Conditional heavy-tail behavior with applications to precipitation and river flow extremes

Abstract

This article deals with the right-tail behavior of a response distribution F_Y conditional on a regressor vector X = x restricted to the heavy-tailed case of Pareto-type conditional distributions F_Y (y| x) = P(Y ≤ y| X = x), with heaviness of the right tail characterized by the conditional extreme value index γ(x) > 0. We particularly focus on testing the hypothesis H_0;tail : γ(x) = γ0 of constant tail behavior for some γ0 > 0 and all possible x. When considering x as a time index, the term trend analysis is commonly used. In the recent past several such trend analyses in extreme value data have been published, mostly focusing on time-varying modeling of location and scale parameters of the response distribution. In many such environmental studies a simple test against trend based on Kendall's tau statistic is applied. This test is powerful when the center of the conditional distribution F_Y (y|x) changes monotonically in x, for instance, in a simple location model μ(x) = μ_0 + x * μ_1, x = (1, x)’, but the test is rather insensitive against monotonic tail behavior, say, μ(x) = η_0 + x * η_1. This has to be considered, since for many environmental applications the main interest is on the tail rather than the center of a distribution. Our work is motivated by this problem and it is our goal to demonstrate the opportunities and the limits of detecting and estimating non-constant conditional heavy-tail behavior with regard to applications from hydrology. We present and compare four different procedures by simulations and illustrate our findings on real data from hydrology: Weekly maxima of hourly precipitation from France and monthly maximal river flows from Germany.