Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization

@article{Daouia2019ExtremeMA,
  title={Extreme M-quantiles as risk measures: From \$L^\{1\}\$ to \$L^\{p\}\$ optimization},
  author={Abdelaati Daouia and St{\'e}phane Girard and Gilles Stupfler},
  journal={Bernoulli},
  year={2019}
}
The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of Lp optimization. These generalized Lp-quantiles steer an advantageous… 
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