Notations

To give a formal description of the scenario, we start with defining the key mathematical notations

• \(\mathcal{P} = \{1, \ldots, N \}\) be a finite population of interest.
• For each \(i \in \mathcal{P}\), let \(y_{i}\) be a value of interest associated with unit \(i\)
• Let \(\mathbf{s} = \{i_{1}, ..., i_{n} \}\) be a subset of distinct elements of \(\mathcal{P}\), which is a sample selected with selection probability \(p(\mathbf{s})\), where \(p\) is known and the value \(y_{i}\) is observed iff \(i \in \mathbf{s}\).
• Sample inclusion probability: \(\pi_{i}\) is the probability that \(i \in \mathbf{s}, i=1,\ldots,N\).
• Population quantity of interest: \(Y = \sum_{i=1}^{N} y_{i}\).

Horvitz-Thompson estimator

The Horvitz-Thompson estimator of the population total is defined as (assuming \(\pi_{i} > 0\) for \(i=1,\ldots,N\))

\[ \hat{Y}_{HT} = \sum_{i=1}^{n} y_{i}/\pi_{i} \] ## Unbiasedness of Horvitz-Thompson estimator Define \(t_{i} = 1\) if \(i \in \mathbf{s}\), and \(0\) otherwise, for \(i=1, \ldots, N\). Therefore we have

\[ \mathbb{E}[\hat{Y}_{HT}] = \mathbb{E}[ \sum_{i=1}^{n} y_{i}/\pi_{i} ] = \mathbb{E}[ \sum_{i=1}^{N} t_{i} y_{i}/\pi_{i} ] = \sum_{i=1}^{N} \mathbb{E}[t_{i} / \pi_{i}] \mathbb{E}[y_{i}] = \sum_{i=1}^{N} y_{i}\] where we take advantage of the linearity of the expectation, the facts that \(t_{i}\) is independent of \(y_{i}\) and \(\mathbb{E}[t_{i}] = \pi_{i}\).