Estimation of a k-Monotone Density: Limit Distribution Theory and the Spline Connection

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a k-monotone density g. at a fixed point x. when k > 2. We find that the jth derivative of the estimators at x. 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 $H_{k}$ 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 $\tau _{n}^{+}-\tau _{n}^{-}$, where $\tau _{n}^{\pm}$ 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.