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

• Published in: Stochastic environmental research and risk assessment Vol. 31; no. 5; pp. 1155 - 1169
• Main Authors: Kinsvater, Paul, Fried, Roland
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-07-2017; Springer Nature B.V
• Subjects: Aquatic Pollution; Chemistry and Earth Sciences; Computational Intelligence; Computer Science; Computer simulation; Earth and Environmental Science; Earth Sciences; Environment; Environmental studies; Estimation; Extreme values; Hydrology; Math. Appl. in Environmental Science; Mathematical models; Maxima; Original Paper; Pareto optimum; Physics; Precipitation; Probability Theory and Stochastic Processes; Random walk theory; Regression analysis; River flow; Rivers; Scale (ratio); Statistics for Engineering; Trend analysis; Waste Water Technology; Water Management; Water Pollution Control; Heavy tails; Relative excesses; Precipitation; Flood frequency; Extreme value index; Regression model
• ISSN: 1436-3240; 1436-3259
• DOI: 10.1007/s00477-016-1345-0

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.