Data-driven selection of the testing index set

Develop a data-driven choice of the testing index set \(\mathcal K_M\) that optimizes power against specified classes of alternatives while accounting for the dependence between the selected set and the test statistic, for example through sample splitting.

Background

The goodness-of-fit test uses a finite index set $\mathcal K_M$ of basis coefficients. The paper establishes uniform Gaussian-approximation and consistency results over deterministic classes of such sets, but notes that selecting $\mathcal K_M$ from the same data creates dependence between the selector and the test statistic. Cross-validation is also unsuitable for the testing objective because it targets prediction error and may select a bounded truncation under the null hypothesis.

The unresolved problem is therefore to construct and theoretically justify a data-driven selector for $\mathcal K_M$ that achieves favorable detection power for specified alternative classes while preserving valid inference, potentially by using sample splitting.