Extreme M-quantiles as risk measures: from L1 to Lp optimization
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, econometri...
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Bernoulli Society for Mathematical Statistics and Probability
2019
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| Online Access: | https://eprints.nottingham.ac.uk/45682/ |
| _version_ | 1848797173385265152 |
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| author | Daouia, Abdelaati Girard, Stéphane Stupfler, Gilles |
| author_facet | Daouia, Abdelaati Girard, Stéphane Stupfler, Gilles |
| author_sort | Daouia, Abdelaati |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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 middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for 1 < p < 2. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate Lp-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme Lp-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme Lp-quantiles and elaborate the choice of p through applications to some simulated and financial real data. |
| first_indexed | 2025-11-14T19:59:40Z |
| format | Article |
| id | nottingham-45682 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:59:40Z |
| publishDate | 2019 |
| publisher | Bernoulli Society for Mathematical Statistics and Probability |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-456822020-05-04T19:05:17Z https://eprints.nottingham.ac.uk/45682/ Extreme M-quantiles as risk measures: from L1 to Lp optimization Daouia, Abdelaati Girard, Stéphane Stupfler, Gilles 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 middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for 1 < p < 2. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate Lp-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme Lp-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme Lp-quantiles and elaborate the choice of p through applications to some simulated and financial real data. Bernoulli Society for Mathematical Statistics and Probability 2019-02-01 Article PeerReviewed Daouia, Abdelaati, Girard, Stéphane and Stupfler, Gilles (2019) Extreme M-quantiles as risk measures: from L1 to Lp optimization. Bernoulli, 25 (1). pp. 264-309. ISSN 1573-9759 Asymptotic normality; Dependent observations; Expectiles; Extrapolation; Extreme values; Heavy tails; Lp optimization; Mixing; Quantiles; Tail risk https://projecteuclid.org/euclid.bj/1544605247 |
| spellingShingle | Asymptotic normality; Dependent observations; Expectiles; Extrapolation; Extreme values; Heavy tails; Lp optimization; Mixing; Quantiles; Tail risk Daouia, Abdelaati Girard, Stéphane Stupfler, Gilles Extreme M-quantiles as risk measures: from L1 to Lp optimization |
| title | Extreme M-quantiles as risk measures: from L1 to Lp optimization |
| title_full | Extreme M-quantiles as risk measures: from L1 to Lp optimization |
| title_fullStr | Extreme M-quantiles as risk measures: from L1 to Lp optimization |
| title_full_unstemmed | Extreme M-quantiles as risk measures: from L1 to Lp optimization |
| title_short | Extreme M-quantiles as risk measures: from L1 to Lp optimization |
| title_sort | extreme m-quantiles as risk measures: from l1 to lp optimization |
| topic | Asymptotic normality; Dependent observations; Expectiles; Extrapolation; Extreme values; Heavy tails; Lp optimization; Mixing; Quantiles; Tail risk |
| url | https://eprints.nottingham.ac.uk/45682/ https://eprints.nottingham.ac.uk/45682/ |