A note on the Hartley-Rao estimator

The Hartley-Rao variance formula was designed to estimate the randomization variance of a Horvitz-Thompson estimator given a systematic probability proportional to size sample from a randomly ordered large population. Using an underappreciated formulation of this variance estimator, one can see that the Hartley-Rao variance estimator is unbiased under a model with a particular error structure given any sample. Moreover, even with a more general error structure, this variance estimator remains nearly model unbiased for a large sample and relatively larger population under mild conditions. A discussion follows concerning an extension of Hartley-Rao variance estimation to linear calibration estimators.