We study global and local robustness properties of several estimators for shape and scale in a generalized Pareto model. The estimators considered in this paper cover maximum likelihood estimators, skipped maximum likelihood estimators, Cram\'er-von-Mises Minimum Distance estimators, and , as a special case of quantile-based estimators, Pickands Estimator. We further consider an estimator matching the population median and an asymmetric, robust estimator of scale (kMAD) to the empirical ones (kMedMAD), which may be tuned to an expected FSBP of 34\%. These estimators are compared to one-step estimators distinguished as optimal in the shrinking neighborhood setting, i.e.; the most bias-robust estimator minimizing the maximal (asymptotic) bias and the estimatorminimizing the maximal (asymptotic) MSE. For each of these  estimators, we determine the finite sample breakdown point, the influence function, as well as  statistical accuracy measured by asymptotic bias, variance, and mean squared error - all evaluated uniformly on shrinking convex contamination neighborhoods. Finally, we check these asymptotic theoretical findings against finite sample behavior by an extensive simulation study  .
We study robustness properties of several procedures for joint estimation of shape and scale in a generalized Pareto model. The estimators we primarily focus on, MBRE and OMSE, are one-step estimators distinguished as optimally-robust in the shrinking neighborhood setting, i.e.; they minimize the maximal bias, respectively, on a specific such neighborhood, the maximal mean squared error. For their initialization, we propose a particular Location-Dispersion (LD) estimator, kMedMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally-robust estimators are compared to maximum likelihood, skipped maximum likelihood, Cramer-von-Mises minimum distance, method of median, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite sample breakdown point, the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE - all evaluated uniformly on shrinking neighborhoods. These asymptotic findings are complemented by an extensive simulation study to assess their finite sample behavior .
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global and local
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estimators for
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considered in this paper cover maximum likelihood estimators, skipped maximum likelihood estimators, Cram\'er-von-Mises Minimum Distance estimators, and , as a special case of quantile-based estimators, Pickands Estimator. We further consider an estimator matching the population median and an asymmetric, robust estimator of scale (kMAD) to the empirical ones (kMedMAD), which may be tuned to an expected FSBP of 34\%. These estimators are compared to
we primarily focus on, MBRE and OMSE, are
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optimal
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the most bias-robust estimator minimizing the maximal (asymptotic) bias and the estimatorminimizing the maximal (asymptotic) MSE. For
they minimize the maximal bias, respectively, on a specific such neighborhood, the maximal mean squared error. For their initialization, we propose a particular Location-Dispersion (LD) estimator, kMedMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally-robust estimators are compared to maximum likelihood, skipped maximum likelihood, Cramer-von-Mises minimum distance, method of median, and Pickands estimators. To quantify their deviation from robust optimality, for
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suboptimal
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the
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mean squared error
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convex contamination neighborhoods. Finally, we check these asymptotic theoretical findings against finite sample behavior
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to assess their finite sample behavior