A Quantile Regression Memoir

Authored by: Gilbert W. Bassett , Roger Koenker

Handbook of Quantile Regression

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

Print ISBN: 9781498725286
eBook ISBN: 9781315120256
Adobe ISBN:

10.1201/9781315120256-1

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Abstract

In the summers of 1972 and 1973 the two of us spent a lot of time playing tennis, in a successful effort to avoid working on our dissertations at the University of Michigan. Gib was working with Lester Taylor on theoretical aspects of l 1 $ l_1 $ regression, and Roger on hierarchical models for longitudinal data. Inevitably our anxiety about work intruded into the tennis conversation and there were frequent discussions of linear programming aspects of the l 1 $ l_1 $ regression problem. Gib had derived conditions under which the l 1 $ l_1 $ estimator was linear in the response vector, which explained some pathological simulation results of Taylor’s. This might be called “breakdown” of the estimator due to influential design points now. More significantly, we frequently mentioned that the l 1 $ l_1 $ estimator seemed to be a regression analogue of the median since it was easily shown that essentially half the regression responses must lie above the fitted l 1 $ l_1 $ regression hyperplane and half must lie below as long as there was an intercept in the model. We also began to ask ourselves the question: if the l 1 $ l_1 $ estimator is a median regression estimator, must there not be other quantile regression estimators?

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