Model-assisted calibration of non-probability sample survey data using adaptive LASSO
Section 6. Conclusion

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In this manuscript, we developed the LASSO calibration estimator of population totals, ${\widehat{T}}_y^{\text{LASSO}}\text{},$ given population auxiliary data. We also derived closed-form variance estimates for ${\widehat{T}}_y^{\text{LASSO}}\text{}.$ Simulation results show that the point estimates are approximately unbiased under simple-random sampling and informative sampling. For sample selections that are related to analysis variables, LASSO was able to significantly reduce sample bias even without the correct design weights. LASSO tends to outperform stepwise-selected working models when covariates are highly collinear. For analysis with many categorical variables, where there are natural correlations between the categories, LASSO calibration estimator can perform better than traditional calibration estimators, even if the effect sizes are small. The improvement is modest in the continuous variable setting, but substantial when the outcome of interest is binary, as shown in simulations and in the NHIS data example. We have demonstrated theoretically and through simulations that LASSO calibration holds great promise in making unbiased inference of population totals from non-probability samples. Although asymptotic closed-form variance estimates did not produce very accurate nominal coverage, the naive bootstrap is a viable alternative approach. In an application to estimate population total of individuals diagnosed with cancer, without correct design weights, the LASSO calibration estimator was able to produce an estimate that is the closest to the estimate based on correct survey weights. LASSO calibration estimator also has the smallest standard error of all the estimators considered, although the bootstrap variance estimate that was used did not fully account for the clustering in the NHIS, which generally increases standard errors. The application shows that LASSO calibration can generate inference to the population for a specific outcome variable, and the inference is both more accurate and precise than traditional calibration estimators.

The question arises when use of LASSO model-assisted calibration should be used instead of traditional calibration methods such as GREG. Both theoretical and empirical results in this paper suggest that there is little to be lost in terms of statistical efficiency to use LASSO model-assisted calibration, it does require additional effort on the part of the analyst to implement. While we cannot give specific cutoffs, our analysis suggests that this effort will be worthwhile when a) there are large numbers of potential calibration variables, b) many of these calibration variables are likely to be highly correlated, and c) the outcome is binary rather than continuous. We believe that conditions a) and b), at least, are increasing likely to be encountered in non-probability settings, where administrative datasets might provide these types of calibration variables and subsets of data obtained through various means will contain the core variable of interest.

While LASSO provides particularly convenient and rapid implementation, there are, of course, other modern regression methods that could be considered in addition to LASSO to develop penalized regression models for high-dimensional model-assisted regression, including approaches such as ridge regression, principle components, or Bayesian additive regression trees (Chipman, George and McCulloch, 2010). These approaches provide opportunity for further research in this area.

Finally, we note that this work is only a part of a larger and rapidly expanding literature on inference from non-probability surveys. In addition to the work of McConville et al. (2017), the “Mr. P” (multi-level regression and poststratification or MRP) approach of Wang et al. (2015) also uses high dimensional covariates to adjust non-probabilities samples, by use of a hierarchical model rather than penalized regression. Quasi-randomization (Elliott, 2009; Elliott, Resler, Flannagan and Rupp, 2010; Elliott and Valliant, 2017) and sample matching (Rivers, 2007; Vavreck and Rivers, 2008) also provide alternatives that use data from either known population quantities or probability sampling estimates to deal with selection bias issues in non-probability samples. Each have their strengths and weaknesses relative to each other and to model-assisted LASSO. The MRP approach makes distributional assumptions that might improve efficiency, but might reduce robustness, and is non-trivial to implement in its fully Bayesian form. Quasi-randomization forfeits the link to a particular outcome variable, making the weights it develops general purpose but likely less effective, while sample matching requires intervention at the design stage to sample elements from the non-probability frame that match elements from the population, ala quota sampling. The decision to use model-assisted LASSO calibration should be made in the context of these tradeoffs.

Acknowledgements

The authors thank Professor Fred M. Feinberg and Associate Research Scientist Sunghee Lee for their helpful review and suggestions, as well as the Editor, Associate Editor, and three referees whose suggestions greatly contributed to improving this paper.

Appendix

Determining estimates for adaptive LASSO

In practice, we do not observe the theoretical rate of growth of ${\lambda }_n,$ which optimize model fit measures such as AIC or BIC, unless we have obtained many samples of the same population with various sample sizes. Given a sample, the choices of ${\lambda }_n$ and $\gamma$ depend on the modeler. In R $glmnet$ implementation (Friedman et al., 2010), a range of ${\lambda }_n$ is determined by the following scheme: 

1. Set $\gamma \mathrm{<mo>=</mo>\; 0.}$
2. Determine ${\lambda }_n^{\text{max}}$ by finding the smallest ${\lambda }_n$ that sets all coefficients to 0.
3. If sample size $n$ is larger than the number of parameters in the regression model, set ${\lambda }_n^{\text{min}}\mathrm{<mo>=</mo>}\text{0}\text{.0001}{\lambda }_n^{\text{max}}\text{}.$ If sample size $n$ is smaller than the number of parameters, set ${\lambda }_n^{\text{min}}\mathrm{<mo>=</mo>}\text{0}\text{.01}{\lambda }_n^{\text{max}}$ (to set parameters to 0 sooner).
4. Generate a grid of ${\lambda }_n,$ typically 100 equally spaced points between ${\lambda }_n^{\text{min}}$ and ${\lambda }_n^{\text{max}}\text{}.$

The initial range of values of ${\lambda }_n$ is determined independently of $\gamma .$ Choices of $\gamma$ are less data-driven. Some modelers choose one of $\gamma \mathrm{<mo>=</mo>}\text{0}\text{.1}\mathrm{,}\text{0}\text{.5}\mathrm{,}\mathrm{1,}2.$ Here we determine $\left({\lambda }_n\mathrm{,}\gamma \right)$ through cross-validation as follows: 

1. Obtain ${\alpha }_j=1/\left|{\widehat{\beta }}_j^{\text{MLE}}\right|.$
2. Determine 100 equally spaced values of ${\lambda }_n$ based on R $glmnet$ ’s implementation.
3. For each pair $\left({\lambda }_n\mathrm{,}\gamma \right),$ ${\lambda }_n$ from Step 2, and $\gamma \mathrm{<mo>=</mo>}\text{0}\text{.1}\mathrm{,}\text{0}\text{.5}\mathrm{,}\text{1}\mathrm{,}\text{2},$ split data into 5 folds. Use 4 folds to obtain $\widehat{\beta }\text{.}$
4. Apply $\widehat{\beta }$ to the last fold not used to estimate $\widehat{\beta }$ and calculate a metric. For continuous $y,$ we calculate the mean-absolute-error (MAE), ${\displaystyle {\sum }_{i\in s_{A\left(k\right)}}}\left|{\widehat{\mu }}_i-y_i\right|.$ For binary $y,$ we calculate the area under curve (AUC) (calculated through R $glmnet\mathrm{<mo>:</mo>\; <mo>:</mo>}auc$ function).
5. Average the 5 metrics for each pair of $\left({\lambda }_n\mathrm{,}\gamma \right),$ and choose the pair with the best average metric: minimum MAE for continuous $y,$ maximum AUC for binary $y.$

The adaptive LASSO coefficient estimates are then obtained by solving equations (3.1) or (3.2) in Section 3.1 given the selected $\left({\lambda }_n\mathrm{,}\gamma \right).$ The R code used to perform cross-validation is provided in the on-line supplemental material.

Asymptotic unbiasedness and variance of model-assisted LASSO calibration estimator of a population total

Lemma 1: Assume the superpopulation model:

$$E_{\xi }\left(y_k|x_k\right)\mathrm{<mo>=</mo>}\mu \left(x_k\mathrm{,}\beta \right)\mathrm{,}{V}_{\xi }\left(y_k|x_k\right)\mathrm{<mo>=</mo>}{\nu }_k^2{\sigma }^2.$$

Let $B$  be the finite-population quasilikelihood estimate of $\beta \text{,}$   $B\to \beta \text{.}$  Under conditions (1)-(5) in Section 3.2, the model-assisted asymptotic estimator of population total is:

$${\widehat{T}}_y^{MC}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left(y_i-{\mu }_i{B}^{MC}\right)+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i{B}^{MC}+o_p\left(\frac{N}{\sqrt{n}}\right)(\text{A}\text{.1})$$

where

$$\begin{array}{ll}{\mu }_i\hfill & \mathrm{<mo>=</mo>}\mu \left(x_i\mathrm{,}B\right)\hfill \\ B^{MC}\hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N\left({\mu }_i-\overline{\mu }\right)\left(y_i-\overline{y}\right)}}{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\left({\mu }_i-\overline{\mu }\right)}^2}}.\hfill \end{array}$$

Proof. The proof is adapted and expanded from the proof of Theorem 1 in Wu and Sitter (2001), with slight modifications in notations to be consistent with this paper. We begin by deriving the asymptotic model-assisted estimator for a population mean, ${\widehat{\overline{y}}}^{\text{MC}}\mathrm{<mo>=</mo>}{N}^{-1}{\widehat{T}}_y^{\text{MC}}$  (see equation (2.7)). By conditions (2) and (3), the second order Taylor series expansion of $\mu \left(x_i\mathrm{,}\widehat{B}\right)$  around $B$  is:

$$\mu \left(x_i\mathrm{,}\widehat{B}\right)\mathrm{<mo>=</mo>}\mu \left(x_i\mathrm{,}B\right)+{\left\{{\frac{\partial \mu \left(x_i\mathrm{,}t\right)}{\partial t}|}_{t\mathrm{<mo>=</mo>}B}\right\}}^T\left(\widehat{B}-B\right)+{\left(\widehat{B}-B\right)}^T\left\{\frac{{\partial }^2\mu \left(x_i\mathrm{,}t\right)}{\partial t\partial t^T}{|}_{t\mathrm{<mo>=</mo>}{B}^{*}}\right\}\left(\widehat{B}-B\right)(\text{A}\text{.2})$$

for $B^{*}\in \left(\widehat{B},B\right)$ or $\left(B\mathrm{,}\widehat{B}\right).$ Let

$$\begin{array}{ll}h\left(x_i\mathrm{,}B\right)\hfill & \mathrm{<mo>=</mo>}\frac{\partial \mu \left(x_i\mathrm{,}t\right)}{\partial t}{|}_{t\mathrm{<mo>=</mo>}B}\hfill \\ k\left(x_i\mathrm{,}{{\rm B}}^{*}\right)\hfill & \mathrm{<mo>=</mo>}{\frac{{\partial }^2\mu \left(x_i\mathrm{,}t\right)}{\partial t\partial t^T}|}_{t\mathrm{<mo>=</mo>}{B}^{*}}\hfill \end{array}$$

Note that $h$  is a vector of length $m$  and $k$  is a matrix of size $m\times m,$  where $m$  is the number of parameters in $\beta \text{.}$  By conditions (2) and (3),

$${\text{max}}_i\left|h\left(x_i\mathrm{,}B\right)\right|\le h\left(x_i\mathrm{,}B\right)(\text{A}\text{.3})$$

$${\text{max}}_{k\text{}\mathrm{,}j}\left|k\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{,}{B}^{*}\mathrm{<mo\; stretchy="false">)</mo>}\right|\le k\left(x_i\mathrm{,}{B}^{*}\right).(\text{A}\text{.4})$$

Conditions (1) and (3) imply that

$$\begin{array}{lll}\mu \left(x_i\mathrm{,}\widehat{B}\right)\hfill & \mathrm{<mo>=</mo>}\mu \left(x_i\mathrm{,}B\right)+O_p\left(1/\sqrt{n}\right)\hfill & (\text{A}\text{.5})\hfill \\ \hfill & \equiv {\mu }_i+O_p\left(1/\sqrt{n}\right).\hfill & (\text{A}\text{.6})\hfill \end{array}$$

By equation (2.2) in Section 2.1 and the boundedness conditions of (2) and (3) in Section 3.2.2 imply

$$\begin{array}{ll}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i\mathrm{,}\widehat{B}\right)\hfill & =N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i\mathrm{,}B\right)+N^{-1}{\left({\displaystyle \sum _{i\in s_A}}{d}_i^{A}h\left(x_i\mathrm{,}B\right)\right)}^T\left(\widehat{B}-B\right)\hfill \\ \hfill & +{\left(\widehat{B}-B\right)}^T{N}^{-1}\left({\displaystyle \sum _{i\in s_A}}{d}_i^{A}k\left(x_i\mathrm{,}{B}^{*}\right)\right)\left(\widehat{B}-B\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i\mathrm{,}B\right)+N^{-1}{\left({\displaystyle \sum _{i\in s_A}}{d}_i^{A}h\left(x_i\mathrm{,}B\right)\right)}^T\left(\widehat{B}-B\right)+O_p\left(\frac{1}{n}\right).(\text{A}\text{.7})\hfill \end{array}$$

By conditions (1), (4), and equation (A.7):

$$\begin{array}{ll}{N}^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_k\mathrm{,}\widehat{B}\right)\hfill & -N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i\mathrm{,}\widehat{B}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_i\mathrm{,}B\right)-N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i\mathrm{,}B\right)+O_p\left(\frac{1}{\sqrt{n}}\right).(\text{A}\text{.8})\hfill \end{array}$$

Using conditions (1) and (3),

$$\begin{array}{ll}\overline{\widehat{\mu }}\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i\mathrm{,}\widehat{B}\right)/{\displaystyle \sum _{i\in s_A}}{d}_i^A\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in s_A}}{d}_i^A\right)}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left(\mu \left(x_i\mathrm{,}B\right)+h^T\left(x_i\mathrm{,}B\right)\left(\widehat{B}-B\right)+{\left(\widehat{B}-B\right)}^{T}k\left(x_i\mathrm{,}{B}^{*}\right)\left(\widehat{B}-B\right)\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\in s_A}}{d}_i^A\right)}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left(\mu \left(x_i\mathrm{,}B\right)+h^T\left(x_i\mathrm{,}B\right)\left(\widehat{B}-B\right)\right)+O_p\left(1/n\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\overline{\mu }+{\left({\displaystyle \sum _{i\in s_A}}{d}_i^A\right)}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A{h}^T\left(x_i\mathrm{,}B\right)\left(\widehat{B}-B\right)+O_p\left(1/n\right)\hfill \\ \hfill & \left(bycondition\left(1\right)and\left(18\right)\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\overline{\mu }+O_p\left(1/\sqrt{n}\right)+O_p\left(1/n\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\overline{\mu }+O_p\left(1/\sqrt{n}\right)(\text{A}\text{.9})\hfill \end{array}$$

for $\overline{\mu }\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_A}}{d}_i^A{\mu }_i/{\displaystyle {\sum }_{i\in s_A}}{d}_i^A\text{}.$

Then from (A.2) and (A.9) and using conditions (1)-(3), we have

$$\begin{array}{ll}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)\hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left(\mu \left(x_i,B\right)+h^T\left(x_i,B\right)\left(\widehat{B}-B\right)+{\left(\widehat{B}-B\right)}^{T}k\left(x_i,B^{*}\right)\left(\widehat{B}-B\right)-\overline{\mu }\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left({\mu }_i-\overline{\mu }\right)+N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A{h}^T\left(x_i,B\right)\left(\widehat{B}-B\right)\hfill \\ \hfill & +N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A{\left(\widehat{B}-B\right)}^{T}k\left(x_i,B^{*}\right)\left(\widehat{B}-B\right)-O_p\left(1/\sqrt{n}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left({\mu }_i-\overline{\mu }\right)+O_p\left(1/\sqrt{n}\right)+O_p\left(1/n\right)-O_p\left(1/\sqrt{n}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left({\mu }_i-\overline{\mu }\right)+O_p\left(1/\sqrt{n}\right).(\text{A}\text{.10})\hfill \end{array}$$

Similarly,

$$N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A{\left({\widehat{\mu }}_i-\overline{\widehat{\mu }}\right)}^2\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A{\left({\mu }_i-\overline{\mu }\right)}^2+O_p\left(1/n\right).(\text{A}\text{.11})$$

From (A.10) and (A.11) we have:

$$\begin{array}{ll}{\widehat{B}}^{MC}\hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s_A}{d}_i^A\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)\left(y_i-\overline{y}\right)}}{{\displaystyle {\sum }_{i\in s_A}{d}_i^A{\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)}^2}}\mathrm{<mo>=</mo>}\frac{N^{-1}{\displaystyle {\sum }_{i\in s_A}{d}_i^A\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)\left(y_i-\overline{y}\right)}}{N^{-1}{\displaystyle {\sum }_{i\in s_A}{d}_i^A{\left({\widehat{\mu }}_i-\widehat{\overline{\mu }}\right)}^2}}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s_A}{d}_i^A\left({\mu }_i-\overline{\mu }\right)\left(y_i-\overline{y}\right)+O_p\left(1/\sqrt{n}\right)}}{{\displaystyle {\sum }_{i\in s_A}{d}_i^A{\left({\mu }_i-\overline{\mu }\right)}^2+O_p\left(1/n\right)}}\hfill \\ \hfill & \to B^{MC}asn\to \infty .(\text{A}\text{.12})\hfill \end{array}$$

Thus ${\widehat{B}}^{MC}\mathrm{<mo>=</mo>}{B}^{MC}+o_p\left(1\right),$ and we have:

$$\begin{array}{ll}{\widehat{\overline{y}}}^{MC}\hfill & \mathrm{<mo>=</mo>}{N}^{-1}{\widehat{T}}_y^{MC}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{d}^{A}y+\left(N^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_k,\widehat{B}\right)+{\displaystyle \sum _{i\in s_A}}{N}^{-1}{d}_i^A\mu \left(x_i,\widehat{B}\right)\right){\widehat{B}}^{MC}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{d}^{A}y+\left(N^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_k,B\right)-N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i,B\right)+O_p\left(\frac{1}{\sqrt{n}}\right)\right)\left(B^{MC}+o_p\left(1\right)\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{N}^{-1}{d}^{A}y+\left(N^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_k,B\right)-N^{-1}{\displaystyle \sum _{i\in s_A}}{d}_i^A\mu \left(x_i,B\right)\right)B^{MC}+o_p\left(\frac{1}{\sqrt{n}}\right).\hfill \end{array}$$

Since $N\mathrm{<mo>=</mo>}{O}_p\left(N\right),$ we have $N\cdot o_P\left(1/\sqrt{n}\right)=O_p\left(N\right)o_p\left(1/\sqrt{n}\right)\mathrm{<mo>=</mo>}{o}_p\left(N/\sqrt{n}\right).$ Thus,

$$\begin{array}{ll}{\widehat{T}}_y^{MC}\hfill & \mathrm{<mo>=</mo>}N{\widehat{\overline{y}}}^{MC}\mathrm{<mo>=</mo>}N\left(N^{-1}{d}^{A}y+\left(N^{-1}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_k,B\right)-N^{-1}{\displaystyle \sum _{i\in s_A}}\mu \left(x_i,B\right)\right)B^{MC}+o_p\left(\frac{1}{\sqrt{n}}\right)\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{d}^{A}y+\left({\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^N}\mu \left(x_k,B\right)-{\displaystyle \sum _{i\in s_A}}\mu \left(x_i,B\right)\right)B^{MC}+o_p\left(\frac{N}{\sqrt{n}}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{iin{s}_A}}{d}_i^A\left(y_i-{\mu }_i{B}^{MC}\right)+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i{B}^{MC}+o_p\left(\frac{N}{\sqrt{n}}\right).(\text{A}\text{.13})\hfill \end{array}$$

Theorem 2: Suppose the parameters in a full regression model have both zero and non-zero components, without loss of generality, let the first $p$  be non-zero and the last $q$  be zero:

$${\beta }^F\mathrm{<mo>=</mo>}\left(\begin{array}{c}{\beta }_{\left(p\times 1\right)}^{\left(1\right)}\\ {\beta }_{\left(q\times 1\right)}^{\left(2\right)}\end{array}\right),{\beta }^{\left(1\right)}=\beta and{\beta }^{\left(2\right)}=0_{\left(q\times 1\right)}.$$

Under conditions (1)-(5), the asymptotic LASSO calibration estimator of total is:

$${\widehat{T}}_y^{LASSO}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left(y_i-{\mu }_i{B}^{MC}\right)+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i{B}^{MC}+o_p\left(\frac{N}{\sqrt{n}}\right).(\text{A}\text{.14})$$

Proof. Under condition (5), the adaptive LASSO regression satisfies the oracle property through Theorems 1 and 4 in Zou (2006):

$$\begin{array}{ll}Pr\left(B^{\left(2\right)}\mathrm{<mo>=</mo>}0\right)\hfill & \to 1\hfill \\ \sqrt{n}\left({\widehat{B}}^{\left(1\right)}-B\right)\hfill & \to N\left(0,C\right)\hfill \\ B\hfill & \to \beta \hfill \end{array}$$

where $C\mathrm{<mo>=</mo>}\Sigma \left(B\right)$  is the covariance matrix of $B^{\left(1\right)}$  under the linear model, and $C\mathrm{<mo>=</mo>}{I}^{-1}\left(B\right)$  is the inverse of Fisher information matrix of $B^{\left(1\right)}$  under generalized linear model. By Slutsky’s theorem, the oracle property implies ${\widehat{B}}^{\left(1\right)}\mathrm{<mo>=</mo>}B+O_p\left(1/\sqrt{n}\right).$  By condition (1) and Lemma 1:

$$\begin{array}{ll}{\widehat{T}}_y^{LASSO}\hfill & \approx {\widehat{T}}_y^{MC}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{iin{s}_A}}{d}_i^A\left(y_i-{\mu }_i{B}^{MC}\right)+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i{B}^{MC}+o_p\left(\frac{N}{\sqrt{n}}\right).\hfill \end{array}$$

Theorem 3: ${\widehat{T}}_y^{LASSO}$  is model-unbiased.  

Proof. Under the assumption of our theoretical framework, the superpopulation parameters are a subset of the full LASSO regression parameters, we can prove the model-unbiasedness of ${\widehat{T}}_y^{LASSO}$  by taking expectations with respect to model $\xi .$  First note that:

$$E_{\xi }\left[B^{MC}\right]\mathrm{<mo>=</mo>}{E}_{\xi }\left[\frac{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N\left({\mu }_i-\overline{\mu }\right)\left(y_i-\overline{y}\right)}}{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\left({\mu }_i-\overline{\mu }\right)}^2}}\right]\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N\left({\mu }_i-\overline{\mu }\right)\left({\mu }_i-\overline{\mu }\right)}}{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\left({\mu }_i-\overline{\mu }\right)}^2}}\mathrm{<mo>=</mo>\; 1.}$$

Thus

$$\begin{array}{ll}{E}_{\xi }\left[{\widehat{T}}_y^{LASSO}-T\right]\hfill & \approx E_{\xi }\left[{\displaystyle \sum _{i\in s_A}}{d}_i^A\left(y_i-{\mu }_i{B}^{MC}\right)+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i{B}^{MC}-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{y}_i\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in s_A}}{d}_i^A\left({\mu }_i-{\mu }_i\right)+{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{\mu }_i\left(since{E}_{\xi }\left[B^{MC}\right]\mathrm{<mo>=</mo>}1\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>\; 0.}\hfill \end{array}$$

Thus, as long as LASSO regression parameters include the superpopulation parameters, ${\widehat{T}}_y^{\text{LASSO}}$ is model-unbiased regardless of design weights. This property is essential in non-probability samples, where there are no initial design weights to guarantee unbiasedness.

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