Nonparametric inference for discretely sampled Lévy processes
Gugushvili, Shota
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 282-307 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/11-AIHP433
Classification:  62G07,  62G20 
@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},
     volume = {48},
     year = {2012},
     pages = {282-307},
     doi = {10.1214/11-AIHP433},
     mrnumber = {2919207},
     zbl = {1235.62121},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 282-307. doi : 10.1214/11-AIHP433. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_1_282_0/

[1] Y. Aït-Sahalia and J. Jacod. Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35 (2007) 355-392. | MR 2332279 | Zbl 1114.62109

[2] M. G. Akritas. Asymptotic theory for estimating the parameters of a Lévy process. Ann. Inst. Statist. Math. 34 (1982) 259-280. | MR 666416 | Zbl 0491.62070

[3] M. G. Akritas and R. A. Johnson. Asymptotic inference in Lévy processes of the discontinuous type. Ann. Statist. 9 (1981) 604-614. | MR 615436 | Zbl 0481.62069

[4] I. V. Basawa and P. J. Brockwell. Inference for gamma and stable processes. Biometrika 65 (1978) 129-133. | MR 494746 | Zbl 0381.62075

[5] I. V. Basawa and P. J. Brockwell. A note on estimation for gamma and stable processes. Biometrika 67 (1980) 234-236. | MR 570526 | Zbl 0425.62067

[6] D. Belomestny and M. Reiß. Spectral calibration of exponential Lévy models. Finance Stoch. 10 (2006) 449-474. | MR 2276314 | Zbl 1126.91022

[7] D. Belomestny and M. Reiß. Spectral calibration of exponential Lévy models [2]. Discussion Paper 2006-035, SFB 649, 2006. | Zbl 1126.91022

[8] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR 1406564 | Zbl 0938.60005

[9] B. M. Bibby and M. Sørensen. A hyperbolic diffusion model for stock prices. Finance Stoch. 1 (1997) 25-41. | Zbl 0883.90010

[10] P. Blæsild and M. Sørensen. HYP - A computer program for analyzing data by means of the hyperbolic distribution. Research Report 248, Dept. Mathematical Statistics, Aarhus Univ., 1992.

[11] S. Borak, W. Härdle and R. Weron. Stable distributions. In Statistical Tools for Finance and Insurance 21-44. P. Cizek, W. Härdle and R. Weron (Eds). Springer, Berlin, 2005. | MR 2156758

[12] L. D. Brown, M. G. Low and L. H. Zhao. Superefficiency in nonparametric function estimation. Ann. Statist. 25 (1997) 2607-2625. | MR 1604424 | Zbl 0895.62043

[13] B. Buchmann. Weighted empirical processes in the nonparametric inference for Lévy processes. Math. Methods Statist. 18 (2009) 281-309. | MR 2608164 | Zbl 1282.60049

[14] B. Buchmann and R. Grübel. Decompounding: An estimation problem for Poisson random sums. Ann. Statist. 31 (2003) 1054-1074. | MR 2001642 | Zbl 1105.62309

[15] B. Buchmann and R. Grübel. Decompounding Poisson random sums: Recursively truncated estimates in the discrete case. Ann. Inst. Statist. Math. 56 (2004) 743-756. | MR 2126809 | Zbl 1078.62020

[16] E. V. Burnaev. Inversion formula for infinitely divisible distributions. Russian Math. Surveys 61 (2006) 772-774. | MR 2278841 | Zbl 1121.60013

[17] C. Butucea and C. Matias. Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11 (2005) 309-340. | MR 2132729 | Zbl 1063.62044

[18] C. Butucea and A. B. Tsybakov. Sharp optimality for density deconvolution with dominating bias, I. Theory Probab. Appl. 52 (2008) 24-39. | MR 2354572 | Zbl 1141.62021

[19] C. Butucea and A. B. Tsybakov. Sharp optimality for density deconvolution with dominating bias, II. Theory Probab. Appl. 52 (2008) 237-249. | MR 2742504 | Zbl 1142.62017

[20] P. Carr, H. Geman, D. B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. J. Bus. 75 (2002) 305-332.

[21] S. X. Chen, A. Delaigle and P. Hall. Nonparametric estimation for a class of Lévy processes. J. Econometrics 157 (2010) 257-271. | MR 2661599

[22] K. L. Chung. A Course in Probability Theory, 3rd edition. Academic Press, San Diego, CA, 2001. | MR 1796326 | Zbl 0345.60003

[23] F. Comte and V. Genon-Catalot. Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stochastic Process. Appl. 119 (2009) 4088-4123. | MR 2565560 | Zbl 1177.62043

[24] F. Comte and V. Genon-Catalot. Nonparametric adaptive estimation for pure jump Lévy processes. Ann. Inst. H. Poincaré Probab. Stat. 46 (2010) 595-617. | Numdam | MR 2682259 | Zbl 1201.62042

[25] F. Comte and V. Genon-Catalot. Non-parametric estimation for pure jump irregularly sampled or noisy Lévy processes. Stat. Neerl. 64 (2010) 290-313. | MR 2683462

[26] F. Comte and V. Genon-Catalot. Estimation for Lévy processes from high frequency data within a long time interval. Ann. Statist. 39 (2011) 803-837. | MR 2816339 | Zbl 1215.62084

[27] F. Comte and C. Lacour. 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 2853732 | Zbl 1226.62034

[28] R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, 2003. | Zbl 1052.91043

[29] R. Cont and P. Tankov. Retrieving Lévy processes from option prices: Regularization of an ill-posed inverse problem. SIAM J. Control Optim. 45 (2006) 1-25. | MR 2225295 | Zbl 1110.49033

[30] A. Delaigle. An alternative view of the deconvolution problem. Statist. Sinica 18 (2008) 1025-1045. | MR 2440402 | Zbl 1149.62025

[31] L. Devroye. On the non-consistency of an estimate of Chiu. Statist. Probab. Lett. 20 (1994) 183-188. | MR 1294102 | Zbl 0802.62041

[32] L. Devroye and L. Györfi. Nonparametric Density Estimation: TheL1 View. Wiley, New York, 1985. | MR 780746 | Zbl 0546.62015

[33] J. Fan. On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 (1991) 1257-1272. | MR 1126324 | Zbl 0729.62033

[34] J. Fan. Deconvolution with supersmooth distributions. Canad. J. Statist. 20 (1992) 155-169. | MR 1183078 | Zbl 0754.62020

[35] E. Figueroa-López. Sieve-based confidence intervals and bands for Lévy densities. Bernoulli 17 (2011) 643-670. | MR 2787609 | Zbl pre06083986

[36] S. Gugushvili. Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process. J. Nonparametr. Stat. 21 (2009) 321-343. | MR 2530929 | Zbl 1158.62023

[37] S. Gugushvili, C. Klaassen and P. Spreij (Eds). Statistical Inference for Lévy Processes with Applications to Finance. Stat. Neerl. 64 (3), 2010. | MR 2683459

[38] S. Gugushvili, B. Van Es and P. Spreij. Deconvolution for an atomic distribution: Rates of convergence. J. Nonparametr. Stat. (2011). To appear. DOI:10.1080/10485252.2011.576763. | MR 2854252 | Zbl 1230.62041

[39] G. Jongbloed and F. H. Van Der Meulen. Parametric estimation for subordinators and induced OU processes. Scand. J. Stat. 33 (2006) 825-847. | MR 2300918 | Zbl 1164.62372

[40] G. Jongbloed, F. H. Van Der Meulen and A. W. Van Der Vaart. Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 11 (2005) 759-791. | MR 2172840 | Zbl 1084.62080

[41] J. Kappus and M. Reiß. Estimation of the characteristics of a Lévy process observed at arbitrary frequency. Stat. Neerl. 64 (2010) 314-328. | MR 2683463

[42] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. | MR 2250061 | Zbl pre06176054

[43] A. Meister. Density estimation with normal measurement error with unknown variance. Statist. Sinica 16 (2006) 195-211. | MR 2256087 | Zbl 1087.62049

[44] R. C. Merton. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976) 125-144. | Zbl 1131.91344

[45] M. H. Neumann. On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Statist. 7 (1997) 307-330. | MR 1460203 | Zbl 1003.62514

[46] M. H. Neumann and M. Reiß. Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 (2009) 223-248. | MR 2546805 | Zbl 1200.62095

[47] J. P. Nolan. 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 1833706 | Zbl 0971.62008

[48] Y.-F. Ren and H.-Y. Liang. On the best constant in Marcinkiewicz-Zygmund inequality. Statist. Probab. Lett. 53 (1999) 227-233. | MR 1841623 | Zbl 0991.60011

[49] T. H. Rydberg. The normal inverse Gaussian Lévy process: Simulation and approximation. Stoch. Models 13 (1997) 887-910. | MR 1482297 | Zbl 0899.60036

[50] K.-I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 2004. | Zbl pre06283558

[51] J. Söhl. Polar sets for anisotropic Gaussian random fields. Statist. Probab. Lett. 80 (2010) 840-847. | MR 2608824 | Zbl 1193.60068

[52] A. B. Tsybakov. Introduction to Nonparametric Estimation. Springer, New York, 2009. | MR 2724359 | Zbl 1029.62034

[53] A. W. Van Der Vaart. Asymptotic Statistics. Cambridge Univ. Press, Cambridge, 1998. | MR 1652247 | Zbl 0910.62001

[54] A. W. Van Der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York, 1996. | MR 1385671 | Zbl 0862.60002

[55] B. Van Es and S. Gugushvili. Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance. J. Korean Statist. Soc. 39 (2010) 102-115. | MR 2655814

[56] B. Van Es, S. Gugushvili and P. Spreij. A kernel type nonparametric density estimator for decompounding. Bernoulli 13 (2007) 672-694. | MR 2348746 | Zbl 1129.62030

[57] M. P. Wand. Finite sample performance of deconvolving density estimators. Statist. Probab. Lett. 37 (1998) 131-139. | MR 1620450 | Zbl 0886.62048

[58] R. N. Watteel and R. J. Kulperger. Nonparametric estimation of the canonical measure for infinitely divisible distributions. J. Stat. Comput. Simul. 73 (2003) 525-542. | MR 1986343 | Zbl 1031.62030

[59] V. M. Zolotarev. One-Dimensional Stable Distributions. American Mathematical Society, Providence, 1986. | MR 854867 | Zbl 0589.60015