Nonparametric Bayesian estimation of the spectral density of a time series
Subhashis Ghosal, May 29, 2002
For a discrete time stationary time series, the most important
aspect is the dependence structure of the observations measured by the
autocovariance function, which can be obtained from the Fourier expansion
of the spectral density f. In this work, we do not assume a parametric
form of the f and take a nonparametric Bayesian approach to estimation.
The prior on f is constructed using the Bernstein polynomial approximation
to a continuous function on an interval. The prior can also be viewed as
a special case of mixtures of beta densities, where a Dirichlet process
prior is put on the mixing distribution. Posterior updating with dependent
data seems to be intractable. Our approach relies on Whittle's
approximate likelihood which is based on the fact \that the spectral
transform of the data, the periodogram at the Fourier frequencies,
are approximately independent (not identical)
exponential random variable with expected values equaling the spectral
density at the corresponding points. We show that the posterior
distribution is consistent.
As the data is not identically distributed, the standard consistency
theorems do not apply. Therefore, we develop a new consistency
theorem for this purpose which is well suited for independent
non-identical data. Finally, we invoke a contiguity argument to show
consistency under the actual dependent data.
As is typical for non-parametric models, the posterior
does not have a closed form expression. We describe a Markov chain Monte
Carlo sampling algorithm to numerically approximate it. We apply the method to
several simulated data and found it satisfactory. Application to the
famous sunspot data will also be discussed.
This is a joint work with Nidhan Chaudhuri and Anindya Roy.
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