<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\varvec{P}}$$</EquationSource> </InlineEquation>-value model selection criteria for exponential families of increasing dimension
Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\mathcal{M }_{\underline{i}}$$</EquationSource> </InlineEquation> be an exponential family of densities on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$[0,1]$$</EquationSource> </InlineEquation> pertaining to a vector of orthonormal functions <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$b_{\underline{i}}=(b_{i_1}(x),\ldots ,b_{i_p}(x))^\mathbf{T}$$</EquationSource> </InlineEquation> and consider a problem of estimating a density <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$f$$</EquationSource> </InlineEquation> belonging to such family for unknown set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${\underline{i}}\subset \{1,2,\ldots ,m\}$$</EquationSource> </InlineEquation>, based on a random sample <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$X_1,\ldots ,X_n$$</EquationSource> </InlineEquation>. Pokarowski and Mielniczuk (<CitationRef CitationID="CR9">2011</CitationRef>) introduced model selection criteria in a general setting based on p-values of likelihood ratio statistic for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$H_0: f\in \mathcal{M }_0$$</EquationSource> </InlineEquation> versus <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$H_1: f\in \mathcal{M }_{\underline{i}}\setminus \mathcal{M }_0$$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\mathcal{M }_0$$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$f$$</EquationSource> </InlineEquation> ultimately belongs to one of them. The results are then generalized to the case when the logarithm of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$f$$</EquationSource> </InlineEquation> has infinite expansion with respect to <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$(b_i(\cdot ))_1^\infty $$</EquationSource> </InlineEquation>. 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