A canonical process for estimation of convex functions: the "invelope" of integrated Brownian motion + t4.
A canonical process for estimation of convex functions: the "invelope" of integrated Brownian motion + t4.
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凸函数估计的规范过程：积分布朗运动 t4 的“包络线”。
DOI：
10.1214/aos/1015345957
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发表时间：
2001
影响因子：
4.5
通讯作者：
J. Wellner
中科院分区：
文献类型：
--
作者：
P. Groeneboom;G. Jongbloed;J. Wellner
引入了与积分布朗运动相关的过程，该过程分别表征了凸函数和凸密度的非参数最小二乘和最大似然估计的极限行为。我们将这个过程称为“包络线”，并表明它几乎肯定是积分布朗运动的唯一定义函数。它的作用类似于布朗运动的最大凸副式加上抛物线漂移在估计单调函数问题中的作用。引入了一种迭代三次样条算法来解决极限情况下的约束最小二乘问题，并显示了应用该算法获得的一些结果来说明该理论。
A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process “the invelope” and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the greatest convex minorant of Brownian motion plus a parabolic drift in the problem of estimating monotone functions. An iterative cubic spline algorithm is introduced that solves the constrained least squares problem in the limit situation and some results, obtained by applying this algorithm, are shown to illustrate the theory.