Abstract
A common statistical problem in hydrology is the estimation of annual maximal river flow distributions and their quantiles, with the objective of evaluating flood protection systems. Typically, record lengths are short and estimators imprecise, so that it is advisable to exploit additional sources of information. However, there is often uncertainty about the adequacy of such information, and a strict decision on whether to use it is difficult. We propose penalized quasi-maximum likelihood estimators to overcome this dilemma, allowing one to push the model towards a reasonable direction defined a priori. We are particularly interested in regional settings, with river flow observations collected at multiple stations. To account for regional information, we introduce a penalization term inspired by the popular Index Flood assumption. Unlike in standard approaches, the degree of regionalization can be controlled gradually instead of deciding between a local or a regional estimator. Theoretical results on the consistency of the estimator are provided and extensive simulations are performed for the reason of comparison with other local and regional estimators. The proposed procedure yields very good results, both for homogeneous as well as for heterogeneous groups of sites. A case study consisting of sites in Saxony, Germany, illustrates the applicability to real data.
Highlights
IntroductionBeing interested in high quantiles (i.e., the right tail), note that the generalized extreme value (GEV) family can handle a wide variety of right tail behaviour, with bounded right tails for ξ < 0, exponential tails for ξ = 0 and arbitrarily heavy tails for ξ > 0
More generally in statistics for extremes in hydrology (Katz et al 2002), one is typically confronted with a version of the following problem: let X1, . . . , Xn denote independent annual maximal river flows observed at a specific site and during the past n years, and let F (x) = P(Xi ≤ x) denote their stationary cumulative distribution function (c.d.f.)
F belongs to the 3-parametric generalized extreme value (GEV) distribution
Summary
Being interested in high quantiles (i.e., the right tail), note that the GEV family can handle a wide variety of right tail behaviour, with bounded right tails for ξ < 0, exponential tails for ξ = 0 and arbitrarily heavy tails for ξ > 0 The drawback of this flexibility shows up in the estimation of the parameter vector θ, by a high estimation variance of the shape ξ resulting in a volatile quantile estimate. Probability weighted moments or L-moments have been proposed as alternatives to moment or maximum likelihood (ML) estimators The former show a superior performance in typical small sample cases (Hosking et al 1985), which has been mainly attributed to their restricted parameter space (Coles and Dixon 1999)
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