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

TL;DR

This paper investigates the Hermite spline conjecture related to k-monotone densities, demonstrating why the conjecture is false and proposing alternative approaches to establish the theoretical properties of estimators.

Contribution

It proves the Hermite spline conjecture cannot be true and offers new methods to develop the limit theory of estimators for k-monotone densities.

Findings

The Hermite spline conjecture is false.

Alternative methods can replace the conjecture in theoretical proofs.

The new approach supports the limit theory for estimators of k-monotone densities.

Abstract

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.