Coordination of spatially balanced samples
Section 2. Notation

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Let $U_1$ and $U_2$ be a population (subject to change over time) at time 1 and time 2, respectively, or consider that $U_1$ and $U_2$ are two overlapping populations. Consider samples $s_1$ and $s_2$ drawn from $U_1$ and $U_2\mathrm{,}$ using the sampling designs $p_1$ and $p_2\mathrm{,}$ respectively. No restriction about the sampling designs $p_1$ and $p_2$ is necessary to introduce the definitions in this section: they can be fixed or random size sampling designs, with or without replacement.

Let $U\mathrm{<mo>=</mo>}{U}_1\cup U_2\mathrm{.}$ We call $U$ the “overall population”. The set of labels of the units in $U$ is $\left\{\mathrm{1,}\mathrm{2,}\dots \mathrm{,}i\mathrm{,}\dots \mathrm{,}N\right\}\mathrm{.}$ We define on $U$ the joint sampling design $p$ used to select a couple $\left(s_1\mathrm{,}{s}_2\right)\mathrm{.}$ The samples $s_1$ and $s_2$ are coordinated if $p\left(s_1\mathrm{,}{s}_2\right)\ne p_1\left(s_1\right)p_2\left(s_2\right),$ that is the samples are not drawn independently (see Cotton and Hesse, 1992; Mach, Reiss and Şchiopu-Kratina, 2006). Let ${\pi }_{i1}\mathrm{<mo>=</mo>}P\left(i\in s_1\right)$ and ${\pi }_{i2}\mathrm{<mo>=</mo>}P\left(i\in s_2\right)$ be the first-order inclusion probabilities of unit $i\in U$ in the first and second sample, respectively. It follows that ${\pi }_{i1}\mathrm{<mo>=</mo>\; 0}$ if $i\notin U_1$ and ${\pi }_{i2}\mathrm{<mo>=</mo>\; 0}$ if $i\notin U_2\mathrm{.}$ Thus, it is not necessary to identify explicitly the subpopulation memberships.

Let ${\pi }_{i\mathrm{,}12}=P\left(i\in s_1\text{}\mathrm{,}i\in s_2\right)$ be the joint inclusion probability of unit $i\in U$ in both samples $s_1$ and $s_2\mathrm{.}$ If the samples $s_1$ and $s_2$ are selected independently, ${\pi }_{i\mathrm{,}12}\mathrm{<mo>=</mo>}{\pi }_{i1}{\pi }_{i2}\text{}\mathrm{,}$ for all $i\in U\mathrm{.}$

Let $c$ be the overlap between $s_1$ and $s_2\text{}\mathrm{,}$ which represents the number of common units of the two samples; it is in most of the cases a random variable. The coordination degree of $s_1$ and $s_2$ is measured by the expected overlap

$$E\left(c\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in U}{\pi }_{i\mathrm{,}12}}\mathrm{,}$$

where ${\pi }_{i\mathrm{,}12}\mathrm{<mo>=</mo>}P\left(i\in s_1\text{}\mathrm{,}i\in s_2\right)\mathrm{.}$ By using the Fréchet bounds of the joint probability ${\pi }_{i\mathrm{,}12}$ it follows that

$${\displaystyle \sum _{i\in U}\max \left(\mathrm{0,}{\pi }_{i1}+{\pi }_{i2}-1\right)}\le E\left(c\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in U}{\pi }_{i\mathrm{,}12}}\le {\displaystyle \sum _{i\in U}\min \left({\pi }_{i1}\text{}\mathrm{,}{\pi }_{i2}\right)}\mathrm{.}(2.1)$$

In negative coordination one wants to achieve the left bound in expression (2.1), that is ${\displaystyle {\sum }_{i\in U}\max \left(\mathrm{0,}{\pi }_{i1}+{\pi }_{i2}-1\right)}\mathrm{<mo>=</mo>}E\left(c\right)\mathrm{,}$ while in positive coordination the right bound, that is $E\left(c\right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}\min \left({\pi }_{i1}\text{}\mathrm{,}{\pi }_{i2}\right)}\mathrm{.}$ Thus, to optimize the sample coordination process, the goal is to achieve these bounds, prior to coordination type, positive or negative. Using the terminology of Matei and Tillé (2005) the left side-part in (2.1) is called the Absolute Lower Bound (ALB) and the right side-part in (2.1) the Absolute Upper Bound (AUB).

The focus here is on sample coordination using PRNs. The PRN method was originally introduced by Brewer et al. (1972) to coordinate Poisson samples. Poisson sampling with PRNs reaches the Fréchet bounds given in equation (2.1). Yet, it results in a random sample size and does not provide spatially balanced samples. In order to achieve spatial balance, the local pivotal method (Grafström et al., 2012) and the spatially correlated Poisson sampling (Grafström, 2012) are used. Both sampling designs provide a good degree of spatial balance (see Grafström et al., 2012, for some empirical results). Moreover, since both are fixed size $\pi \text{ps}$ sampling designs (probability proportional to size sampling, see Särndal, Swensson and Wretman, 1992, page 90), the precision of the estimators is in general improved compared to Poisson sampling.

In what follows, we consider the sampling designs $p_1$ and $p_2$ to be without replacement, and the expected sample sizes of $s_1$ and $s_2$ are denoted by $n_1$ and $n_2\text{}\mathrm{,}$ respectively.

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