Soit un échantillon d'un processus de Lévy X = (Xt)t≥0 à activité finie observé en temps discret, le problème d'estimation non-paramétrique de la densité de Lévy ρ est étudié. Un estimateur de ρ est proposé basé sur une inversion de Fourier de la formule de Lévy-Khintchine et un principe de plug-in. Les principaux résultats de cet article portent sur la majoration du risque de l'estimateur de ρ pour des classes de triplets de Lévy. La minoration du risque est aussi discutée.
Given a sample from a discretely observed Lévy process X = (Xt)t≥0 of the finite jump activity, the problem of nonparametric estimation of the Lévy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Lévy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.
Mots clés : empirical characteristic function, empirical process, Fourier inversion, Lévy density, Lévy process, maximal inequality, mean square error
@article{AIHPB_2012__48_1_282_0, author = {Gugushvili, Shota}, title = {Nonparametric inference for discretely sampled {L\'evy} processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {282--307}, publisher = {Gauthier-Villars}, volume = {48}, number = {1}, year = {2012}, doi = {10.1214/11-AIHP433}, mrnumber = {2919207}, zbl = {1235.62121}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP433/} }
TY - JOUR AU - Gugushvili, Shota TI - Nonparametric inference for discretely sampled Lévy processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 282 EP - 307 VL - 48 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP433/ DO - 10.1214/11-AIHP433 LA - en ID - AIHPB_2012__48_1_282_0 ER -
%0 Journal Article %A Gugushvili, Shota %T Nonparametric inference for discretely sampled Lévy processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 282-307 %V 48 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP433/ %R 10.1214/11-AIHP433 %G en %F AIHPB_2012__48_1_282_0
Gugushvili, Shota. Nonparametric inference for discretely sampled Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 282-307. doi : 10.1214/11-AIHP433. http://www.numdam.org/articles/10.1214/11-AIHP433/
[1] Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35 (2007) 355-392. | MR | Zbl
and .[2] Asymptotic theory for estimating the parameters of a Lévy process. Ann. Inst. Statist. Math. 34 (1982) 259-280. | MR | Zbl
.[3] Asymptotic inference in Lévy processes of the discontinuous type. Ann. Statist. 9 (1981) 604-614. | MR | Zbl
and .[4] Inference for gamma and stable processes. Biometrika 65 (1978) 129-133. | MR | Zbl
and .[5] A note on estimation for gamma and stable processes. Biometrika 67 (1980) 234-236. | MR | Zbl
and .[6] Spectral calibration of exponential Lévy models. Finance Stoch. 10 (2006) 449-474. | MR | Zbl
and .[7] Spectral calibration of exponential Lévy models [2]. Discussion Paper 2006-035, SFB 649, 2006. | Zbl
and .[8] Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl
.[9] A hyperbolic diffusion model for stock prices. Finance Stoch. 1 (1997) 25-41. | Zbl
and .[10] HYP - A computer program for analyzing data by means of the hyperbolic distribution. Research Report 248, Dept. Mathematical Statistics, Aarhus Univ., 1992.
and .[11] Stable distributions. In Statistical Tools for Finance and Insurance 21-44. P. Cizek, W. Härdle and R. Weron (Eds). Springer, Berlin, 2005. | MR
, and .[12] Superefficiency in nonparametric function estimation. Ann. Statist. 25 (1997) 2607-2625. | MR | Zbl
, and .[13] Weighted empirical processes in the nonparametric inference for Lévy processes. Math. Methods Statist. 18 (2009) 281-309. | MR | Zbl
.[14] Decompounding: An estimation problem for Poisson random sums. Ann. Statist. 31 (2003) 1054-1074. | MR | Zbl
and .[15] Decompounding Poisson random sums: Recursively truncated estimates in the discrete case. Ann. Inst. Statist. Math. 56 (2004) 743-756. | MR | Zbl
and .[16] Inversion formula for infinitely divisible distributions. Russian Math. Surveys 61 (2006) 772-774. | MR | Zbl
.[17] Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11 (2005) 309-340. | MR | Zbl
and .[18] Sharp optimality for density deconvolution with dominating bias, I. Theory Probab. Appl. 52 (2008) 24-39. | MR | Zbl
and .[19] Sharp optimality for density deconvolution with dominating bias, II. Theory Probab. Appl. 52 (2008) 237-249. | MR | Zbl
and .[20] The fine structure of asset returns: An empirical investigation. J. Bus. 75 (2002) 305-332.
, , , and .[21] Nonparametric estimation for a class of Lévy processes. J. Econometrics 157 (2010) 257-271. | MR
, and .[22] A Course in Probability Theory, 3rd edition. Academic Press, San Diego, CA, 2001. | MR | Zbl
.[23] Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stochastic Process. Appl. 119 (2009) 4088-4123. | MR | Zbl
and .[24] Nonparametric adaptive estimation for pure jump Lévy processes. Ann. Inst. H. Poincaré Probab. Stat. 46 (2010) 595-617. | Numdam | MR | Zbl
and .[25] Non-parametric estimation for pure jump irregularly sampled or noisy Lévy processes. Stat. Neerl. 64 (2010) 290-313. | MR
and .[26] Estimation for Lévy processes from high frequency data within a long time interval. Ann. Statist. 39 (2011) 803-837. | MR | Zbl
and .[27] Data driven density estimation in presence of additive noise with unknown distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. (2011). To appear. DOI:10.1111/j.1467-9868.2011.00775.x. | MR | Zbl
and .[28] Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, 2003. | Zbl
and .[29] Retrieving Lévy processes from option prices: Regularization of an ill-posed inverse problem. SIAM J. Control Optim. 45 (2006) 1-25. | MR | Zbl
and .[30] An alternative view of the deconvolution problem. Statist. Sinica 18 (2008) 1025-1045. | MR | Zbl
.[31] On the non-consistency of an estimate of Chiu. Statist. Probab. Lett. 20 (1994) 183-188. | MR | Zbl
.[32] Nonparametric Density Estimation: TheL1 View. Wiley, New York, 1985. | MR | Zbl
and .[33] On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 (1991) 1257-1272. | MR | Zbl
.[34] Deconvolution with supersmooth distributions. Canad. J. Statist. 20 (1992) 155-169. | MR | Zbl
.[35] Sieve-based confidence intervals and bands for Lévy densities. Bernoulli 17 (2011) 643-670. | MR
.[36] Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process. J. Nonparametr. Stat. 21 (2009) 321-343. | MR | Zbl
.[37] Statistical Inference for Lévy Processes with Applications to Finance. Stat. Neerl. 64 (3), 2010. | MR
, and (Eds).[38] Deconvolution for an atomic distribution: Rates of convergence. J. Nonparametr. Stat. (2011). To appear. DOI:10.1080/10485252.2011.576763. | MR | Zbl
, and .[39] Parametric estimation for subordinators and induced OU processes. Scand. J. Stat. 33 (2006) 825-847. | MR | Zbl
and .[40] Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 11 (2005) 759-791. | MR | Zbl
, and .[41] Estimation of the characteristics of a Lévy process observed at arbitrary frequency. Stat. Neerl. 64 (2010) 314-328. | MR
and .[42] Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. | MR
.[43] Density estimation with normal measurement error with unknown variance. Statist. Sinica 16 (2006) 195-211. | MR | Zbl
.[44] Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976) 125-144. | Zbl
.[45] On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Statist. 7 (1997) 307-330. | MR | Zbl
.[46] Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 (2009) 223-248. | MR | Zbl
and .[47] Maximum likelihood estimation and diagnostics for stable distributions. In Lévy Processes: Theory and Applications 379-400. O. E. Barndorff-Nielsen, T. Mikosch, and S. I. Resnick (Eds). Birkhäuser, Boston, 2001. | MR | Zbl
.[48] On the best constant in Marcinkiewicz-Zygmund inequality. Statist. Probab. Lett. 53 (1999) 227-233. | MR | Zbl
and .[49] The normal inverse Gaussian Lévy process: Simulation and approximation. Stoch. Models 13 (1997) 887-910. | MR | Zbl
.[50] Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 2004.
.[51] Polar sets for anisotropic Gaussian random fields. Statist. Probab. Lett. 80 (2010) 840-847. | MR | Zbl
.[52] Introduction to Nonparametric Estimation. Springer, New York, 2009. | MR | Zbl
.[53] Asymptotic Statistics. Cambridge Univ. Press, Cambridge, 1998. | MR | Zbl
.[54] Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York, 1996. | MR | Zbl
and .[55] Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance. J. Korean Statist. Soc. 39 (2010) 102-115. | MR
and .[56] A kernel type nonparametric density estimator for decompounding. Bernoulli 13 (2007) 672-694. | MR | Zbl
, and .[57] Finite sample performance of deconvolving density estimators. Statist. Probab. Lett. 37 (1998) 131-139. | MR | Zbl
.[58] Nonparametric estimation of the canonical measure for infinitely divisible distributions. J. Stat. Comput. Simul. 73 (2003) 525-542. | MR | Zbl
and .[59] One-Dimensional Stable Distributions. American Mathematical Society, Providence, 1986. | MR | Zbl
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