  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.
