Estimating conditional quantiles with the help of the pinball loss

Details Estimating conditional quantiles with the help of the pinball loss Estimating conditional quantiles with the help of the pinball loss

2011-02-10

arxiv.org/abs/1102.2101v1

Bernoulli 2011, Vol. 17, No. 1, 211-225

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ267 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ267

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.

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