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Data Collectionhard
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In survey sampling, the Horvitz-Thompson estimator for the population total YYY is Y^HT=∑i∈syiπi\hat{Y}_{HT} = \sum_{i \in s} \frac{y_i}{\pi_i}Y^HT​=∑i∈s​πi​yi​​. The Sen-Yates-Grundy (SYG) variance estimator is given by V^SYG=∑i<j∈sπiπj−πijπij(yiπi−yjπj)2\hat{V}_{SYG} = \sum_{i < j \in s} \frac{\pi_i \pi_j - \pi_{ij}}{\pi_{ij}} \left( \frac{y_i}{\pi_i} - \frac{y_j}{\pi_j} \right)^2V^SYG​=∑i<j∈s​πij​πi​πj​−πij​​(πi​yi​​−πj​yj​​)2. What condition must hold for this variance estimator to be non-negative for all possible samples sss?