**Sankhya:
The Indian Journal of Statistics**

2005, Volume 67, Pt. 2, 227--252

**Two-step Regression Quantiles**

By

Jana Jure\v{c}kov\'a, Charles
University in Prague, Czech Republic

Jan Picek, Technical University in Liberec, Czech Republic

SUMMARY. We propose a new version of the regression $\alpha$-quantile in the linear regression model, ordering the residuals with respect to an initial R-estimate of the slope parameter. In this way we obtain a consistent estimator of $(\beta_0+F^{-1}(\alpha),\beta_1,\ldots,\beta_p)^{\prime},$ asymptotically equivalent to the regression $\alpha$-quantile of Koenker and Bassett. The result is extended to the extreme regression quantiles. Similarly we construct a version of the autoregression quantile in the linear AR($p$) model. We also propose an estimate of the extreme error in the linear regression and autoregression models, using the initial R-estimate of the slope. A simulation experiment illustrates a very small difference between the original regression quantiles and their new versions, and a very good approximation of the extreme errors.

*AMS (1991) subject classification. *62J05, 62G32, 62G35.

*Key words and phrases. *Autoregression quantile, extreme value,
extreme regression quantile, regression quantile, R-estimator.