A note on density model size testing

Let (F-k) (kgreater than or equal to1). be a nested family of parametric classes of densities with finite Vapnik-Chervonenkis dimension. Let f be a probability density belonging to F-k*, where k* is the unknown smallest integer such that f is an element of F-k. Given a random sample X-1,..., X-n drawn from f, an integer k(o) greater than or equal to 1 and a real number alpha is an element of (0, 1), we introduce a new, simple, explicit alpha-level consistent testing procedure of the null hypothesis {H-0 : k* = k(0)} versus the alternative {H-1 : k* not equal k(0)}. Our method is inspired by the combinatorial tools developed in Devroye and Lugosi [1] and it includes a wide range of density models, such as mixture models, neural networks, or exponential families.