In model assisted survey estimation, one typically uses the generalized difference estimator:

$$ \hat{t}_{ma} = \sum_{k \in U} \hat{m}(\mathbf{x}_k) + \sum_{k \in S} \frac{y_k - \hat{m}(\mathbf{x}_k)}{\pi_k}, $$

where $U$ is the target population, $S$ is a probabilistic sample where each unit $k \in U$ has a probability of $\pi_k$ to be in the sample $S$. Here, $y_k$ are some characteristics of interest that is measured only for the units in the sample (i.e. one has access to the ground truth $y_k$ for $k \in S$), and $\hat{m}: \mathcal{X} \rightarrow \mathcal{Y}$ is a model that predicts the values of $y_k$ for the unit $k$ with a feature vector $\mathbf{k} \in \mathcal{X}$. We assume that the $\mathbf{x}_k$ feature vectors are auxiliary $d$-dimensional known vectors for the entire population $U$. The model $\hat{m}$ is an estimation of the unknown function $m: \mathcal{X} \rightarrow \mathcal{Y}$ on the entire population and the model-assisted survey estimation satisfies some properties that are out of the scope of the question here.

Now, the question is if one wants to conformal prediction on the sample (with train, calibration and validation sets) and then finds the prediction sets $\hat{C}_{k}$ for the entire population, i.e. $k \in U$, what would the aggregated prediction set look like for the entire population ($\hat{C}_U$)? Let's first assume that the pairs $(\mathbf{x}_k, y_k)$ are i.i.d. for $k \in U$ and then relax it, because in reality this assumption does not hold.



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