On Extreme Regression Quantiles

Smith (1994) introduced the idea of extreme regression quantiles and he developed some asymptotic results for algebraically tailed error distributions. The results provided a close analogy to standard extreme value theory for one-sample extremes. Here we obtain the following generalizations. First, an extreme value distribution theory is developed in the exponentially tailed case, where the extreme slope estimates need not diverge to infinity and may actually be consistent. The design conditions of Smith (1994) are also generalized. Second, the tail behavior measure of Jurecčkova´ (1981) and He et al. (1990) is considered for extreme quantiles. For algebraically tailed error distributions, the “average” right extreme regression fit acts like a one-sample right extreme; while in the exponentially tailed case, the tail behavior is more like that of a slightly more central order statistic.