Almost sure rate of convergence of maximum likelihood estimators for multidimensional diffusions

  • Dasha Loukianova
  • Oleg Loukianov
Part of the Lecture Notes in Statistics book series (LNS, volume 187)


Brownian Motion Maximum Likelihood Estimation Invariant Measure Quadratic Variation Entropy Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BP80]
    I.V. Basawa, B.L.S. Prakasa Rao, Statistical Inference for Stochastic Processes, Academic Press, London (1980)zbMATHGoogle Scholar
  2. [DKu03]
    H.M. Dietz, Yu.A. Kutoyants, Parameter estimation for some nonrecurrent solutions of SDE, Statistics and Decisions 21, 29–45 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  3. [F76]
    P.D. Feigin, Maximum likelihood estimation for continuous-time stochastic processes, Advances in Applied Probability 8, 712–736 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  4. [JKh97]
    A. Jankunas, R.Z. Khasminskii, Estimation of parameters of linear homogeneous stochastic differential equations, Stochastic Processes and their Applications 72, 205–219 (1997)MathSciNetCrossRefGoogle Scholar
  5. [HKu03]
    R. Höpfner, Yu.A. Kutoyants, On a problem of statistical inference in null recurrent diffusion, Statistical Inference for Stochastic Processes 6(1), 25–42 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  6. [KS99]
    U. Küchler, M. Sørensen, A note on limit theorems for multivariate martingales, Bernoulli 5(3), 483–493 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  7. [Ku03]
    Yu.A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer Series in Statistics, New York (2003)zbMATHGoogle Scholar
  8. [L03]
    D. Loukianova, Remark on semigroup techniques and the maximum likelihood estimation, Statistics and Probability Letters 62, 111–115 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [LL05]
    D. Loukianova, O. Loukianov, Uniform law of large numbers and consistency of estimators for Harris diffusions, to appear in Statistics and Probability Letters Google Scholar
  10. [N99]
    Y. Nishiyama, A maximal inequality for continuous martingales and Mestimation in gaussian white noise model, The Annals of Statistics 27(2), 675–696 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  11. [RY94]
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Second Edition, Springer-Verlag, Berlin, Heidelberg (1994)zbMATHGoogle Scholar
  12. [Se00]
    R. Senoussi, Uniform iterated logarithm laws for martingales and their application to functional estimation in controlled Markov chains, Stochastic Processes and their Applications 89, 193–211 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. [GS91]
    S.A. van de Geer, L. Stougie, On rates of convergence and asymptotic normality in the multiknapsack problem, Mathematical Programming 51, 349–358 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  14. [G00]
    S.A. van de Geer, Empirical Processes in M-estimation, Cambridge University Press, Cambridge (2000)Google Scholar
  15. [VW96]
    A.W. van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes, Springer-Verlag (1996)Google Scholar
  16. [VZ05]
    A. van der Vaart, H. van Zanten, Donsker theorems for diffusions: necessary and sufficient conditions, to appear in The Annals of Probability (2005)Google Scholar
  17. [Z99]
    H. van Zanten, On the Uniform Convergence of Local Time and the Uniform Consistency of Density Estimators for Ergodic Diffusions, Probability, Networks and Algorithms PNA-R9909 (1999)Google Scholar
  18. [Z03a]
    H. van Zanten, On empirical processes for ergodic diffusions and rates of convergence of M-estimators, Scandinavian Journal of Statistics 30(3), 443–458 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  19. [Z03b]
    H. van Zanten, On uniform laws of large numbers for ergodic diffusions and consistency of estimators, Statistical Inference for Stochastic Processes 6(2), 199–213 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  20. [Z05]
    H. van Zanten, On the rate of convergence of the maximum likelihood estimator in Brownian semimartingale models, Bernoulli 11(4), 643–664 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Y90]
    N. Yoshida, Asymptotic behavior of M-estimators and related random field for diffusion process, Ann. Inst. Statist. Math. 42(2), 221–251 (1990)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Dasha Loukianova
    • 1
  • Oleg Loukianov
    • 2
  1. 1.Département de MathématiquesUniversité d’Evry-Val d’EssonneFrance
  2. 2.Département d’InformatiqueIUT de FontainebleauFrance

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