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Unbiasedness in Least Quantile Regression

by Dirk Tasche of the Technische Universität München

September 6, 2001

Abstract: We develop an abstract notion of regression which allows for a non-parametric formulation of unbiasedness. We prove then that least quantile regression is unbiased in this sense even in the heteroscedastic case if the error distribution has a continuous, symmetric, and uni-modal density. An example shows that unbiasedness may break down even for smooth and symmetric but not uni-modal error distributions. We compare these results to those for least MAD and least squares regression.

Keywords: Least Quantile, Regression, Unbiasedness, Fisher consistency, Quantile Derivative, Lord's paradox.

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