Bayesian Inference for Continuous-Time Arma Models Driven by Non-Gaussian LÉVY Processes

In this paper we present methods for estimating the parameters of a class of non-Gaussian continuous-time stochastic process, the continuous-time auto regressive moving average (CARMA) model driven by symmetric alpha-stable (SalphaS) Levy processes. In this challenging framework we are not able to evaluate the likelihood function directly, and instead we use a distretized approximation to the likelihood. The parameters are then estimated from this approximating model using a Bayesian Monte Carlo scheme, and employing a Kalman filter to marginalize and sample the trajectory of the state process. An efficient exploration of the parameter space is achieved through a novel reparameterization in terms of an equivalent mechanical system. Simulations demonstrate the potential of the methods

[1]  Simon J. Godsill,et al.  On-line Bayesian estimation of signals in symmetric /spl alpha/-stable noise , 2006, IEEE Transactions on Signal Processing.

[2]  Simon J. Godsill,et al.  Estimation of CAR processes observed in noise using Bayesian inference , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[3]  N. Shephard,et al.  The simulation smoother for time series models , 1995 .

[4]  T. Söderström,et al.  Least squares parameter estimation of continuous-time ARX models from discrete-time data , 1997, IEEE Trans. Autom. Control..

[5]  S. Godsill,et al.  Bayesian inference for time series with heavy-tailed symmetric α-stable noise processes , 1999 .

[6]  Erik G. Larsson,et al.  Cramer-Rao bounds for continuous-time AR parameter estimation with irregular sampling , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[7]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[8]  Bengt Carlsson,et al.  Estimation of continuous-time AR process parameters from discrete-time data , 1999, IEEE Trans. Signal Process..

[9]  S. Godsill.  MCMC and EM-based methods for inference in heavy-tailed processes with /spl alpha/-stable innovations , 1999, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics. SPW-HOS '99.

[10]  D. Wilkinson,et al.  Bayesian Inference for Stochastic Kinetic Models Using a Diffusion Approximation , 2005, Biometrics.

[11]  Richard A. Davis,et al.  Introduction to time series and forecasting , 1998 .

[12]  G. Roberts,et al.  On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm , 2001 .

[13]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[14]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter. , 1991 .

[15]  Philip Protter,et al.  The Euler scheme for Lévy driven stochastic differential equations , 1997 .

[16]  R. Shanmugam.  Introduction to Time Series and Forecasting , 1997 .

[17]  N. Shephard,et al.  Non-Gaussian OU based models and some of their uses in financial economics , 2000 .

[18]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[19]  C. L. Nikias,et al.  Signal processing with alpha-stable distributions and applications , 1995 .

[20]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[21]  C. Mallows,et al.  A Method for Simulating Stable Random Variables , 1976 .

[22]  Richard H. Jones,et al.  Fitting Multivariate Models to Unequally Spaced Data , 1984 .

[23]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[24]  G. Roberts,et al.  Bayesian inference for non‐Gaussian Ornstein–Uhlenbeck stochastic volatility processes , 2004 .

[25]  N. Shephard,et al.  Likelihood INference for Discretely Observed Non-linear Diffusions , 2001 .

[26]  Masahito Yamada,et al.  Structural Time Series Models and the Kalman Filter , 1989 .

[27]  Richard H. Jones.  FITTING A CONTINUOUS TIME AUTOREGRESSION TO DISCRETE DATA , 1981 .

[28]  Andrew Harvey,et al.  Forecasting, structural time series models and the Kalman filter: Selected answers to exercises , 1990 .