Can the jackknife be used with a two-phase sample?
The jackknife variance estimator has been shown to have desirable properties when used with smooth estimators based on stratified multi-stage samples. This paper focuses on the use of the jackknife given a particular two-phase sampling design: a stratified with-replacement probability cluster sample is drawn, elements from sampled clusters are then retratified, and simple random subsamples are selected within each second-phase stratum. It turns out that the jackknife can behave reasonably well as an estimator for the variance for one common "expansion" estimator but not for another. Extensions to more complex estimation strategies are then discussed. A Monte Carlo study supports our principal findings.