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Since

the estimator is unbiased if and only if for all

This result enables us to construct two variance estimators. The first one is called the Horvitz‐Thompson estimator. It is obtained by applying Result 2.7 to Expression (2.3):

(2.6)

This estimator can take negative values, but is unbiased.

The second one is called Sen–Yates–Grundy estimator (see Sen, 1953; Yates & Grundy, 1953). It is unbiased only for designs with fixed sample size s. It is obtained by applying Result 2.7 to Expression (2.4):

(2.7)

This estimator can also take negative values, but when for all then the estimator is always positive. This condition is called the Yates–Grundy condition.

The Yates–Grundy condition is a special case of the negative correlation property defined as follows: