Determine when the standard penalty is preferable under joint design–error dependence

Determine whether, in one-dimensional nonparametric regression with random design and dependent errors, the standard dimension-proportional penalty is preferable to the long-range-dependence penalty whenever a condition involving both the dependence structure of the design process and that of the error process is satisfied.

Background

The paper derives a penalty proportional to a power of the model dimension for long-range-dependent errors whose conditional autocovariances decay polynomially. Through simulations, however, the authors observe that this penalty can be suboptimal for certain random designs, including designs independent of the errors, where the standard dimension-proportional penalty may perform better.

To explain this behavior, Section 2.3 studies regular regressograms with stationary Gaussian errors independent of the design and proves a refined bound under a rho-dependent design assumption. The resulting heuristic suggests a penalty combining a dimension term and a logarithmic long-memory term. The broader conjecture stated in the introduction concerns identifying the dependence condition under which the standard penalty is preferable, and is not established in general by the paper.