Adaptive Quantile Trend Filtering
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We study quantile trend filtering, a recently proposed method for one-dimensional nonparametric quantile regression. We show that the penalized version of quantile trend filtering attains minimax rates, off by a logarithmic factor, for estimating the vector of quantiles when its kth discrete derivative belongs to the class of bounded variation signals. Our results also show that the constrained version of trend filtering attains minimax rates in the same class of signals. Furthermore, we show that if the true vector of quantiles is piecewise polynomial, then the constrained estimator attains optimal rates up to a logarithmic factor. All of our results hold based on a robust metric and under minimal assumptions of the data generation mechanism. We also illustrate how our technical arguments can be used for analysing other shape constrained problems with quantile loss. Finally, we provide extensive experiments that show that quantile trend filtering can perform well, based on mean squared error criteria, under Gaussian, Cauchy, and t-distributed errors.
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