  This paper aims at formulating the issue of ranking multivariate unlabeled
observations depending on their degree of abnormality as an unsupervised
statistical learning task. In the 1-d situation, this problem is usually
tackled by means of tail estimation techniques: univariate observations are
viewed as all the more `abnormal' as they are located far in the tail(s) of the
underlying probability distribution. It would be desirable as well to dispose
of a scalar valued `scoring' function allowing for comparing the degree of
abnormality of multivariate observations. Here we formulate the issue of
scoring anomalies as a M-estimation problem by means of a novel functional
performance criterion, referred to as the Mass Volume curve (MV curve in
short), whose optimal elements are strictly increasing transforms of the
density almost everywhere on the support of the density. We first study the
statistical estimation of the MV curve of a given scoring function and we
provide a strategy to build confidence regions using a smoothed bootstrap
approach. Optimization of this functional criterion over the set of piecewise
constant scoring functions is next tackled. This boils down to estimating a
sequence of empirical minimum volume sets whose levels are chosen adaptively
from the data, so as to adjust to the variations of the optimal MV curve, while
controling the bias of its approximation by a stepwise curve. Generalization
bounds are then established for the difference in sup norm between the MV curve
of the empirical scoring function thus obtained and the optimal MV curve.
