Horvitz–Thompson Unequal Sampling

The Horvitz-Thompson estimator provides an unbiased estimate of a finite population total when elements are sampled with unequal probabilities. Instead of treating each sample unit equally, we weight each unit by the inverse of its inclusion probability. This ensures that units less likely to be selected contribute proportionately more when they do appear, balancing the expected value across all possible samples.

Worked Example

Consider a population of three elements with values y₁=1, y₂=2, y₃=3. The total population is 6. Assume an unequal sampling design where exactly one element is chosen with inclusion probabilities p₁=1/2, p₂=1/3, and p₃=1/6.

The possible Horvitz-Thompson estimates for the total are:

If unit 1 is selected: HT = y₁ / p₁ = 1 / (1/2) = 2
If unit 2 is selected: HT = y₂ / p₂ = 2 / (1/3) = 6
If unit 3 is selected: HT = y₃ / p₃ = 3 / (1/6) = 18

The expected value of the HT estimator is the sum of (Estimate × Probability):

E(HT) = (2 × 1/2) + (6 × 1/3) + (18 × 1/6) = 1 + 2 + 3 = 6.

This matches the actual population total, confirming the estimator is unbiased. In contrast, an unweighted expansion estimator (multiplying the selected unit by the population size 3) would give estimates of 3, 6, and 9, leading to an expected value of (3 × 1/2) + (6 × 1/3) + (9 × 1/6) = 1.5 + 2 + 1.5 = 5. This is biased because it ignores the unequal inclusion probabilities.

Experiment

Use the Horvitz–Thompson Estimator module to interact with different finite population values and sampling designs.

Set the Design control to "Unequal".
Set Value 1 to 1, Value 2 to 2, and Value 3 to 3.
Set the Selected unit to 3 to see the realization if the third unit is drawn.
Observe that the selected HT Estimate is 18.0, but the overall HT Expectation remains exactly 6.0.
Change the Design to "Uniform" and observe how all inclusion probabilities become 1/3, and the variance drops to 0 if all values are equal.

Limitations

The variance of the Horvitz-Thompson estimator depends on the joint inclusion probabilities (the probability that both unit i and unit j are included). Because our design samples exactly one unit (sample size 1), the joint inclusion probabilities for any two distinct units are strictly zero.

Consequently, standard variance estimation formulas that require strictly positive joint inclusion probabilities for all pairs will fail or be inapplicable here. Furthermore, estimating the variance from a single drawn sample is impossible without bias.

References

The principles of unequal probability sampling and the derivation of the Horvitz-Thompson estimator are detailed in standard survey sampling texts.

PSU STAT 506: Lesson 3 discusses unequal probability sampling and provides the foundational theory for using inclusion probabilities to estimate population totals without bias.

Horvitz–Thompson Unequal Sampling