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

For $\vartheta =(q,t)$ use the estimating equationIts empirical mean vanishes exactly at $(\hat{p},\hat{\tau })$. The population equation has the unique root $(p,{\tau }^{\ast })$, so the Z-estimator is consistent. Linearizing the equation, or simplifying the corresponding sandwich covariance, gives the influence functionHence