Horvitz-Thompson Estimator

The Horvitz-Thompson estimator is an estimator for population totals and means.

Description

While a population total Y can be determined by measuring all members of the population, an estimate can be calculated using a smaller sample s. For each unit i of the sample, there is a corresponding measurement y_i, and the population total is estimated as:

where π_i is the probability of inclusion in the sample for unit i. Equivalently, let w_i be a weight equal to the inverse of that probability, i.e. w_i = π_i^-1.

It should be clear that the population mean is then estimated as:

where n(s) is the sample size.

Domain Estimator

Let 1_d be an indicator function for a domain d.

A direct estimator for a domain mean is:

The numerator can be considered an estimator for the population total within the domain (Y_d), denoted Ŷ_d. And the denominator can be considered an estimator for the domain total (t_d), denoted t̂_d. Therefore this estimator is a function of two estimators.

Sometimes this estimator is written as a summation over s_d, i.e. the subset of s that is in d, instead. That is:

This formulation ignores measurements from the rest of the sample. Importantly it hides the fact that the estimator is a ratio (i.e. is not linear). As such, the variance is generally approximated using linearization.

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