How should we estimate inverse probability weights with possibly misspecified propensity score models?

Inverse probability weighting is a common remedy for missing data issues, notably in causal inference. Despite its prevalence, practical application is prone to bias from propensity score model misspecification. Recently proposed methods try to rectify this by balancing some moments of covariates between the target and weighted groups. Yet, bias persists without knowledge of the true outcome model. Drawing inspiration from the quasi maximum likelihood estimation with misspecified statistical models, I propose an estimation method minimizing a distance between true and estimated weights with possibly misspecified models. This novel approach mitigates bias and controls mean squared error by minimizing their upper bounds. As an empirical application, it gives new insights into the study of foreign occupation and insurgency in France.