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.