Optimal model selection with V-fold cross-validation: how should V be chosen?

V-fold cross-validation is a simple and efficient method for estimator selection when the goal is to minimize the final prediction error. Nevertheless, few theoretical 
results exist about the influence of V on the risk of the final estimator, in particular results taking into account the variability of the cross-validation criterion 
as a function of V.

We will focus on the case-example of model selection for least-squares density estimation. First, a non-asymptotic oracle inequality holds for V-fold cross-validation 
(and its bias corrected version, V-fold penalization), with a leading constant 1+o(1) for V-fold penalization, and the second  order term in the constant decreases when 
V increases. Second, making an exact variance computation allows to quantify the improvement we can expect when V increases. In particular, this computation enlightens 
why the improvement is larger when V goes from 2 to 10 than when V goes from 10 to 100, for instance. Simulation experiments confirm these theoretical results for 
realistic values of the sample size.

This talk is based upon a collaboration with Matthieu Lerasle (CNRS - Universite de Nice, France).
Preprint: http://arxiv.org/abs/1210.5830