3. Robust estimation based on the conditional bias

Cyril Favre Martinoz, David Haziza and Jean-François Beaumont

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To guard against the undue influence of certain units, it is advisable to construct robust estimators of the total $t,$ that is, estimators that reduce the impact of the most influential units. We consider a class of estimators of the form

$${\widehat{t}}_R=\widehat{t}+\Delta \mathrm{,}(3.1)$$

where $\Delta$ is a certain random variable. As we will see in Section 4, the winsorized estimators considered can be written in form (3.1). As in Beaumont et al. (2013), we want to determine the value of $\Delta$ that minimizes the maximum estimated conditional bias of ${\widehat{t}}_R$ in the sample. Formally, we are seeking the value of $\Delta$ that minimizes

$$\underset{i\in S}{{\displaystyle \max }}\left\{\left|{\widehat{B}}_{1i}^R\right|\right\}\mathrm{,}(3.2)$$

where ${\widehat{B}}_{1i}^R$ denotes the estimated conditional bias of ${\widehat{t}}_R$ associated with sampled unit $i.$ This conditional bias is given by

$$\begin{array}{ll}{B}_{1i}^R\hfill & =E_p\left({\widehat{t}}_R|I_i=1\right)-t\hfill \\ \hfill & =B_{1i}^{\text{HT}}+E_p\left(\Delta |I_i=1\right)(3.3)\hfill \end{array}$$

which is estimated by

$${\widehat{B}}_{1i}^R={\widehat{B}}_{1i}^{\text{HT}}+\Delta \mathrm{,}(3.4)$$

where ${\widehat{B}}_{1i}^{\text{HT}}$ is a conditionally unbiased estimator of $B_{1i}^{\text{HT}}.$ If we note that $\Delta$ is a conditionally unbiased estimator of $E_p\left(\Delta |I_i=1\right),$ it follows that the estimator of the conditional bias (3.4) is conditionally unbiased for $B_{1i}^R.$ In other words, we have $E_p\left\{{\widehat{B}}_{1i}^R|I_i=1\right\}=B_{1i}^R.$

Beaumont et al. (2013) showed that the value of $\Delta$ that minimizes (3.2) is given by

$${\Delta }_{\text{opt}}=-\frac{1}{2}\left({\widehat{B}}_{\text{min}}+{\widehat{B}}_{\text{max}}\right)\mathrm{,}$$

where ${\widehat{B}}_{\text{min}}={\min }_{i\in S}\left({\widehat{B}}_{1i}^{\text{HT}}\right)$ and ${\widehat{B}}_{\text{max}}={\max }_{i\in S}\left({\widehat{B}}_{1i}^{\text{HT}}\right).$ Estimator (3.1) then becomes

$${\widehat{t}}_R=\widehat{t}-\frac{1}{2}\left({\widehat{B}}_{\text{min}}+{\widehat{B}}_{\text{max}}\right)\mathrm{.}(3.5)$$

Beaumont et al. (2013) demonstrated that under certain regularity conditions, the estimator (3.5) is design-consistent; that is, ${\widehat{t}}_R-t=O_p\left(N/\sqrt{n}\right)\mathrm{.}$

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