Robust estimation methods with application to flood statistics
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Date
2017
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Abstract
Robust statistics and the use of robust estimators have come more and more into focus during the last couple of years. In the context of flood statistics, robust estimation methods are used to obtain stable estimations of e.g. design floods. These are estimations that do not change from one year to another just because one large flood occurred.
A problem which is often ignored in flood statistics is the underlying dependence structure of the data. When considering discharge data with high time-resolution, short range dependent behaviour can be detected within the time series. To take this into account, in this thesis a limit theorem for the class of GL-statistics is developed under the very general assumption of near epoch dependent processes on absolutely regular random variables, which is a well known concept of short range dependence. GL-statistics form a very general class of statistics and can be used to represent many robust and non-robust estimators, such as Gini's mean difference, the Qn-estimator or the generalized Hodges-Lehmann estimator. In a direct application the limit distribution of L-moments and their robust extension, the trimmed L-moments, is derived. Moreover, a long-run variance estimator is developed. For all these results, the use of U-statistics and U-processes proves to be the key tool, such that a Central Limit Theorem for multivariate U-statistics as well as an invariance principle for U-processes and the convergence of the remaining term of the Bahadur-representation for U-quantiles is shown. A challenge for proving these results pose the multivariate kernels that are considered to be able to represent very general estimators and statistics.
A concrete application in the context of flood statistics, in particular in the estimation of design floods, the classification of homogeneous groups and the modelling of short range dependent discharge series, is given. Here, well known models (peak-over-thresholds) as well as newly developed ones, for example mixing models using the distinction of floods according to their timescales, are combined with robust estimators and the advantages and disadvantages under consideration of stability and efficiency are investigated. The results show that the use of the new models, that take more information into account by enlarging the data basis, in combination with robust estimators leads to a very stable estimation of design floods, even in high quantiles. Whereas a lot of the classical estimators, like Maximum-Likelihood estimators or L-moments, are affected by single extraordinary extreme events and need a long time to stabilise, the robust methods approach the same level of stabilisation rather fast. Moreover, the newly developed mixing model cannot only be used for flood estimation but also for regionalisation, that is the modelling of ungauged basins. Here, especially when needing a classification of flood events and homogeneous groups of gauges, the use of robust estimators proves to result in stable estimations, too.
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Keywords
Flood statistics, Short-range dependence, GL-statistics, Mixing models