Poisson Sampling, Regression Estimation, and the Delete-a-Group Jackknife

When coupled with the simple expansion estimator, Poisson sampling leads to estimators with higher-than-necessary variances. That problem vanishes when the expansion estimator is replaced by a randomization-consistent regression estimator. A simultaneous estimator for the model variance and randomization mean squared error of this estimation strategy is developed. It is nearly identical to the weighted residual variance estimator, but can be slightly better at estimated the model variance when finite population correction matters. When finite population correction can be ignored, an appropriately-defined delete-a-group jackknife variance estimator is shown to have desirable asymptotic properties making it a practical alternative in many applications.