Estimation of unimodal densities without smoothness assumptions
Estimation of unimodal densities without smoothness assumptions
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无平滑度假设的单峰密度估计
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
L. Birgé
中科院分区：
文献类型：
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作者：
L. Birgé
已知的降低密度的降低密度的手榴弹估计量被认为是经验c.d.f.的凹形信封的衍生物，这是一个非常好的估计量，当时不假定该密度不知道该密度在半线R +上的降低密度非常好。保持平稳。它的确是最大似然估计器，当通过密度之间的L 1距离测量损失时，它的风险可以精确。此外，如果一个人将自己限制为紧凑的亚集，那就是在[0，l]上受支持的H界的降低密度降低，则该估计器的风险在固定因素的固定因素上，这是最小型风险的固定因素。如果一个具有已知模式的单峰密度的最大似然估计量，也是如此。当该模式未知时，最大似然估计器不再存在。我们将提供一个通用估计器（以及计算算法）来估计非平滑的单峰密度。它的风险与基于真实模式的知识和一些较低订单期限的知识与Grenander估算器的风险相同。它还可以应对偏离单形态的小小的。
The Grenander estimator of a decreasing density, which is defined as the derivative of the concave envelope of the empirical c.d.f., is known to be a very good estimator of an unknown decreasing density on the half-line R + when this density is not assumed to be smooth. It is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the L 1 -distance between densities. Moreover, if one restricts oneself to the compact subsets of decreasing densities bounded by H with support on [0, L] the risk of this estimator is within a fixed factor of the minimax risk. The same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode. When the mode is unknown, the maximum likelihood estimator does not exist any more. We shall provide a general purpose estimator (together with a computational algorithm) for estimating nonsmooth unimodal densities. Its risk is the same as the risk of the Grenander estimator based on the knowledge of the true mode plus some lower order term. It can also cope with small departures from unimodality.