Optimal Control Variates for Survey Sampling and Causal Inference
Jinglong Zhao
econ.EM, math.OC, stat.ME
2026-08-16
Unifies Hajek, normalized IPW, and AIPW as control variate estimators with the inverse-probability basis, then derives the optimal basis: 45.6% SE reduction on Swiss survey data and 66.75% variance reduction in network interference simulations versus Horvitz-Thompson.
Inverse probability weighting is the workhorse estimator of survey sampling and causal inference. Its simplest form, the Horvitz-Thompson estimator, is unbiased but notoriously noisy: when sampling, treatment, or exposure probabilities are small, the weights explode and the variance grows so large that a study loses all power. Network experiments are hit hardest, because a unit's exposure probability depends on how many neighbors it has, which makes the probabilities heterogeneous and small.
The existing fixes (Hajek, normalized IPW, augmented IPW) look like a scattered toolkit. The paper's first contribution is showing they are one thing: the first-order approximations of Hajek and normalized IPW, and AIPW itself, are all control variate estimators that use the inverse probabilities as their basis. A control variate is a classical Monte Carlo trick where a random variable with known expectation cancels part of the randomness in the estimator.
With the unified frame in place, the question becomes whether any basis beats 1/πi. The author formulates basis selection as a decision problem under uncertainty: model the unknown outcomes as i.i.d. draws and minimize the expected finite-sample variance of the control variate estimator. The answer is spectral.
The relation to AIPW is stated crisply: for a fixed basis and coefficient, the control variate estimator is an AIPW with a one-dimensional working outcome model. The difference is not the presence of augmentation but the information used to pick its direction: classical AIPW estimates outcome regressions from covariates and targets robustness to misspecification, while this paper treats the probabilities as known and optimizes finite-sample variance directly against the randomized design.
Swiss Environmental Panel survey (14,961 units, share of food waste attributed to households):
| Estimator | Estimate | SE | SE vs. HT |
| Horvitz-Thompson | 21.64 | 0.41 | n/a |
| Hajek | 21.64 | 0.22 | −45.6% |
| CV, sample splitting | 21.63 | 0.22 | −45.6% |
Point estimates barely move while standard errors are cut nearly in half.
On the Chinese insurance network experiment (Cai et al. 2015, spillover of one extra friend attending the intensive session), across three shrinking subsamples the sample-splitting CV cuts SE by 12.2%, 27.0%, and 32.0% versus HT. In the smallest panel (n = 168) the Hajek estimator's SE is 94.3% higher than HT's, while CV still delivers a 32% reduction.
Synthetic simulations (n = 100, 10,000 replications): without interference, the oracle optimal basis cuts variance by 22.65% and the feasible sample-splitting version by 21.39%, within a point of the oracle. Under network interference, sample splitting cuts variance by 66.75% versus Hajek's 39.46%. The cost is bias: percent bias rises from Hajek's 12.47% to 54.93% in the network setting, though MSE still favors CV across the board.
For anyone running experiments or surveys, this is a way to tighten confidence intervals without collecting a single additional sample — power translated directly into money. The sample-splitting version is plug-and-play for small-probability weighting regimes: online A/B tests, network experiments, marketing lift studies.
The theoretical value is a clean unification: a family of classic estimators placed in one lineage, plus a quantitative account of when the IPW basis is asymptotically optimal and how far from optimal it sits in finite sample. A long-standing recommendation from the simulation literature (Hesterberg 1988; Owen 2013), to use importance weights as control variates, gets refined: the right object is not the weights themselves but their optimal deformation.
Honestly, in the applications the variance reduction mostly matches Hajek (the two tie on the Swiss data); the incremental gain shows up in small samples with tiny exposure probabilities. This is an incremental improvement, not a free lunch.
The authors list three. The optimal bases depend on moments of the outcome distribution that are unknown in practice, so sample splitting has to estimate them — and the covariance between the two potential outcomes is not identifiable, forcing a zero-covariance assumption with a surrogate matrix. The probabilities are taken as known, which fails in observational studies. And no auxiliary covariates are used, leaving the combination of design-induced control variates with covariates to future work.
One more concern from reading the tables: in the network simulation, bias grows alongside the variance reduction (percent bias 57.57% for sample splitting). The authors' own explanation is that with very small exposure probabilities, the asymptotics need a larger n before they kick in. So the headline variance numbers in small-sample network settings are partly bought with bias, and nominal confidence interval coverage may not hold. The authors themselves flag Panel C (n = 168) as needing cautious interpretation. Both empirical applications reuse classic public datasets, so no point estimate is revised, only the intervals get tighter.