Bernoulli

  • Bernoulli
  • Volume 4, Number 2 (1998), 185-201.

Discretely observing a white noise change-point model in the presence of blur

Jacques Istas and Henrik Stryhn

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Abstract

In discretely observed diffusion models, inference about unknown parameters in a smooth drift function has attracted much interest of late. This paper deals with a diffusion-type change-point model where the drift has a discontinuity across the point of change, analysed in detail in continuous time by Ibragimov and Hasminskii. We consider discrete versions of this model with integrated or blurred observations at a regular lattice. Asymptotic convergence rates and limiting distributions are given for the maximum likelihood change-point estimator when the observation noise and the lattice spacing simultaneously decrease. In particular, it is shown that the continuous and discrete model convergence rates are generally equal only up to a constant; under specific conditions on the blurring function this constant equals unity, and the normalized difference between the estimators tends to zero in the limit.

Article information

Source
Bernoulli, Volume 4, Number 2 (1998), 185-201.

Dates
First available in Project Euclid: 26 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1174937294

Mathematical Reviews number (MathSciNet)
MR1632983

Zentralblatt MATH identifier
1066.62536

Keywords
asymptotic distribution blur maximum likelihood orthogonalization

Citation

Istas, Jacques; Stryhn, Henrik. Discretely observing a white noise change-point model in the presence of blur. Bernoulli 4 (1998), no. 2, 185--201. https://projecteuclid.org/euclid.bj/1174937294


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References

  • [1] Adler, R.J. (1990) An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS Lecture Notes Monograph Ser. Hayward, CA: Institute of Mathematical Statistics.
  • [2] Bhattacharya, P.K. and Brockwell, P.J. (1976) The minimum of an additive process with applications to signal estimation and storage theory. Z. Wahrscheinlichkeitstheorie Verw. Geb., 37, 51-75.
  • [3] Bibby, B.M. and Sørensen, M. (1995) Martingale estimation functions for discretely observed diffusion processes. Bernoulli, 1, 17-39.
  • [4] Daubechies, I. (1988) Orthonormal basis of compactly supported wavelets. Comm. Pure Appl. Math., 41, 909-996.
  • [5] Genon-Catalot, V. and Jacod, J. (1993) On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. H. Poincaré, Probab. Statist., 29, 119-151.
  • [6] Hasminskii, R.Z. and Lebedev, V.S. (1990) On the properties of parametric estimators for areas of a discontinuous image. Probl. Control Inform. Theory, 19, 375-385.
  • [7] Hinkley, D.V. (1970) Inference about the change-point in a sequence of random variables. Biometrika, 57, 1-17.
  • [8] Ibragimov, I.A. and Hasminskii, R.Z. (1981) Statistical Estimation: Asymptotic Theory. New York: Springer-Verlag.
  • [9] Korostelev, A.P. (1987) On minimax estimation of a discontinuous signal. Theory Probab. Appl., 32, 727-730.
  • [10] Kutoyants, Yu.A. (1984) Parameter Estimation for Stochastic Processes. Berlin: Heldermann Verlag.
  • [11] Laredo, C.F. (1990) A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process. Ann. Statist., 18, 1158-1171.
  • [12] Meyer, Y. (1990) Ondelettes: Ondelettes et Opérateurs I. Paris: Hermann.
  • [13] Neuhaus, G. (1971) On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist., 42, 1285-1295.
  • [14] Pedersen, A.R. (1995) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist., 22, 55-71.
  • [15] Rosenfeld, A. and Kak, A.C. (1982) Digital Picture Processing, 2nd edn. London: Academic Press.
  • [16] Rubin, H. and Song, K.-S. (1995) Exact computation of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise. Ann. Statist., 23, 732-739.
  • [17] Rudemo, M. and Stryhn, H. (1994) Boundary estimation for star-shaped objects. In E. Carlstein, H.-G. Müller and D. Siegmund (eds), Change-Point Problems IMS Lecture Notes Monograph Ser. Hayward, CA: Institute of Mathematical Statistics.
  • [18] Stryhn, H. (1994) Spatial change point models applied to image segmentation. PhD thesis, The Royal Veterinary and Agricultural University, Copenhagen.
  • [19] Yao, Y.-C. (1987) Approximating the distribution of the maximum likelihood estimate of the changepoint in a sequence of independent random variables. Ann. Statist., 15, 1321-1328.