Abstract (EN)

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 $k > 2$. In \mycite{balabwell:04a}, it was proved that both estimators exist and are splines of degree $k-1$ with simple knots. These knots, which are also the jump points of the $(k-1)-$st derivative of the estimators, cluster around a point $x_0 > 0$ under the assumption that $g_0$ has a continuous $k$-th derivative in a neighborhood of $x_0$ and $(-1)^k g^{(k)}_0(x_0) > 0$. If $\tau^{-}_n$ and $\tau^{+}_n$ are two successive knots, we prove that the random ``gap'' \ $\tau^{+}_n - \tau^{-}_n $ is $O_p(n^{-1/(2k+1)})$ for any $k > 2$ if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds. Based on the order of the gap, the asymptotic distribution of the Maximum Likelihood and Least Squares estimators can be established. We find that the $j-$th derivative of the estimators at $x_0$ converges at the rate $n^{-(k-j)/(2k+1)}$ for $j=0, \ldots, k-1$. 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).