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 g0 at a fixed point x0 when k > 2. We find that the j th 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 τ+ n − τ− n , where τ± n 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.