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

Solution

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

An estimator is doubly robust when it remains consistent if either of two nuisance models is correctly specified, commonly an outcome-regression model or a propensity-score model.


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