3. Desirable constraints

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

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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 $[]$ 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 $j$ :

$$\begin{array}{rrrrr}\hfill b_{ijk}& \hfill \ge & \hfill 0& \hfill & \hfill (3.1)\\ \hfill \left|b_{ijk}-a_{ij}\right|& \hfill <& \hfill 1& \hfill & \hfill (3.2)\\ \hfill \left|b_{i.k}-a_{i.}\right|& \hfill <& \hfill 1& \hfill \text{and}& \hfill (3.3)\\ \hfill \left|b_{.jk}-a_{.j}\right|& \hfill <& \hfill 1,& \hfill & \hfill (3.4)\end{array}$$

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

Consider the set of all possible arrays, $\mathfrak{B}=\left\{B_k,k=1,\dots ,L\right\}$ , 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 $B_k(\in \mathfrak{B})$ are especially important.

$$E\left(b_{ijk}|i,j\right)={\displaystyle \sum _{B_k\in \mathfrak{B}}{b}_{ijk}p\left(B_k\right)=a_{ij},i=1,\dots ,R,\text{ and }j=1,\dots ,C}(3.5)$$

and

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

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 ${\displaystyle {\sum }_{i=1}^R{\displaystyle {\sum }_{j=1}^{C}E\left(b_{ijk}|i,j\right)}}=a_{\mathrm{..}}{\displaystyle {\sum }_{B_k\in \mathfrak{B}}p\left(B_k\right)}=a_{\mathrm{..}}$ , (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

$$E\left(b_{ijk}/X_{ijq}|i,j\right)=a_{ij}/X_{ijq}=n/X_q,(3.7)$$

which indicates the equal allocation for each cell.

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