A Bayesian Analysis of Spectral ARMA Model

Bezerra et al. (2008) proposed a new method, based on Yule-Walker equations, to estimate the ARMA spectral model. In this paper, a Bayesian approach is developed for this model by using the noninformative prior proposed by Jeffreys (1967). The Bayesian computations, simulation via Markov Monte Carlo (MCMC) is carried out and characteristics of marginal posterior distributions such as Bayes estimator and confidence interval for the parameters of the ARMA model are derived. Both methods are also compared with the traditional least squares and maximum likelihood approaches and a numerical illustration with two examples of the ARMA model is presented to evaluate the performance of the procedures.

[1]  Charles W. Therrien,et al.  Discrete Random Signals and Statistical Signal Processing , 1992 .

[2]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[3]  Torsten Söderström,et al.  An efficient linear method for ARMA spectral estimation , 1994, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[4]  Piet M. T. Broersen,et al.  Finite sample criteria for autoregressive order selection , 2000, IEEE Trans. Signal Process..

[5]  Anthony N. Pettitt,et al.  A Bayesian Approach to Selecting Covariates for Prediction , 2001 .

[6]  Y. Iano,et al.  Evaluating some Yule-Walker Methods with the Maximum-Likelihood Estimator for the Spectral ARMA Model , 2008 .

[7]  Steven Kay,et al.  Modern Spectral Estimation: Theory and Application , 1988 .

[8]  David R. Cox,et al.  Time Series Analysis , 2012 .

[9]  B. Friedlander,et al.  The Modified Yule-Walker Method of ARMA Spectral Estimation , 1984, IEEE Transactions on Aerospace and Electronic Systems.

[10]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[11]  J. Cadzow,et al.  High performance spectral estimation--A new ARMA method , 1980 .

[12]  B. Porat,et al.  Digital Spectral Analysis with Applications. , 1988 .

[13]  Leland B. Jackson.  Frequency-domain Steiglitz-McBride method for least-squares IIR filter design, ARMA modeling, and periodogram smoothing , 2008, IEEE Signal Processing Letters.

[14]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[15]  Manoel I. Silvestre Bezerra.  Precision of yulewalkermethods for the arma spectral model , 2004, Circuits, Signals, and Systems.

[16]  H. Jeffreys,et al.  Theory of probability , 1896 .

[17]  Dimitris N. Politis,et al.  Computer-intensive methods in statistical analysis , 1998, IEEE Signal Process. Mag..

[18]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[19]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .