# Estimating a Centered Ornstein-Uhlenbeck Process under Measurement Errors

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The problem of estimating the two parameters of a stationary process satisfying the differential equation , where follows a standard Wiener process, from observations at equidistant points of the interval , has been well studied. This is also the classical problem of fitting an autoregressive time series of order 1 (AR1), the case " large" yielding the "near unit root" situation. This Demonstration considers the important case where the observations may have additive measurement errors: we assume that these errors are independent, normal random variables with known variance .

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Contributed by: Didier A. Girard (July 2014)

(CNRS-LJK and Univ. Grenoble Alpes)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: Selecting as the true diffusion coefficient (the value of from which the non-noisy data is simulated, being fixed) and choosing , this setting may be thought of as "close" to a case where the noise could be forgotten; this could be confirmed by moving only from to (the underlying is then unchanged but the measurement errors are eliminated) and observing that all the estimates are almost unchanged (up to three digits). Concerning the diffusion coefficient, you can observe that the two estimation methods produce very close results. Such closeness is less pronounced for the two estimates of the variance. By changing the seed (and thus a new and new measurement errors used to generate the data) you can be convinced that this is not an accident. Furthermore, a rather large variability, from seed to seed, for the two estimates of the variance is also observed; it is much larger than the variability of the estimates of the diffusion coefficient; notice that this observation is well in agreement with the known theory about the estimation of the variance, the inverse-range parameter, and their product (see [1] and the references therein). By moving from seed to seed, neither of the two methods seems a clear winner in this "small noise" setting. Let us now consider a higher noise level. Another important point to note is that the estimates are not very influenced by the noise perturbing the data, provided the noise level stays lower than . However, by increasing to 0.05, a clear degradation of the neglecting-errors-ML is observed. In contrast, CGEM-EV still produces reasonable estimates of . Here ; however, you can change from to and the conclusions remain similar.

Snapshot 2: Staying at and selecting as true diffusion coefficient (so that we are close to the "near unit root" situation), a noise with can no longer be considered as a negligible noise. Indeed, we can observe that by diminishing the amplitude of the present noise from to , we restore significantly the quality of the estimates of using neglecting-errors-ML. And by moving from to and trying several seeds, one can be convinced that a noise-to-signal ratio of order or less is required if we want to trust the neglecting-errors-ML estimator (since we can be content with three accurate digits in the estimates). By increasing to 0.05, the neglecting-errors-ML becomes meaningless. In contrast, CGEM-EV still produces reasonable estimates of .

Snapshot 3: Selecting an "intermediate" value as true diffusion coefficient, similar conclusions can be drawn, except that the approximate upper bound on required to trust the neglecting-errors-ML estimator of seems also intermediate between and .

References

[1] D. A. Girard, "Asymptotic Near-Efficiency of the 'Gibbs-Energy and Empirical-Variance' Estimating Functions for Fitting Matérn Models to a Dense (Noisy) Series." https://hal.archives-ouvertes.fr/hal-00413693.

[2] A. Gloter and J. Jacod, "Diffusions with Measurements Errors, II-Optimal Estimators,"
*ESAIM-Probability and Statistics,* 5, 2001 pp. 243–260.

[3] Y. Zhang, H. Yu, and A. Ian McLeod, "Developments in Maximum Likelihood Unit Root Tests," *Communications in Statistics—Simulation and Computation*, 42(5), 2013 pp. 1088–1103.

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