Tests for Gumbel domain of attraction based on regression quantiles

If we are interested in such events as the extreme intensity of the wind, high flood levels of the rivers or extreme values of environmental indicators, or maximal or minimal performance of foreign exchange rates or share prices, we should take an interest in the tails of the underlying probability distribution rather than in its central part. Many authors were have dealt with an estimation of the tails of the distribution. However, besides the point and interval estimation, a typical and important part of statistical inference and modelling is the testing of hypotheses. Many authors have developed methods for location model, i.e. they consider an i.i.d. sample, from an underlying distribution function with unknown shape, location and scale parameters, belonging to some max-domain of attraction. They tested the problem of Gumbel domain against Frechet or Weibull domains. Neves, Picek and Alves (2006) based the testing decision on the ratio between the maximum and the mean of the top sample excesses above some random threshold. The present paper deals with a linear model Y = X beta + E, where the errors are again from an underlying distribution function with unknown shape, location and scale parameters, belonging to some max-domain of attraction. We study a generalization of test as above based on the regression quantilesfor the null hypothesis that the distribution comes from the Gumbel domain of attraction. The regression quantiles were introduced as a generalization of usual quantiles to linear regression model. The key idea in generalizing the quantiles is the fact that we can expressed the problem of finding the sample quantile as the solution to a simple optimization problem. This leads, naturally, to more general method of estimating of conditional quantiles fuctions. The optimization problem may be reformulated as a linear program and the simplex approach may be used to computing regression quantiles. Dienstbier (2009) showed that location and scale invariant smooth functionals of the standardized intercepts of the highest order regression quantiles have the same asymptotic distribution as the same functionals based on the empirical tail quantile function of the underlying sample of errors. We generalize the tests on the basis of the exceedances over high quantile regression threshold. The type I error and power of the test are studied for finite sample sizes by simulation.