Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 4 / b / ii / Solution

If both outcome models are correct, each residual term in the preceding conditional expectations has mean zero regardless of the working propensity model. Thus ${\tilde{\mu }}_z^{\mathrm{dr}}(X)={\mu }_z(X)$ for $z=0,1$, and their difference is again $\tau (X)$. This proves double robustness.