Let $W=Y-\theta\sim G_0$ and suppose $a=\mathbb E_{G_0}[\psi'(W)]\ne0$. A first-order expansion of the <estimating equation> $n^{-1}\sum_i\psi(Y_i-T_n)=0$ gives the <asymptotic linear representation>
$$
\sqrt n(T_n-\theta)
=\frac1a\frac1{\sqrt n}\sum_{i=1}^n\psi(W_i)+o_p(1).
$$
The <central limit theorem> therefore yields
$$
\sqrt n(T_n-\log\sigma^2)
\xrightarrow d
N\!\left(0,
\frac{\mathbb E_{G_0}[\psi(W)^2]}
{\{\mathbb E_{G_0}[\psi'(W)]\}^2}
\right).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 1 / c / Solution

An estimator has an asymptotic linear representation with influence function $\psi$ when
$$
\sqrt n(\widehat\theta-\theta)
=\frac1{\sqrt n}\sum_{i=1}^n\psi(X_i)+o_p(1).
$$


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