Veronika Skrivankova
Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. Extremal quantile regression deals with a problem of sparsity of data in tails of the response variable and attempts to estimate extreme conditional quantiles by employing the extreme value theory.
Under the assumption that a distribution function F is in the maximum domain of attraction (MDA) of some extreme value distribution (EVD), the extremal domain condition can be derived. It identifies which of the three types of EVD; Fréchet, Gumbel or Weibull, contains F in their MDA. The “Peak-Over-Threshold” method employs the generalized Pareto distribution to approximate the tail of the distribution function F. All presented methods for determination of the tail heaviness and estimators for the extreme value index are illustrated on real data.
Empirical conditional quantiles are represented by regression quantile lines. Since each conditional quantile function is nondecreasing, the regression quantile lines are not supposed to cross. Under the assumption that our data follow the heteroscedastic model, we follow a procedure which not only yields non-crossing regression quantile lines, but also enables us to estimate the extreme regression quantiles.
