The Horvitz-Thompson (H-T) estimator is widely used for estimating various types of average treatment effects under network interference. We systematically investigate the optimality properties of H-T estimator under network interference, by embedding it in the class of all linear estimators. In particular, we show that in presence of any kind of network interference, H-T estimator is in-admissible in the class of all linear estimators when using a completely randomized and a Bernoulli design. We also show that the H-T estimator becomes admissible under certain restricted randomization schemes termed as ``fixed exposure designs''. We give examples of such fixed exposure designs. It is well known that the H-T estimator is unbiased when correct weights are specified. Here, we derive the weights for unbiased estimation of various causal effects, and illustrate how they depend not only on the design, but more importantly, on the assumed form of interference (which in many real world situations is unknown at design stage), and the causal effect of interest.

翻译：霍维茨-汤普森（H-T）估计量被广泛用于估计网络干扰下的各类平均处理效应。我们通过将该估计量嵌入所有线性估计量类中，系统研究了其在网络干扰下的最优性质。特别地，我们证明在存在任何形式的网络干扰时，当采用完全随机化和伯努利设计时，H-T估计量在所有线性估计量类中不可容许。同时我们证明，在称为“固定暴露设计”的特定限制性随机化方案下，H-T估计量变为可容许。本文给出了此类固定暴露设计的实例。众所周知，当指定正确权重时，H-T估计量具有无偏性。在此，我们推导了各类因果效应无偏估计的权重，并阐明这些权重不仅取决于实验设计，更关键的是取决于假定的干扰形式（在许多现实情境中，设计阶段未知）以及目标因果效应。