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

Let $$\mathcal{M }_{\underline{i}}$$ be an exponential family of densities on $$[0,1]$$ pertaining to a vector of orthonormal functions $$b_{\underline{i}}=(b_{i_1}(x),\ldots ,b_{i_p}(x))^\mathbf{T}$$ and consider a problem of estimating a density $$f$$ belonging to such family for unknown set $${\underline{i}}\subset \{1,2,\ldots ,m\}$$ , based on a random sample $$X_1,\ldots ,X_n$$ . Pokarowski and Mielniczuk ( 2011 ) introduced model selection criteria in a general setting based on p-values of likelihood ratio statistic for $$H_0: f\in \mathcal{M }_0$$ versus $$H_1: f\in \mathcal{M }_{\underline{i}}\setminus \mathcal{M }_0$$ , where $$\mathcal{M }_0$$ is the minimal model. In the paper we study consistency of these model selection criteria when the number of the models 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 expansion with respect to $$(b_i(\cdot ))_1^\infty $$ . Moreover, it is shown how the 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 post-model-selection estimators with analogous entities based on Schwarz’s rule as well as their greedy counterparts. Copyright The Author(s) 2014