Resampling Methods

Authored by: Xuming He

Handbook of Quantile Regression

Print publication date:  October  2017
Online publication date:  October  2017

Print ISBN: 9781498725286
eBook ISBN: 9781315120256
Adobe ISBN:

10.1201/9781315120256-2

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Abstract

Regression quantile estimators solve a linear program and can be computed efficiently. The finite-sample distributions of regression quantiles can be characterized (Koenker, 2005), but they are difficult to use for statistical inference. Suppose that we have data { ( X i , Y i ) $ \{(X_i, Y_i ) $ , i = 1 , … , n } $ i=1, \ldots , n\} $ , where the conditional quantile of Y given X is of interest and assumed to be linear. The asymptotic distributions of regression quantiles are normal under mild conditions, but the asymptotic variance depends on the conditional densities of Y given X = X i $ X=X_i $ , which are generally unknown. Statistical inference based on the asymptotic variance is arguably difficult for a simple reason. That is, one needs to use a nonparametric estimate of the asymptotic variance that requires the choice of a smoothing parameter, and such estimates can be quite unstable. Even if the asymptotic variance is well estimated, the accuracy of its approximation to the finite-sample variance depends on the design matrix as well as the quantile level. Resampling methods provide a reliable approach to inference for quantile regression analysis under a wide variety of settings.

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