Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T09:08:54.850Z Has data issue: false hasContentIssue false
  • English
  • Français

Local asymptotic normality for normal inverse Gaussian Lévyprocesses with high-frequency sampling∗

Published online by Cambridge University Press:  06 December 2012

Reiichiro Kawai  and
Hiroki Masuda
Reiichiro Kawai
Affiliation:
School of Mathematics and Statistics, University of Sydney NSW 2006, Australia.. [email protected]
Hiroki Masuda
Affiliation:
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.; [email protected]
Get access

Abstract

We prove the local asymptotic normality for the full parameters of the normal inverseGaussian Lévy process X, when we observe high-frequency data XΔn,X2Δn,...,XnΔnwith sampling mesh Δn → 0 and the terminalsampling time nΔn → ∞. Therate of convergence turns out to be (√nΔn, √nΔn, √n, √n) for the dominating parameter(α,β,δ,μ), where α stands for the heaviness of thetails, β the degree of skewness, δ the scale, andμ the location. The essential feature in our study is that the suitablynormalized increments of X in small time is approximatelyCauchy-distributed, which specifically comes out in the form of the asymptotic Fisherinformation matrix.

Keywords

High-frequency samplinglocal asymptotic normalitynormal inverse Gaussian Lévy process.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 13 - 32
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

M. Abramowitz and I.A. Stegun Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Reprint of the 1972 edition, Dover Publications, Inc., New York (1992)
Aït-Sahalia, Y. and Jacod, J., Fisher’s information for discretely sampled Lévy processes. Econometrica 76 (2008) 727–761. Google Scholar
Akritas, M.G. and Johnson, R.A., Asymptotic inference in Lévy processes of the discontinuous type. Ann. Stat. 9 (1981) 604–614. Google Scholar
Asmussen, S. and Rosiński, J., Approximations of small jumps of Levy processes with a view towards simulation. J. Appl. Probab. 38 (2001) 482–493. Google Scholar
Barndorff-Nielsen, O.E., Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. A 353 (1977) 401–419. Google Scholar
O.E. Barndorff-Nielsen, Normal inverse Gaussian processes and the modelling of stock returns. Research report 300, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus (1995)
Barndorff-Nielsen, O.E., Processes of normal inverse Gaussian type. Finance Stoch. 2 (1998) 41–68. Google Scholar
Cox, D.R. and Reid, N., Parameter orthogonality and approximate conditional inference. With a discussion. J. R. Stat. Soc., Ser. B 49 (1987) 1–39. Google Scholar
J. Jacod, Inference for stochastic processes, in Handbook of Financial Econometrics, edited by Y. Aït-Sahalia and L.P. Hansen, Amsterdam, North-Holland (2010)
Jørgensen, B. and Knudsen, S.J., Parameter orthogonality and bias adjustment for estimating functions. Scand. J. Statist. 31 (2004) 93–114. Google Scholar
O. Kallenberg, Foundations of Modern Probability. 2nd edition, Springer-Verlag, New York (2002)
Karlis, D., An EM type algorithm for maximum likelihood estimation of the normal-inverse Gaussian distribution. Stat. Probab. Lett. 57 (2002) 43–52. Google Scholar
Karlis, D. and Lillestöl, J., Bayesian estimation of NIG models via Markov chain Monte Carlo methods. Appl. Stoch. Models Bus. Ind. 20 (2004) 323–338. Google Scholar
Kawai, R. and Masuda, H., On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling. Stat. Probab. Lett. 81 (2011) 460–469. Google Scholar
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. Stat. 3 (1960) 37–98. Google Scholar
L. Le Cam and G.L. Yang, Asymptotics in Statistics. Some Basic Concepts. 2nd edition, Springer-Verlag, New York (2000)
Masuda, H., Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling. Ann. Inst. Stat. Math. 61 (2009) 181–195. Google Scholar
Masuda, H., Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density. J. Japan Stat. Soc. 39 (2009) 49–75. Google Scholar
K. Prause, The Generalized Hyperbolic Model : Estimation, Financial Derivatives, and Risk Measures. Ph.D. thesis, University of Freiburg (1999). Available at http://www.freidok.uni-freiburg.de/volltexte/15/
S. Raible, Lévy Processes in Finance : Theory, Numerics, and Empirical Facts. Ph.D. thesis, University of Freiburg (2000). Available at http://www.freidok.uni-freiburg.de/volltexte/51/
K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
A.N. Shiryaev, Probability. 2nd edition, Springer-Verlag, New York (1996)
H. Strasser, Mathematical Theory of Statistics. Statistical Experiments and Asymptotic Decision Theory. Walter de Gruyter & Co., Berlin (1985)
A.W. van der Vaart, Asymptotic Statistics. Cambridge University Press, Cambridge (1998)
J.H.C. Woerner, Statistical Analysis for Discretely Observed Lévy Processes. Ph.D. thesis, University of Freiburg (2001). Available at http://www.freidok.uni-freiburg.de/volltexte/295/
Woerner, J.H.C., Estimating the skewness in discretely observed Lévy processes. Econ. Theory 20 (2004) 927–942. Google Scholar