A method to find an efficient and robust sampling strategy under model uncertainty
Section 5. The risk measure under the Generalized Regression Estimator

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The difference estimator (2.1) requires that ${\delta }_1$ is fully specified in order to calculate $f\left(x_k|{\delta }_1\right),$ which is undesirable from a practical standpoint. The generalized regression (GREG) estimator is an alternative that allows for the estimation of all or some of the components of ${\delta }_1$ at the cost of introducing a small bias. In this section we adapt the material in Sections 2 to 4 to strategies using the GREG estimator.

We define the generalized regression estimator in a more general way than in Särndal et al. (1992) as follows. Let $a_k$ $\left(k\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,N\right)$ be a weight defined by the statistician and ${\delta }_1\mathrm{<mo>=</mo>}\left({\delta }_1^{*},{\delta }_1^{**}\right)$ where ${\delta }_1^{*}$ is fixed and ${\delta }_1^{**}$ is to be estimated. Let also

$${\widehat{\delta }}_{1s}^{**}\mathrm{<mo>=</mo>}{\text{argmin}}_{{\delta }_1^{**}}{\displaystyle \sum _s\frac{{\left(y_k-f\left(x_k|{\delta }_1\right)\right)}^2}{a_k{\pi }_k}}$$

and ${\widehat{\delta }}_{1s}\mathrm{<mo>=</mo>}\left({\delta }_1^{*}\mathrm{,}{\widehat{\delta }}_{1s}^{**}\right).$ The GREG estimator is

$${\widehat{t}}_{\text{greg}}\mathrm{<mo>=</mo>}\left({\displaystyle \sum _{U}f\left(x_k|{\widehat{\delta }}_{1s}\right)}-{\displaystyle \sum _s\frac{f\left(x_k|{\widehat{\delta }}_{1s}\right)}{{\pi }_k}}\right)+{\displaystyle \sum _s\frac{y_k}{{\pi }_k}}.(5.1)$$

An approximation to the design MSE of the GREG estimator is of the form (2.2) with $e_k\mathrm{<mo>=</mo>}{y}_k-f\left(x_k|{\widehat{\delta }}_{1U}\right)$ where ${\widehat{\delta }}_{1U}\mathrm{<mo>=</mo>}\left({\delta }_1^{*}\mathrm{,}{\widehat{\delta }}_{1U}^{**}\right)$ and

$${\widehat{\delta }}_{1U}^{**}\mathrm{<mo>=</mo>}{\text{argmin}}_{{\delta }_1^{**}}{\displaystyle \sum _U\frac{{\left(y_k-f\left(x_k|{\delta }_1\right)\right)}^2}{a_k}}.$$

Example 1. Let us consider the case where $f\left(x_k|{\delta }_1\right)\mathrm{<mo>=</mo>}{\delta }_{\mathrm{1,}1}{x}_{1k}^{{\delta }_{\mathrm{1,}J+1}}+{\delta }_{\mathrm{1,}2}{x}_{2k}^{{\delta }_{\mathrm{1,}J+2}}+\dots +{\delta }_{\mathrm{1,}J}{x}_{Jk}^{{\delta }_{\mathrm{1,}2J}}.$ Let ${\delta }_1^{*}\mathrm{<mo>=</mo>}\left({\delta }_{\mathrm{1,}J+1}\mathrm{,}\dots ,{\delta }_{\mathrm{1,}2J}\right),$ ${\delta }_1^{**}\mathrm{<mo>=</mo>}{\left({\delta }_{\mathrm{1,}1}\mathrm{,}\dots ,{\delta }_{\mathrm{1,}J}\right)}^{\prime }$ and $x_k^{\delta }\mathrm{<mo>=</mo>}\left(x_{1k}^{{\delta }_{\mathrm{1,}J+1}}\mathrm{,}\dots ,x_{Jk}^{{\delta }_{\mathrm{1,}2J}}\right).$ In this case we obtain

$${\widehat{\delta }}_{1s}^{**}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _s\frac{x_k^{{\delta }^{{}^{\prime }}}{x}_k^{\delta }}{a_k{\pi }_k}}\right)}^{-1}{\displaystyle \sum _s\frac{x_k^{{\delta }^{{}^{\prime }}}{y}_k}{a_k{\pi }_k}}\text{and}{\widehat{\delta }}_{1U}^{**}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _U\frac{x_k^{{\delta }^{{}^{\prime }}}{x}_k^{\delta }}{a_k}}\right)}^{-1}{\displaystyle \sum _U\frac{x_k^{{\delta }^{{}^{\prime }}}{y}_k}{a_k}}\mathrm{.}$$

Letting the exponents ${\delta }_1^{*}\mathrm{<mo>=</mo>}\left({\delta }_{\mathrm{1,}J+1}\mathrm{,}\dots ,{\delta }_{\mathrm{1,}2J}\right)\mathrm{<mo>=</mo>}\left(\mathrm{1,}\dots ,1\right),$ we obtain the classical expression of the GREG estimator found in Särndal et al. (1992).

Example 2. The case with only one auxiliary variable, i.e., $f\left(x_k|{\delta }_1\right)\mathrm{<mo>=</mo>}{\delta }_{10}+{\delta }_{11}{x}_k^{{\delta }_{12}}$ with $a_k\mathrm{<mo>=</mo>}1,$ ${\delta }_1^{*}\mathrm{<mo>=</mo>}{\delta }_{12}$ and ${\delta }_1^{**}\mathrm{<mo>=</mo>}{\left({\delta }_{10}\mathrm{,}{\delta }_{11}\right)}^{\prime }$ is known as the regression estimator. In this case we obtain the well known result that the design MSE can be approximated by expression (2.2) with $e_k\mathrm{<mo>=</mo>}{y}_k-f\left(x_k|{\widehat{\delta }}_{1U}\right)$ where $f\left(x_k|{\widehat{\delta }}_{1U}\right)\mathrm{<mo>=</mo>}{\widehat{\delta }}_{10}+{\widehat{\delta }}_{11}{x}_k^{{\delta }_{12}}$ and

$${\widehat{\delta }}_{11}\mathrm{<mo>=</mo>}\frac{N{\displaystyle {\sum }_U{x}_k^{{\delta }_{12}}{y}_k}-{\displaystyle {\sum }_U{x}_k^{{\delta }_{12}}}{\displaystyle {\sum }_U{y}_k}}{N{\displaystyle {\sum }_U{x}_k^{2{\delta }_{12}}}-{\left({\displaystyle {\sum }_U{x}_k^{{\delta }_{12}}}\right)}^2}\text{and}{\widehat{\delta }}_{10}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U{y}_k}-{\widehat{\delta }}_{11}\frac{1}{N}{\displaystyle \sum _U{x}_k^{{\delta }_{12}}}\mathrm{.}$$

The misspecified model

Let us consider again the situation where the statistician uses the working model (2.4) but the true model is of the form (3.1) with ${\beta }_1\mathrm{<mo>=</mo>}\left({\beta }_1^{*}\mathrm{,}{\beta }_1^{**}\right),$ where ${\beta }_1^{*}$ is the counterpart of the fixed component ${\delta }_1^{*}.$ The following result states a condition under which Result 1 is valid for the GREG estimator.

Result 2. If ${\xi }_0$ is assumed when $\xi$ is the true superpopulation model, ${\widehat{\delta }}_{1s}^{**}\to {\widehat{\delta }}_{1U}^{**}$ as $n\to \infty$ and ${\widehat{\delta }}_{1U}^{**}$ converges to some ${\delta }_1^{**}$ as $N\to \infty ,$ then

$${\text{MSE}}_{\xi \text{p}}\left({\widehat{t}}_{\text{greg}}\right)\to {\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{f\left(x_k|{\beta }_1\right)-f\left(x_k|{\delta }_1\right)}{{\pi }_k}}\right)+{\sigma }^2{\displaystyle \sum _U\left(\frac{1}{{\pi }_k}-1\right)g{\left(x_k|{\beta }_2\right)}^2}(5.2)$$

where ${\delta }_1\mathrm{<mo>=</mo>}\left({\delta }_1^{*}\mathrm{,}{\delta }_1^{**}\right).$

Proof. Note that if ${\widehat{\delta }}_{1s}^{**}\to {\delta }_{1U}^{**},$ then ${\widehat{\delta }}_{1s}\mathrm{<mo>=</mo>}\left({\delta }_1^{*}\mathrm{,}{\widehat{\delta }}_{1s}^{**}\right)\to \left({\delta }_1^{*}\mathrm{,}{\widehat{\delta }}_{1U}^{**}\right)\mathrm{<mo>=</mo>}{\widehat{\delta }}_{1U}.$ Thus, $f\left(x_k|{\widehat{\delta }}_{1s}\right)\to f\left(x_k|{\widehat{\delta }}_{1U}\right).$ In turn, if ${\widehat{\delta }}_{1U}^{**}\to {\delta }_1^{**},$ then ${\widehat{\delta }}_{1U}\mathrm{<mo>=</mo>}\left({\delta }_1^{*}\mathrm{,}{\widehat{\delta }}_{1U}^{**}\right)\to \left({\delta }_1^{*}\mathrm{,}{\delta }_1^{**}\right)\mathrm{<mo>=</mo>}{\delta }_1.$ Thus $f\left(x_k|{\widehat{\delta }}_{1U}\right)\to f\left(x_k|{\delta }_1\right).$ Therefore,

$$\begin{array}{ll}{\text{MSE}}_{\xi \text{p}}\left({\widehat{t}}_{\text{greg}}\right)\hfill & \mathrm{<mo>=</mo>}{\text{MSE}}_{\xi \text{p}}\left(\left({\displaystyle \sum _{U}f\left(x_k|{\widehat{\delta }}_{1s}\right)}-{\displaystyle \sum _s\frac{f\left(x_k|{\widehat{\delta }}_{1s}\right)}{{\pi }_k}}\right)+{\displaystyle \sum _s\frac{y_k}{{\pi }_k}}\right)\hfill \\ \hfill & \to {\text{MSE}}_{\xi \text{p}}\left(\left({\displaystyle \sum _{U}f\left(x_k|{\delta }_1\right)}-{\displaystyle \sum _s\frac{f\left(x_k|{\delta }_1\right)}{{\pi }_k}}\right)+{\displaystyle \sum _s\frac{y_k}{{\pi }_k}}\right)\mathrm{,}\hfill \end{array}$$

which, by Result 1, is (5.2).

Example 3 (Continuation of Example 1). Let the working model be as in Example 1 and the true model be $f\left(x_k|{\beta }_1\right)\mathrm{<mo>=</mo>}{\beta }_{\mathrm{1,}1}{x}_{1k}^{{\beta }_{\mathrm{1,}J+1}}+{\beta }_{\mathrm{1,2}}{x}_{2k}^{{\beta }_{\mathrm{1,}J+2}}+\dots +{\beta }_{\mathrm{1,}J}{x}_{Jk}^{{\beta }_{\mathrm{1,}2J}}.$ Let also ${\beta }_1^{*}\mathrm{<mo>=</mo>}\left({\beta }_{\mathrm{1,}J+1}\mathrm{,}\dots ,{\beta }_{\mathrm{1,}2J}\right),$ ${\beta }_1^{**}\mathrm{<mo>=</mo>}{\left({\beta }_{\mathrm{1,}1}\mathrm{,}\dots ,{\beta }_{\mathrm{1,}J}\right)}^{\prime }$ and $x_k^{\beta }\mathrm{<mo>=</mo>}\left(x_{1k}^{{\beta }_{\mathrm{1,}J+1}}\mathrm{,}\dots ,x_{Jk}^{{\beta }_{\mathrm{1,}2J}}\right).$ In this case, ${\widehat{\delta }}_{1U}^{**}\to A{\beta }_1^{**},$ where

$$A\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _U\frac{x_k^{{\delta }^{{}^{\prime }}}{x}_k^{\delta }}{a_k}}\right)}^{-1}{\displaystyle \sum _U\frac{x_k^{{\delta }^{{}^{\prime }}}{x}_k^{\beta }}{a_k}}\mathrm{,}$$

and (5.2) becomes

$${\text{MSE}}_{\xi \text{p}}\left({\widehat{t}}_{\text{greg}}\right)\to {\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{\left(x_k^{\beta }-x_k^{\delta }A\right){\beta }_1^{**}}{{\pi }_k}}\right)+{\sigma }^2\left({\displaystyle \sum _U\frac{g{\left(x_k|{\beta }_2\right)}^2}{{\pi }_k}}-{\displaystyle \sum _{U}g{\left(x_k|{\beta }_2\right)}^2}\right)\mathrm{.}(5.3)$$

Example 4 (Continuation of Example 2). Let the working model be as in Example 2 and the true model be $f\left(x_k|{\beta }_1\right)\mathrm{<mo>=</mo>}{\beta }_{10}+{\beta }_{11}{x}_k^{{\beta }_{12}}$ with ${\beta }_1^{*}\mathrm{<mo>=</mo>}{\beta }_{12}$ and ${\beta }_1^{**}\mathrm{<mo>=</mo>}{\left({\beta }_{10}\mathrm{,}{\beta }_{11}\right)}^{\prime }.$ It can be shown that (5.2) becomes

$${\text{MSE}}_{\xi \text{p}}\left({\widehat{t}}_{\text{greg}}\right)\to {\beta }_{11}^2{\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{v_k}{{\pi }_k}}\right)+{\sigma }^2\left({\displaystyle \sum _U\frac{g{\left(x_k|{\beta }_2\right)}^2}{{\pi }_k}}-{\displaystyle \sum _{U}g{\left(x_k|{\beta }_2\right)}^2}\right)(5.4)$$

with

$$v_k\mathrm{<mo>=</mo>}\left(x_k^{{\beta }_{12}}-{\overline{x}}^{{\beta }_{12}}\right)-\left(x_k^{{\delta }_{12}}-{\overline{x}}^{{\delta }_{12}}\right)\frac{S_{\beta \mathrm{,}\delta }}{S_{\delta \mathrm{,}\delta }}\mathrm{,}(5.5)$$

and

$$\begin{array}{ll}{\overline{x}}^{{\beta }_{12}}=\frac{1}{N}{\displaystyle \sum _U{x}_k^{{\beta }_{12}}}\hfill & S_{\beta \mathrm{,}\delta }=\frac{1}{N-1}{\displaystyle \sum _U\left(x_k^{{\beta }_{12}}-{\overline{x}}^{{\beta }_{12}}\right)\left(x_k^{{\delta }_{12}}-{\overline{x}}^{{\delta }_{12}}\right)}\hfill \\ {\overline{x}}^{{\delta }_{12}}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U{x}_k^{{\delta }_{12}}}\hfill & S_{\delta \mathrm{,}\delta }\mathrm{<mo>=</mo>}\frac{1}{N-1}{\displaystyle \sum _U{\left(x_k^{{\delta }_{12}}-{\overline{x}}^{{\delta }_{12}}\right)}^2}.\hfill \end{array}$$

Note that (5.4) does not depend on ${\beta }_{10}.$

For the particular case developed in Examples 2 and 4, where $f\left(x_k|\beta \right)\mathrm{<mo>=</mo>}{\beta }_{10}+{\beta }_{11}{x}_k^{{\beta }_{12}}$ and $f\left(x_k|\delta \right)\mathrm{<mo>=</mo>}{\delta }_{10}+{\delta }_{11}{x}_k^{{\delta }_{12}},$ an alternative approximation of ${\sigma }^2$ is (Proof in the Appendix)

$${\sigma }^2\approx {\beta }_{11}^2{F}_0\text{with}{F}_0\mathrm{<mo>=</mo>}\frac{1}{{\overline{x}}^{2{\beta }_2}}\frac{S_{\mathrm{1,}\beta }^2}{S_{\mathrm{1,}1}}\left(\frac{1}{R_{x\mathrm{,}y}^2}-\frac{1}{R_{\mathrm{1,}\beta }^2}\right)(5.6)$$

where

$${\overline{x}}^{2{\beta }_2}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U{x}_k^{2{\beta }_2}}{S}_{\mathrm{1,}\beta }\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U\left(x_k-\overline{x}\right)\left(x_k^{{\beta }_{12}}-{\overline{x}}^{{\beta }_{12}}\right)}{S}_{\mathrm{1,}1}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U{\left(x_k-\overline{x}\right)}^2}$$

with $\left|R_{x\mathrm{,}y}\right|\le \left|R_{\mathrm{1,}\beta }\right|$ and $R_{\mathrm{1,}\beta }$ and $R_{x\mathrm{,}y}$ are, respectively, the correlation coefficients between $x$ and $x^{{\beta }_{12}}$ and between $x$ and $y.$ The latter is unknown but often some decent guess about it is available.

The approximation of ${\sigma }^2$ in (5.6) is more convenient than the one in (4.2) as now we have that (5.4) is approximated by

$${\text{MSE}}_{\xi \text{p}}\left({\widehat{t}}_{\text{greg}}\right)\approx {\beta }_{11}^2\left[{\text{MSE}}_{\text{p}}\left({\displaystyle \sum _s\frac{v_k}{{\pi }_k}}\right)+F_0\left({\displaystyle \sum _U\frac{g{\left(x_k|\beta \right)}^2}{{\pi }_k}}-{\displaystyle \sum _{U}g{\left(x_k|\beta \right)}^2}\right)\right](5.7)$$

with $v_k$ given by (5.5). This expression depends neither on the intercept ${\beta }_{01}$ nor the parameter $\sigma ,$ and the slope ${\beta }_{11}$ becomes a proportionality constant that can be ignored.

The risk measure

As in Section 4, the asymptotic model expected MSE of the GREG estimator given by Result 2 can be seen as the loss incurred by assuming that $\delta$ is the true parameter when it is, in fact, $\beta .$ Assuming a prior distribution on $\beta ,$ the risk (4.1) can be calculated.

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