Integration of data from probability surveys and big found data for finite population inference using mass imputation
Section 8. Discussion

• Previous
• Next

Mass imputation is an important technique for survey data integration. When the training dataset for imputation is obtained from a probability sample, the theory of Kim and Rao (2012) can be directly applied. If the training dataset is a non-probability sample and its size is huge, we have shown in this paper that various non-parametric methods can be used for mass imputation, and the estimation error in the imputation model can be safely ignored, under the assumption that the sampling mechanism for training data is missing at random in the sense of Rubin (1976). If the sampling mechanism is believed to be missing not at random, imputation techniques can be developed under the strong structural assumptions for the sampling mechanism (e.g., Riddles, Kim and Im, 2016; Morikawa and Kim, 2018) or the outcome model (e.g., Yang, Zeng and Wang, 2020). Also, when the training dataset has a hierarchical structure, multi-level models can be used to develop mass imputation. This is closely related to unit-level small area estimation in survey sampling (Rao and Molina, 2015).

The mass imputation estimator is not necessarily efficient. In Section 5, we have described a method of using calibration weighting as a tool for efficient data integration with big data. The calibration weighting requires correct matching between two data sources, as investigated by Kim and Tam (2020). Also, if the fraction of big data in the finite population is not substantial, the efficiency gain will be limited. Instead, one could improve the efficiency by combining the mass imputation estimator with the inverse propensity weighting estimator in the big data (Yang, Kim and Song, 2020). However, the correct specification of the propensity score model will be challenging. These are topics for future research.

Acknowledgements

We thank two anonymous referees and the associated editor for very constructive comments. Dr. Yang is partially supported by NSF grant DMS 1811245 and NIA grant 1R01AG066883. Dr. Kim is partially supported by NSF grant MMS 1733572 and the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa.

Appendix

A.1   Proof for Theorem 1

For a given $X_i\mathrm{<mo>=</mo>}x$ in Sample A, we show that $X_{i(1)}$ converges to $x$ in probability as $N_B\to \infty \mathrm{.}$ Consider for any $\epsilon \mathrm{<mo>></mo>}0,$ we show that

$$P\{d(X_{i(1)}\mathrm{,}x)\mathrm{<mo>></mo>}\epsilon \}\mathrm{<mo>=</mo>}P\{d(X_j\mathrm{,}x)\mathrm{<mo>></mo>}\epsilon \mathrm{,}\forall j\in B\}(\text{A}\text{.1})$$

converges to zero, and therefore $X_{i(1)}$ converges to $x$ in probability as $N_B\to \infty ,$ where the probability is induced by the sampling process of Sample B of size $N_B.$ We show this fact by contradiction. Assume that for some $\epsilon \mathrm{<mo>></mo>}0,$ $P\{d(X_{i(1)}\mathrm{,}x)\mathrm{<mo>></mo>}\epsilon \}$ does not coverage to zero as $N_B\to \infty .$ Define the region $R_{\epsilon }\mathrm{<mo>=</mo>}\left\{X\text{:}d\left(X\mathrm{,}x\right)\le \epsilon \right\}.$ Then, we must have $f\left(X|{\delta }_B\mathrm{<mo>=</mo>}1\right)\mathrm{<mo>=</mo>}0$ for $X\in R_{\epsilon };$ otherwise, there exists $\tilde{X}\in R_{\epsilon }$ with a positive probability in Sample B as $N_B\to \infty \mathrm{,}$ and therefore $P\{d(X_{i(1)}\mathrm{,}x)\mathrm{<mo>></mo>}\epsilon \}\mathrm{<mo>=</mo>}0$ as $N_B\to \infty .$ But the claim that $f\left(X|{\delta }_B\mathrm{<mo>=</mo>}1\right)\mathrm{<mo>=</mo>}0$ for $X\in R_{\epsilon }$ implies that $R_{\epsilon }$ is a non-overlap region of the distribution of $X$ between Sample A (and also the population) and Sample B, violating Assumption 2.

Given $X_i\mathrm{<mo>=</mo>}x$ in Sample A, for any continuous and bounded $g\left(y\right),$

$$\begin{array}{ll}E\{g(Y_{i\mathrm{<mo\; stretchy="false">(</mo>1<mo\; stretchy="false">)</mo>}})|X_i=x\mathrm{,}i\in A\}\hfill & \mathrm{<mo>=</mo>}E[E\{g(Y_{i(1)})|X_{i(1)}\mathrm{,}{X}_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A\}|X_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A]\hfill \\ \hfill & \mathrm{<mo>=</mo>}E[E\mathrm{<mo>\{</mo>}g(Y_{i(1)})|X_{i(1)}\mathrm{<mo>\}</mo>}|X_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A]\hfill \\ \hfill & \mathrm{<mo>=</mo>}E\{{\mu }_g(X_{i(1)})|X_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A\}\to E\{{\mu }_g(X_i)|X_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}E\{g(Y_i)|X_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A\}\mathrm{,}\hfill \end{array}$$

in probability as $N_B\to \infty ,$ where $\to$ follows from the fact that ${\mu }_g\left(\text{}x\text{}\right)$ is bounded and continuous. Then, by Portmanteau Lemma (Klenke, 2006), $Y_{i(1)}\to Y_i|(X_i\mathrm{<mo>=</mo>}x\mathrm{,}i\in A)$ in distribution as $N_B\to \infty .$ By Assumption 1, $g(Y_{i(1)})|(X_i\mathrm{,}i\in A)\to {\mu }_g(X_i)+e_g^{*}(X_i)$ in distribution as $N_B\to \infty ,$ where $e_g^{*}\left(X_i\right)$ has the same distribution as $\left\{g\left(Y_i\right)|\left(X_i\mathrm{,}i\in A\right)\right\}-{\mu }_g\left(X_i\right).$

We now show that for $i\ne j\in A,$ $e_g^{*}\left(X_i\right)$ and $e_g^{*}\left(X_j\right)$ are conditionally independent, given data $O_A$ . It is sufficient to show that $P\left\{i\left(\text{}1\text{}\right)\mathrm{<mo>=</mo>}j\left(\text{}1\text{}\right)\right\}\to 0$ as $N_B\to \infty ;$ in other words, the same unit can not be matched for unit $i$ and unit $j$ with probability 1. This can be shown using (A.1) with $\epsilon \mathrm{<mo>=</mo>}{{\displaystyle \min }}_{i\ne j\in A}\Vert X_i-X_j\Vert .$

Therefore, conditional on data $O_A,$ we have

$${\widehat{\mu }}_{g\mathrm{,}\text{nni}}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}g(Y_{i(1)})}\to \frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}g(Y_i)}\mathrm{<mo>=</mo>}{\widehat{\mu }}_{g\mathrm{,}\text{HT}}$$

in distribution as $N_B\to \infty .$ This completes the proof for Theorem 1.

Let

$${\tilde{V}}_{\text{nni}}\mathrm{<mo>=</mo>}\frac{n}{N^2}{\displaystyle \sum _{i\in A}{\displaystyle \sum _{j\in A}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\frac{g\left(Y_i\right)}{{\pi }_i}\frac{g\left(Y_j\right)}{{\pi }_j}}}\mathrm{.}(\text{A}\text{.2})$$

Then, ${\tilde{V}}_{\text{nni}}$ is consistent for $V_{\text{nni}}.$

Similar to the above argument, for $i\mathrm{,}j\in A,$ conditional on data $O_A,$ $g(Y_{i(1)})g(Y_{j(1)})\to g(Y_i)g(Y_j)$ as $N_B\to \infty .$ Therefore, conditional on data $O_A,$

$${\widehat{V}}_{\text{nni}}\mathrm{<mo>=</mo>}\frac{n}{N^2}{\displaystyle \sum _{i\in A}{\displaystyle \sum _{j\in A}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\frac{g(Y_{i(1)})}{{\pi }_i}\frac{g(Y_{j(1)})}{{\pi }_j}}}\to {\tilde{V}}_{\text{nni}}\mathrm{,}(\text{A}\text{.3})$$

in distribution as $N_B\to \infty .$ Combining (A.2) and (A.3), ${\widehat{V}}_{\text{nni}}$ is consistent for $V_{\text{nni}}.$

A.2   Proof for Theorem 2

To investigate the asymptotic properties of ${\widehat{\mu }}_{g\mathrm{,}\text{knn}},$ we re-express

$${\widehat{\mu }}_g\left(\text{}x\text{}\right)\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{j\in B}{K}_{R_x}\left(x-X_j\right)g\left(Y_j\right)}}{{\displaystyle {\sum }_{j\in B}{K}_{R_x}\left(x-X_j\right)}}\mathrm{,}$$

where

$$K_h\left(\text{}u\text{}\right)\mathrm{<mo>=</mo>}\frac{1}{h^p}K\left(\frac{u}{h}\right)\mathrm{,}K\left(\text{}u\text{}\right)\mathrm{<mo>=</mo>}\text{0}\text{.5}I\left(\Vert u\Vert \le 1\right)\mathrm{,}$$

and the bandwidth $h\mathrm{<mo>=</mo>}{R}_x$ is the random distance between $x$ and its furthest among the $k$ nearest neighbors. Therefore, ${\widehat{\mu }}_{g\mathrm{,}\text{knn}}$ can be viewed as a kernel estimator incorporating a data-driven bandwidth.

In the literature, asymptotic properties of the $k$ nearest neighbor imputation estimator have been studied extensively. The result shown in the following lemma on $k$ nearest neighbor imputation is extracted from Mack (1981).

Lemma 1. Under Assumptions 1-3,

$$N^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N{\delta }_{B\mathrm{,}j}{K}_{R_x}\left(x-X_j\right)g\left(Y_j\right)}\mathrm{<mo>=</mo>}f\left(\text{}x\text{}\right){\pi }_B\left(\text{}x\text{}\right){\mu }_g\left(\text{}x\text{}\right)+O_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\mathrm{.}(\text{A}\text{.4})$$

We now express

$${\widehat{\mu }}_{g\mathrm{,}\text{knn}}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\delta }_{A\mathrm{,}i}{\mu }_g\left(X_i\right)}+\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\delta }_{A\mathrm{,}i}\left\{{\widehat{\mu }}_g\left(X_i\right)-{\mu }_g\left(X_i\right)\right\}}\mathrm{.}$$

Let $T_N\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\delta }_{A\mathrm{,}i}\left\{{\widehat{\mu }}_g\left(X_i\right)-{\mu }_g\left(X_i\right)\right\}}.$ To study the properties for $T_N,$ we first look at ${\widehat{\mu }}_g\left(\text{}x\text{}\right),$ which can be expressed as

$${\widehat{\mu }}_g\left(\text{}x\text{}\right)\mathrm{<mo>=</mo>}\frac{h_N(x)}{f_N(x)}\mathrm{,}$$

where $h_N\left(\text{}x\text{}\right)\equiv N^{-1}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^N{\delta }_{B\mathrm{,}j}{K}_{R_x}\left(x-X_j\right)g\left(Y_j\right)}$ and $f_N\left(\text{}x\text{}\right)\equiv N^{-1}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^N{\delta }_{B\mathrm{,}j}{K}_{R_x}\left(x-X_j\right)}.$ By the result in Lemma 1, we obtain

$$h_N\left(\text{}x\text{}\right)\mathrm{<mo>=</mo>}f\left(\text{}x\text{}\right){\pi }_B\left(\text{}x\text{}\right){\mu }_g\left(\text{}x\text{}\right)+O_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}$$

$$f_N\left(\text{}x\text{}\right)\mathrm{<mo>=</mo>}f\left(\text{}x\text{}\right){\pi }_B\left(\text{}x\text{}\right)+O_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\mathrm{.}$$

Now, by a Taylor expansion, we obtain

$$\begin{array}{ll}{\widehat{\mu }}_g(x)-{\mu }_g(x)\hfill & \mathrm{<mo>=</mo>}\frac{h_N(x)}{f_N(x)}-{\mu }_g(x)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{1}{f(x){\pi }_B(x)}\{h_N(x)-f(x){\pi }_B(x){\mu }_g(x)\}\hfill \\ \hfill & -\frac{f(x){\pi }_B(x){\mu }_g(x)}{{\left\{f(x){\pi }_B(x)\right\}}^2}\{f_N(x)-f(x){\pi }_B(x)\}+O_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{1}{f(x){\pi }_B(x)}\{h_N(x)-f_N(x){\mu }_g(x)\}+O_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\mathrm{.}\hfill \end{array}$$

Therefore, we obtain

$$T_N\mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\frac{{\delta }_{A\mathrm{,}i}}{{\pi }_i}\frac{1}{f\left(X_i\right){\pi }_B\left(X_i\right)}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N{\delta }_{B\mathrm{,}j}{K}_{R_{X_i}}\left(X_i-X_j\right)\left\{g\left(Y_j\right)-{\mu }_g\left(X_i\right)\right\}+O_{\text{p}}}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\mathrm{.}$$

Under the assumption in Theorem 2, it is easy to derive that ${\left(k/N\right)}^{2/p}+1/k\mathrm{<mo>=</mo>}o\left(\text{}{n}^{-1/2}\text{}\right)\mathrm{,}$ and therefore,

$$T_N\mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\frac{{\delta }_{A\mathrm{,}i}}{{\pi }_i}\frac{1}{f\left(X_i\right){\pi }_B\left(X_i\right)}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N{\delta }_{B\mathrm{,}j}{K}_{R_{X_i}}\left(X_i-X_j\right)\left\{g\left(Y_j\right)-{\mu }_g\left(X_i\right)\right\}+o_{\text{p}}\left(\text{}{n}^{-1/2}\text{}\right)}}\mathrm{.}$$

We then express $T_N$ in a form of U-statistics (van der Vaart, 2000; Chapter 12):

$$T_N\mathrm{<mo>=</mo>}\frac{1}{N\left(N-1\right)}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\displaystyle \sum _{j\ne i}h\left(Z_i\mathrm{,}{Z}_j\right)+o_{\text{p}}\left(\text{}{n}^{-1/2}\text{}\right)}}\mathrm{,}$$

where $Z_i\mathrm{<mo>=</mo>}\left(X_i\mathrm{,}{Y}_i\mathrm{,}{\delta }_{A\mathrm{,}i}\mathrm{,}{\delta }_{B\mathrm{,}i}\right)$ and

$$\begin{array}{ll}h\left(Z_i\mathrm{,}{Z}_j\right)\hfill & \mathrm{<mo>=</mo>}\frac{1}{2}[\frac{{\delta }_{A\mathrm{,}i}{\delta }_{B\mathrm{,}j}}{{\pi }_i}\frac{1}{f\left(X_i\right){\pi }_B\left(X_i\right)}{K}_{R_{X_i}}\left(X_i-X_j\right)\left\{g\left(Y_j\right)-{\mu }_g\left(X_i\right)\right\}\hfill \\ \hfill & +\frac{{\delta }_{A\mathrm{,}j}{\delta }_{B\mathrm{,}i}}{{\pi }_j}\frac{1}{f\left(X_j\right){\pi }_B\left(X_j\right)}{K}_{R_{X_j}}\left(X_j-X_i\right)\left\{g\left(Y_i\right)-{\mu }_g\left(X_j\right)\right\}]\hfill \\ \hfill & \equiv \frac{1}{2}\left({\zeta }_{ij}+{\zeta }_{ji}\right)\mathrm{.}\hfill \end{array}$$

Now, by Lemma 1, we obtain

$$\begin{array}{ll}E\left({\zeta }_{ij}|Z_i\right)\hfill & \mathrm{<mo>=</mo>}E\left[\frac{{\delta }_{A\mathrm{,}i}{\delta }_{B\mathrm{,}j}}{{\pi }_i}\frac{1}{f\left(X_i\right){\pi }_B\left(X_i\right)}{K}_{R_{X_i}}\left(X_i-X_j\right)\left\{g\left(Y_j\right)-{\mu }_g\left(X_i\right)\right\}|Z_i\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{{\delta }_{A\mathrm{,}i}}{{\pi }_i}\frac{1}{f\left(X_i\right){\pi }_B\left(X_i\right)}E\left[{\delta }_{B\mathrm{,}j}{K}_{R_{X_i}}\left(X_i-X_j\right)\left\{g\left(Y_j\right)-{\mu }_g\left(X_i\right)\right\}|Z_i\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}{O}_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\mathrm{,}\hfill \end{array}$$

and

$$\begin{array}{ll}E\left({\zeta }_{ji}|Z_i\right)\hfill & \mathrm{<mo>=</mo>}E\left[\frac{{\delta }_{A\mathrm{,}j}{\delta }_{B\mathrm{,}i}}{{\pi }_j}\frac{1}{f\left(X_j\right){\pi }_B\left(X_j\right)}{K}_{R_{X_j}}\left(X_j-X_i\right)\left\{g\left(Y_i\right)-{\mu }_g\left(X_j\right)\right\}|Z_i\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\delta }_{B\mathrm{,}i}E\left(E\left[\frac{{\delta }_{A\mathrm{,}j}}{{\pi }_j}\frac{1}{f\left(X_j\right){\pi }_B\left(X_j\right)}{K}_{R_{X_j}}\left(X_j-X_i\right)\left\{g\left(Y_i\right)-{\mu }_g\left(X_j\right)\right\}|R_{X_j}\text{}\mathrm{,}{Z}_i\right]|Z_i\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{{\delta }_{B\mathrm{,}i}}{{\pi }_B\left(X_i\right)}\left\{g\left(Y_i\right)-{\mu }_g\left(X_i\right)\right\}+O_{\text{p}}\left\{{\left(\frac{k}{N}\right)}^{2/p}+\frac{1}{k}\right\}\mathrm{.}\hfill \end{array}$$

Therefore, by the theory of U-statistics, we obtain

$$\begin{array}{ll}{T}_N\hfill & \mathrm{<mo>=</mo>}\frac{2}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{N}E\left\{h\left(Z_i\mathrm{,}{Z}_j\right)|Z_i\right\}+o_{\text{p}}\left(\text{}{n}^{-1/2}\text{}\right)}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\frac{{\delta }_{B\mathrm{,}i}}{{\pi }_B\left(X_i\right)}\left\{g\left(Y_i\right)-{\mu }_g\left(X_i\right)\right\}+o_{\text{p}}\left(\text{}{n}^{-1/2}\text{}\right)}\mathrm{.}\hfill \end{array}$$

Combining the above results leads to

$$\begin{array}{ll}{\widehat{\mu }}_{g\mathrm{,}\text{knn}}-{\mu }_g\hfill & \mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\left\{{\pi }_i^{-1}{\delta }_{A\mathrm{,}i}{\mu }_g\left(X_i\right)-{\mu }_g\left(X_i\right)\right\}}\hfill \\ \hfill & +\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\left\{\frac{{\delta }_{B\mathrm{,}i}}{{\pi }_B\left(X_i\right)}-1\right\}\left\{g\left(Y_i\right)-{\mu }_g\left(X_i\right)\right\}+o_{\text{p}}\left(\text{}{n}^{-1/2}\text{}\right)}\mathrm{.}(\text{A}\text{.5})\hfill \end{array}$$

Then, the asymptotic results in Theorem 2 follow by Assumptions 1-4 and (A.5).

A.3   Proof for Theorem 3

The consistency and asymptotic normality of $n^{1/2}{\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ follow by the standard arguments under Assumptions 1-4. The remaining is to show that the asymptotic variance of $n^{1/2}{\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ is $V_{\text{nni}}\mathrm{.}$

Using the distance function $G\left({\omega }_i\mathrm{,}{d}_i\right)\mathrm{<mo>=</mo>}{d}_i{\left({\omega }_i/d_i-1\right)}^2$ in (5.1), the minimum distance estimation leads to generalized regression estimation (Park and Fuller, 2012). Therefore, we express

$$\begin{array}{ll}{n}^{1/2}{\widehat{\mu }}_g\hfill & \mathrm{<mo>=</mo>}\frac{n^{1/2}}{N}{\displaystyle \sum _{i\in A}{\omega }_{i}g(Y_{i(1)})}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{n^{1/2}}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}g(Y_{i(1)})}-\frac{n^{1/2}}{N}\left({\displaystyle \sum _{i\in A}{\pi }_i^{-1}{h}_i^{*\text{T}}{\beta }_N}-{\displaystyle \sum _{i\mathrm{<mo>=</mo>1}}^N{h}_i^{*\text{T}}{\beta }_N}\right)+o_{\text{p}}(n^{-1/2})\mathrm{.}(\text{A}\text{.6})\hfill \end{array}$$

Similar to the argument in the proof for Theorem 1, we express

$$\begin{array}{ll}{n}^{1/2}{\widehat{\mu }}_g\hfill & \mathrm{<mo>=</mo>}\frac{n^{1/2}}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}g(Y_{i(1)})}-\frac{n^{1/2}}{N}\left({\displaystyle \sum _{i\in A}{\pi }_i^{-1}{h}_i^{*\text{T}}{\beta }_N}-{\displaystyle \sum _{i\mathrm{<mo>=</mo>1}}^N{h}_i^{*\text{T}}{\beta }_N}\right)+o_{\text{p}}(n^{-1/2})\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{n^{1/2}}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}\{g(Y_{i(1)})-h_i^{*\text{T}}{\beta }_N\}}+\frac{n^{1/2}}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>1}}^N{h}_i^{*\text{T}}{\beta }_N+o_{\text{p}}(n^{-1/2})}\mathrm{.}(\text{A}\text{.7})\hfill \end{array}$$

It is straightforward to show the variance of the second term in (A.7) is negligible given $n{N}^{-1}\mathrm{<mo>=</mo>}o\left(\text{}1\text{}\right).$ Following the arguments in the proof for Theorems 1 and 2, $g\left(Y_{i(1)}\right)$ and $h_i^{*}$ have the asymptotic distribution as $g\left(Y_i\right)$ and $h_i$ given the data $O_A$ from Sample A, respectively. Therefore, the asymptotic variance of $n^{1/2}{\widehat{\mu }}_g$ is

$$V_{\text{RC}}\mathrm{<mo>=</mo>}\underset{n\to \infty }{{\displaystyle \lim }}\text{var}\left[\frac{n^{1/2}}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}\left\{g\left(Y_i\right)-h_i^{\text{T}}{\beta }_N\right\}}\right]\mathrm{.}$$

References

Abadie, A., and Imbens, G.W. (2006). Large sample properties of matching estimators for average treatment effects. Econometrica, 74, 235-267.

Baker, R., Brick, J.M., Bates, N.A., Battaglia, M., Couper, M.P., Dever, J.A., Gile, K.J. and Tourangeau, R. (2013). Summary report of the AAPOR task force on non-probability sampling. Journal of Survey Statistics and Methodology, 1, 90-143.

Beaumont, J.-F. (2020). Are probability surveys bound to disappear for the production of official statistics? Survey Methodology, 46, 1, 1-28. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/2020001/article/00001-eng.pdf.

Belloni, A., Chernozhukov, V., Chetverikov, D. and Kato, K. (2015). Some new asymptotic theory for least squares series: Pointwise and uniform results. Journal of Econometrics, 186, 345-366.

Beretta, L., and Santaniello, A. (2016). Nearest neighbor imputation algorithms: A critical evaluation. BMC Medical Informatics and Decision Making, 16, 198-208.

Bethlehem, J. (2016). Solving the nonresponse problem with sample matching? Social Science Computer Review, 34, 59-77.

Breidt, F.J., and Opsomer, J.D. (2017). Model-assisted survey estimation with modern prediction techniques. Statistical Science, 32, 190-205.

Breidt, F.J., McVey, A. and Fuller, W.A. (1996). Two-phase estimation by imputation. Journal of the Indian Society of Agricultural Statistics, 49, 79-90.

Breiman, L. (2001). Random forests. Machine Learning, 45, 5-32.

Brookhart, M.A., Schneeweiss, S., Rothman, K.J., Glynn, R.J., Avorn, J. and Stürmer, T. (2006). Variable selection for propensity score models. American Journal of Epidemiology, 163, 1149-1156.

Buchanan, A.L., Hudgens, M.G., Cole, S.R., Mollan, K.R., Sax, P.E., Daar, E.S., Adimora, A.A., Eron, J.J. and Mugavero, M.J. (2018). Generalizing evidence from randomized trials using inverse probability of sampling weights. Journal of the Royal Statistical Society, Series A, (Statistics in Society), 181, 1193-1209.

Buelens, B., Burger, J. and van den Brakel, J.A. (2018). Comparing inference methods for nonprobability samples. International Statistical Review, 86, 322-343.

Chen, Y., Li, P. and Wu, C. (2020). Doubly robust inference with nonprobability survey samples. Journal of the American Statistical Association, 115, 2011-2021.

Cheng, P.E. (1994). Nonparametric estimation of mean functionals with data missing at random. Journal of American Statistical Association, 89, 81-87.

Chipperfield, J., Chessman, J. and Lim, R. (2012). Combining household surveys using mass imputation to estimate population totals. Australian & New Zealand Journal of Statistics, 54, 223-238.

Citro, C.F. (2014). From multiple modes for surveys to multiple data sources for estimates. Survey Methodology, 40, 2, 137-161. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2014002/article/14128-eng.pdf.

Cochran, W.G. (2007). Sampling Techniques, New York: John Wiley & Sons, Inc.

Couper, M.P. (2013). Is the sky falling? new technology, changing media, and the future of surveys. Survey Research Methods, 7, 145-156.

Deville, J.-C., and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376-382.

DiSogra, C., Cobb, C., Chan, E. and Dennis, J.M. (2011). Calibrating non-probability internet samples with probability samples using early adopter characteristics. Joint Statistical Meetings (JSM), Survey Research Methods, 4501-4515.

Dong, L., Yang, S., Wang, X., Zeng, D. and Cai, J. (2020). Integrative analysis of randomized clinical trials with real world evidence studies, arXiv preprint arXiv:2003.01242.

Eilers, P.H., and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11, 89-102.

Elliott, M.R., and Valliant, R. (2017). Inference for nonprobability samples. Statistical Science, 32, 249-264.

Fuller, W.A. (2009). Sampling Statistics, Wiley, Hoboken.

Hastie, T., and Tibshirani, R. (1990). Generalized Additive Models, NY: Chapman and Hall, Inc.

Imbens, G., Johnson, P. and Spady, R.H. (1998). Information theoretic approaches to inference in moment condition models. Econometrica, 66, 333-357.

Kalton, G., and Kish, L. (1984). Some efficient random imputation methods. Communications in Statistics-Theory and Methods, 13, 1919-1939.

Kang, J.D., and Schafer, J.L. (2007). Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical Science, 22, 523-539.

Keiding, N., and Louis, T.A. (2016). Perils and potentials of self-selected entry to epidemiological studies and surveys. Journal of the Royal Statistical Society, Series A (Statistics in Society), 179, 319-376.

Kim, J.K., and Fuller, W.A. (2004). Fractional hot deck imputation. Biometrika, 91, 559-578.

Kim, J.K., and Rao, J.N.K. (2012). Combining data from two independent surveys: A model-assisted approach. Biometrika, 99, 85-100.

Kim, J.K., and Tam, S.M. (2020). Data integration by combining big data and survey sample data for finite population inference. International Statistical Review. Accepted (Available at https://doi.org/10.1111/insr.12434).

Kim, J.K., and Wang, Z. (2019). Sampling techniques for big data analysis. International Statistical Review, 87, 177-191.

Kitamura, Y., and Stutzer, M. (1997). An information-theoretic alternative to generalized method of moments estimation. Econometrica, 65, 861-874.

Klenke, A. (2006). Probability Theory, Springer-Verlag: Heidelberg.

Kott, P.S. (2006). Using calibration weighting to adjust for nonresponse and coverage errors. Survey Methodology, 32, 2, 133-142. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/2006002/article/9547-eng.pdf.

Kweon, S., Kim, Y., jin Jang, M., Kim, Y., Kim, K., Choi, S., Chun, C., Khang, Y.-H. and Oh, K. (2014). Data resource profile: The Korea National Health and Nutrition Examination Survey (KNHANES). International Journal of Epidemiology, 43, 69-77.

Lee, S., and Valliant, R. (2009). Estimation for volunteer panel web surveys using propensity score adjustment and calibration adjustment. Sociological Methods & Research, 37, 319-343.

Lohr, S.L., and Raghunathan, T.E. (2017). Combining survey data with other data sources. Statistical Science, 32, 293-312.

Mack, Y.-P. (1981). Local properties of k-NN regression estimates. SIAM Journal on Algebraic Discrete Methods, 2, 311-323.

Mack, Y., and Rosenblatt, M. (1979). Multivariate k-nearest neighbor density estimates. Journal of Multivariate Analysis, 9, 1-15.

McCullagh, P. (1984). Generalized linear models. European Journal of Operational Research, 16, 285-292.

McRoberts, R.E., Tomppo, E.O. and Næsset, E. (2010). Advances and emerging issues in national forest inventories. Scandinavian Journal of Forest Research, 25, 368-381.

Morikawa, K., and Kim, J.K. (2018). A note on the equivalence of two semiparametric estimation methods for nonignorable nonresponse. Statistics & Probability Letters, 140, 1-6.

Mulry, M.H., Oliver, B.E. and Kaputa, S.J. (2014). Detecting and treating verified influential values in a Monthly Retail Trade Survey. Journal of Official Statistics, 30, 721-747.

National Health Insurance Data Sharing Service (2014). National health screening data. https://nhiss.nhis.or.kr/bd/ab/bdabf006cv.do, [Accessed: 2019-07-11].

Nelder, J.A., and Baker, R.J. (1972). Generalized Linear Models, New York: John Wiley & Sons, Inc.

Newey, W.K., and Smith, R.J. (2004). Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica, 72, 219-255.

O’Muircheartaigh, C., and Hedges, L.V. (2014). Generalizing from unrepresentative experiments: A stratified propensity score approach. Journal of the Royal Statistical Society, Series C (Applied Statistics), 63, 195-210.

Palmer, J.R., Espenshade, T.J., Bartumeus, F., Chung, C.Y., Ozgencil, N.E. and Li, K. (2013). New approaches to human mobility: Using mobile phones for demographic research. Demography, 50, 1105-1128.

Park, M., and Fuller, W.A. (2012). Generalized regression estimators. Encyclopedia of Environmetrics, 2, 1162-1166.

Pfeffermann, D., Eltinge, J.L. and Brown, L.D. (2015). Methodological issues and challenges in the production of official statistics: 24^th Annual Morris Hansen Lecture. Journal of Survey Statistics and Methodology, 3, 425-483.

Rao, J.N.K. (2020). On making valid inferences by integrating data from surveys and other sources. Sankhyā B. pp. DOI 10.1007/s13571-020-00227-w.

Rao, J.N.K., and Molina, I. (2015). Small Area Estimation, New York: John Wiley & sons, Inc.

Riddles, M.K., Kim, J.K. and Im, J. (2016). A propensity-score-adjustment method for nonignorable nonresponse. Journal of Survey Statistics and Methodology, 4, 215-245.

Rivers, D. (2007). Sampling for web surveys. Joint Statistical Meetings.

Rosenbaum, P.R., and Rubin, D.B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70, 41-55.

Rubin, D.B. (1976). Inference and missing data. Biometrika, 63, 581-592.

Ruppert, D., Wand, M.P. and Carroll, R.J. (2009). Semiparametric regression during 2003-2007. Electronic Journal of Statistics, 3, 1193-1256.

Särndal, C.-E., Swensson, B. and Wretman, J. (2003). Model-Assisted Survey Sampling. Springer Science & Business Media, New York: Springer-Verlag.

Schennach, S.M. (2007). Point estimation with exponentially tilted empirical likelihood. Annals of Statistics, 35, 634-672.

Stuart, E.A., Bradshaw, C.P. and Leaf, P.J. (2015). Assessing the generalizability of randomized trial results to target populations. Prevention Science, 16, 475-485.

Stuart, E.A., Cole, S.R., Bradshaw, C.P. and Leaf, P.J. (2011). The use of propensity scores to assess the generalizability of results from randomized trials. Journal of the Royal Statistical Society, Series A (Statistics in Society), 174, 369-386.

Tam, S.-M., and Clarke, F. (2015). Big data, official statistics and some initiatives by the Australian Bureau of Statistics. International Statistical Review, 83, 436-448.

Tam, S.-M., and Kim, J.-K. (2018). Big data ethics and selection-bias: An official statistician’s perspective. Statistical Journal of the IAOS, 34, 577-588.

Tourangeau, R., Conrad, F.G. and Couper, M.P. (2013). The Science of Web Surveys, New York: Oxford University Press.

Valliant, R., and Dever, J.A. (2011). Estimating propensity adjustments for volunteer web surveys. Sociological Methods & Research, 40, 105-137.

van der Vaart, A.W. (2000). Asymptotic Statistics, Cambridge University Press, Cambridge, MA.

Vavreck, L., and Rivers, D. (2008). The 2006 cooperative congressional election study. Journal of Elections, Public Opinion and Parties, 18, 355-366.

Wood, S.N. (2006). Generalized Additive Models: An Introduction with R, Chapman and Hall/CRC.

Yang, S., and Ding, P. (2020). Combining multiple observational data sources to estimate causal effects. Journal of the American Statistical Association, 115, 1540-1554.

Yang, S., and Kim, J.K. (2020). Statistical data integration in survey sampling: A review. Japanese Journal of Statistics and Data Science, 3, 625-650.

Yang, S., Kim, J.K. and Song, R. (2020). Doubly robust inference when combining probability and non-probability samples with high-dimensional data. Journal of the Royal Statistical Society, Series B, (Statistical Methodology), 82, 445-465.

Yang, S., Zeng, D. and Wang, X. (2020). Improved inference for heterogeneous treatment effects using real-world data subject to hidden confounding, arXiv preprint arXiv:2007.12922.

• Previous
• Article main page
• Next