Starting from the requirement that risk measures of financial portfolios
should be based on their losses, not their gains, we define the notion of
loss-based risk measure and study the properties of this class of risk
measures. We characterize loss-based risk measures by a representation theorem
and give examples of such risk measures. We then discuss the statistical
robustness of estimators of loss-based risk measures: we provide a general
criterion for qualitative robustness of risk estimators and compare this
criterion with sensitivity analysis of estimators based on influence functions.
Finally, we provide examples of statistically robust estimators for loss-based
risk measures.Comment: 40 page
