Asymptotic Expansions for Long-Memory Stationary Gaussian Processes

Abstract

This paper surveys results recently obtained by the authors on higher-order asymptotic expansions for stationary Gaussian processes with long memory, that is, with a hyperbolically decaying autocovariance function. Such processes have been used to model time series data in various fields. Frequentist-type results presented include the following: an Edgeworth expansion for the sample autocovariance function, an Edgeworth expansion for the log-likelihood derivatives and the maximum likelihood estimator in parametric time series models, and a Bartlett corrected likelihood ratio test for the fractional integration parameter in the ARFIMA model. Bayesian-type results presented include the following: an Edgeworth expansion for the posterior density of the parameter vector in parametric models, identification of matching priors under which frequentist and Bayesian inferences approximately agree, and identification of approximate reference priors in the sense of Bernardo, which carry minimum initial information on the parameter vector in a certain Kullback-Leibler sense. The key tools are theorems concerning the limiting behavior of the trace of the product of certain Toeplitz matrices and a general theorem of Durbin on Edgeworth expansions for dependent data. The results and proofs are briefly sketched, with references to the original papers for further details.KeywordsAsymptotic ExpansionMaximum Likelihood EstimatorSpectral Density FunctionToeplitz MatriceAuto Covariance FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.