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

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 , t a i l : γ ( x ) = γ 0 of constant tail behavior for some γ 0 > 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 or 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, γ ( 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.