Bayesian analysis of time series using Lindley's approximation

The autoregressive model of order p and moving-average model of order q are analyzed when the parameters and the precision of the error term are random variables. In the analysis the squared error (SE) and linear exponential (LINEX) loss functions are utilized. Using four different priors, the Bayes estimators of the parameters are derived. Under independent truncated normal priors for the parameters and gamma or improper priors for the precision of the error term, the Bayes estimators are found not to be in a closed form. Therefore, Lindley's approximation is used to obtain the approximate estimators. A computer simulation study compares the maximum likelihood and the Bayes estimates obtained using Lindley's approximation and Markov chain Monte Carlo (MCMC) techniques, in particular, Gibbs sampler. Under independent Uniform priors for the parameters and gamma or improper priors for the precision of the error term, the posterior distributions of the model parameters and the precision of the error term have a closed form, however using the LINEX loss function, the Bayes estimators of the model parameters are intractable. In this case, the truncated normal approximation is used to derive the approximate Bayes estimators. A computer simulation study is employed to compare the maximum likelihood and the Bayes estimates. Examples are given to illustrate the findings.