### 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^{−1/2}Δ_{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

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

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