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Non-parametric Quantile Regression via the K-NN Fused Lasso

Abstract

Quantile regression is a statistical method for estimating conditional quantiles of a response variable. In addition, for mean estimation, it is well known that quantile regression is more robust to outliers than l2l_2-based methods. By using the fused lasso penalty over a KK-nearest neighbors graph, we propose an adaptive quantile estimator in a non-parametric setup. We show that the estimator attains optimal rate of n1/dn^{-1/d} up to a logarithmic factor, under mild assumptions on the data generation mechanism of the dd-dimensional data. We develop algorithms to compute the estimator and discuss methodology for model selection. Numerical experiments on simulated and real data demonstrate clear advantages of the proposed estimator over state of the art methods.

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