Electronic Journal of Statistics

Statistical inference across time scales

Céline Duval and Marc Hoffmann

Full-text: Open access


We consider a compound Poisson process with symmetric Bernoulli jumps, observed at times iΔ for i=0,1, over [0,T], for different sizes of Δ=ΔT relative to T in the limit T. We quantify the smooth statistical transition from a microscopic Poissonian regime (when ΔT0) to a macroscopic Gaussian regime (when ΔT). The classical quadratic variation estimator is efficient for estimating the intensity of the Poisson process in both microscopic and macroscopic scales but surprisingly, it shows a substantial loss of information in the intermediate scale ΔTΔ(0,). This loss can be explicitly related to Δ. We provide an estimator that is efficient simultaneously in microscopic, intermediate and macroscopic regimes. We discuss the implications of these findings beyond this idealised framework.

Article information

Electron. J. Statist., Volume 5 (2011), 2004-2030.

First available in Project Euclid: 30 December 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62B15: Theory of statistical experiments
Secondary: 62B10: Information-theoretic topics [See also 94A17] 62M99: None of the above, but in this section

Discretely observed random process LAN property information loss


Duval, Céline; Hoffmann, Marc. Statistical inference across time scales. Electron. J. Statist. 5 (2011), 2004--2030. doi:10.1214/11-EJS660. https://projecteuclid.org/euclid.ejs/1325264855

Export citation


  • [1] Baricz, Á. (2008). Functional inequalities involving Bessel and modified Bessel functions of the first kind., Expositiones Mathematicae 26, 279–293.
  • [2] Bauwens, L. and Hautsch, N. (2006). Modelling high frequency financial data using point processes., Discussion paper.
  • [3] Billingsley, P. (1999)., Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics.
  • [4] Bøgsted, M. and Pitts, S. (2010). Decompounding random sums: a nonparametric approach., Ann Inst Stat Math 62, 855–872.
  • [5] Buchmann, B. and Grübel, R. (2003). Decompounding: an estimation problem for Poisson random sums., Ann. Statist 31, 1054–1074.
  • [6] Buchmann, B. and Grübel, R. (2004). Decompounding Poisson random sums: recursively truncated estimates in the discrete case., Ann. Inst. Math 56, 743–756.
  • [7] Comte, F. and Genon-Catalot, V. (2009). Nonparametric estimation for pure jump Lévy processes based on high frequency data., Stochastic Processes and their Applications 119, 4088–4123.
  • [8] Comte, F. and Genon-Catalot, V. (2010). Nonparametric adaptive estimation for pure jump Lévy processes., Annales de l’I.H.P., Probability and Statistics 46, 595–617.
  • [9] Ibragimov, I.A. and Hasminskii, R.Z (1981)., Statistical Estimation. Asymptotic Theory. Springer-Verlag.
  • [10] Masoliver, J., Montero, M., Perelló, J. and Weiss, G.H. (2008). Direct and inverse problems with some generalizations and extensions. Arxiv preprint, 0308017v2.
  • [11] Le Cam, L. and Yang, L.G. (2000), Asymptotics in Statistics: Some Basic Concepts. 2nd edition. New York: Springer-Verlag.
  • [12] Nasell, I. (1974). Inequalities for Modified Bessel Functions., Math. Comp. 28, 253–256.
  • [13] Neumann, M. and Reiß, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations., Bernoulli 15, 223–248.
  • [14] Sato, K-I. (1999)., Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
  • [15] van der Vaart, A.W. (1998)., Asymptotic Statistics. Cambridge University Press.
  • [16] van Es, B., Gugushvili, S. and Spreij, P. (2007). A kernel type nonparametric density estimator for decompounding., Bernoulli 13, 672–694.
  • [17] Watson, G.N. (1922)., A Treatise on the Theory of Bessel Functions. Cambridge University Press.