5 Extension to some balanced sampling designs

Jae Kwang Kim and Changbao Wu

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We now consider sampling designs which are balanced in $x_i$ in the sense that ${\widehat{t}}_x={\displaystyle {\sum }_{i\in S}{x}_i/{\pi }_i}=t_x$ holds exactly or nearly exactly, where $x_i$ is a $q\text{–}$ dimensional vector and $t_x={\displaystyle {\sum }_{i\in U}{x}_i}$ is known. We assume that the first element of $x_i$ is equal to ${\pi }_i\text{},$ which implicitly assumes that the survey design has fixed sample size. Tillé (2006) provides a comprehensive account of balanced sampling designs.

Deville and Tillé (2005) argue that ${\widehat{t}}_y={\displaystyle {\sum }_{i\in S}{y}_i/{\pi }_i}$ under balanced sampling has a variance that can be approximated by its variance under conditional Poisson sampling. Breidt and Chauvet (2011) using the same approximation derived

$$v({\widehat{t}}_y)=\frac{n}{n-q}{\displaystyle \sum _{i\in S}(1-{\pi }_i){\left(\frac{y_i}{{\pi }_i}-\frac{{\tilde{y}}_i}{{\pi }_i}\right)}^2},\text{ (5.1)}$$

where ${\tilde{y}}_i={x^{\prime }}_i{\widehat{\beta }}_P$ and ${\widehat{\beta }}_P={\left\{{\displaystyle {\sum }_{i\in S}(1-{\pi }_i){\pi }_i^{-2}{x}_i{x^{\prime }}_i}\right\}}^{-1}{\displaystyle {\sum }_{i\in S}(1-{\pi }_i){\pi }_i^{-2}{x}_i{y}_i.}$ Roughly speaking, the variance formula (5.1) can be interpreted as approximating ${\widehat{t}}_y$ under the balanced sampling design by a generalized regression estimator under Poisson sampling. That is, $V({\widehat{t}}_y)\doteq V({\widehat{t}}_P),$ where ${\widehat{t}}_P={\widehat{t}}_y+(t_x-{\widehat{t}}_x{)}^{\prime }{\widehat{\beta }}_P.$ For a formal justification on this approximation, see Fuller (2009b).

The variance formula (5.1) can be used to derive replication weights. To see this, we re-express (5.1) as a jackknife replication variance estimator

$v_J({\widehat{t}}_y)={\displaystyle \sum _{k=1}^n{c}_k}{({\tilde{t}}_y^{(k)}-{\widehat{t}}_y)}^2,$(5.2)

where ${\tilde{t}}_y^{(k)}={\widehat{t}}_y^{(k)}+(t_x-{\widehat{t}}_x^{(k)}{)}^{\prime }{\widehat{\beta }}_P^{(k)},({\widehat{t}}_x^{(k)},{\widehat{t}}_y^{(k)})={\displaystyle {\sum }_{i\in S^{(k)}}{\pi }_i^{-1}}({x^{\prime }}_i,y_i),$

${\widehat{\beta }}_P^{(k)}={\left\{{\displaystyle \sum _{i\in S^{(k)}}(1-{\pi }_i){\pi }_i^{-2}{x}_i{x^{\prime }}_i}\right\}}^{-1}{\displaystyle \sum _{i\in S^{(k)}}(1-{\pi }_i){\pi }_i^{-2}{x}_i{y}_i,}$

$c_k=(1-{\pi }_k)n/(n-q),$ and $S^{(k)}=S\backslash \{k\}.$ To show the asymptotic equivalence between (5.1) and (5.2), we first note that

${\tilde{t}}_y^{(k)}-{\widehat{t}}_y=({\widehat{t}}_y^{(k)}-{\widehat{t}}_y)+(t_x-{\widehat{t}}_x^{(k)}{)}^{\prime }{\widehat{\beta }}_P+(t_x-{\widehat{t}}_x^{(k)}{)}^{\prime }({\widehat{\beta }}_P^{(k)}-{\widehat{\beta }}_P).$

Under certain regularity conditions, we have ${\widehat{\beta }}_P^{(k)}={\widehat{\beta }}_P+O_p(n^{-1})$ and $t_x-{\widehat{t}}_x^{(k)}=x_k/{\pi }_k=O_p(n^{-1}N).$ Here we used the condition $t_x={\widehat{t}}_x$ under the balanced sampling design. It follows that ${\tilde{t}}_y^{(k)}-{\widehat{t}}_y=-{\pi }_k^{-1}(y_k-{x^{\prime }}_k{\widehat{\beta }}_P)+O_p(n^{-2}N),$ and $v_J({\widehat{t}}_y)$ in (5.2) is asymptotically equivalent to ${\displaystyle {\sum }_{k=1}^n{c}_k{\pi }_k^{-2}}{(y_k-{\tilde{y}}_k)}^2,$ which equals $v({\widehat{t}}_y)$ given by (5.1). The variance formula (5.2) is quite useful because it makes the construction of the replication weights quite straightforward for balanced sampling designs. When $n$ is large, the number of replicates can be reduced by using the weight-calibration method described in Section 3.2. Simulation results based on the rejective Poisson sampling of Fuller (2009b), not reported here to save space, showed that the proposed replication variance estimator performs very well.

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