On nonparametric tests of positivity/monotonicity/convexity
On nonparametric tests of positivity/monotonicity/convexity
复制标题
关于正性/单调性/凸性的非参数检验
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
A. Nemirovski
中科院分区：
文献类型：
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作者：
A. Juditsky;A. Nemirovski
我们考虑在白噪声模型中观测到的未知信号到正/单调/凸函数的凸锥的距离估计问题。我们表明，当未知函数属于赫尔德（Holder）类时，估计从信号到一个锥的\(L^r\)距离（\(1\leq r<\infty\)）的风险与估计信号本身的风险本质上相同（至多相差一个对数因子）。对于未知信号的正性、单调性和凸性的检验，同样的风险界也成立。我们还提供了到正函数锥的距离的一个估计，其风险比插件估计的风险小一个对数因子。
We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Holder class, the risk of estimating the L r -distance, 1 ≤ r < ∞, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal. We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the plug-in estimate.