Likelihood asymptotics of stationary Gaussian arrays

TL;DR

This paper establishes an asymptotic likelihood framework for stationary Gaussian arrays, enabling efficient parameter estimation in complex high-frequency and autoregressive models with non-uniform spectral properties.

Contribution

It introduces conditions for local asymptotic normality in Gaussian arrays, accommodating non-diagonal rate matrices and spectral densities with diverse behaviors.

Findings

Maximum likelihood estimators are asymptotically efficient.

The theory applies to high-frequency sampled Gaussian processes.

It covers autoregressive models near unit root.

Abstract

This paper develops an asymptotic likelihood theory for triangular arrays of stationary Gaussian time series depending on a multidimensional unknown parameter. We give sufficient conditions for the associated sequence of statistical models to be locally asymptotically normal in Le Cam's sense, which in particular implies the asymptotic efficiency of the maximum likelihood estimator. Unique features of the array setting covered by our theory include potentially nondiagonal rate matrices as well as spectral densities that satisfy different power-law bounds at different frequencies and may fail to be uniformly integrable. To illustrate our theory, we study efficient estimation for Gaussian processes sampled at high frequency and for a class of autoregressive models with moderate deviations from a unit root.