Asymptotic Statistics

van der Vaart (1998) — Asymptotic Statistics

Result

Theorem 3.1 (delta method): if $r_n(T_n-\theta)\Rightarrow T$ and $\phi$ is differentiable at $\theta$, then $r_n(\phi(T_n)-\phi(\theta))\Rightarrow\phi'_\theta(T)$. Chapter 3 works the sample variance under $E X^4<\infty$; Chapter 5 (Theorem 5.23) gives the M-estimator sandwich and the plug-in variance principle.

Use

O6.2 is this theorem applied to the sample-mean vector of $(\mathbf 1_{Z=b}, S\mathbf 1_{Z=b}, S^2)$; O6.3's consistency argument is the plug-in principle made explicit for a polynomial functional.

Verification

Theorem 3.1 read in primary text (the author's lecture notes with identical numbering; book contents corroborate); the sample-variance example is Example 3.2 (primary text); Chapter 20 is the functional delta method (primary text, contents); Theorem 5.23 sits in §5.3 but its number is secondary (audit 5 Sep 2026).