Summary: We consider the problem of nonparametric estimation of a concave regression function \(F\). We show that the supremum distance between the least squares estimator and \(F\) on a compact interval is typically of order \((\log(n)/n)^{2/5}\). This entails rates of convergence for the estimator’s derivative. Moreover, we discuss the impact of additional constraints on \(F\) such as monotonicity and pointwise bounds. Then we apply these results to the analysis of current status data, where the distribution function of the event times is assumed to be concave.