The Horvitz-Thompson Estimator

The Horvitz-Thompson estimator is a general estimator for a population total, which can be used for any probability sampling plan. This includes both sampling with and without replacement. • Let π i be the probability that the i th unit of the population is included in the sample (inclusion probability). • On each unit i, we measure a response y i , and typically seek to estimate: τ = N i=1 y i (population total) OR µ = 1 N N i=1 y i (population mean). Definition: The Horvitz-Thompson (H-T) estimator of τ is given by: τ π = v i=1 y i π i where the sum is taken only over the v distinct units in the sample. • The value v is sometimes referred to as the " effective " sample size. • The higher the probability of selection, π i , of a unit i to the sample, the less weight the corresponding response y i is given. In this way the H-T estimator, like the Hansen-Hurwitz estimator, uses probability to weight the responses in estimating the total. • The primary difference between the H-T and H-H estimator is the fact that the former uses the inclusion probability (π i) of the units to the sample, whereas the latter uses the probability of selection (p i) of a unit for a single draw. The H-H estimator is restricted to random sampling with replacement while the H-T estimator can be used in much wider range of sampling plans.