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The SURVEYSELECT Procedure |

Notation in the remainder of this section drops the stratum
subscript *h* for simplicity, but selection is still done
independently within strata if you specify a stratified design.
For a stratified design,
*n* now denotes the sample size for the current stratum,
*N* denotes the stratum population size, and *M*_{i} denotes the
size measure for unit *i* in the stratum.
If the design is not stratified, this notation applies
to the entire sampling frame.

According to the Hanurav-Vijayan algorithm, PROC SURVEYSELECT
first orders units within the stratum in ascending order by
size measure, so that
.Then the procedure selects the PPS sample of *n* observations
as follows:

- The procedure randomly chooses one of the integers
1, 2, ... ,
*n*with probability , where*Z*_{j}=*M*_{j}/*M*, , and, by definition,*Z*_{N+1}= 1/*n*to ensure that . - If
*i*is the integer selected in step 1, the procedure includes the last (*n*-*i*) units of the stratum in the sample, where the units are ordered by size measure as described previously. The procedure then selects the remaining*i*units according to steps 3 through 6 below. - The procedure defines new normed size measures for
the remaining (
*N*-*n*+*i*) stratum units that were not selected in steps 1 and 2, - The procedure selects the next unit from the first (
*N*-*n*+1) stratum units with probability proportional to*a*_{j}(1), where*P*_{k}=*M*_{k}/ (*M*_{k+1}+*M*_{k+2}+ ... +*M*_{N-n+i}) . - If stratum unit
*j*is the unit selected in step 4, then the procedure selects the next unit from units_{1}*j*+1 through_{1}*N*-*n*+2 with probability proportional to*a*_{j}(2,*j*), where_{1} - The procedure repeats step 5 until all
*n*sample units are selected.

If you request the JTPROBS option, PROC SURVEYSELECT computes
the joint selection probabilities for all pairs of selected units
in each stratum. The joint selection probability for units *i* and
*j* in the stratum equals

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