2. Notation and objective criterion

David G. Steel and Robert Graham Clark

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Consider a finite population, $U$ containing $N$ units, consisting of values $Y_i$ for $i\in U.$ A sample $s\in U$ is to be selected using an unequal probability sampling scheme with positive probability of selection ${\pi }_i=P\left[i\in s\right]$ for all units $i\in U.$ A vector of auxiliary variables $x_i$ is assumed to be available either for the whole population, or for all units $i\in s$ with the population total, $t_x={\displaystyle {\sum }_{i\in U}{x}_i}$, also known. The auxiliary variables could consist of, for example, industry, region and size in a business survey, or age, sex and region in a household survey.

In the model-assisted approach (see for example Särndal, Swensson and Wretman 1992), the relationship between a variable of interest and the auxiliary variables is captured in a model, typically of the following form in single-stage surveys:

$$\begin{array}{l}{E}_M\left[Y_i\right]={\beta }^T{x}_i\hfill \\ va{r}_M\left[Y_i\right]={\sigma }^2{z}_i\hfill \\ Y_i\text{ independent of }{Y}_j\text{ for all }i\ne j\hfill \end{array}\}(2.1)$$

where $E_M$ and $va{r}_M$ denote expectation and variance under the model, $\beta$ is a vector of unknown regression parameters, ${\sigma }^2$ is an unknown variance parameter, and $x_i$ and $z_i$ are assumed to be known for all $i\in U.$ Let $E_p$ and $va{r}_p$ denote expectation and variance under repeated probability sampling with all population values held fixed.

The generalized regression estimator is a widely used model-assisted estimator of $t_y:$

$${\widehat{t}}_y={\displaystyle \sum _{i\in s}}{\pi }_i^{-1}\left(y_i-{\widehat{\beta }}^T{x}_i\right)+{\widehat{\beta }}^T{t}_x(2.2)$$

where $\widehat{\beta }$ may be a weighted or unweighted least squares estimate of the regression coefficients of $y_i$ on $x_i$ using sample data. Estimators can also be constructed for nonlinear extensions to model (2.1), but in practice the linear model is almost always used.

The anticipated variance of ${\widehat{t}}_y$ is defined by $E_{M}va{r}_p\left[{\widehat{t}}_y-t_y\right],$ and is approximated by

$$E_{M}va{r}_p\left[{\widehat{t}}_y\right]\approx {\sigma }^2{\displaystyle \sum _{i\in U}}\left({\pi }_i^{-1}-1\right)z_i(2.3)$$

for large samples (Särndal et al. 1992, formula 12.2.12, p. 451) under model (2.1). Model-assisted designs and estimators should minimise $E_{M}va{r}_p\left[{\widehat{t}}_y\right]$ subject to approximate design unbiasedness, $E_p\left[{\widehat{t}}_y\right]=t_y.$ Even if the model is incorrect, (2.2) remains approximately design-unbiased, although it will no longer have the lowest possible large sample anticipated variance. The anticipated variance has been used to motivate model-assisted sample designs in one stage (Särndal et al. 1992) and two stage sampling (Clark and Steel 2007; Clark 2009). One advantage of using the anticipated variance for this purpose is that it depends only on the selection probabilities and a small number of model parameters, which can be roughly estimated when designing the sample. In contrast, $va{r}_p\left[{\widehat{t}}_y\right]$ typically depends on the population values of $y_i$ and on joint probabilities of selection, both of which are difficult to quantify in advance.

The cost of enumerating a sample is assumed to be $C={\displaystyle {\sum }_{i\in s}}{c}_i$ where $c_i$ is the cost of surveying a particular unit $i.$ The values of $c_i$ are usually assumed to be known. Typically $c_i$ are also assumed to be constant for all units in the population, or constant within strata. With the generalization that $c_i$ may be different for every unit $i,$ the cost $C$ depends on the particular sample $s$ selected. The expected cost is $E_p\left[C\right]={\displaystyle {\sum }_{i\in U}{\pi }_i{c}_i}.$ The aim is to minimise the anticipated variance (2.3) subject to a constraint on the expected enumeration cost,

$${\displaystyle \sum _{i\in U}}{\pi }_i{c}_i=C_f\mathrm{.}(2.4)$$

There will also be fixed costs that are not affected by the sample design and so do not have to be included here.

Some notation for population variances and covariances is needed. Consider the pairs $\left(u_i\mathrm{,}{v}_i\right),$ and let $S_{uv}=N^{-1}{\displaystyle {\sum }_{i\in U}}\left(u_i-\overline{u}\right)\left(v_i-\overline{v}\right)$ denote their population covariance, and $S_u^2=N^{-1}{\displaystyle {\sum }_{i\in U}}{\left(u_i-\overline{u}\right)}^2$ denote the population variance of $u_i$ $(i=1,\dots \mathrm{,}N).$ Let $\overline{u}$ and $\overline{v}$ be the population means of $u_i$ and $v_i.$ The population coefficient of variation of $u_i$ is $C_u=S_u/\overline{u}.$ The population relative covariance of $\left(u_i\mathrm{,}{v}_i\right)$ is $C_{u\mathrm{,}v}=S_{uv}/\overline{u}\overline{v}.$ A useful result is

$${\displaystyle \sum _{i\in U}}{u}_i{v}_i=N\overline{u}\overline{v}\left(1+C_{u\mathrm{,}v}\right)\mathrm{.}(2.5)$$

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