Integration of data from probability surveys and big found data for finite population inference using mass imputation
Section 5. Regression calibration

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In practice, especially for government agencies, one nearest neighbor may be preferred because of its simplicity in implementation and data storage. We now consider another strategy to improve the efficiency for ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ when additionally the membership to Sample B can be determined throughout Sample A with the indicator ${\delta }_B.$ In some situation, we can obtain ${\delta }_B$ by matching the membership to Sample B (i.e., data linkage). We focus on the ideal setting without linkage errors. The key insight is that the subsample of units in Sample A with ${\delta }_B\mathrm{<mo>=</mo>}1$ constitutes a second-phase sample from Sample B, where Sample B acts as a new population. Standard regression calibration requires all calibration variables to be observed in Sample A and Sample B, and thus rules out the possibility of using $Y$ as the calibration variable due to lack of the outcome data from Sample B. One of the advantages of mass imputations is that we can leverage the imputed outcomes to facilitate calibration of $Y.$

Let $h\left({\delta }_B\mathrm{,}X\mathrm{,}Y\right)$ be a multi-dimensional function of ${\delta }_B,$ ${\delta }_{B}X$ and ${\delta }_{B}Y,$ e.g., $h\left({\delta }_B\mathrm{,}X\mathrm{,}Y\right)\mathrm{<mo>=</mo>}{\left({\delta }_B\mathrm{,}1-{\delta }_B\mathrm{,}{\delta }_{B}X\mathrm{,}{\delta }_{B}Y\right)}^{\text{T}}.$ For simplicity of notation, we use $h_i$ to denote $h\left({\delta }_{Bi}\mathrm{,}{X}_i\mathrm{,}{Y}_i\right)$ and $h_i^{*}$ to denote $h({\delta }_{Bi}\mathrm{,}{X}_i\mathrm{,}{Y}_{i(1)}).$ We can calculate the population quantity $H\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{h}_i}$ from Sample B. This insight enables the typical calibration weighting in survey sampling with known marginal totals. In Sample A, we treat the imputed values as observed values, and the design weighted estimator of $H$ is ${\widehat{H}}_A\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\in A}{\pi }_i^{-1}{h}_i^{*}}\mathrm{.}$ In general, ${\widehat{H}}_A$ is not equal to $H.$ We can use the known information $H$ to improve the efficiency of ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}.$

This suggests the following calibration strategy. We modify the original design weights $\left\{d_i\text{:}i\in A\right\}$ in ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ to a new set of weights $\left\{{\omega }_i\text{:}i\in A\right\}$ by minimizing a distance function

$${\displaystyle \sum _{i\in A}G\left({\omega }_i\mathrm{,}{d}_i\right)}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in A}{d}_i{\left(\frac{{\omega }_i}{d_i}-1\right)}^2}\mathrm{,}(5.1)$$

subject to the calibration constraints $N^{-1}{\displaystyle {\sum }_{i\in A}{\omega }_i{h}_i^{*}}\mathrm{<mo>=</mo>}H\mathrm{.}$ By Lagrange multiplier, the solution to the constraint minimization problem is

$${\omega }_i\mathrm{<mo>=</mo>}{d}_i+{\left(N\times H-{\displaystyle \sum _{k\in A}{d}_k{h}_k^{*}}\right)}^{\text{T}}{\left({\displaystyle \sum _{k\in A}{d}_k{h}_k^{*}{h}_k^{*\text{T}}}\right)}^{-1}{d}_i{h}_i^{*}\mathrm{,}$$

for $i\in A.$ The resulting weights $\left\{{\omega }_i\text{:}i\in A\right\}$ can be called generalized regression weights.

The proposed estimator utilizing the new set of weights is

$${\widehat{\mu }}_{g\mathrm{,}\text{RC}}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}{\omega }_{i}g(Y_{i(1)})}\mathrm{,}(5.2)$$

which is asymptotically equivalent to a generalized regression estimator (Park and Fuller, 2012). Following Yang and Ding (2020), one can show that ${\widehat{\mu }}_{g\mathrm{,}\text{RC}}$ is the optimal estimator among the class of $\{{\widehat{\mu }}_{g\mathrm{,}\text{nni}}+{\left(N\times H-{\displaystyle {\sum }_{k\in A}{d}_k{h}_k^{*}}\right)}^{\text{T}}\gamma \text{:}\gamma \in R^{\text{dim(}h\text{)}}\}.$

We derive the asymptotic theory for ${\widehat{\mu }}_{g\mathrm{,}\text{RC}}$ in the following theorem and defer its proof to the Supplementary Material.

Theorem 3. Under Assumptions 1-4,

$$n^{1/2}\left({\widehat{\mu }}_{g\mathrm{,}\text{RC}}-{\mu }_g\right)\to N\left(\mathrm{0,}{V}_{\text{RC}}\right)\mathrm{,}(5.3)$$

in distribution, as $n\to \infty ,$  where

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

and ${\beta }_N\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{h}_i{h}_i^{\text{T}}}\right)}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{h}_{i}g\left(Y_i\right)}.$

The calibrated estimator ${\widehat{\mu }}_{g\mathrm{,}\text{RC}}$ improves the efficiency of ${\widehat{\mu }}_{g\mathrm{,}\text{nni}}$ in the sense that $V_{\text{RC}}$ is at most as large as $V_{\text{nni}}$ given in Theorem 1. If $h_i$ explains a proportion of the variability of $g\left(Y_i\right),$ $V_{\text{RC}}$ is strictly less than $V_{\text{nni}}$ and the efficiency gain does not require any parametric model assumption.

Remark 2 (Choice of distance functions). Different distance functions in (5.1) can be considered. If we choose $G\left({\omega }_i\mathrm{,}{d}_i\right)\mathrm{<mo>=</mo>}-d_i\log \left({\omega }_i/d_i\right),$  it leads to empirical likelihood estimation (Newey and Smith, 2004). If we choose the Kullback-Leibler distance function $G\left({\omega }_i\mathrm{,}{d}_i\right)\mathrm{<mo>=</mo>}{\omega }_i\log \left(d_i/{\omega }_i\right),$  it leads to exponential tilting estimation (Kitamura and Stutzer, 1997; Imbens, Johnson and Spady, 1998; Schennach, 2007; Dong et al., 2020). Under mild conditions, these procedures provide a set of weights that is asymptotically equivalent to the set of regression weights (Deville and Särndal, 1992; Breidt and Opsomer, 2017).

For variance estimation, by Theorem (3), we construct a consistent variance estimator for ${\widehat{\mu }}_{g\mathrm{,}\text{RC}}$ as ${\widehat{V}}_{\text{RC}}/n,$ where

$${\widehat{V}}_{\text{RC}}\mathrm{<mo>=</mo>}\frac{n}{N^2}{\displaystyle \sum _{i\in A}{\displaystyle \sum _{j\in A}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_{ij}}\frac{{\widehat{e}}_i}{{\pi }_i}\frac{{\widehat{e}}_j}{{\pi }_j}}}\mathrm{,}$$

with ${\widehat{e}}_i\mathrm{<mo>=</mo>}g(Y_{i(1)})-h_i^{*\text{T}}\widehat{\beta },$

$$\widehat{\beta }\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>1}}^N{h}_i^{*}{h}_i^{*\text{T}}}\right)}^{-1}\left(\begin{array}{l}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^N{\delta }_{Bi}g(Y_i)}\hfill \\ {\displaystyle {\sum }_{i\in A}{\pi }_i^{-1}(1-{\delta }_{Bi})g(Y_{i(1)})}\hfill \\ {\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^N{\delta }_{Bi}{X}_{i}g(Y_i)}\hfill \\ {\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^N{\delta }_{Bi}{Y}_{i}g(Y_i)}\hfill \end{array}\right)\mathrm{.}$$

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