Statistical inference with non-probability survey samples
Section 4. Propensity scores based approach

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The propensity scores ${\pi }_i^A=P\left(R_i\mathrm{<mo>=</mo>}1|x_i\mathrm{,}{y}_i\right)$ for the non-probability survey sample $S_A$ are theoretically defined for all the units in the target population. Estimation of the propensity scores for units in $S_A,$ which plays the most crucial role for propensity scores based methods, requires an assumed model on the propensity scores and auxiliary information at the population level. In this section, we first discuss estimation procedures for the propensity scores under the setting and assumptions described in Section 2, and then provide an overview of estimation methods proposed in the recent literature on the finite population mean ${\mu }_y$ involving the estimated propensity scores.

4.1  Estimation of propensity scores

Under assumption A1, the propensity scores ${\pi }_i^A\mathrm{<mo>=</mo>}P\left(R_i\mathrm{<mo>=</mo>}1|x_i\right)=\pi \left(x_i\right)$ are a function of the auxiliary variables $x_i$ but the functional form can be complicated and is completely unknown. Three popular parametric forms ${\pi }_i^A=\pi \left(x_i\mathrm{,}\alpha \right)$ in dealing with a binary response can be considered: (i) the inverse logit function ${\pi }_i^A=1-{\left\{1+\exp \left(x_i^{\prime }\alpha \right)\right\}}^{-1}\text{};$ (ii) the inverse probit function ${\pi }_i^A\mathrm{<mo>=</mo>}\Phi \left(x_i^{\prime }\alpha \right),$ where $\Phi (\cdot )$ is the cumulative distribution function of $N(\mathrm{0,}1);$ and (iii) the inverse complementary log-log function ${\pi }_i^A\mathrm{<mo>=</mo>}1-\exp \left\{-\exp \left(x_i^{\prime }\alpha \right)\right\}.$ Nonparametric techniques without assuming an explicit functional form for $\pi \left(x\right)$ are attractive alternatives for the estimation of propensity scores.

4.1.1   The pseudo maximum likelihood method

Let ${\pi }_i^A=\pi (x_i\mathrm{,}\alpha )$ be a specified parametric form with unknown model parameters $\alpha .$ Under the ideal situation where the complete auxiliary information $\left\{x_1\mathrm{,}{x}_2\mathrm{,}\dots \mathrm{,}{x}_N\right\}$ is available and with the independence assumption A3, the full log-likelihood function on $\alpha$ can be written as (Chen et al., 2020)

$$\text{ℓ}\left(\alpha \right)=\log \left\{{\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^N{\left({\pi }_i^A\right)}^{R_i}{\left(1-{\pi }_i^A\right)}^{1-R_i}}\right\}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in S_A}\log \left(\frac{{\pi }_i^A}{1-{\pi }_i^A}\right)}+{\displaystyle \sum _{i=1}^N\log \left(1-{\pi }_i^A\right)}.(4.1)$$

The maximum likelihood estimator of $\alpha$ is the maximizer of $\text{ℓ}\left(\alpha \right).$ Under the current setting where the population auxiliary information is supplied by the reference probability sample $S_B,$ we replace $\text{ℓ}\left(\alpha \right)$ by the pseudo log-likelihood function (Chen et al., 2020)

$${\text{ℓ}}^{*}\left(\alpha \right)={\displaystyle \sum _{i\in S_A}\log \left(\frac{{\pi }_i^A}{1-{\pi }_i^A}\right)}+{\displaystyle \sum _{i\in S_B}{d}_i^B\log \left(1-{\pi }_i^A\right)}.(4.2)$$

The maximum pseudo-likelihood estimator $\widehat{\alpha }$ is the maximizer of ${\text{ℓ}}^{*}\left(\alpha \right)$ and can be obtained as the solution to the pseudo score equations given by $U(\alpha )=\partial {\text{ℓ}}^{*}(\alpha )/\partial \alpha =0.$ If the inverse logit function is assumed for ${\pi }_i^A,$ the pseudo score functions are given by

$$U\left(\alpha \right)={\displaystyle \sum _{i\in S_A}{x}_i}-{\displaystyle \sum _{i\in S_B}{d}_i^B\pi \left(x_i\mathrm{,}\alpha \right)x_i}.(4.3)$$

In general, the pseudo score functions $U\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ at the true values of the model parameters ${\alpha }_0$ are unbiased under the joint $qp$ randomization in the sense that $E_{qp}\left\{U({\alpha }_0)\right\}=0,$ which implies that the estimator $\widehat{\alpha }$ is $qp$ -consistent for ${\alpha }_0$ (Tsiatis, 2006).

Valliant and Dever (2011) made an earlier attempt to estimate the propensity scores by pooling the non-probability sample $S_A$ with the reference probability sample $S_B.$ Let $S_{AB}=S_A\cup S_B$ be the pooled sample without removing any potential duplicated units. Let $R_i^{*}=1$ if $i\in S_A$ and $R_i^{*}=0$ if $i\in S_B.$ Valliant and Dever (2011) proposed to fit a survey weighted logistic regression model to the pooled dataset $\left\{(R_i^{*}\mathrm{,}{x}_i\mathrm{,}{d}_i)\mathrm{,}i\in S_{AB}\right\},$ where the weights are defined as $d_i=1$ if $i\in S_A$ and $d_i=d_i^B\left(1-n_A/{\widehat{N}}_B\right)$ if $i\in S_B.$ The key motivation behind the creation of the weights $d_i$ is that the total weight ${\displaystyle {\sum }_{i\in S_{AB}}}{d}_i={\displaystyle {\sum }_{i\in S_B}}{d}_i^B={\widehat{N}}_B$ for the pooled sample matches the estimated population size, and the hope is that the survey weighted logistic regression model would lead to valid estimates for the propensity scores. It was shown by Chen et al. (2020) that the pooled sample approach of Valliant and Dever (2011) does not lead to consistent estimators for the parameters of the propensity scores model unless the non-probability sample $S_A$ is a simple random sample from the target population.

The method of Valliant and Dever (2011) reveals a fundamental difficulty with approaches based on the pooled sample $S_{AB}.$ If the units in the non-probability sample $S_A$ are treated as exchangeable in the pooled sample $S_{AB},$ which was reflected by the equal weights $d_i\mathrm{<mo>=</mo>}1$ used in the method of Valliant and Dever (2011), the resulting estimates for the propensity scores will be invalid unless $S_A$ is a simple random sample. This observation has implications to the validity of nonparametric methods or regression tree-based methods to be discussed in Section 4.1.3.

In a recent paper, Wang, Valliant and Li (2021) proposed an adjusted logistic propensity (ALP) weighting method. The method involves two steps for computing the estimated propensity scores. The initial estimates, denoted as ${\widehat{p}}_i$ for $i\in S_A,$ are obtained by fitting the survey weighted logistic regression model to the pooled sample $S_{AB}$ similar to Valliant and Dever (2011), with the weights defined as $d_i\mathrm{<mo>=</mo>}1$ if $i\in S_A$ and $d_i=d_i^B$ if $i\in S_B.$ The final estimated propensity scores are computed as ${\widehat{\pi }}_i^A={\widehat{p}}_i/\left(1-{\widehat{p}}_i\right).$ The key theoretical argument is the equation ${\pi }_i^A=p_i/\left(1-p_i\right)$ where ${\pi }_i^A=P\left(i\in S_A|U\right),$ $p_i=P\left(i\in S_A^{*}|S_A^{*}\cup U\right),$ and $S_A^{*}$ is a copy of $S_A$ but is viewed as a different set. However, there are conceptual issues with the arguments since the probabilities ${\pi }_i^A=P\left(i\in S_A|U\right)$ are defined under the assumed propensity scores model with the given finite population $U,$ and the assumed model does not lead to a meaningful interpretation of the probabilities $p_i=P\left(i\in S_A^{*}|S_A^{*}\cup U\right).$ The latter require a different probability space and are conditional on the given $S_A.$ As a matter of fact, one can easily argue that under the assumed propensity scores model and conditional on the given $S_A,$ we have $p_i\mathrm{<mo>=</mo>}1$ if $i\in S_A$ and $p_i=0$ otherwise.

4.1.2  Estimating equations based methods

The pseudo score equations $U(\alpha )=0$ derived from the pseudo likelihood function ${\text{ℓ}}^{*}(\alpha )$ may be replaced by a system of general estimating equations. Let $h(x,\alpha )$ be a user-specified vector of functions with the same dimension of $\alpha .$ Let

$$G\left(\alpha \right)={\displaystyle \sum _{i\in S_A}h\left(x_i\mathrm{,}\alpha \right)}-{\displaystyle \sum _{i\in S_B}{d}_i^B\pi \left(x_i\mathrm{,}\alpha \right)h\left(x_i\mathrm{,}\alpha \right).}(4.4)$$

It follows that $E_{qp}\left\{G\mathrm{<mo\; stretchy="false">(</mo>}{\alpha }_0\mathrm{<mo\; stretchy="false">)</mo>}\right\}\mathrm{<mo>=</mo>}0$ for any chosen $h(x,\alpha ).$ In principle, an estimator $\widehat{\alpha }$ of $\alpha$ can be obtained by solving $G(\alpha )=0$ with the chosen parametric form ${\pi }_i^A=\pi (x_i\mathrm{,}\alpha )$ and the chosen functions $h(x,\alpha ),$ and the estimator $\widehat{\alpha }$ is consistent.

The estimator $\widehat{\alpha }$ using arbitrary user-specified functions $h(x,\alpha )$ is typically less efficient than the one based on the pseudo score functions, due to the optimality of the maximum likelihood estimator (Godambe, 1960). Some limited empirical results also show that the solution to $G(\alpha )=0$ can be unstable for certain choices of $h(x,\alpha ).$ Nevertheless, the estimating equations based methods provide a useful tool for the estimation of the propensity scores under more restricted scenarios. For instance, if we let $h(x,\alpha )=x/\pi (x,\alpha ),$ the estimating functions given in (4.4) reduce to

$$G\left(\alpha \right)={\displaystyle \sum _{i\in S_A}\frac{x_i}{\pi \left(x_i\mathrm{,}\alpha \right)}}-{\displaystyle \sum _{i\in S_B}{d}_i^B{x}_i}.(4.5)$$

The form of $G(\alpha )$ in (4.5) looks like a “distorted” version of the pseudo score functions given in (4.3) under a logistic regression model for the propensity scores. The most practically important difference between the two versions, however, is the fact that the $G(\alpha )$ given in (4.5) only requires the estimated population totals for the auxiliary variables $x.$ There are scenarios where the population totals of the auxiliary variables $x$ can be accessed or estimated from an existing source but values of $x$ at the unit level for the entire population or even a probability sample are not available. The use of estimating functions $G(\alpha )$ given (4.5) makes it possible to obtain valid estimates of the propensity scores for units in the non-probability sample. Section 6.3 describes an example where the estimating equations based approach leads to a valid variance estimator for the doubly robust estimator of the population mean.

4.1.3  Nonparametric methods and regression-tree based methods

The propensity scores ${\pi }_i^A=P\left(R_i=1|x_i\right)$ are the mean function $E_q\left(R_i|x_i\right)=\pi \left(x_i\right)$ for the binary response $R_i.$ Nonparametric methods for estimating $\pi (x)$ can be an attractive alternative. The major challenge is to develop estimation procedures which provide valid estimates of the propensity scores. As noted in Section 4.1.1, estimation methods based on the pooled sample $S_{AB}=S_A\cup S_B$ may lead to invalid estimates. Strategies similar to the one used by Chen et al. (2020) can be theoretically justified under the joint $qp$ framework, where the estimation procedures are first derived using data from the entire finite population and unknown population quantities are then replaced by estimates obtained from the reference probability sample.

We consider the kernel regression estimator of ${\pi }_i^A=\pi \left(x_i\right).$ Suppose that the dataset $\left\{(R_i\mathrm{,}{x}_i)\mathrm{,}i=\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N\right\}$ is available for the finite population. Let $K_h\left(t\right)=K\left(t/h\right)$ be a chosen kernel with a bandwidth $h.$ The Nadaraya-Watson kernel regression estimator (Nadaraya, 1964; Watson, 1964) of $\pi \left(x\right)$ is given by

$$\tilde{\pi }\left(x\right)=\frac{{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^N{K}_h\left(x-x_j\right)R_j}}{{\displaystyle {\sum }_{j=1}^N{K}_h\left(x-x_j\right)}}.(4.6)$$

A kernel estimator in the form of $\tilde{\pi }(x)$ given in (4.6) usually has no practical values since we do not have complete auxiliary information for the finite population. It turns out that for the estimation of propensity scores the numerator in (4.6) only requires observations from the non-probability sample due to the binary variable $R_j,$ and the denominator is a population total and can be estimated by using the reference probability sample. The nonparametric kernel regression estimator of the propensity scores is given by (Yuan, Li and Wu, 2022)

$${\widehat{\pi }}_i^A=\widehat{\pi }\left(x_i\right)=\frac{{\displaystyle {\sum }_{j\in S_A}{K}_h\left(x_i-x_j\right)}}{{\displaystyle {\sum }_{j\in S_B}{d}_j^B{K}_h\left(x_i-x_j\right)}}\mathrm{,}i\in S_A.(4.7)$$

The estimator ${\widehat{\pi }}_i^A$ given in (4.7) is consistent under the joint $qp$ framework and the $q$ -model for the propensity scores is very flexible due to the nonparametric assumption on $\pi (x).$ The estimated propensity scores are easy to compute when the dimension of $x$ is not too high. Issues with high dimensional $x$ and the choices of the kernel $K_h(\cdot )$ and the bandwidth $h$ remain as in general applications of kernel-based estimation methods. Simulation results reported by Yuan et al. (2022) show that the kernel estimation method provides robust results for the propensity scores using the normal kernel and popular choices for the bandwidth.

Chu and Beaumont (2019) considered regression-tree based methods for estimating the propensity scores. Their proposed TrIPW method is a variant of the CART algorithm (Breiman, Friedman, Olshen and Stone, 1984) and uses data from the combined sample of the non-probability sample and the reference probability sample. The method aims to construct a classification tree with the terminal nodes of the final tree treated as homogeneous groups in terms of the propensity scores. The estimator of ${\mu }_y$ is constructed based on the final tree and post-stratification. Section 5 contains further details on poststratified estimators.

Statistical learning techniques such as classification and regression trees and random forests have been developed primarily for the purpose of prediction. Their use for estimating the propensity scores of non-probability samples requires further research. It is not a desirable approach to naively apply the methods over the pooled sample $S_{AB}$ without theoretical justifications on the consistency of the final estimators. Further research towards this direction should be encouraged.

4.2  Inverse probability weighting

Let ${\widehat{\pi }}_i^A$ be an estimate of ${\pi }_i^A=P\left(i\in S_A|x_i\right)$ under a chosen method for the estimation of the propensity scores. Two versions of the inverse probability weighted (IPW) estimator of ${\mu }_y$ are constructed as

$${\widehat{\mu }}_{\text{IPW}1}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in S_A}\frac{y_i}{{\widehat{\pi }}_i^A}}\text{and}{\widehat{\mu }}_{\text{IPW}2}\mathrm{<mo>=</mo>}\frac{1}{{\widehat{N}}^A}{\displaystyle \sum _{i\in S_A}\frac{y_i}{{\widehat{\pi }}_i^A}},(4.8)$$

where $N$ is the population size and ${\widehat{N}}^A={\displaystyle {\sum }_{i\in S_A}{\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\pi }}_i^A\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}}$ is the estimated population size. The estimator ${\widehat{\mu }}_{\text{IPW}1}$ is a version of the Horvitz-Thompson estimator and ${\widehat{\mu }}_{\text{IPW}2}$ corresponds to the Hájek estimator as discussed in design-based estimation theory. There are ample evidences from both theoretical justifications and practical observations that the Hájek estimator ${\widehat{\mu }}_{\text{IPW}2}$ performs better than the Horvitz-Thompson estimator and should be used in practice even if the population size $N$ is known.

The validity of the IPW estimators ${\widehat{\mu }}_{\text{IPW}1}$ and ${\widehat{\mu }}_{\text{IPW}2}$ depends on the validity of the estimated propensity scores. Under the assumptions A1 and A2 and the parametric model ${\pi }_i^A=\pi (x_i\mathrm{,}{\alpha }_0),$ the consistency of ${\widehat{\mu }}_{\text{IPW}1}$ follows a standard two-step argument. Let ${\tilde{\mu }}_{\text{IPW}}=N^{-1}{\displaystyle {\sum }_{i\in S_A}{y}_i/{\pi }_i^A},$ which is not a computable estimator but an analytic tool useful for asymptotic purposes. It follows that $E_q\left({\tilde{\mu }}_{\text{IPW}}\right)={\mu }_y$ and the order $V_q\left({\tilde{\mu }}_{\text{IPW}}\right)=O\left(n_A^{-1}\right)$ holds under the condition that $n_A{\pi }_i^A/N$ is bounded away from zero. As a consequence, we have ${\tilde{\mu }}_{\text{IPW}}\to {\mu }_y$ in probability as $n_A\to \infty .$ Under the correctly specified model ${\pi }_i^A=\pi \left(x_i\mathrm{,}{\alpha }_0\right)$ for the propensity scores, the typical root- $n$ order $\widehat{\alpha }-{\alpha }_0=O_p\left(n_A^{-1/2}\right)$ holds for commonly encountered scenarios. We can show by treating ${\widehat{\mu }}_{\text{IPW}1}$ as a function of $\widehat{\alpha }$ and using a Taylor series expansion that ${\widehat{\mu }}_{\text{IPW}1}={\tilde{\mu }}_{\text{IPW}}+O_p\left(n_A^{-1/2}\right)$ under some mild finite moment conditions. The consistency of ${\widehat{\mu }}_{\text{IPW2}}$ can be established using standard arguments for a ratio estimator (Section 5.3, Wu and Thompson, 2020) where $N^{-1}{\displaystyle {\sum }_{i\in S_A}}{\left({\pi }_i^A\right)}^{-1}\mathrm{<mo>=</mo>}1+o_p(1).$

4.3  Doubly robust estimation

The dependence of the IPW estimator on the validity of the assumed propensity score model is viewed as a weakness of the method. The issue is not unique to the IPW estimators and is faced by many other approaches involving an assumed statistical model. Robust estimation procedures which provide certain degrees of protection against model misspecifications have been pursued by researchers, and the so-called doubly robust estimators have been a successful story since the work of Robins, Rotnitzky, and Zhao (1994).

The doubly robust (DR) estimator of ${\mu }_y$ is constructed using both the propensity score model $q$ and the outcome regression model $\xi .$ The DR estimator with the given propensity scores ${\pi }_i^A,$ $i\in S_A$ and the mean responses $m_i=E_{\xi }\left(y_i|x_i\right),$ $i=\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N$ has the following general form,

$${\tilde{\mu }}_{\text{DR}}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in S_A}\frac{y_i-m_i}{{\pi }_i^A}}+\frac{1}{N}{\displaystyle \sum _{i=1}^N{m}_i}.(4.9)$$

The second term on the right hand side of (4.9) is the model-based prediction of ${\mu }_y.$ The first term is a propensity score based adjustment using the errors ${\epsilon }_i=y_i-m_i$ from the outcome regression model. The magnitude of the adjustment term is negatively correlated to the “goodness-of-fit” of the outcome regression model. It can be shown that ${\tilde{\mu }}_{\text{DR}}$ is an exactly unbiased estimator of ${\mu }_y$ if one of the two models $q$ and $\xi$ is correctly specified and hence it is doubly robust. The estimator ${\tilde{\mu }}_{\text{DR}}$ has an identical structure to the generalized difference estimator of Wu and Sitter (2001). It is important to note that the double robustness property of ${\tilde{\mu }}_{\text{DR}}$ does not require the knowledge of which of the two models being correctly specified. It is also apparent that the estimator ${\tilde{\mu }}_{\text{DR}}$ given in (4.9) is not computable in practical applications.

Let ${\widehat{\pi }}_i^A$ and ${\widehat{m}}_i$ be respectively the estimators of ${\pi }_i^A$ and $m_i$ under the assumed models $q$ and $\xi .$ Under the two-sample setting described in Section 2, the two DR estimators of ${\mu }_y$ proposed by Chen et al. (2020) are given by

$${\widehat{\mu }}_{\text{DR}1}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in S_A}\frac{y_i-{\widehat{m}}_i}{{\widehat{\pi }}_i^A}}+\frac{1}{N}{\displaystyle \sum _{i\in S_B}{d}_i^B{\widehat{m}}_i}(4.10)$$

and

$${\widehat{\mu }}_{\text{DR}2}\mathrm{<mo>=</mo>}\frac{1}{{\widehat{N}}^A}{\displaystyle \sum _{i\in S_A}\frac{y_i-{\widehat{m}}_i}{{\widehat{\pi }}_i^A}}+\frac{1}{{\widehat{N}}^B}{\displaystyle \sum _{i\in S_B}{d}_i^B{\widehat{m}}_i},(4.11)$$

where $d_i^B$ are the design weights for the probability sample $S_B,$ ${\widehat{N}}^A\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_A}{\left({\widehat{\pi }}_i^A\right)}^{-1}}$ and ${\widehat{N}}^B\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_B}}{d}_i^B.$ The estimator ${\widehat{\mu }}_{\text{DR}2}$ using the estimated population size has better performance in terms of bias and mean squared error and should be used in practice.

The probability survey design $p$ is an integral part of the theoretical framework for assessing the two estimators ${\widehat{\mu }}_{\text{DR}1}$ and ${\widehat{\mu }}_{\text{DR}2}.$ It is assumed that $S_A$ and $S_B$ are selected independently, which implies that $E_p\left({\displaystyle {\sum }_{i\in S_B}{d}_i^B{\widehat{m}}_i}\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i=1}^N{\widehat{m}}_i}.$ Consistency of the estimators ${\widehat{\mu }}_{\text{DR}1}$ and ${\widehat{\mu }}_{\text{DR}2}$ can be established under either the $qp$ or the $\xi p$ framework. It should be noted that even if the non-probability sample $S_A$ is a simple random sample with ${\pi }_i^A=n_A/N,$ the doubly robust estimator in the form of (4.9) does not reduce to the model-based prediction estimator ${\widehat{\mu }}_{y2}$ given in (3.3).

4.4  The pseudo empirical likelihood approach

The pseudo empirical likelihood (PEL) methods for probability survey samples have been under development over the past two decades. Two early papers on the topic are Chen and Sitter (1999) on point estimation incorporating auxiliary information and Wu and Rao (2006) on PEL ratio confidence intervals. The PEL approaches are further used for multiple frame surveys (Rao and Wu, 2010a) and Bayesian inferences with survey data (Rao and Wu, 2010b; Zhao, Ghosh, Rao and Wu, 2020b). Using the PEL methods for general inferential problems with complex surveys has been studied in two recent papers (Zhao and Wu, 2019; Zhao, Rao and Wu, 2020a).

Chen, Li, Rao and Wu (2022) showed that the PEL provides an attractive alternative approach to inference with non-probability survey samples. Let ${\widehat{\pi }}_i^A,$ $i\in S_A$ be the estimated propensity scores under an assumed parametric or non-parametric model, $q.$ The PEL function for the non-probability survey sample $S_A$ is defined as

$${\text{ℓ}}_{\text{PEL}}\mathrm{<mo\; stretchy="false">(</mo>}p\mathrm{<mo\; stretchy="false">)</mo>}=n_A{\displaystyle \sum _{i\in S_A}{\tilde{d}}_i^A\log \left(p_i\right)},(4.12)$$

where $p\mathrm{<mo>=</mo>}(p_1\mathrm{,}\dots \mathrm{,}{p}_{n_A})$ is a discrete probability measure over the $n_A$ selected units in $S_A,$ ${\tilde{d}}_i^A\mathrm{<mo>=</mo>}{({\widehat{\pi }}_i^A)}^{-1}/{\widehat{N}}^A$ and ${\widehat{N}}^A\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in S_A}{({\widehat{\pi }}_j^A)}^{-1}}$ which is defined earlier in Section 4. Without using any additional information, maximizing ${\text{ℓ}}_{\text{PEL}}\left(p\right)$ under the normalization constraint

$${\displaystyle \sum _{i\in S_A}{p}_i}\mathrm{<mo>=</mo>}1(4.13)$$

leads to ${\widehat{p}}_i\mathrm{<mo>=</mo>}{\tilde{d}}_i^A,$ $i\in S_A.$ The maximum PEL estimator of ${\mu }_y$ is given by ${\widehat{\mu }}_{\text{PEL}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_A}{\widehat{p}}_i{y}_i},$ which is identical to the IPW estimator ${\widehat{\mu }}_{\text{IPW}2}$ given in (4.8).

The PEL approach to non-probability survey samples provides flexibilities in combining information through additional constraints and constructing confidence intervals and conducting hypothesis tests using the PEL ratio statistic. The maximum PEL estimator ${\widehat{\mu }}_{\text{PEL}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in S_A}{\widehat{p}}_i{y}_i}$ is doubly robust if $\left({\widehat{p}}_1\mathrm{,}\dots \mathrm{,}{\widehat{p}}_{n_A}\right)$ is the maximizer of ${\text{ℓ}}_{\text{PEL}}\left(p\right)$ under both the normalization constraint and the model-calibration constraint given by

$${\displaystyle \sum _{i\in S_A}{p}_i{\widehat{m}}_i}={\overline{m}}^B,(4.14)$$

where ${\overline{m}}^B={\left({\widehat{N}}^B\right)}^{-1}{\displaystyle {\sum }_{i\in S_B}{d}_i^B{\widehat{m}}_i}$ is computed using the fitted values ${\widehat{m}}_i,$ $i\in S_B$ from an assumed outcome regression model, $\xi .$ The equation (4.14) is a modified version of the original model-calibration constraint of Wu and Sitter (2001) using the probability sample $S_B$ . Chen et al. (2022) contain further details on the asymptotic distributions of the PEL ratio statistic and simulation studies on the performances of PEL ratio confidence intervals on a finite population proportion.

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