Smooth estimation of a monotone density
Smooth estimation of a monotone density
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单调密度的平滑估计
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
M. J. Laan
中科院分区：
文献类型：
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作者：
A. Vaart;M. J. Laan
我们研究了光滑性和单调性假设在从观察样本估计密度时的相互作用。正半直线上密度递减的非参数极大似然估计，如果密度在t处有负导数，则在固定点t达到[公式：见正文]的收敛速度。带宽的核估计[公式：见正文]获得相同的收敛速度，但极限分布是不同的。如果密度在t处是可微的，并且已知是单调的，那么第三个估计量是通过核估计量的同质化得到的。我们证明了这再次达到了收敛速度[公式：见正文]，并比较了三种估计量的极限分布。结果表明，同位素化和光滑化都会导致更集中的极限分布，并研究了其对带宽中比例常数的依赖关系。我们还表明，同位素化不会改变核估计的极限行为。
We investigate the interplay of smoothness and monotonicity assumptions when estimating a density from a sample of observations. The nonparametric maximum likelihood estimator of a decreasing density on the positive half line attains a rate of convergence of [Formula: See Text] at a fixed point t if the density has a negative derivative at t. The same rate is obtained by a kernel estimator of bandwidth [Formula: See Text], but the limit distributions are different. If the density is both differentiable at t and known to be monotone, then a third estimator is obtained by isotonization of a kernel estimator. We show that this again attains the rate of convergence [Formula: See Text], and compare the limit distributions of the three types of estimators. It is shown that both isotonization and smoothing lead to a more concentrated limit distribution and we study the dependence on the proportionality constant in the bandwidth. We also show that isotonization does not change the limit behaviour of a kernel esti...