×

On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling. (English) Zbl 1208.62037

Summary: We discuss the local asymptotic behavior of the likelihood function associated with all the four characterizing parameters \((\alpha ,\beta ,\delta ,\mu )\) of the Meixner Lévy process under a high-frequency sampling scheme. We derive the optimal rate of convergence for each parameter and the Fisher information matrix in a closed form. The skewness parameter \(\beta \) exhibits a slower rate alone, relative to the other three parameters free of sampling rate. An unusual aspect is that the Fisher information matrix is constantly singular for full joint estimation of the four parameters. This is a particular phenomenon in the regular high-frequency sampling setting and is of essentially different nature from low-frequency sampling. As soon as either \(\alpha \) or \(\delta \) is fixed, the Fisher information matrix becomes diagonal, implying that the corresponding maximum likelihood estimators are asymptotically orthogonal.

MSC:

62F12 Asymptotic properties of parametric estimators
62M09 Non-Markovian processes: estimation
60G51 Processes with independent increments; Lévy processes
62E20 Asymptotic distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Aït-Sahalia, Y.; Jacod, J., Fisher’s information for discretely sampled Lévy processes, Econometrica, 76, 727-761 (2008) · Zbl 1144.62070
[2] Grigelionis, B., Processes of Meixner type, Lithuanian Mathematical Journal, 39, 1, 33-41 (1999) · Zbl 0959.60034
[3] Grigoletto, M.; Provasi, C., Simulation and estimation of the Meixner distribution, Communications in Statistics—Simulation and Computation, 38, 1, 58-77 (2009) · Zbl 1161.62004
[4] Houdré, C.; Kawai, R., On layered stable processes, Bernoulli, 13, 1, 252-278 (2007) · Zbl 1121.60052
[5] Kawai, R., 2010. Likelihood ratio gradient estimation for Meixner distribution and Levy processes. Available at:http://sites.google.com/site/reiichirokawai/Home/mxsen.pdf; Kawai, R., 2010. Likelihood ratio gradient estimation for Meixner distribution and Levy processes. Available at:http://sites.google.com/site/reiichirokawai/Home/mxsen.pdf · Zbl 1304.65044
[6] Kawai, R., Masuda, H., 2010. Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling. Available at: http://hdl.handle.net/2324/17018; Kawai, R., Masuda, H., 2010. Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling. Available at: http://hdl.handle.net/2324/17018
[7] 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., 3, 37-98 (1960)
[8] Le Cam, L.; Yang, G. L., (Asymptotics in Statistics. Some Basic Concepts. Asymptotics in Statistics. Some Basic Concepts, Springer Series in Statistics (1990), Springer-Verlag: Springer-Verlag New York) · Zbl 0719.62003
[9] Masuda, H., Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density, Journal of the Japan Statistical Society, 39, 49-75 (2009)
[10] Masuda, H., Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling, Annals of the Institute of Statistical Mathematics, 61, 181-195 (2009) · Zbl 1294.62039
[11] Rosiński, J., Tempering stable processes, Stochastic Processes and their Applications, 117, 6, 677-707 (2007) · Zbl 1118.60037
[12] Sato, K., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0973.60001
[13] Schoutens, W.; Teugels, J. L., Lévy processes, polynomials and martingales, Communications in Statistics: Stochastic Models, 14, 335-349 (1998) · Zbl 0895.60050
[14] van der Vaart, A. W., Asymptotic Statistics (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0910.62001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.