Parameter estimation for tempered power law distributions

Tail estimates are developed for power law probability distributions with exponential tempering using a conditional maximum likelihood approach based on the upper order statistics. The method is demonstrated on simulated data from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.

[1]  D. Benson,et al.  Multidimensional advection and fractional dispersion. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  V. Zolotarev,et al.  Chance and Stability, Stable Distributions and Their Applications , 1999 .

[3]  J. Rosínski.  Tempering stable processes , 2007 .

[4]  Mark M. Meerschaert,et al.  Limit theorems for continuous-time random walks with infinite mean waiting times , 2004, Journal of Applied Probability.

[5]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

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

[7]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

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

[9]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[10]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

[11]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[12]  Mark M. Meerschaert,et al.  Tempered stable Lévy motion and transient super-diffusion , 2010, J. Comput. Appl. Math..

[13]  P. Hall.  On Some Simple Estimates of an Exponent of Regular Variation , 1982 .

[14]  D. Benson,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[15]  Rina Schumer,et al.  Multiscaling fractional advection‐dispersion equations and their solutions , 2003 .

[16]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[17]  Thomas Augustin,et al.  Foundations of Probability , 2011, International Encyclopedia of Statistical Science.

[18]  G. Shorack.  Probability for Statisticians , 2000 .

[19]  장윤희,et al.  Y. , 2003, Industrial and Labor Relations Terms.

[20]  R. Mantegna,et al.  Scaling behaviour in the dynamics of an economic index , 1995, Nature.

[21]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[22]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[23]  Stanley,et al.  Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. , 1994, Physical review letters.

[24]  Peter Richmond,et al.  Waiting time distributions in financial markets , 2002 .

[25]  M. Meerschaert,et al.  Tempered anomalous diffusion in heterogeneous systems , 2008 .