Coherent estimation of risk measures

Abstract:<\strong> We develop a statistical framework for risk estimation, inspired by the axiomatic theory of risk measures. Coherent risk estimators—functionals of P&L samples inheriting the economic properties of risk measures—are defined and characterized through robust representations linked to L-estimators. The framework provides a canonical methodology for constructing estimators with sound financial and statistical properties, unifying risk measure theory, principles for capital adequacy, and practical statistical challenges in market risk. A numerical study illustrates the approach, focusing on expected shortfall estimation under both i.i.d. and overlapping samples relevant for regulatory FRTB model applications.