The classes of monotone or convex (and necessarily monotone) densities on ℝ(+) can be viewed as special cases of the classes of k-monotone densities on ℝ(+). These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on ℝ(+). In this paper we consider non-parametric maximum likelihood and least squares estimators of a k-monotone density g(0).We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k - 1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives[Formula: see text], at a fixed point x(0) under the assumption that [Formula: see text].

中文翻译：

---

ak-单调密度的估计：特征、一致性和极小极大下界

ℝ(+) 上的单调或凸（必然是单调）密度的类别可以看作是ℝ(+) 上的 k-单调密度类别的特例。这些类别弥补了渐近结果已知的单调（1-单调）和凸递减（2-单调）密度类别与ℝ（+）上的完全单调（∞-单调）密度类别之间的差距。在本文中，我们考虑了 k 单调密度 g(0) 的非参数最大似然估计和最小二乘估计。我们证明了估计的存在并给出了表征。我们还建立了一致性属性，并表明估计量是 k - 1 度的样条曲线，带有简单的结点。我们进一步提供了用于估计导数的渐近最小最大风险下限[公式：见正文]，在一个固定点 x(0) 假设 [公式：