Let \(X_1,\dots,X_{n}\) be independent observations from a common density, and suppose that it is required to test whether this density is equal to a specified \(f_0\), against the nonparametric alternative that it is not. To test the hypothese that a density \(f\) is equal to a specified \(f_0\) one knows by the Neyman-Pearson lemma the form of the optimal test at a specified alternative \(f_1\). Any nonparametric density estimation scheme allows an estimate of \(f\). This leads to estimated likelihood ratios. The author consider estimators constructed via log-linear expansions. Let \[ f_{S}(x\,|\,a)=f_0(x)c_{S}^{-1}(a)\exp\left\{\sum_{j\in S}a_{j}\psi_{j}(x)\right\} \] for \(x\) in the interval of interest, where the functions \(\psi_{j}\) are chosen to be orthogonal and normalized w.r.t. \(f_0\), \(S\) is a subset of the natural integers. The authors paper deals with the statistics \[ Z_{n}^{*}=2n\left\{\sum_{j\in S_{n}^{*}}\hat a_{j}\bar\psi_{j}-\log c_{S_{n}^{*}}(\hat a)\right\}\quad \text{and}\quad T_{n}^{*}=\sum_{j\in S_{n}^{*}}n\bar\psi_{j}^2, \] where \(\hat a\) is arrived at via maximum likelihood in the particular model indexed by the selected set \(S_{n}^{*}\); \(\bar\psi_{j}=n^{-1}\sum_{i=1}^{n}\psi_{j}(X_{i})\). It is proven that \(Z_{n}^{*}\) and \(T_{n}^{*}\) are asymptotically equivalent tests, but only under local alternatives circumstances.
The behaviour of \(Z_{n}^{*}\) and \(T_{n}^{*}\) is studied, where the index set \(S_{n}^{*}\) is chosen in data-driven ways. The limit behaviour of the statistics \(Z_{n}^{*}\) and \(T_{n}^{*}\) is characterized not only for Akaike information criterion and Bayesian information criterion type selected subsets, but also for much broader classes. Applications of the general theory to specific models, including testing for multivariate normality, are described. Results of some simulation studies are presented.