AN ASYMMETRIC MULTIVARIATE LAPLACE DISTRIBUTION

We present a class of multivariate laws which is an extension of the symmetric multivariate Laplace distributions and of the univariate asymmetric Laplace distributions. The extension retains natural, asymmetric and multivariate, properties characterizing these two subclasses. The results include characterizations, mixture representations, formulas for densities and moments, and a simulation algorithm. The new family can be viewed as a subclass of hyperbolic distributions.

[1]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[2]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[3]  W. J. McGill,et al.  Random fluctuations of response rate , 1962 .

[4]  J. F. Wagner,et al.  Elliptically symmetric distributions , 1968, IEEE Trans. Inf. Theory.

[5]  La Hire,et al.  Memoires de mathematique et de physique , 1970 .

[6]  D. F. Andrews,et al.  Robust Estimates of Location , 1972 .

[7]  On Classes of Limit Distributions for Sums of a Random Number of Identically Distributed Independent Random Variables , 1973 .

[8]  F. Olver.  Asymptotics and Special Functions , 1974 .

[9]  O. Barndorff-Nielsen.  Exponentially decreasing distributions for the logarithm of particle size , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  D. Hinkley,et al.  Estimation of the Pareto law from underreported data: A further analysis , 1977 .

[11]  Grady Miller.  Properties of certain symmetric stable distributions , 1978 .

[12]  D. A. Hsu,et al.  Long-tailed Distributions for Position Errors in Navigation , 1979 .

[13]  Ole E. Barndorff-Nielsen,et al.  Hyperbolic Distributions and Ramifications: Contributions to Theory and Application , 1981 .

[14]  A. C. Atkinson,et al.  The Simulation of Generalized Inverse Gaussian and Hyperbolic Random Variables , 1982 .

[15]  F. Mosteller,et al.  Understanding robust and exploratory data analysis , 1985 .

[16]  Uttarayan Bagchi,et al.  THE HERMITE DISTRIBUTION AS A MODEL OF DEMAND DURING LEAD TIME FOR SLOW‐MOVING ITEMS , 1983 .

[17]  Michael Stuart,et al.  Understanding Robust and Exploratory Data Analysis , 1984 .

[18]  Lev B. Klebanov,et al.  A Problem of Zolotarev and Analogs of Infinitely Divisible and Stable Distributions in a Scheme for Summing a Random Number of Random Variables , 1985 .

[19]  R. N. Pillai.  Semi - α - laplace distributions , 1985 .

[20]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[21]  R. Marks,et al.  Detector Relative Efficiencies in the Presence of Laplace Noise , 1987, IEEE Transactions on Aerospace and Electronic Systems.

[22]  M. E. Johnson,et al.  Multivariate Statistical Simulation , 1988 .

[23]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[24]  A. El-Shaarawi,et al.  ARMA Models with Double-exponentially Distributed Noise , 1989 .

[25]  W. Chan,et al.  Unimodality, convexity, and applications , 1989 .

[26]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[27]  Svetlozar T. Rachev,et al.  GEOMETRIC STABLE DISTRIBUTIONS AND LAPLACE-WEIBULL MIXTURES , 1992 .

[28]  A characterization of gamma mixtures of stable laws motivated by limit theorems , 1992 .

[29]  Dale N. Anderson.  A multivariate Linnik distribution , 1992 .

[30]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[31]  THE INNER CHARACTERIZATION OF GEOMETRIC STABLE LAWS , 1994 .

[32]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[33]  M. Steel,et al.  Modeling and Inference with υ-Spherical Distributions , 1995 .

[34]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[35]  Tomasz J. Kozubowski.  CHARACTERIZATION OF MULTIVARIATE GEOMETRIC STABLE DISTRIBUTIONS , 1997 .

[36]  V. Kalashnikov,et al.  Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing , 1997 .

[37]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[38]  Multivariate Geometric Stable Laws , 1999 .

[39]  Adrian F. M. Smith,et al.  On Robust Bayesian Analysis for Location and Scale Parameters , 1999 .

[40]  Tomasz J. Kozubowski,et al.  A CLASS OF ASYMMETRIC DISTRIBUTIONS , 1999 .

[41]  T. Kozubowski,et al.  Multivariate geometric stable distributions in financial applications , 1999 .

[42]  Svetlozar T. Rachev,et al.  Univariate Geometric Stable Laws , 1999 .

[43]  R. Wolpert.  Lévy Processes , 2000 .

[44]  Samuel Kotz,et al.  Asymmetric Laplace Distributions , 2001 .

[45]  K. Podgórski,et al.  Asymmetric laplace laws and modeling financial data , 2001 .

[46]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[47]  Statistical Valuation of Diamondiferous Deposits , 2010 .

[48]  Mark E. Johnson,et al.  Multivariate Statistical Simulation , 1989, International Encyclopedia of Statistical Science.

[49]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[50]  Maximum likelihood characterizations of distributions: a generalization of Gauss' result , 2012 .

[51]  I. V. Zolotukhin.  MULTIVARIATE GENERALIZED LAPLACE DISTRIBUTION , 2015 .

[52]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .