A method to find an efficient and robust sampling strategy under model uncertainty
Section 3. Robustness under a misspecified model

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If the finite population is a realization of the superpopulation model (2.4), and if $f,$ $g$ and $\delta$ were known, then an optimal strategy could be defined. In this section we study the robustness of this strategy when the model is misspecified.

We begin by defining how “misspecification” shall be understood in this paper. The working model ${\xi }_0$ reflects the beliefs the statistician has about the relation between the auxiliary variables $x$ and the study variable $y$ at the design stage. We shall assume that a true, unknown model $\xi$ exists. Any deviation of ${\xi }_0$ with respect to $\xi$ is a misspecification of the model. In order to keep the analysis tractable, we limit ourselves to the situation when the working model is of the form (2.4) and the true model, $\xi ,$ is

$$Y_k\mathrm{<mo>=</mo>}f\left(x_k|{\beta }_1\right)+{\epsilon }_k$$

with

$${\text{E}}_{\xi }\left({\epsilon }_k\right)\mathrm{<mo>=</mo>}\mathrm{0,}{\text{V}}_{\xi }\left({\epsilon }_k\right)\mathrm{<mo>=</mo>}{\sigma }^{2}g{\left(x_k|{\beta }_2\right)}^2\text{and}{\text{E}}_{\xi }\left({\epsilon }_k{\epsilon }_l\right)\mathrm{<mo>=</mo>}0\forall k\ne l(3.1)$$

where $\beta \mathrm{<mo>=</mo>}\left({\beta }_1\mathrm{,}{\beta }_2\right)$ is a vector of parameters, $f$ and $g$ as in (2.4) and $\beta \ne \delta .$ The random sample $s$ and the errors ${\epsilon }_k$ are assumed to be independent.

Result 1. If ${\xi }_0$ is assumed when $\xi$ is the true superpopulation model, the model expected value of the design MSE in (2.2), under the difference estimator satisfying condition 1 above, becomes

$${\text{MSE}}_{\xi \text{p}}\left({\widehat{t}}_y\right)\mathrm{<mo>=</mo>}{\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{f\left(x_k|{\beta }_1\right)-f\left(x_k|{\delta }_1\right)}{{\pi }_k}}\right)+{\sigma }^2{\displaystyle \sum _U\left(\frac{1}{{\pi }_k}-1\right)g{\left(x_k|{\beta }_2\right)}^2.}(3.2)$$

The result is proven by noting that $f\left(x_k|{\delta }_1\right)$ takes the role of $z_k$ in (2.5) and by taking into account that $e_k\mathrm{<mo>=</mo>}f\left(x_k|{\beta }_1\right)-f\left(x_k|{\delta }_1\right)+{\epsilon }_k,$ therefore ${\text{E}}_{\xi }{e}_k\mathrm{<mo>=</mo>}f\left(x_k|{\beta }_1\right)-f\left(x_k|{\delta }_1\right)$ and ${\text{E}}_{\xi }{e}_k^2\mathrm{<mo>=</mo>}{\left(f\left(x_k|{\beta }_1\right)-f\left(x_k|{\delta }_1\right)\right)}^2+{\sigma }^{2}g{\left(x_k|{\beta }_2\right)}^2.$ As the model is misspecified, we have deliberately avoided the use of the adjective “anticipated” in Result 1.

Using Result 1, it can be seen that for a design that satisfies condition 2 we obtain

$$\begin{array}{ll}{\text{MSE}}_{\xi \mathrm{,}\pi \text{ps}}\left({\widehat{t}}_y\right)\hfill & \mathrm{<mo>=</mo>}{\left(\frac{{\displaystyle {\sum }_{U}g\left(x_k|{\delta }_2\right)}}{n}\right)}^2{\text{MSE}}_{\pi \text{ps}}\left({\displaystyle \sum _s\frac{f\left(x_k|{\beta }_1\right)-f\left(x_k|{\delta }_1\right)}{g\left(x_k|{\delta }_2\right)}}\right)\hfill \\ \hfill & +{\sigma }^2{\displaystyle \sum _U\left(\frac{n{\displaystyle {\sum }_{U}g\left(x_k|{\delta }_2\right)}}{g\left(x_k|{\delta }_2\right)}-1\right)g{\left(x_k|{\beta }_2\right)}^2}\mathrm{.}(\text{3}\text{.3})\hfill \end{array}$$

It is now possible to see that, even under a mild misspecification as the one considered here, the strategy $\pi \text{ps}\left({\delta }_2\right)-\text{diff}\left({\delta }_1\right)$ is not optimal anymore, as its MSE (3.3) can be greater than the MSE obtained under other designs (3.2). In particular, the strategy using the correct model, i.e., $z_k\mathrm{<mo>=</mo>}f\left(x_k|{\beta }_1\right)$ into the estimator and a design such that ${\pi }_k\propto g\left(x_k|{\beta }_2\right),$ would be more efficient than $\pi \text{ps}\left({\delta }_2\right)-\text{diff}\left({\delta }_1\right).$

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