Estimating conditional quantiles with the help of the pinball loss
Estimating conditional quantiles with the help of the pinball loss
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DOI：
10.3150/10-bej267
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发表时间：
2011-02-01
期刊：
影响因子：
1.5
通讯作者：
Christmann, Andreas
中科院分区：
文献类型：
--
作者：
Steinwart, Ingo;Christmann, Andreas
所谓的弹球损失用于估计条件分位数，是统计学和机器学习中的一个众所周知的工具。然而，到目前为止，在量化该工具对非参数方法的效率方面只做了很少的工作。我们通过建立描述弹球风险最小化与相应条件分位数的近似程度的不等式来填补这一空白。这些不等式在数据生成分布的温和假设下成立，然后被用来建立所谓的方差界限，最近被证明在(正则化的)经验风险最小化方法的统计分析中发挥了重要作用。最后，我们使用这两种类型的不等式为使用弹球损失的支持向量机建立了一个预言不等式。在条件分位数的一些标准正则性假设下，所得到的学习率是最小-最大最优的。
The so-called pinball loss for estimating conditional quantiles is a well-known tool in both statistics and machine learning. So far, however, only little work has been done to quantify the efficiency of this tool for nonparametric approaches. We fill this gap by establishing inequalities that describe how close approximate pinball risk minimizers are to the corresponding conditional quantile. These inequalities, which hold under mild assumptions on the data-generating distribution, are then used to establish so-called variance bounds, which recently turned out to play an important role in the statistical analysis of (regularized) empirical risk minimization approaches. Finally, we use both types of inequalities to establish an oracle inequality for support vector machines that use the pinball loss. The resulting learning rates are min-max optimal under some standard regularity assumptions on the conditional quantile.