Asymptotic stochastic dominance rules for sums of i.i.d. random variables

In this paper, we deal with stochastic dominance rules under the assumption that the random variables are stable distributed. The stable Paretian distribution is generally used to model a wide range of phenomena. In particular, its use in several applicative areas is mainly justified by the generalized central limit theorem, which states that the sum of a number of i.i.d. random variables with heavy tailed distributions tends to a stable Paretian distribution. We show that the asymptotic behavior of the tails is fundamental for establishing a dominance in the stable Paretian case. Moreover, we introduce a new weak stochastic order of dispersion, aimed at evaluating whether a random variable is more "risky" than another under condition of maximum uncertainty, and a stochastic order of asymmetry, aimed at evaluating whether a random variable is more or less asymmetric than another. The theoretical results are confirmed by a financial application of the obtained dominance rules. The empirical analysis shows that the weak order of risk introduced in this paper is generally a good indicator for the second order stochastic dominance.

[1]  Anna K. Panorska,et al.  Generalized stable models for financial asset returns , 1996 .

[2]  Svetlozar T. Rachev,et al.  The basics of financial econometrics : tools, concepts, and asset management applications , 2014 .

[3]  S. O. Lozza,et al.  Dominance among financial markets , 2014 .

[4]  H. L. MacGillivray,et al.  Skewness and Asymmetry: Measures and Orderings , 1986 .

[5]  Michel Denuit,et al.  Actuarial Theory for Dependent Risks: Measures, Orders and Models , 2005 .

[6]  Svetlozar T. Rachev,et al.  Safety-first analysis and stable paretian approach to portfolio choice theory , 2001 .

[7]  Svetlozar T. Rachev,et al.  Financial Econometrics: From Basics to Advanced Modeling Techniques , 2006 .

[8]  J. Nolan,et al.  Maximum likelihood estimation and diagnostics for stable distributions , 2001 .

[9]  E. Fama.  Mandelbrot and the Stable Paretian Hypothesis , 1963 .

[10]  S. Rachev,et al.  Stable Paretian Models in Finance , 2000 .

[11]  Stoyan Stoyanov,et al.  Portfolio selection with heavy tailed distributions , 2005 .

[12]  Mark E. J. Newman,et al.  Power-Law Distributions in Empirical Data , 2007, SIAM Rev..

[13]  Jan Dhaene,et al.  Comonotonic approximations for a generalized provisioning problem with application to optimal portfolio selection , 2011, J. Comput. Appl. Math..

[14]  J. L. Nolan,et al.  Numerical calculation of stable densities and distribution functions: Heavy tails and highly volatil , 1997 .

[15]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[16]  E. Fama.  Portfolio Analysis in a Stable Paretian Market , 1965 .

[17]  Stefan Mittnik,et al.  Computing the probability density function of the stable Paretian distribution , 1999 .

[18]  B. Mandelbrot.  The Variation of Some Other Speculative Prices , 1967 .

[19]  Svetlozar T. Rachev,et al.  Orderings and Risk Probability Functionals in Portfolio Theory , 2008 .

[20]  Svetlozar T. Rachev,et al.  Approximation of skewed and leptokurtic return distributions , 2012 .

[21]  J. Shanthikumar,et al.  Multivariate Stochastic Orders , 2007 .

[22]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[23]  T. Lando,et al.  Statistical functionals consistent with a weak relative majorization ordering: applications to the mimimum divergence estimation , 2014 .

[24]  Barry C. Arnold,et al.  Skewness and kurtosis orderings: an introduction , 1992 .

[25]  H. Levy,et al.  The Efficiency Analysis of Choices Involving Risk1 , 1975 .

[26]  Howard M. Taylor,et al.  On the Distribution of Stock Price Differences , 1967, Oper. Res..

[27]  K. Chung,et al.  Limit Distributions for Sums of Independent Random Variables. , 1955 .

[28]  竹中 茂夫.  G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance , 1996 .

[29]  Quang Huy Nguyen,et al.  New efficient estimators in rare event simulation with heavy tails , 2014, J. Comput. Appl. Math..

[30]  B. Mandelbrot.  New Methods in Statistical Economics , 1963, Journal of Political Economy.

[31]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[32]  Gennady Samorodnitsky,et al.  Computing the Portfolio Conditional Value-at-Risk in the Alpha-Stable Case , 2005 .

[33]  R. H. Rimmer,et al.  Stable Distributions in Mathematica , 2005 .

[34]  Jean-Marie Dufour,et al.  Editors’ introduction: Heavy tails and stable Paretian distributions in econometrics , 2014 .

[35]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[36]  S. Rachev,et al.  The Methods of Distances in the Theory of Probability and Statistics , 2013 .

[37]  Eckhard Platen,et al.  Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices , 2007 .