Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 28K (Principles of Statistics)

Each leave-one-out statistic has the same distribution as $T_{n-1}$, soConsequently the bias of the jackknife bias correction isThe assumed expansion givesandTheir difference is therefore $O(n^{-2})$, as required.

Since ${\overline{X}}_n\sim N(\mu ,1/n)$,soEvery leave-one-out mean has variance $1/(n-1)$, and henceTherefore

The central limit theorem givesApplying the delta method to $g(x)=x^2$ yieldsFor $\mu =0$ this denotes the degenerate distribution at zero, which is the correct limit at the $\sqrt{n}$ scale.

The given identities implyThus, writingwe have the exact identityThe sample variance satisfies $s_n^2\stackrel{P}{\to }1$, soThe Slutsky theorem and part (c) now give