Maximin estimation of multidimensional boundaries
Maximin estimation of multidimensional boundaries
复制标题
多维边界的最大最小估计
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
K. Song
中科院分区：
文献类型：
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作者：
H. Müller;K. Song
我们考虑在光滑的多维回归函数中估计不连续点的位置和大小的问题。假定不连续的边界或位置是一个封闭的曲线各自的表面，我们的目的是估计这个封闭集。我们的方法利用多元核估计的一致收敛性来求解方向极限。在光滑假设下，这些极限的差收敛于零，并收敛于沿不连续的跳跃大小。这导致了一个极大估计量的提出，它选择边界，在该边界上所有点之间的方向差估计最小是最大的。结果表明，该估计边界几乎肯定被包围在真实边界周围的一系列缩小的邻域中，并得到了相应的收敛速度。
We consider the problem of estimating the location and size of a discontinuity in an otherwise smooth multidimensional regression function. The boundary or location of the discontinuity is assumed to be a closed curve respective surface, and we aim to estimate this closed set. Our approach utilizes the uniform convergence of multivariate kernel estimators for directional limits. Differences of such limits converge to zero under smoothness assumptions, and to the jump size along the discontinuity. This leads to the proposal of a maximin estimator, which selects the boundary for which the minimal estimated directional difference among all points belonging to this boundary is maximized. It is shown that this estimated boundary is almost surely enclosed in a sequence of shrinking neighborhoods around the true boundary, and corresponding rates of convergence are obtained.