Discussion
If \(\pi_{i}\) and \(\pi_{ij}\) are respectively the first order and second order inclusion probabilities of a sampling design in PPSWOR, then which of the following relation is true ?
Answer: B
The true relation is \(\sum_{j}\pi_{ij} = (n - 1)\pi_{i}\), the sum over \(j \ne i\).
For any fixed-size design of size \(n\), count the pairs containing unit \(i\). Whenever \(i\) is in the sample, there are exactly \(n - 1\) other units with it, so
\[ \begin{aligned} \sum_{j \ne i}\pi_{ij} &= E\left[I_{i}\sum_{j \ne i}I_{j}\right] \\ &= (n - 1)\pi_{i} \end{aligned} \]
where \(I_{i}\) indicates that unit \(i\) is selected. This identity is used to derive the Yates-Grundy form of the Horvitz-Thompson variance. Summing it over \(i\) gives \(n(n - 1)\), which is the total over all ordered pairs, not the sum for one \(i\) (D); A and C contradict the identity.
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