3. Estimation with common first-stage selection

Guillaume Chauvet and Guylène Tandeau de Marsac

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Here we are interested in the case of two samples selected using a two-stage design, with common first-stage selection. Population $U$ is partitioned to obtain a population $U_I=\left\{u_1\mathrm{,}\dots \mathrm{,}{u}_M\right\}$ of $M$ primary sampling units. In the first stage, a sample $S_I$ of primary sampling units (PSU) is selected, with a selection probability ${\pi }_{Ii}$ for a PSU $u_i$. In the second stage, in each primary sampling unit $u_i\in S_I$, the following is selected: a sample $S_i^A$ in $u_i^A\equiv u_i\cap U_A$, with a (conditional) selection probability ${\pi }_{k|i}^A>0$ for $k\in u_i^A$; a sample $S_i^B$ in $u_i^B\equiv u_i\cap U_B$, with a (conditional) selection probability ${\pi }_{k|i}^B>0$ for unit $k\in u_i^B$. We make the following hypotheses, which are common for two-stage selection: the second stage of selection in the primary sampling unit $u_i$ depends only on $i$; between two primary sampling units $u_i\ne u_j\in S_I$, the samples $S_i^A$ and $S_j^A$ (respectively, $S_i^B$ and $S_j^B$) are conditionally independent to $S_I$ (property of independence). We also assume that within each primary sampling unit $u_i\in S_I$, the sub-samples $S_i^A$ and $S_i^B$ are conditionally independent to $S_I$.

For a domain $d_1\subset U_A$, the sub-total $Y_{d_1}$ is estimated by ${\widehat{Y}}_{d_1}^A={\displaystyle {\sum }_{u_i\in S_I}}{d}_{Ii}{\widehat{Y}}_{d_1\mathrm{,}i}^A$ with $d_{Ii}={\left({\pi }_{Ii}\right)}^{-1}$ the sampling weight of the primary sampling unit $u_i$, ${\widehat{Y}}_{d_1\mathrm{,}i}^A={\displaystyle {\sum }_{k\in S_i^A}}{d}_{k|i}^A{y}_{k}1\left(k\in d_1\right)$ the estimator of the sub-total $Y_{d_1\mathrm{,}i}={\displaystyle {\sum }_{k\in u_i}}{y}_{k}1\left(k\in d_1\right)$ over $d_1\cap u_i$, and $d_{k|i}^A={\left({\pi }_{k|i}^A\right)}^{-1}$ the sampling weight of $k$ in $u_i^A$. For a domain $d_2\subset U_B$, the sub-total $Y_{d_2}$ is estimated by ${\widehat{Y}}_{d_2}^B={\displaystyle {\sum }_{u_i\in S_I}}{d}_{Ii}{\widehat{Y}}_{d_2\mathrm{,}i}^B$ with ${\widehat{Y}}_{d_2\mathrm{,}i}^B={\displaystyle {\sum }_{k\in S_i^B}}{d}_{k|i}^B{y}_{k}1\left(k\in d_2\right)$ the estimator of the sub-total $Y_{d_2\mathrm{,}i}$ and $d_{k|i}^B={\left({\pi }_{k|i}^B\right)}^{-1}$ the sampling weight of $k$ in $u_i^B$. This yields in particular the estimators

$$\begin{array}{ll}{\widehat{Y}}_{ab}^A={\displaystyle \sum _{u_i\in S_I}}{d}_{Ii}{\widehat{Y}}_{ab\mathrm{,}i}^A\text{ where }{\widehat{Y}}_{ab\mathrm{,}i}^A={\displaystyle \sum _{k\in S_i^A}}{d}_{k|i}^A{y}_{k}1\left(k\in ab\right)\mathrm{,}& \text{ (3}\text{.1)}\\ {\widehat{Y}}_b^A={\displaystyle \sum _{u_i\in S_I}}{d}_{Ii}{\widehat{Y}}_{b\mathrm{,}i}^A\text{ where }{\widehat{Y}}_{b\mathrm{,}i}^A={\displaystyle \sum _{k\in S_i^A}}{d}_{k|i}^A{y}_{k}1\left(k\in b\right)\mathrm{,}& \text{ (3}\text{.2)}\\ {\widehat{Y}}_{ab}^B={\displaystyle \sum _{u_i\in S_I}}{d}_{Ii}{\widehat{Y}}_{ab\mathrm{,}i}^B\text{ where }{\widehat{Y}}_{ab\mathrm{,}i}^B={\displaystyle \sum _{k\in S_i^B}}{d}_{k|i}^B{y}_{k}1\left(k\in ab\right)\mathrm{.}& \text{ (3}\text{.3)}\end{array}$$

3.1 Hartley estimator

The Hartley estimator given in (2.1) may be re-expressed as

$${\widehat{Y}}_{\theta }={\displaystyle \sum _{u_i\in S_I}}{d}_{Ii}{\widehat{Y}}_{\theta \mathrm{,}i}\text{ (3}\text{.4)}$$

with ${\widehat{Y}}_{\theta \mathrm{,}i}={\widehat{Y}}_{a\mathrm{,}i}^A+\theta {\widehat{Y}}_{ab\mathrm{,}i}^A+\left(1-\theta \right){\widehat{Y}}_{ab\mathrm{,}i}^B+{\widehat{Y}}_{b\mathrm{,}i}^B$ the Hartley estimator of sub-total $Y_i$ over unit primary sampling unit $u_i$. We get $E\left({\widehat{Y}}_{\theta }|S_I\right)={\displaystyle {\sum }_{i\in S_I}}{d}_{Ii}{Y}_i$, then

$$V\left({\widehat{Y}}_{\theta }\right)=V\left({\displaystyle \sum _{i\in S_I}}{d}_{Ii}{Y}_i\right)+EV\left({\widehat{Y}}_{\theta }|S_I\right)\mathrm{.}\text{ (3}\text{.5)}$$

In (3.5), the first term of the right member does not depend on $\theta$ . Hartley’s optimal estimator can, therefore, be calculated by minimizing the second term only. This gives:

$${\theta }_{opt|S_I}=\frac{EV\left({\widehat{Y}}_{ab}^B|S_I\right)+ECov\left({\widehat{Y}}_{ab}^B\mathrm{,}{\widehat{Y}}_b^B|S_I\right)-ECov\left({\widehat{Y}}_a^A\mathrm{,}{\widehat{Y}}_{ab}^A|S_I\right)}{EV\left({\widehat{Y}}_{ab}^A|S_I\right)+EV\left({\widehat{Y}}_{ab}^B|S_I\right)}\mathrm{,}\text{ (3}\text{.6)}$$

which can be estimated by

$${\widehat{\theta }}_{opt}=\frac{\widehat{V}\left({\widehat{Y}}_{ab}^B\right)+\widehat{Cov}\left({\widehat{Y}}_{ab}^B\mathrm{,}{\widehat{Y}}_b^B\right)-\widehat{Cov}\left({\widehat{Y}}_a^A\mathrm{,}{\widehat{Y}}_{ab}^A\right)}{\widehat{V}\left({\widehat{Y}}_{ab}^A\right)+\widehat{V}\left({\widehat{Y}}_{ab}^B\right)}\text{ (3}\text{.7)}$$

by replacing each variance and covariance term with an unbiased estimator conditional on the first stage.

3.2 Kalton and Anderson estimator

With the sample design considered, we get $d_k^A=d_{Ii}{d}_{k|i}^A$ for any unit $k\in u_i^A$, and $d_k^B=d_{Ii}{d}_{k|i}^B$ for any unit $k\in u_i^B$. Therefore, the Kalton and Anderson estimator given in (2.4) can be re-expressed as

$${\widehat{Y}}_{KA}={\displaystyle \sum _{i\in S_I}}{d}_{Ii}{\widehat{Y}}_{KA\mathrm{,}i}\text{ (3}\text{.8)}$$

with ${\widehat{Y}}_{KA\mathrm{,}i}={\displaystyle {\sum }_{k\in S^A}}{d}_{k|i}^A{m}_{k|i}^A{y}_k+{\displaystyle {\sum }_{k\in S^B}}{d}_{k|i}^B{m}_{k|i}^B{y}_k$ the Kalton and Anderson estimator of the sub-total $Y_i$, where

$$m_{k|i}^A=\{\begin{array}{ll}1\hfill & \text{if }k\in a\cap u_i\mathrm{,}\hfill \\ \frac{d_{k|i}^B}{d_{k|i}^A+d_{k|i}^B}\hfill & \text{if }k\in ab\cap u_i\mathrm{,}\hfill \end{array}\text{ and }{m}_{k|i}^B=\{\begin{array}{ll}1\hfill & \text{if }k\in b\cap u_i\mathrm{,}\hfill \\ \frac{d_{k|i}^A}{d_{k|i}^A+d_{k|i}^B}\hfill & \text{if }k\in ab\cap u_i\mathrm{.}\hfill \end{array}$$

3.3 Bankier estimator

With the sampling design considered, we get ${\pi }_k^{HT}={\pi }_{Ii}\left({\pi }_{k|i}^A+{\pi }_{k|i}^B-{\pi }_{k|i}^A{\pi }_{k|i}^B\right)$ for any $k\in u_i$. Therefore, the Bankier estimator given in (2.5) can be re-expressed as

$${\widehat{Y}}_{HT}={\displaystyle \sum _{i\in S_I}}{d}_{Ii}{\widehat{Y}}_{HT\mathrm{,}i}\text{ (3}\text{.9)}$$

with ${\widehat{Y}}_{HT\mathrm{,}i}={\displaystyle {\sum }_{k\in S_i^A\cup S_i^B}}\left(y_k/{\pi }_{k|i}^{HT}\right)$ the Bankier estimator for the sub-total $Y_i$, and ${\pi }_{k|i}^{HT}={\pi }_{k|i}^A$ if $k\in a$, ${\pi }_{k|i}^{HT}={\pi }_{k|i}^B$ if $k\in b$, ${\pi }_{k|i}^{HT}={\pi }_{k|i}^A+{\pi }_{k|i}^B-{\pi }_{k|i}^A{\pi }_{k|i}^B$ if $k\in ab$.

Each of the three estimators examined is obtained by applying the estimation method PSU by PSU, conditional on the first stage. This result is particularly attractive for Hartley’s optimal method, since the optimal coefficient estimator given in (3.7) only requires variance estimators conditional on the first stage.

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