Asymptotic distribution of law-invariant risk functionals

Law-invariant or version-independent coherent risk or acceptability functionals do not explicitly depend on the underlying probability space and can be considered as functionals of the distribution function. In this paper, we consider estimates of these functionals based on the empirical distribution function and investigate their asymptotic properties.

[1]  Alexander Shapiro,et al.  Coherent risk measures in inventory problems , 2007, Eur. J. Oper. Res..

[2]  C. Acerbi.  Spectral measures of risk: A coherent representation of subjective risk aversion , 2002 .

[3]  英敦 塚原.  Aad W. van der Vaart and Jon A. Wellner: Weak Convergence and Empirical Processes: With Applications to Statistics, Springer,1996年,xvi + 508ページ. , 2009 .

[4]  Galen R. Shorack,et al.  Functions of Order Statistics , 1972 .

[5]  Werner Römisch,et al.  Delta Method, Infinite Dimensional , 2006 .

[6]  Bruce L. Jones,et al.  Empirical Estimation of Risk Measures and Related Quantities , 2003 .

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

[8]  Pushpa L. Gupta,et al.  Failure rate of the minimum and maximum of a multivariate normal distribution , 2001 .

[9]  Olivier Scaillet,et al.  Nonparametric Estimation of Conditional Expected Shortfall , 2004, Assurances et gestion des risques.

[10]  A. V. D. Vaart,et al.  Asymptotic Statistics: U -Statistics , 1998 .

[11]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[12]  Alexander Shapiro,et al.  Optimization of Convex Risk Functions , 2006, Math. Oper. Res..

[13]  Tom Fischer,et al.  Risk Capital allocation by coherent risk measures based on one-sided moments , 2003 .

[14]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[15]  Enrico G. De Giorgi,et al.  A Note on Portfolio Selections under Various Risk Measures , 2002 .

[16]  Philippe Artzner.  Application of Coherent Risk Measures to Capital Requirements in Insurance , 1999 .

[17]  Bruce L. Jones,et al.  Risk measures, distortion parameters, and their empirical estimation , 2007 .

[18]  A. V. D. Vaart,et al.  Asymptotic Statistics: Frontmatter , 1998 .

[19]  G. Pflug,et al.  Modeling, Measuring and Managing Risk , 2008 .

[20]  Steven E. Shreve,et al.  Satisfying convex risk limits by trading , 2005, Finance Stochastics.

[21]  Christian Gourieroux,et al.  Sensitivity Analysis of Distortion Risk Measures , 2006 .

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

[23]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[24]  D. Mason,et al.  Some characterizations of almost sure bounds for weighted multidimensional empirical distributions and a Glivenko-Cantelli theorem for sample quantiles , 1982 .

[25]  Miklós Csörgő.  Review: Galen R. Shorack and Jon A. Wellner, Empirical processes with applications to statistics , 1987 .

[26]  Robert A. Jarrow.  Put Option Premiums and Coherent Risk Measures , 2002 .

[27]  Georg Ch. Pflug,et al.  Subdifferential representations of risk measures , 2006, Math. Program..

[28]  Pflug Georg Ch..  On distortion functionals , 2006 .

[29]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[30]  A. S. Cherny,et al.  Weighted V@R and its Properties , 2006, Finance Stochastics.

[31]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

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