On the Hermite spline conjecture and its connection to k-monotone densities

The k-monotone classes of densities defined on (0, \infty) have been known in the mathematical literature but were for the first time considered from a statistical point of view by Balabdaoui and Wellner (2007, 2010). In these works, the authors generalized the results established for monotone (k=1) and convex (k=2) densities by giving a characterization of the Maximum Likelihood and Least Square estimators (MLE and LSE) and deriving minimax bounds for rates of convergence. For k strictly larger than 2, the pointwise asymptotic behavior of the MLE and LSE studied by Balabdaoui and Wellner (2007) would show that the MLE and LSE attain the minimax lower bounds in a local pointwise sense. However, the theory assumes that a certain conjecture about the approximation error of a Hermite spline holds true. The main goal of the present note is to show why such a conjecture cannot be true. We also suggest how to bypass the conjecture and rebuild the key proofs in the limit theory of the estimators.