Horvitz-Thompson Estimator

The Horvitz-Thompson (HT) estimator of a population total $Y={\sum }_{i=1}^N{y}_i$ is ${\hat{Y}}_{HT}={\sum }_{i\in s}\frac{y_i}{{\pi }_i}$, summing over the sampled units $s$, where ${\pi }_i$ is unit $i$'s known inclusion probability under the design. The HT estimator is design-unbiased for the population total under any probability sampling design, regardless of how unequal the inclusion probabilities are, because each sampled unit's contribution $y_i/{\pi }_i$ has expectation $y_i$ over repeated sampling (a unit sampled with probability ${\pi }_i$ is expected to represent $1/{\pi }_i$ population units like itself); under simple random sampling, where ${\pi }_i=n/N$ for every unit, the HT estimator reduces exactly to the familiar $N\bar{y}$. The HT estimator's variance depends not only on each unit's marginal inclusion probability but also on the joint inclusion probabilities of every pair of units, ${\pi }_{ij}$, which makes its variance formula and estimation considerably more involved than for SRS, and is one of the main reasons complex survey data analysis requires specialized software rather than standard formulas.

• The HT estimator's design-unbiasedness holds for any probability sampling design whatsoever, no matter how complex, as long as every sampled unit's inclusion probability πᵢ is known and strictly positive, this generality is exactly why it underlies estimation for stratified, cluster, multistage, and PPS designs alike, rather than requiring a separate estimator for each.
• Because the HT estimator's variance depends on joint inclusion probabilities πᵢⱼ for every pair of sampled units, not just marginal probabilities, exact variance calculation can be complex or infeasible for intricate multistage designs, in practice survey statisticians commonly use approximations such as Taylor series linearization or replication methods (jackknife, BRR) instead of the exact HT variance formula.
• The HT estimator can produce unstable estimates, with unexpectedly high variance, when some units have very small inclusion probabilities (very large weights), a known practical weakness that has motivated alternative and hybrid estimators like generalized regression (GREG) estimators, which use auxiliary information to reduce variance relative to the raw HT estimator.