# Generalized Quantiles as Risk Measures

@article{Bellini2013GeneralizedQA, title={Generalized Quantiles as Risk Measures}, author={Fabio Bellini and Bernhard Klar and Alfred M{\"u}ller and Emanuela Rosazza Gianin}, journal={Microeconomics: Decision-Making under Risk \& Uncertainty eJournal}, year={2013} }

In the statistical and actuarial literature several generalizations of quantiles have been considered, by means of the minimization of a suitable asymmetric loss function. All these generalized quantiles share the important property of elicitability, which has received a lot of attention recently since it corresponds to the existence of a natural backtesting methodology. In this paper we investigate the case of M-quantiles as the minimizers of an asymmetric convex loss function, in contrast to…

## 231 Citations

### Assessing coherent Value-at-Risk and expected shortfall with extreme expectiles

- Mathematics
- 2015

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…

### Estimation of extreme regression risk measures

- Mathematics
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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…

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

- MathematicsBernoulli
- 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…

### Nonparametric extreme conditional expectile estimation

- MathematicsScandinavian Journal of Statistics
- 2020

The fact that the expectiles of a distribution F are in fact the quantiles of another distribution E explicitly linked to F is exploited in order to construct nonparametric kernel estimators of extreme conditional expectiles, which are analyzed in the context of conditional heavy-tailed distributions.

### Extreme $$L^p$$-quantile Kernel Regression

- MathematicsAdvances in Contemporary Statistics and Econometrics
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Quantiles are recognized tools for risk management and can be seen as minimizers of an \(L^1\)-loss function, but do not define coherent risk measures in general. Expectiles, meanwhile, are…

### Backtesting VaR and expectiles with realized scores

- MathematicsStat. Methods Appl.
- 2019

This paper suggests a procedure to test the accuracy of a quantile or expectile forecasting model in an absolute sense, as in the original Basel I backtesting procedure of value-at-risk, and studies the asymptotic and finite-sample distributions of empirical scores for normal and uniform i.i.d. samples.

### Backtesting VaR and expectiles with realized scores

- MathematicsStatistical Methods & Applications
- 2018

Several statistical functionals such as quantiles and expectiles arise naturally as the minimizers of the expected value of a scoring function, a property that is called elicitability (see Gneiting…

### An $Lp$ −quantile methodology for estimating extreme expectiles

- Mathematics
- 2020

Quantiles are a fundamental concept in extreme-value theory. They can be obtained from a minimization framework using an absolute error loss criterion. The companion notion of expectiles, based on…

### A New Family of Expectiles and its Properties

- Computer Science
- 2020

Comparison of new expectiles with quantile and CVaR for a set of distributions shows that the proposed expectiles can be closer to the quantile than the standard expectile.

### ASYMPTOTIC EXPANSIONS OF GENERALIZED QUANTILES AND EXPECTILES FOR EXTREME RISKS

- MathematicsProbability in the Engineering and Informational Sciences
- 2015

Generalized quantiles of a random variable were defined as the minimizers of a general asymmetric loss function, which include quantiles, expectiles and M-quantiles as their special cases. Expectiles…

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