**Sankhya:
The Indian Journal of Statistics**

2007, Volume 69, Pt. 1, 87--100

**Remark on Extreme Regression Quantile**

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

SUMMARY. We consider the extreme regression quantile in the linear
regression model and derive its asymptotic distribution under a
density ($f$) of the errors with exponential tail, and under mild
conditions on the regressors. The method is based on the fact that
the slope components of the extreme regression quantile can be
interpreted as a special $R$-estimator. Hence, we can use the
methods based on ranks, in view of the uniform asymptotic linearity
of the H\'ajek (1965) rank scores process. This $R$-estimator can
even estimate the slope parameters consistently under some
conditions on the tails of $f.$

The intercept component is the maximum of the pertaining residuals;
its asymptotic distribution differs from that of the sample extreme
in the i.i.d. case only in that it involves the hat matrix of the
regressors. The rate of convergence is $f(F^{-1}(1-1/n))$ both for
the slopes and the intercept, which is in conformity with the
maximal domain of attraction of the errors.

*AMS (2000) subject classification. *62J05, 62G32, 62G35.

*Key words and phrases. *Extreme regression quantile, R-estimator, uniform asymptotic linearity of H\'ajek's rank scores process.