$${\varvec{P}}$$ -value model selection criteria for exponential families of increasing dimension

Abstract

Let Mi be an exponential family of densities on [0, 1] pertaining to a vector of orthonormal functions bi = (bi1 (x), . . . , bi p (x))T and consider a problem of estimating a density f belonging to such family for unknown set i ⊂ {1, 2, . . . ,m}, based on a random sample X1, . . . , Xn. Pokarowski and Mielniczuk (2011) introduced model selection criteria in a general setting based on p-values of likelihood ratio statistic for H0 : f ∈M0 versus H1 : f ∈Mi\M0,whereM0 is theminimal model. In the paper we study consistency of thesemodel selection criteria when the number of themodels is allowed to increase with a sample size and f ultimately belongs to one of them. The results are then generalized to the case when the logarithm of f has infinite expansionwith respect to (bi (·)) ∞ 1 .Moreover, it is shown howthe results can be applied to study convergence rates of ensuing post-model-selection estimators of the density with respect to Kullback–Leibler distance.We also present results of simulation study comparing small sample performance of the discussed selection criteria and the postmodel- selection estimators with analogous entities based on Schwarz’s rule as well as their greedy counterparts.