Publication Cover
Statistics
A Journal of Theoretical and Applied Statistics
Volume 49, 2015 - Issue 1
171
Views
8
CrossRef citations to date
0
Altmetric
Original Articles

On the likelihood function of small time variance Gamma Lévy processes

Pages 63-83 | Received 29 Aug 2013, Accepted 14 Apr 2014, Published online: 20 May 2014

References

  • Wilson EB. First and second laws of error. J Am Statist Assoc. 1923;18(143):841–851. doi: 10.1080/01621459.1923.10502116
  • Kotz S, Kozubowski TJ, Podgórski K. The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Boston, MA: Birkhäuser; 2001.
  • Kawai R, Masuda H. On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling. Stat Probab Lett. 2011;81(4):460–469. doi: 10.1016/j.spl.2010.12.011
  • Kawai R, Masuda H. Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling. ESAIM: Probab Stat. 2013;17:13–32. doi: 10.1051/ps/2011101
  • Podgórski K, Wegener J. Estimation for stochastic models driven by Laplace motion. Commun Stat – Theory Methods. 2011;40:3281–3302. doi: 10.1080/03610926.2010.499051
  • Madan D, Carr P, Chang E. The variance gamma process and option pricing. Eur Finance Rev. 1998;2:79–105. doi: 10.1023/A:1009703431535
  • Mathai AM. Generalized Laplace distribution with applications. J Appl Statist Sci. 1993;1(2):169–178.
  • Hinkley DV, Revankar NS. Estimation of the Pareto law from underreported data. J Econometrics. 1977;5:1–11. doi: 10.1016/0304-4076(77)90031-8
  • Kozubowski TJ, Podgórski K. Asymmetric Laplace distributions. Math Sci. 2000;25:37–46.
  • Laplace PS. Mémoire sur la probabilité des causes par les événements. Mémoires de Mathématique et de Physique, Presentés à l'Académie Royale des Sciences, par divers Savans et lŝ dans ses Assemblées. 1774;6:621–656. (English translation Mem Probab Causes Events Statist Sci. 1986;1(3):364–378.)
  • Bondesson L. On simulation from infinitely divisible distributions. Adv Appl Probab. 1982;14(4):855–869. doi: 10.2307/1427027
  • Imai J, Kawai R. Quasi-Monte Carlo methods for infinitely divisible random vectors via series representations. SIAM J Sci Comput. 2010;32(4):1879–1897. doi: 10.1137/090752365
  • Harter HL, Moore AH. Maximum-likelihood estimation of the parameters of gamma and Weibull populations from complete and from censored samples. Technometrics. 1965;7(4):639–643. doi: 10.1080/00401706.1965.10490304
  • Podgórski K, Wallin J. Maximizing leave-one-out likelihood for the location parameter of unbounded densities. Ann Inst Statist Math; 2013, in press. 10.1007/s10463-013-0437-6.
  • Kawai R, Kohatsu-Higa A. Computation of Greeks and multidimensional density estimation for asset price models with time-changed Brownian motion. Appl Math Financ. 2010;17(4):301–321. doi: 10.1080/13504860903336429
  • Kawai R, Takeuchi A. Greeks formulas for an asset price model with gamma processes. Mathematical Finance. 2011;21(4):723–742.
  • Kawai R, Petrovskii S. Multiscale properties of random walk models of animal movement: lessons from statistical inference. Proc R Soc A. 2012;468(2141):1428–1451. doi: 10.1098/rspa.2011.0665
  • Le Cam L. Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses. Univ California Publ Statist. 1960;3:37–98.
  • Le Cam L, Yang GL. Asymptotics in statistics. Some basic concepts. 2nd ed. Springer Series in Statistics. New York: Springer-Verlag; 1990.
  • van der Vaart AW. Asymptotic statistics. Cambridge: Cambridge University Press; 1998.
  • Kawai R. On singularity of Fisher information matrix for stochastic processes under high frequency sampling. In: Cangiani A, Davidchack RL, Georgoulis E, Gorban AN, Levesley J, Tretyakov MV. Numerical mathematics and advanced applications 2011. Springer-Verlag; 2013. p. 841–849.
  • Woerner J. Estimating the skewness in discretely observed Lévy processes. Econom Theory. 2004;20(5):927–942.
  • Bickel PJ, Doksum KA. Mathematical statistics: basic ideas and selected topics. Englewood Cliffs, NJ: Prentice-Hall; 1977.
  • Figueroa-López JE, Lancette SR, Lee K, Mi Y. Estimation of NIG and VG models for high frequency financial data. In: Viens F, Mariani MC, Florescu I, editors. Handbook of modeling high-frequency data in finance. Wiley; 2012. 1–25.
  • Scott DJ, Dong CY. VarianceGamma, Version 0.2-1, The variance gamma distribution [Software] CRAN, the Comprehensive R Archive Network; 2000.
  • Sato K. Lévy Processes and infinitely divisible distributions. Cambridge: Cambridge University Press; 1999.
  • Brychkov YA, Geddes KO. On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec Funct. 2005;16(3):187–198. doi: 10.1080/10652460410001727572
  • Cramér H. Mathematical methods of statistics. Princeton, NJ: Princeton University Press; 1946.
  • Ferguson TS. A course in large sample theory. Chapman & Hall/ CRC; 1996.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.