3. Sampling

Piero Demetrio Falorsi and Paolo Righi

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Let $z_k$  be a vector of auxiliary variables available for all $k\in U.$  A sampling design $p\left(s\right)$  is said to be balanced on the auxiliary variables if and only if it satisfies the following balancing equations

$${\displaystyle {\sum }_{k\in s}\frac{z_k}{{\pi }_k}}={\displaystyle {\sum }_{k\in U}{z}_k}(3.1)$$

for each sample $s$  such that $p\left(s\right)>0$  (Deville and Tillé 2004). Depending on the auxiliary variables and the inclusion probabilities, equation (3.1) can be exactly or approximately satisfied in each possible sample; therefore, a balanced sampling design does not always exist. By specifying

$$z_k={\pi }_k{\delta }_k,(3.2)$$

equations (3.1) become

$${\displaystyle {\sum }_{k\in s}{\delta }_k}={\displaystyle {\sum }_{k\in U}{\pi }_k{\delta }_k}.(3.3)$$

In this case, the balancing equations state that the sample size achieved in each subpopulation $U_h$  is equal to the expected size. In different contexts, Ernst (1989) and Deville and Tillé (2004; page 905 Section 7.3), have proved that, (i) with the specification (3.2) and (ii) if the vector of the expected sample sizes, given by $n={\displaystyle {\sum }_{k\in U}{\pi }_k{\delta }_k},$  includes only integer numbers, then a balanced sampling design always exists. Specification (3.2) defines sampling designs that guarantee equation (2.4), upon which we wish to focus on. Deville and Tillé (2004, pages 895 and 905), Deville and Tillé (2005, page 577) and Tillé (2006, page 168) have shown that several customary sampling designs may be considered as special cases of balanced sampling, by properly defining the vectors $\pi$  and ${\delta }_k$  of equation (3.2). These issues are illustrated in Remark 4.2 and in Section 6. Balanced samples may be drawn by means of the Cube method (Deville and Tillé 2004). This strongly facilitates the sample selection of incomplete stratified sampling designs that overcome the computational drawbacks of methods based on linear programming algorithms (Lu and Sitter 2002). The Cube method satisfies (3.1) exactly when (3.2) holds and $n$  is a vector of integers. In the cases of SRSWOR and SSRSWOR, the standard sample selection methods can be used, as well as the Cube method. Deville and Tillé (2005) propose as approximation of the variance for the HT estimator, in the balanced sampling

$$E_p{\left({\widehat{t}}_{\left(dr\right)}-t_{\left(dr\right)}\right)}^2\cong \left[N/\left(N-H\right)\right]\left[{\displaystyle {\sum }_{k\in U}\left(1/{\pi }_k-1\right){\eta }_{\left(dr\right)k}^2}\right](3.4)$$

where $E_p$  denotes the sampling expectation and

$${\eta }_{\left(dr\right)k}=y_{rk}{\gamma }_{dk}-{\pi }_k{{\delta }^{\prime }}_k{\left[A\left(\pi \right)\right]}^{-1}{\displaystyle {\sum }_{j\in U}{\pi }_j\left(1/{\pi }_j-1\right){\delta }_j{y}_{rk}{\gamma }_{dk}}(3.5)$$

with

$$A\left(\pi \right)={\displaystyle {\sum }_{j\in U}{\delta }_j{{\delta }^{\prime }}_j{\pi }_j\left(1-{\pi }_j\right)}.(3.6)$$

Recently, the simulation results in Breidt and Chauvet (2011) confirm that equation (3.4) represents a good approximation of the sampling variance when the balanced equations are satisfied exactly. Variance estimation is studied in Deville and Tillé (2005).

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