We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a k-monotone density g0 at a fixed point x0 when k>2. We find that the jth derivative of the estimators at x0 converges at the rate n−(k−j)/(2k+1) for j=0, …, k−1. The limiting distribution depends on an almost surely uniquely defined stochastic process Hk that stays above (below) the k-fold integral of Brownian motion plus a deterministic drift when k is even (odd). Both the MLE and LSE are known to be splines of degree k−1 with simple knots. Establishing the order of the random gap τn+−τn−, where τn± denote two successive knots, is a key ingredient of the proof of the main results. We show that this “gap problem” can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.
"Estimation of a k-monotone density: Limit distribution theory and the spline connection." Ann. Statist. 35 (6) 2536 - 2564, December 2007. https://doi.org/10.1214/009053607000000262