  We study the asymptotic behavior of the Maximum Likelihood and Least Squares
Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$.
We find that the $j$th derivative of the estimators at $x_0$ 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.
