• Gene–environment independence;
  • Gene–environment interaction;
  • Generalized odds ratio;
  • Double robustness;
  • Local efficiency

Summary We propose a semiparametric case-only estimator of multiplicative gene–environment or gene–gene interactions, under the assumption of conditional independence of the two factors given a vector of potential confounding variables. Our estimator yields valid inferences on the interaction function if either but not necessarily both of two unknown baseline functions of the confounders is correctly modeled. Furthermore, when both models are correct, our estimator has the smallest possible asymptotic variance for estimating the interaction parameter in a semiparametric model that assumes that at least one but not necessarily both baseline models are correct.