O_{p}\mathbf {(}n^{\mathbf {-}\frac{1}{2}}\mathbf {)}. \end{aligned}$$

(A4)

Subtracting Eq. (A3) from Eq. (A2) and using the Eq. (A4), we get

$$\begin{aligned} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) } -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}O_{p}\left( n^{\mathbf {-}\frac{1}{2}} \right) . \end{aligned}$$

(A5)

Now, since, both $\hat{B}_{N}=O_{p}\mathbf {(}1\mathbf {)}$ and $\hat{B}_{N}^{*}=O_{p}\mathbf {(}1\mathbf {)}$, using the results of the Eq. (A4) in Eq. (6), we can finally write

$$\begin{aligned} \hat{Y}_{MC,1}=\hat{Y}_{HT}\quad + \quad O_{p}\mathbf {(}n^{\mathbf {-}\frac{1}{2}}\mathbf {)}\quad \hbox {and} \quad \hat{Y}_{MC,2}=\hat{Y}_{HT} \quad + \quad O_{p}\left( n^{-\frac{1}{2}} \right) . \end{aligned}$$

(A6)

Since, $\hat{Y}_{HT}$ is a design-unbiased estimator for population total Y, hence, both$\hat{Y}_{MC,1}$ and $\hat{Y}_{MC,2}$ are also asymptotically design-unbiased.

Now, if $\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \varvec{\rightarrow \beta }\left( u_{i} \right)$, where $\varvec{\beta }\left( u_{i} \right)$ is the true superpopulation parameter, then

$$\begin{aligned} E_{\xi }\left( \hat{B}_{N} \right) =\frac{\sum \nolimits _{i\in S} {d_{i}q_{i}E_{\xi }\left( y_{i} \right) \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) } }{\sum \nolimits _{i\in S} {d_{i}q_{i}\left( \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \right) ^{2}} }\cong \frac{\sum \nolimits _{i\in S} {d_{i}q_{i}\left( \varvec{x}_{i}^{T}\varvec{\beta }\left( u_{i} \right) \right) ^{2}} }{\sum \nolimits _{i\in S} {d_{i}q_{i}\left( \varvec{x}_{i}^{T}\varvec{\beta }\left( u_{i} \right) \right) ^{2}} }=1 \end{aligned}$$

Similarly, $E_{\xi }\left( \hat{B}_{N}^{*} \right) =1$, where the expectation is taken over superpopulation model $\xi$.

Hence, using assumption (ii), we get

$$\begin{aligned} E_{\xi }\left( \hat{Y}_{MC,1}-Y \right)= & {} \sum \nolimits _{i\in S} {d_{i}E_{\xi }(} y_{i})+\left( \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) } \right. \\{} & {} \left. -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \right) -\sum \nolimits _{i\in U} E_{\xi } (y_{i})\\\cong & {} \sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\varvec{\beta }\left( u_{i} \right) + \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\varvec{\beta }\left( u_{i} \right) }\\{} & {} -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\varvec{\beta }\left( u_{i} \right) -\sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\varvec{\beta }\left( u_{i} \right) }\\= & {} \quad 0. \end{aligned}$$

Similarly, $E_{\xi }\left( \hat{Y}_{MC,2}-Y \right) \quad =$ 0.

Hence, both the estimators $\hat{Y}_{MC,1}$ and $\hat{Y}_{MC,2}$ are model-unbiased. Therefore, theorem 1 is proved. $\square$

Proof of Theorem 2

By using second-order Taylor series approximation to $\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right)$ at $\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right)$, we get

$$\begin{aligned} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right)= & {} \varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{+}\left\{ \left. \frac{\partial }{\partial \varvec{t}_{\varvec{i}}}\varvec{x}_{i}^{T}\varvec{t}_{\varvec{i}} \right| _{\varvec{t}_{\varvec{i}}\varvec{=}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } \right\} ^{T}\left[ \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] \nonumber \\{} & {} +\left[ \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] ^{T} \left\{ \left. \frac{\partial ^{2}}{\partial \varvec{t}_{\varvec{i}}\partial \varvec{t}_{\varvec{i}}\varvec{'}}\varvec{x}_{i}^{T}\varvec{t}_{\varvec{i}} \right| _{\varvec{t}_{\varvec{i}}\varvec{=}{\varvec{\beta }\left( u_{i} \right) }^{{*}}} \right\} \nonumber \\{} & {} \quad \left[ \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] \end{aligned}$$

(A7)

where ${\varvec{\beta }\left( u_{i} \right) }^{{*}}{\in } \{\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) ,\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \}$ or $\{\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) ,\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \}$.

Using assumption (iv) in above expression, we get

$$\begin{aligned} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) }= & {} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } +\left\{ \sum \nolimits _{i\in U} {\varvec{K}\left( \varvec{x}_{\varvec{i}}, \bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) } \right\} ^{\varvec{T}}\\{} & {} \times \left[ \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] \quad + \quad O_{p}\left( \frac{1}{n} \right) \quad \hbox {and}\\ \sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right)= & {} \varvec{ }\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{+}\left\{ \sum \nolimits _{i\in S} d_{i} \varvec{K}\left( \varvec{x}_{\varvec{i}}, \bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) \right\} ^{T}\\{} & {} \times \left( \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) \varvec{+}O_{p}\left( \frac{1}{n} \right) \end{aligned}$$

where $\varvec{K}\left( \varvec{x}_{\varvec{i}}, \bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) \varvec{=}\left. \frac{\partial }{\partial \varvec{t}_{i}}\varvec{x}_{i}^{T}\varvec{t}_{i} \right| _{\varvec{t}_{\varvec{i}}\varvec{=}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) }$

By taking difference of the above two equations, we get,

$$\begin{aligned}{} & {} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) } -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \nonumber \\{} & {} \quad =\left[ \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] \varvec{ +}\left\{ \sum \nolimits _{i\in U} {\varvec{K}\left( \varvec{x}_{\varvec{i}}, \bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) }\right. \nonumber \\{} & {} \quad \left. -\sum \nolimits _{i\in S} d_{i} \varvec{K}\left( \varvec{x}_{\varvec{i}}, \bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) \right\} ^{T}\times \left[ \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right] \varvec{+}O_{p}\left( \frac{1}{n} \right) \end{aligned}$$

(A8)

By assumption (ii), we get

$$\begin{aligned} \hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \mathbf {-}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{=}\textrm{ }O_{p}\left( n^{\mathbf {-}\frac{1}{2}} \right) . \end{aligned}$$

(A9)

By assumption (i), we get,

$$\begin{aligned} \sum \nolimits _{i\in S} d_{i} \varvec{K}\left( \varvec{x}_{\varvec{i}},\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) \mathbf {-}\sum \nolimits _{i\in U} {\varvec{K}\left( \varvec{x}_{\varvec{i}},\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \right) } =O_{p}\left( n^{\mathbf {-}\frac{1}{2}} \right) . \end{aligned}$$

(A10)

Thus, by using results of Eqs. (A9) and (A10) in Eq. (A8), we get

$$\begin{aligned}{} & {} \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) } -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\hat{\varvec{\beta }}^{\varvec{gwr}}\left( u_{i} \right) \\{} & {} \quad =\sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } -\sum \nolimits _{i\in S} d_{i} \varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) \varvec{+}O_{p}\left( \frac{1}{n} \right) . \end{aligned}$$

Following Wu and Sitter (2001), we can write $\hat{B}_{N}= B_{N}+o_{p}\mathbf {(}1\mathbf {)}$ and $\hat{B}_{N}^{*}= B_{N}^{*}+o_{p}\mathbf {(}1\mathbf {)}$.

Hence, the proposed estimators can be linearized as

$$\begin{aligned} \hat{Y}_{MC,1}= & {} \sum \nolimits _{i\in S} d_{i} Z_{i}+\left( \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } \right) B_{N}+o_{p}\left( n^{\mathbf {-}\frac{1}{2}} \right) \quad \hbox { and }\nonumber \\ \hat{Y}_{MC,2}= & {} \sum \nolimits _{i\in S} d_{i} Z_{i}^{'}+\left( \sum \nolimits _{i\in U} {\varvec{x}_{i}^{T}\bar{\varvec{\beta } }^{\varvec{gwr}}\left( u_{i} \right) } \right) B_{N}^{*}+o_{p}\left( n^{\mathbf {-}\frac{1}{2}} \right) \end{aligned}$$

(A11)

where $Z_{i}=y_{i}-\mu _{i}B_{N}$ and $Z_{i}^{'}=y_{i}-\mu _{i}B_{N}^{*}$.

Thus, following Särndal et al. (1992), the asymptotic design variances of $\hat{Y}_{MC,1}$ and $\hat{Y}_{MC,2}$ are expressed as

$$\begin{aligned} AV\left( \hat{Y}_{MC,1} \right)= & {} \sum \nolimits _{i=1}^N \sum \nolimits _{j>i}^N {\left( \pi _{i}\pi _{j}-\pi _{ij} \right) \left( \frac{Z_{i}}{\pi _{i}}-\frac{Z_{j}}{\pi _{j}} \right) ^{2}} \quad \hbox { and} \end{aligned}$$

(A12)

$$\begin{aligned} AV\left( \hat{Y}_{MC,2} \right)= & {} \sum \nolimits _{i=1}^N \sum \nolimits _{j>i}^N {\left( \pi _{i}\pi _{j}-\pi _{ij} \right) \left( \frac{Z_{i}^{'}}{\pi _{i}}-\frac{Z_{j}^{'}}{\pi _{j}} \right) ^{2}}. \end{aligned}$$

(A13)

Following Särndal et al. (1992), the estimates of the variances of the proposed model calibration estimators are expressed as

$$\begin{aligned} v\left( \hat{Y}_{MC,1} \right)= & {} \sum \nolimits _{i=1}^n \sum \nolimits _{j>i}^n {\left( \frac{\pi _{i}\pi _{j}-\pi _{ij}}{\pi _{ij}} \right) \left( \frac{z_{i}}{\pi _{i}}-\frac{z_{j}}{\pi _{j}} \right) ^{2}} \end{aligned}$$

(A14)

$$\begin{aligned} v\left( \hat{Y}_{MC,2} \right)= & {} \sum \nolimits _{i=1}^n \sum \nolimits _{j>i}^n {\left( \frac{\pi _{i}\pi _{j}-\pi _{ij}}{\pi _{ij}} \right) \left( \frac{z_{i}^{'}}{\pi _{i}}-\frac{z_{j}^{'}}{\pi _{j}} \right) ^{2}} \end{aligned}$$

(A15)

where $z_{i}=y_{i}-\hat{\mu }_{i}\hat{B}_{N}$, $z_{i}^{'}=y_{i}-\hat{\mu }_{i}\hat{B}_{N}^{*}$.

Hence, theorem 2 is proved. $\square$