A Weissman-type estimator of the conditional marginal expected shortfall

The marginal expected shortfall is an important risk measure in finance, which has been extended recently in the case where the random variables of main interest (Y^{(1)}, Y^{(2)}) are observed together with a covariate X \in R^d. This leads to the concept of conditional marginal expected shortfall. It is defined as \theta_p(x_0)=E[Y^{(1)} | Y^{(2)} \geq Q_{Y^{(2)}}(1-p|x_0); x_0], where p is small and Q_{Y^{(2)}} denotes the quantile function of Y^{(2)}. In this paper, we propose an estimator for \theta_p(x_0) allowing extrapolation outside the Y^{(2)} - data range, i.e., valid for p < 1/n. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory.

[1]  Holger Drees,et al.  Extreme quantile estimation for dependent data with applications to finance , 2003 .

[2]  Stéphane Girard,et al.  Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models , 2021, The Annals of Statistics.

[3]  A. Guillou,et al.  A local moment type estimator for the extreme value index in regression with random covariates , 2014 .

[4]  A. Guillou,et al.  Bias-corrected estimation for conditional Pareto-type distributions with random right censoring , 2019, Extremes.

[5]  A. Guillou,et al.  Local robust estimation of the Pickands dependence function , 2018, The Annals of Statistics.

[6]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[7]  S. Girard,et al.  Kernel estimators of extreme level curves , 2011 .

[8]  A. Guillou,et al.  Local Estimation of the Conditional Stable Tail Dependence Function , 2018 .

[9]  Jonathan El Methni,et al.  Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions , 2014 .

[10]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[11]  Jun Cai,et al.  Conditional tail expectations for multivariate phase-type distributions , 2005 .

[12]  Areski Cousin,et al.  On multivariate extensions of Value-at-Risk , 2011, J. Multivar. Anal..

[13]  L. Pedersen,et al.  Measuring Systemic Risk , 2010 .

[14]  Hélène Cossette,et al.  TVaR-based capital allocation with copulas , 2009 .

[15]  Chen Zhou,et al.  Estimation of the marginal expected shortfall: the mean when a related variable is extreme , 2012 .

[16]  S. Girard,et al.  On kernel smoothing for extremal quantile regression , 2012, 1312.5123.

[17]  S. Girard,et al.  A moving window approach for nonparametric estimation of the conditional tail index , 2008, 1104.0763.

[18]  A. Guillou,et al.  Local robust and asymptotically unbiased estimation of conditional Pareto-type tails , 2014 .

[19]  A. Guillou,et al.  Nonparametric regression estimation of conditional tails: the random covariate case , 2014 .

[20]  Emiliano A. Valdez,et al.  Tail Conditional Expectations for Elliptical Distributions , 2003 .

[21]  Vytaras Brazauskas,et al.  Estimating conditional tail expectation with actuarial applications in view , 2008 .

[22]  Armelle Guillou,et al.  Conditional marginal expected shortfall , 2021, Extremes.

[23]  Bikramjit Das,et al.  Risk contagion under regular variation and asymptotic tail independence , 2016, J. Multivar. Anal..

[24]  K. S. Tan,et al.  Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures , 2007, ASTIN Bulletin.

[25]  Jonathan El Methni,et al.  Kernel estimation of extreme regression risk measures , 2018 .

[26]  G. Stupfler,et al.  A moment estimator for the conditional extreme-value index , 2013 .

[27]  Vicky Fasen-Hartmann,et al.  Conditional excess risk measures and multivariate regular variation , 2019, Statistics & Risk Modeling.

[28]  Testing asymptotic independence in bivariate extremes , 2009 .

[29]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[30]  Q. Yao.  Conditional Predictive Regions For Stochastic Processes , 1996 .

[31]  C. Prieur,et al.  Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets , 2018, Environmetrics.

[32]  Johan Segers,et al.  Second-order refined peaks-over-threshold modelling for heavy-tailed distributions , 2009, 0901.1518.