Optimal transport (OT) distances are increasingly used as loss functions for\nstatistical inference, notably in the learning of generative models or\nsupervised learning. Yet, the behavior of minimum Wasserstein estimators is\npoorly understood, notably in high-dimensional regimes or under model\nmisspecification. In this work we adopt the viewpoint of projection robust (PR)\nOT, which seeks to maximize the OT cost between two measures by choosing a\n$k$-dimensional subspace onto which they can be projected. Our first\ncontribution is to establish several fundamental statistical properties of PR\nWasserstein distances, complementing and improving previous literature that has\nbeen restricted to one-dimensional and well-specified cases. Next, we propose\nthe integral PR Wasserstein (IPRW) distance as an alternative to the PRW\ndistance, by averaging rather than optimizing on subspaces. Our complexity\nbounds can help explain why both PRW and IPRW distances outperform Wasserstein\ndistances empirically in high-dimensional inference tasks. Finally, we consider\nparametric inference using the PRW distance. We provide an asymptotic guarantee\nof two types of minimum PRW estimators and formulate a central limit theorem\nfor max-sliced Wasserstein estimator under model misspecification. To enable\nour analysis on PRW with projection dimension larger than one, we devise a\nnovel combination of variational analysis and statistical theory.\n