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On parameter estimation of the hidden Gaussian process in perturbed SDE. (English) Zbl 1460.62139

The authors propose a new estimator for the parameters of a partially observed linear Gaussian process. Let’s look at the problem under consideration. Let \((\Omega,F,P)\) be a probability space, \(T>0\), \(X\), \(Y\), \(W\), and \(V\) real valued stochastic processes defined over \(\Omega\times[0,T]\), \(\Theta=\{(\alpha,\beta)/|\alpha|+|\beta|<\infty\}\) such that \[ \begin{aligned} dX_{t} &= f(\theta,t)\cdot Y_{t}\cdot dt+\varepsilon\cdot\sigma(t)\cdot dW_{t},\quad X_0=0,\\ dY_{t} &= a(\theta,t)\cdot Y_{t}\cdot dt+\varepsilon\cdot b(t)\cdot dV_{t},\quad Y_0=y_0\neq 0, \end{aligned} \] where \(W\) and \(V\) are independent Wiener processes, \(f\) and \(a\) are (known) real functions defined over \(\Theta\times[0,T]\), \(\sigma\) and \(b\) are (known) real functions defined over \([0,T]\), and \(\varepsilon\) is a positive constant. Here the problem is to estimate the parameter \(\theta\) from observations of \(X\). There are already estimators of the referred parameter, as maximum likelihood estimator (MLE) and Bayesian estimator (BE), but they are hard to calculate.
In this paper, the authors propose a new estimator the computation of which is simpler and the consistency and efficiency properties are as reliable as those of the existing estimators. The last section of the paper shows the application of the proposed method in two examples with simulated data. Unfortunately, no results of applications to examples with real data are shown. Most of the main theorems and claims are rigorously and carefully proved. However, understanding this work requires a good knowledge of the mathematics of stochastic differential equations.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62M20 Inference from stochastic processes and prediction
62F12 Asymptotic properties of parametric estimators
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
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References:

[1] Arato, M. (1983), Linear Stochastic Systems with Constant Coefficients A Statistical Approach. Lecture Notes in Control and Inform. Sci., 45, New York: Springer-Verlag. · Zbl 0544.93060
[2] Cappé, O., Moulines, E. and Rydén, T. (2005), Inference in Hidden Markov Models. Springer, N.Y. · Zbl 1080.62065
[3] Chigansky, P. (2009) Maximum likelihood estimation for hidden Markov models in continuous time., Statist. Inference Stoch. Processes, 12, 2, 139-163. · Zbl 1205.62120
[4] Elliott, R.J., Aggoun, L. and Moor, J.B. (1995), Hidden Markov Models. Springer, N.Y. · Zbl 0823.60062
[5] Ephraim, Y., Mehrav, N. (2002) Hidden Markov processes., IEEE Trans. Inform. Theory, 48, 6, 1518-1569. · Zbl 1061.94560
[6] Fisher, R.A. (1925), Theory of Statistical Estimation. Proc. Camb. Phil. Soc., 22, 700-725. · JFM 51.0385.01
[7] Gill, R.D. and Levit, B.Ya. (1995) Application of the van Trees inequality: a Bayesian Cramer-Rao bound., Bernoulli, 1, 59-79. · Zbl 0830.62035
[8] Golubev, G.K. (1984) Fisher’s method of scoring in the problem of frequency estimation., J. of Soviet Math., 25, 3, 1125-1139. · Zbl 0549.62055
[9] Gustafsson, F. (2000), Adaptive Filtering and Change Detection. J. Wiley&Sons, N.Y.
[10] Hide, C., Moore, T. and Smith, M. (2003) Adaptive Kalman filtering for low cost ING/GPS., The Journal of Navigation, 56, 1, 143-152.
[11] Hu, C., Chen, W., Chen, Y. and Liu, D. (2003) Adaptive Kalman filtering for vehicule navigation., J. Global Positioning systems, 2, 1, 42-47.
[12] Kamatani, K. and Uchida, M. (2015) Hybrid multi-step estimators for stochastic differential equations based on sampled data., Statist. Inference Stoch. Processes. 18, 2, 177-204. · Zbl 1329.62110
[13] Khasminskii, R. (2005) Nonlinear filtering of smooth signals., Stochastics and Dynamics, 5, 1, 27-35. · Zbl 1059.62088
[14] Khasminskii, R.Z. and Kutoyants, Yu.A. (2018) On parameter estimation of hidden telegraph process., Bernoulli, 24, 3, 2064-2090. · Zbl 1414.62348
[15] Kutoyants, Yu.A. (1984), Parameter Estimation for Stochastic Processes. Heldermann, Berlin.
[16] Kutoyants, Yu.A. (1994), Identification of Dynamical Systems with Small Noise. Kluwer Academic Publisher, Dordrecht. · Zbl 0831.62058
[17] Kutoyants, Yu.A. (2004), Statistical Inference for Ergodic Diffusion Processes. Springer, London. · Zbl 1038.62073
[18] Kutoyants, Yu.A. (2017) On the multi-step MLE-process for ergodic diffusion., Stochastic Process. Appl., 127, 2243-2261. · Zbl 1422.62274
[19] Kutoyants, Yu.A. (2019) On parameter estimation of the hidden Ornstein-Uhlenbeck process., J. Multivar. Analysis, 169, 1, 248-269. · Zbl 1408.62143
[20] Kutoyants, Yu.A. (2019) On parameter estimation of the hidden ergodic Ornstein-Uhlenbeck process., Electronic J. of Statistics, 13, 4508-4526. · Zbl 1442.62190
[21] Kutoyants, Yu.A. and Motrunich, A. (2016) On multi-step MLE-process for Markov sequences., Metrika, 79, 705-724. · Zbl 1347.62044
[22] Kutoyants, Yu.A. and Zhou, L. (2014) On approximation of the backward stochastic differential equation., J. Stat. Plann. Infer. 150, 111-123. · Zbl 1287.62017
[23] Liptser, R.S. and Shiryayev, A.N. (2001), Statistics of Random Processes, I. General Theory. 2nd Ed., Springer, N.Y. · Zbl 0364.60004
[24] Rutan, S.C. (1991) Adaptive Kalman filtering., Anal. Chem., 63 (22), 1103A-1109A.
[25] Zhou, L.
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