Discussion
For a sampling design \((\mathfrak{U}, \mathfrak{S}, P)\), the Horvitz-Thompson estimator for population total is given by :
Answer: B
The Horvitz-Thompson estimator of the population total is \(\sum_{i=1}^{N} \frac{y_{i} T_{i}}{\pi_{i}}\).
Here \(T_{i}\) is the indicator that unit i is in the sample and \(\pi_{i} = P(T_{i} = 1)\) its inclusion probability. Each sampled value is weighted by \(\frac{1}{\pi_{i}}\), and since \(E(T_{i}) = \pi_{i}\),
\[ \begin{aligned} E\left(\sum_{i=1}^{N} \frac{y_{i} T_{i}}{\pi_{i}}\right) &= \sum_{i=1}^{N} y_{i} \\ &= Y \end{aligned} \]
so the estimator is unbiased for the total.
Dividing by N (option A) estimates the mean instead. Options C and D drop the indicator \(T_{i}\), so they sum over every unit of the population, sampled or not, which is not a statistic.
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