Abstract
We investigate shrinkage priors on power spectral densities for complex-valued circular-symmetric autoregressive processes. We construct shrinkage predictive power spectral densities, which asymptotically dominate (i) the Bayesian predictive power spectral density based on the Jeffreys prior and (ii) the estimative power spectral density with the maximum likelihood estimator, where the Kullback-Leibler divergence from the true power spectral density to a predictive power spectral density is adopted as a risk. Furthermore, we propose general constructions of objective priors for Kähler parameter spaces by utilizing a positive continuous eigenfunction of the Laplace-Beltrami operator with a negative eigenvalue. We present numerical experiments on a complex-valued stationary autoregressive model of order 1.
Highlights
W E INVESTIGATE the time fluctuation of a single particle having a circular-symmetric distribution in a two-dimensional space
The complex plane C is often used for the representation of the process, and an observation of a single particle at different time points can be represented as a complex-valued vector
We focus on complex-valued circular-symmetric discrete Gaussian processes, which are defined as complex-valued processes having finite-dimensional marginal distributions that are complex normal distributions
Summary
W E INVESTIGATE the time fluctuation of a single particle having a circular-symmetric distribution in a two-dimensional space. ODA AND KOMAKI: SHRINKAGE PRIORS ON COMPLEX-VALUED CIRCULAR -SYMMETRIC AUTOREGRESSIVE PROCESSES underlying true power spectral density of the process under the loss given by (2) based on the Kullback–Leibler divergence. BAYESIAN PREDICTIVE POWER SPECTRAL DENSITIES FOR COMPLEX-VALUED GAUSSIAN PROCESSES. We present the asymptotic expansion of Bayesian predictive power spectral densities Sπ(N) for complex-valued autoregressive moving average processes. This asymptotic expansion is a basic tool for assessing the performance of the choice of a prior π. The asymptotic expansion of a Bayesian predictive power spectral density (4) of a complex-valued. The only remaining problem is to find a reasonable prior π
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