Estimation of tail quantities, such as expected shortfall or Value at Risk, is a difficult problem. We show how the theory of nonlinear expectations, in particular the Data-robust expectation introduced in [ 4 ], can assist in the quantification of statistical uncertainty for these problems. However, when we are in a heavy-tailed context (in particular when our data are described by a Pareto distribution, as is common in much of extreme value theory), the theory of [ 4 ] is insufficient, and requires an additional regularization step which we introduce. By asking whether this regularization is possible, we obtain a qualitative requirement for reliable estimation of tail quantities and risk measures, in a Pareto setting.
Estimation of tail quantities, such as expected shortfall or Value at Risk, is a difficult problem. We show how the theory of nonlinear expectations, in particular the Data-robust expectation introduced in [ 5 ], can assist in the quantification of statistical uncertainty for these problems. However, when we are in a heavy-tailed context (in particular when our data are described by a Pareto distribution, as is common in much of extreme value theory), the theory of [ 5 ] is insufficient, and requires an additional regularization step which we introduce. By asking whether this regularization is possible, we obtain a qualitative requirement for reliable estimation of tail quantities and risk measures, in a Pareto setting.
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