• Bernoulli
  • Volume 21, Number 4 (2015), 2430-2456.

On ADF goodness-of-fit tests for perturbed dynamical systems

Yury A. Kutoyants

Full-text: Open access


We consider the problem of construction of goodness-of-fit tests for diffusion processes with a small noise. The basic hypothesis is composite parametric and our goal is to obtain asymptotically distribution-free tests. We propose two solutions. The first one is based on a change of time, and the second test is obtained using a linear transformation of the “natural” statistics.

Article information

Bernoulli, Volume 21, Number 4 (2015), 2430-2456.

Received: January 2014
Revised: June 2014
First available in Project Euclid: 5 August 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

asymptotically distribution free test Cramér–von Mises tests diffusion processes goodness of fit test perturbed dynamical systems


Kutoyants, Yury A. On ADF goodness-of-fit tests for perturbed dynamical systems. Bernoulli 21 (2015), no. 4, 2430--2456. doi:10.3150/14-BEJ650.

Export citation


  • [1] Dachian, S. and Kutoyants, Y.A. (2008). On the goodness-of-fit tests for some continuous time processes. In Statistical Models and Methods for Biomedical and Technical Systems (F. Vonta, M. Nikulin, N. Limnios and C. Huber-Carol, eds.). Stat. Ind. Technol. 385–403. Boston, MA: Birkhäuser.
  • [2] Darling, D.A. (1955). The Cramér–Smirnov test in the parametric case. Ann. Math. Statist. 26 1–20.
  • [3] Freidlin, M.I. and Wentzell, A.D. (1998). Random Perturbations of Dynamical Systems, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. New York: Springer. Translated from the 1979 Russian original by Joseph Szücs.
  • [4] Hitsuda, M. (1968). Representation of Gaussian processes equivalent to Wiener process. Osaka J. Math. 5 299–312.
  • [5] Iacus, S.M. and Kutoyants, Yu.A. (2001). Semiparametric hypotheses testing for dynamical systems with small noise. Math. Methods Statist. 10 105–120.
  • [6] Khmaladze, È.V. (1981). A martingale approach in the theory of goodness-of-fit tests. Theory Probab. Appl. 26 240–257.
  • [7] Kleptsyna, M. and Kutoyants, Y.A. (2014). On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes. Stat. Inference Stoch. Process. To appear. Available at arXiv:1305.3382.
  • [8] Kutoyants, Yu. (1994). Identification of Dynamical Systems with Small Noise. Mathematics and Its Applications 300. Dordrecht: Kluwer Academic.
  • [9] Kutoyants, Y.A. (2011). Goodness-of-fit tests for perturbed dynamical systems. J. Statist. Plann. Inference 141 1655–1666.
  • [10] Kutoyants, Yu.A. (2014). On ADF goodness-of-fit tests for stochastic processes. In New Perspectives on Stochastic Modeling and Data Analysis (J. Bozeman, V. Girardin and C. Skiadas, eds.). To appear.
  • [11] Kutoyants, Yu.A. (2014). On score-function processes and goodness of fit tests for stochastic processes. Available at arXiv:1403.7715.
  • [12] Kutoyants, Y.A. (2014). On asymptotic distribution of parameter free tests for ergodic diffusion processes. Stat. Inference Stoch. Process. 17 139–161.
  • [13] Kutoyants, Y.A. and Zhou, L. (2014). On approximation of the backward stochastic differential equation. J. Statist. Plann. Inference 150 111–123.
  • [14] Liptser, R. and Shiryaev, A. (2001). Statistics of Random Processes. Vols. I, II, 2nd ed. Berlin: Springer.
  • [15] Maglaperidze, N.O., Tsigroshvili, Z.P. and van Pul, M. (1998). Goodness-of-fit tests for parametric hypotheses on the distribution of point processes. Math. Methods Statist. 7 60–77.
  • [16] Shepp, L.A. (1966). Radon–Nikodým derivatives of Gaussian measures. Ann. Math. Statist. 37 321–354.
  • [17] Yoshida, N. (1993). Asymptotic expansion of Bayes estimators for small diffusions. Probab. Theory Related Fields 95 429–450.
  • [18] Yoshida, N. (1996). Asymptotic expansions for perturbed systems on Wiener space: Maximum likelihood estimators. J. Multivariate Anal. 57 1–36.