Given samples from a probability distribution, anomaly detection is the problem of determining if a given point lies in a low-density region. This paper concerns calibrated anomaly detection, which is the practically relevant extension where we additionally wish to produce a confidence score for a point being anomalous. Building on a classification framework for anomaly detection, we show how minimisation of a suitably modified proper loss produces density estimates only for anomalous instances. We then show how to incorporate quantile control by relating our objective to a generalised version of the pinball loss. Finally, we show how to efficiently optimise the objective with kernelised scorer, by leveraging a recent result from the point process literature. The resulting objective captures a close relative of the one-class SVM as a special case.