Massart Concentration Inequalities

Model selection is a classical topic in statistics. The idea of selecting a model

via penalizing a log-likelihood type criterion goes back to the early seventies

with the pioneering works of Mallows and Akaike. One can find many consis-

tency results in the literature for such criteria. These results are asymptotic

in the sense that one deals with a given number of models and the number of

observations tends to infinity. We shall give an overview of a nonasymptotic

theory for model selection which has emerged during these last ten years. In

various contexts of function estimation it is possible to design penalized log-

likelihood type criteria with penalty terms depending not only on the number

of parameters defining each model (as for the classical criteria) but also on the

? complexity ? of the whole collection of models to be considered. The perfor-

mance of such a criterion is analyzed via non asymptotic risk bounds for the

corresponding penalized estimator which express that it performs almost as

well as if the ? best model ? (i.e. with minimal risk) were known. For practical

relevance of these methods, it is desirable to get a precise expression of the

penalty terms involved in the penalized criteria on which they are based. This

is why this approach heavily relies on concentration inequalities, the proto-

type being Talagrand’s inequality for empirical processes. Our purpose will be

to give an account of the theory and discuss some selected applications such

as variable selection or change points detection.