Theory and Methods of Statistics

10.7.2 Heuristic Justification for the Jackknife and the Bootstrap

Let us first justify the validity of the bootstrap estimate. In the arguments given here we denote n^−1/2Δ_n = F_n − F by D_n and $F_n^{*}-F_n$ by $D_n^{*}$. We can simplify notations by writing $\int T^{\prime }\left(x;F\right)d{D}_n(x)$ by $L_F\left(D_n\right)$ and $\int T^{\prime \prime }\left(x_1,x_2;F\right)d{D}_n\left(x_1\right)d{D}_n\left(x_2\right)$ by $Q_F\left(D_n\right)$. If we expand $T\left(F_n^{*}\right)$ about F_n, then we have

$\begin{array}{l}\hfill T\left(F_n^{*}\right)\end{array}$