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

For $\vartheta=(q,t)$ use the <estimating equation>
$$
\psi(Y,Z;\vartheta)=
\binom{Z-q}{ZY/q-(1-Z)Y/(1-q)-t}.
$$
Its empirical mean vanishes exactly at $(\widehat p,\widehat\tau)$. The population equation has the unique root $(p,\tau^*)$, so the <Z-estimator> is consistent. Linearizing the equation, or simplifying the corresponding sandwich covariance, gives the influence function
$$
\phi(Y,Z)
=\frac Zp(Y-\mu_1)-\frac{1-Z}{1-p}(Y-\mu_0).
$$
Hence
$$
\sqrt n(\widehat\tau-\tau^*)
\xrightarrow{d}N(0,V_{\mathrm{estimated}}),
\qquad
V_{\mathrm{estimated}}
=\frac{\operatorname{Var}(Y\mid Z=1)}p
+\frac{\operatorname{Var}(Y\mid Z=0)}{1-p}.
$$

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 2 / b / Solution

An estimating equation sets an empirical average $n^{-1}\sum_i\psi(X_i;\theta)$ equal to zero to define an estimator of $\theta$.


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