Horvitz–Thompson vs Hájek

Monte Carlo of two IPW estimators; shift the outcome level and only the normalised (Hájek) version stays stable.

Day 44: IPW & Hájek Estimators of the ATE

Confounding / weakness of overlap1.5Outcome level (baseline spending, $)20Clip propensity scores to [c, 1−c]0.010Sample size300

Horvitz–Thompson

sd 3.497

bias -0.132

Hájek

sd 0.485

bias 0.049

effective sample size

167

of n = 300

sd ratio HT ÷ Hájek

7.20

Horvitz–ThompsonHájek (normalised weights)

Weighting by 1/e(x) recreates a randomised comparison. Horvitz–Thompson divides by n; Hájek divides by the sum of the weights, so adding a constant to every outcome cannot move it. Raise the outcome level and watch HT spread out while Hájek stays put. True effect +1.

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