Abstract
In this paper, we consider projection estimates for Lévy densities in high-frequency setup. We give a unified treatment for different sets of basis functions and focus on the asymptotic properties of the maximal deviation distribution for these estimates. Our results are based on the idea to reformulate the problems in terms of Gaussian processes of some special type and to further analyze these Gaussian processes. In particular, we construct a sequence of excursion sets, which guarantees the convergence of the deviation distribution to the Gumbel distribution. We show that the exact rates of convergence presented in previous articles on this topic are logarithmic and construct the sequence of accompanying laws, which approximate the deviation distribution with polynomial rate.
Similar content being viewed by others
References
Belomestny, D.: Statistical inference for time-changed Lévy processes via composite characteristic function estimation. Ann. Stat. 39(4), 2205–2242 (2011)
Bickel, P., Rosenblatt, M.: On some global measures of the deviations of density function estimates. Ann. Stat. 1(6), 1071–1095 (1973)
Chen, S.X., Delaigle, A., Hall, P.: Nonparametric estimation for a class of Lévy processes. J. Econ. 157, 257–271 (2010)
Comte, F., Genon-Catalot, V.: Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stoch. Process. Appl. 119, 4088–4123 (2009)
Comte, F., Genon-Catalot, V.: Nonparametric estimation for pure jump irregularly sampled or noisy Lévy processes. Statistica Neerlandica 64, 290–313 (2010a)
Comte, F., Genon-Catalot, V.: Nonparametric adaptive estimation for pure jump Lévy processes. Ann. l’Inst. Henri Poincaré - Prob. Stat. 46(3), 595–617 (2010b)
Comte, F., Genon-Catalot, V.: Estimation for Lévy processes from high frequency data within a long time interval. Ann. Stat. 39, 803–837 (2011)
Figueroa-López, J.E.: Nonparametric Estimation of Lévy Processes with a View Towards Mathematical Finance. PhD thesis, Georgia Institute of Technology (2004)
Figueroa-López, J.E.: Sieve-based confidence intervals and bands for Lévy densities. Bernoulli 17(2), 643–670 (2011)
Fisher, M., Nappo, G.: On the moments of the modulus of continuity of Ito Processes. Stochast. Process. Appl. 28(1), 103–122 (2010)
Gradshtein, I., Ryzhik I.: Table of integrals, Series and Products. Academic Press (1996)
Gugushvili, S.: Nonparametric inference for discretely sampled Lévy processes. Ann. l’Inst. Henri Poincaré - Probab. Stat. 48(1), 282–307 (2012)
Hall, P.: On convergences rates of suprema. Probab. Theory Relat. Fields 89 (447-455) (1991)
Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent rv’s and the sample DF. Z. Wahrscheinlichkeitstheorie Verw Geb. 32, 111–131 (1975)
Konakov, V., Piterbarg, V.: On the convergence rates of maximal deviation distribution for kernel regression estimates. J. Multivar. Anal. 15, 279–294 (1984)
Konakov, V., Panov, V.: Convergence rates of maximal deviation distribution for projection estimates of Lévy densities (2016). arXiv:1411.4750v3
Kuo, H.-H.: Introduction to stochastic integration. Springer (2006)
Michna, Z.: Remarks on Pickands theorem (2009). arXiv:0904.3832v1
Neumann, M., Reiss, M.: Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15(1), 223–248 (2009)
Nickl, R., Reiss, M.: A Donsker theorem for Lévy measures. J. Funct. Anal. 253, 3306–3332 (2012)
Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. AMS, Providence (1996)
Piterbarg, V.I.: Twenty Lectures about Gaussian Processes. Atlantic Financial Press, London (2015)
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Springer-Verlag (1999)
Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge University Press (1999)
Suetin, P.: Classical orthogonal polynomials (in Russian). Fizmatlit (2005)
Van Es, B., Gugushvili, S., Spreij, P.: A kernel type nonparametric density estimator for decompounding. Bernoulli 13, 672–694 (2007)
Wörner, J.: Variational sums and power variations: a unifying approach to model selection and estimation in semimartingale models. Stat. Decis. 21, 47–68 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
The article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.
Rights and permissions
About this article
Cite this article
Konakov, V., Panov, V. Sup-norm convergence rates for Lévy density estimation. Extremes 19, 371–403 (2016). https://doi.org/10.1007/s10687-016-0246-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-016-0246-4