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
denote one such solution, whose internal
entries
are the replacement of the real numbers
in the controlled selection problem
by the adjacent nonnegative integers. The
entry,
,
equals either
or
,
where
is the greatest integer function. If
is a nonnegative integer,
for all
.
The same rule is applied to the marginal expectations. As noted by Jessen
(1970) and Causey et al. (1985), we primarily pay attention to
that simultaneously satisfy the following constraints for all
and
:
where
equals either
or
,
equals either
or
,
Consider the set of all possible arrays,
,
satisfying (3.1) - (3.4). Since
is the expectation of the sample allocation to
each cell in
,
the following constraints (3.5) and
(3.6) on
in
are especially important.
and
where
,
which depends on a specified algorithm for solving the controlled selection
problem, is the selection probability of the array
and
.
Note that (3.5) and (3.6)
will define a rigorous probability sampling method when randomly selecting any array in
.
Also, note that since
,
(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
,
(3.5) yields
which
indicates the equal allocation for each cell.
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