  Various events in the nature, economics and in other areas force us to
combine the study of extremes with regression and other methods. A useful tool
for reducing the role of nuisance regression, while we are interested in the
shape or tails of the basic distribution, is provided by the averaged
regression quantile and namely by the average extreme regression quantile. Both
are weighted means of regression quantile components, with weights depending on
the regressors. Our primary interest is the averaged extreme regression
quantile (AERQ), its structure, qualities and its applications, e.g. in
investigation of a conditional loss given a value exogenous economic and market
variables. AERQ has several interesting equivalent forms: While it is
originally defined as an optimal solution of a specific linear programming
problem, hence is a weighted mean of responses corresponding to the optimal
base of the pertaining linear program, we give another equivalent form as a
maximum residual of responses from a specific R-estimator of the slope
components of regression parameter. The latter form shows that while AERQ
equals to the maximum of some residuals of the responses, it has minimal
possible perturbation by the regressors. Notice that these finite-sample
results are true even for non-identically distributed model errors, e.g. under
heteroscedasticity. Moreover, the representations are formally true even when
the errors are dependent - this all provokes a question of the right
interpretation and of other possible applications.
