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.

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