# 3. Desirable constraints

Sun Woong Kim, Steven G. Heeringa and Peter W. Solenberger

Each controlled selection problem of the form illustrated in Table 2.1 has many possible integer solutions. Let ${B}_{k}$ denote one such solution, whose internal entries ${b}_{ijk}$ are the replacement of the real numbers ${a}_{ij}$ in the controlled selection problem $A$ by the adjacent nonnegative integers. The entry, ${b}_{ijk}$ , equals either $\left[{a}_{ij}\right]$ or $\left[{a}_{ij}\right]+1$ , where $\left[\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\right]$ is the greatest integer function. If ${a}_{ij}$ is a nonnegative integer, ${b}_{ijk}={a}_{ij}$ for all $k$ . The same rule is applied to the marginal expectations. As noted by Jessen (1970) and Causey et al. (1985), we primarily pay attention to ${B}_{k}$ that simultaneously satisfy the following constraints for all $i$ and $j$ :

where ${b}_{i.k}={\sum }_{j=1}^{C}{b}_{ijk}$ equals either $\left[{a}_{i.}\right]$ or $\left[{a}_{i.}\right]+1$ , ${b}_{.jk}={\sum }_{i=1}^{R}{b}_{ijk}$ equals either $\left[{a}_{.j}\right]$ or $\left[{a}_{.j}\right]+1$ ,

Consider the set of all possible arrays, $\mathfrak{B}=\left\{{B}_{k},\text{\hspace{0.17em}}\text{ }k=1,\dots ,L\right\}\text{ }\text{ }$ , satisfying (3.1) - (3.4). Since ${a}_{ij}$ is the expectation of the sample allocation to each cell in $A$ , the following constraints (3.5) and (3.6) on ${b}_{ijk}$ in ${B}_{k}\left(\in \mathfrak{B}\right)$ are especially important.

and

$\sum _{{B}_{k}\in \mathfrak{B}}p\left({B}_{k}\right)=1,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.6\right)$

where $p\left({B}_{k}\right)$ , which depends on a specified algorithm for solving the controlled selection problem, is the selection probability of the array ${B}_{k}$ and $p\left({B}_{k}\right)\ge 0$ .

Note that (3.5) and (3.6) will define a rigorous probability sampling method when randomly selecting any array in $\mathfrak{B}$ . Also, note that since ${\sum }_{i=1}^{R}{\sum }_{j=1}^{C}E\left({b}_{ijk}|i,j\right)={a}_{..}{\sum }_{{B}_{k}\in \mathfrak{B}}p\left({B}_{k}\right)={a}_{..}$ , (3.5) implies (3.6) for any controlled selection problem such as those described in Problems 2.1 through 2.4. In addition, as an illustration, when applied to Problem 2.3, where ${a}_{ij}=n{X}_{ijq}/{X}_{q}$ , (3.5) yields

$E\left({b}_{ijk}/{X}_{ijq}|i,j\right)={a}_{ij}/{X}_{ijq}=n/{X}_{q},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.7\right)$

which indicates the equal allocation for each cell.

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