The asymptotic distribution of the delete-dd Jackknife variance estimator for smooth functionals

The delete-d   Jackknife estimate of variance is commonly used in survey sampling. We derive the asymptotic distribution of this estimator for a class of second-order differentiable statistical functionals, including the useful case of estimating the variance of an estimated coefficient in logistic regression. The asymptotic distribution of the variance estimator depends on the quadratic term of the functional through its Hajék projection. We relate these results to existing results in which the variance estimator is seen to have the asymptotic distribution of a “linearized” version. A practical consequence is that d=1d=1 is the best choice for variance estimates of many common statistics.