Comments on “Statistical inference with non-probability survey samples”
Section 3. Uniform calibration approach

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Calibration is commonly used to improve the representativeness of a non-probability sample, but existing methods, including the information projection approach mentioned in Section 2, are based on calibrating a set of pre-specified functions. However, it is hard to correctly specify them for calibration in practice. In this section, we propose a general framework for uniformly calibrating functions in an RKHS. Instead of considering a parametric form for $E_{\xi }\mathrm{<mo\; stretchy="false">(</mo>}Y|x\mathrm{<mo\; stretchy="false">)</mo>}$ in (3.1), we only assume $E_{\xi }\mathrm{<mo\; stretchy="false">(</mo>}{y}_i|x_i\mathrm{<mo\; stretchy="false">)</mo>}=m\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>},$ where $m\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}$ is a smooth function satisfying certain conditions.

We still consider (2.1) under the assumption A1. Instead of assuming a set of pre-specified functions $b\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>},$ we propose to estimate $\left\{r_i\mathrm{<mo>:</mo>}i\in S_A\right\}$ by the following optimization,

$$\widehat{\gamma }=\underset{\gamma \ge 0}{\text{argmin}}\left[\underset{u\in H}{{\displaystyle \sup }}\left\{\frac{S\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo>}}{{\Vert u\Vert }_2^2}-{\lambda }_1\frac{{\Vert u\Vert }_H^2}{{\Vert u\Vert }_2^2}\right\}+{\lambda }_2{Q}_A\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{<mo\; stretchy="false">)</mo>}\right]\mathrm{,}(3.1)$$

where $\gamma \mathrm{<mo>=</mo>}\left(r_1\mathrm{,}\dots \mathrm{,}{r}_N\right),$ $r_i\mathrm{<mo>=</mo>}0$ for $i\notin S_A,$ $\gamma \ge 0$ is equivalent to $r_i\ge 0$ for $i=\mathrm{1,}\dots \mathrm{,}N,$ $H$ is an RKHS,

$$S\left(\gamma \mathrm{,}u\right)={\left[N^{-1}{\displaystyle \sum _{i\in S_A}}\left\{1+\left(\frac{N}{n_A}-1\right)r_i\right\}u\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}-N^{-1}{\displaystyle \sum _{i\in S_B}}{d}_i^{B}u\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}\right]}^2\mathrm{,}(3.2)$$

${\Vert u\Vert }_2^2={\mathrm{<mo\; stretchy="false">(</mo>}{n}_A+n_B\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}{\displaystyle {\sum }_{i\in S_A\cup S_B}}u{\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}}^2,$ ${\Vert u\Vert }_H$ is the norm associated with the RKHS, $Q_A\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{<mo\; stretchy="false">)</mo>}$ is a general penalty on $\gamma$ to avoid overfitting, and ${\lambda }_1$ and ${\lambda }_2$ are two tuning parameters; see Wahba (1990) for a detailed introduction about the RKHS.

The intuition for the optimization (3.1) is briefly discussed. First, if $r_i$ approximates the true density ratio $r\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ well, the bias of the first term in (3.1) is negligible for estimating $N^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^N}u\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ for $u\in H.$ Besides, $N^{-1}{\displaystyle {\sum }_{i\in S_B}}{d}_i^{B}u\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ is design-unbiased. Thus, $S\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo>}$ balances two estimators for $N^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^N}u\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>},$ and it is small if $r_i$ approximately equals $r\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ for $i\in S_A.$ However, $S\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo>}$ is not scale invariant, and we have $S\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{,}cu\mathrm{<mo\; stretchy="false">)</mo>}=c^{2}S\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{,}u\mathrm{<mo\; stretchy="false">)</mo>}$ for $c\in ℝ.$ Thus, we use ${\Vert u\Vert }_2^2$ to make it scale-invariant. The term ${\lambda }_1{\Vert u\Vert }_H^2$ is used to penalize the smoothness of the function $u$ for $u\in H.$ There exist different choices for $Q_A\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{<mo\; stretchy="false">)</mo>}$. For example, $Q_A\mathrm{<mo\; stretchy="false">(</mo>}\gamma \mathrm{<mo\; stretchy="false">)</mo>}={\displaystyle {\sum }_{i\in S_A}}{\left\{1+\mathrm{<mo\; stretchy="false">(</mo>}N{n}_A^{-1}-\mathrm{1<mo\; stretchy="false">)</mo>}{r}_i\right\}}^2$ corresponds to penalizing extreme values for the sampling weights, and Wong and Chan (2018) investigated a similar problem assuming the availability of $\mathrm{<mo>\{</mo>}{x}_i\mathrm{<mo>:</mo>}i=\mathrm{1,}\dots \mathrm{,}N\mathrm{<mo>\}</mo>.}$ The optimization (3.1) can be viewed as a “minmax” problem, and if $m\in H,$ the estimated density ratios $\left\{{\widehat{r}}_i\mathrm{<mo>:</mo>}i\in S_A\right\}$ may lead to a reasonably good estimator

$${\widehat{\mu }}_{uc}=N^{-1}{\displaystyle \sum _{i\in S_A}}\left\{1+\left(\frac{N}{n_A}-1\right){\widehat{r}}_i\right\}{y}_i\mathrm{.}(3.3)$$

Uniform calibration is a new method for non-probability sampling, and there are some technical challenges in (3.1). For example, how to incorporate the design properties of $S_B$ when establishing the theoretical properties of (3.3) has not be fully investigated, and we have finished a working paper about this topic (Wang, Mao and Kim, 2022). The kernel-based method is computationally expensive, especially when the sample sizes are large. It may be interesting to propose a more computationally efficient algorithm for the uniform calibration problem. One possible answer is to consider some other functional spaces, such as the one spanned by B-splines. In addition, it is also of interest to consider how to incorporate more than one reference probability sample, and how to formulate a uniform calibration if we have different covariates in different reference probability samples.

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