A note on density model size testing

Let (F_k)_k≥1 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∈F_k. Given a random sample X₁,...,X_n drawn from f, an integer k₀≥1 and a real number α∈(0,1), we introduce a new, simple, explicit α-level consistent testing procedure of the hypothesis {H₀:k^*=k₀} versus the alternative {H₁:k^*≠k₀}. Our method is inspired by the combinatorial tools developed in Devroye and Lugosi and it includes a wide range of density models, such as mixture models, neural networks, or exponential families.