Neumann, Michael H.; Reiß, Markus Nonparametric estimation for Lévy processes from low-frequency observations. (English) Zbl 1200.62095 Bernoulli 15, No. 1, 223-248 (2009). Summary: We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the Lévy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific \(C^{2}\)-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line. Cited in 86 Documents MSC: 62M05 Markov processes: estimation; hidden Markov models 62G05 Nonparametric estimation 62M09 Non-Markovian processes: estimation Keywords:deconvolution; density estimation; Lévy-Khinchine characteristics; minimum distance estimator × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Basawa, I.V. and Brockwell, P.J. (1982). Non-parametric estimation for non-decreasing Lévy processes., J. Roy. Statist. Soc. Ser. B 44 262-269. JSTOR: · Zbl 0491.62069 [2] Belomestny, D. and Reiß, M. (2006). Spectral calibration of exponential Lévy models., Finance Stoch. 10 449-474. · Zbl 1126.91022 · doi:10.1007/s00780-006-0021-5 [3] Cont, R. and Tankov, P. (2004)., Financial Modelling with Jump Processes . Boca Raton: Chapman and Hall. · Zbl 1052.91043 [4] Chung, K.L. (1974)., A Course in Probability Theory , 2nd ed. San Diego: Academic Press. · Zbl 0345.60003 [5] Csörgő, S. and Totik, V. (1983). On how long interval is the empirical characteristic function uniformly consistent., Acta Sci. Math. ( Szeged ) 45 141-149. · Zbl 0518.60039 [6] Dudley, R.M. (1989)., Real Analysis and Probability . Belmont: Wadsworth. · Zbl 0686.60001 [7] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems., Ann. Statist. 19 1257-1272. · Zbl 0729.62033 · doi:10.1214/aos/1176348248 [8] Feuerverger, A. and McDunnough, P. (1981a). On the efficiency of empirical characteristic function procedures., J. Roy. Statist. Soc. Ser. B 43 20-27. JSTOR: · Zbl 0454.62034 [9] Feuerverger, A. and McDunnough, P. (1981b). On some Fourier methods for inference., J. Amer. Statist. Assoc. 76 379-387. JSTOR: · Zbl 0463.62030 · doi:10.2307/2287839 [10] Figueroa-López, J. and Houdré, C. (2006). Risk bounds for the nonparametric estimation of Lévy processes. In, High Dimensional Probability. IMS Lecture Notes 51 96-116. Beachwood, OH: IMS. · Zbl 1117.62085 · doi:10.1214/074921706000000789 [11] Gnedenko, B.V. and Kolmogorov, A.N. (1968)., Limit Distributions for Sums of Independent Random Variables , 2nd ed. Reading, MA: Addison-Wesley. · Zbl 0056.36001 [12] Goldenshluger, A. and Pereverzev, S.V. (2003). On adaptive inverse estimation of linear functionals in Hilbert scales., Bernoulli 9 783-807. · Zbl 1055.62034 · doi:10.3150/bj/1066418878 [13] Gugushvili, S. (2007). Decompounding under Gaussian noise. Available at, [14] Hall, P. and Yao, Q. (2003). Inference in components of variance models with low replication., Ann. Statist. 31 414-441. · Zbl 1039.62065 · doi:10.1214/aos/1051027875 [15] Jacod, J. and Shiryaev, A. (2002)., Limit Theorems for Stochastic Processes , 2nd ed. Grundlehren 288 . Berlin: Springer. · Zbl 0635.60021 [16] Jongbloed, G., van der Meulen, F.H. and van der Vaart, A.W. (2005). Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes., Bernoulli 11 759-791. · Zbl 1084.62080 · doi:10.3150/bj/1130077593 [17] Katznelson, Y. (1976)., An Introduction to Harmonic Analysis , 2nd ed. New York: Dover. · Zbl 0352.43001 [18] Kolmogorov, A.N. (1932). Sulla formula generale di un processo stochastico omogeneo (Un problema di Bruno de Finetti)., Rendiconti della R. Accademia Nazionale dei Lincei ( Ser. VI ) 15 866-869. (In Italian.) [19] Korostelev, A.P. and Tsybakov, A.B. (1993)., Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82 . New York: Springer. · Zbl 0833.62039 [20] Küchler, U. and Tappe, S. (2008). On the shapes of bilateral Gamma densities., Statist. Probab. Lett. 78 2478-2484. · Zbl 1146.62309 · doi:10.1016/j.spl.2008.02.039 [21] Mainardi, F. and Rogosin, S. (2006). The origin of infinitely-divisible distributions: from de Finetti’s problem to Lévy-Khinchine formula., Math. Methods Econ. Finance 1 37-55. [22] Nishiyama, Y. (2008). Nonparametric estimation and testing time-homogeneity for processes with independent increments., Stochastic. Process. Appl. 118 1043-1055. · Zbl 1144.62025 · doi:10.1016/j.spa.2007.07.011 [23] van Es, B., Gugushvili, S. and Spreij P. (2007). A kernel-type nonparametric density estimator for decompounding., Bernoulli 13 672-694. · Zbl 1129.62030 · doi:10.3150/07-BEJ6091 [24] van der Vaart, A. (1998)., Asymptotic Statistics . Cambridge: Cambridge Univ. Press. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256 [25] Watteel, R.N. and Kulperger, R.J. (2003). Nonparametric estimation of the canonical measure for infinitely divisible distributions., J. Statist. Comput. Simul. 73 525-542. · Zbl 1031.62030 · doi:10.1080/0094965021000015477 [26] Yukich, J.E. (1985). Weak convergence of the empirical characteristic function., Proc. Amer. Math. Soc. 95 470-473. JSTOR: · Zbl 0607.60003 · doi:10.2307/2045821 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.