{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 $H_0: fin mathcal{M }_0$ versus $H_1: fin mathcal{M }_{underline{i}}setminus mathcal{M }_0$ , 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.