On the Asymptotic Properties of the Jackknife Histogram

We study the asymptotic normality of the jackknife histogram. For one sample mean, it holds if and only if $r$, the number of observations retained, and $d (= n - r)$, the number of observations deleted, both diverge to infinity. The best convergence rate $n^{-1/2}$ is obtained when $r = O(n)$ and $d = O(n)$. For $U$ statistics of degree 2 and nonlinear statistics admitting the expansion (3.1), similar results are obtained under conditions on $r$ and $d$. A second order approximation based on the Edgeworth expansion is discussed briefly.