The local asymptotic normality (LAN) property is established for multivariate ARMA models with a linear trend or, equivalently, for multivariate general linear models with ARMA error term. In contrast with earlier univariate results, the central sequence here is correlogram-based, i.e. expressed in terms of a generalized concept of residual cross-covariance function.

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Local asymptotic normality of multivariate ARMA processes with a linear trend

We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the same mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, the studentized sample mean, to reflect econometric interest in autocorrelation-consistent or long-run variance estimation. Our main focus is on stationary Gaussian series, though we discuss relaxation of the Gaussianity assumption. Only smoothness conditions on the spectral density that are local to the frequency of interest are imposed. We deduce empirical expansions from our Edgeworth expansions designed to improve on the normal approximation in practice, and also a feasible rule of bandwidth choice.

Edgeworth expansions for spectral density estimates and studentized sample mean

Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem (2) on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and nally of the empirical spectral repartition function. R esum e. Sous des hypoth eses de r egularit e convenables, on etablit un principe de grandes d eviations pr ecises pour des formes quadratiques de processus gaussiens stationnaires. Notre r esultat est l'analogue du th eor eme de Bahadur-Rao (2) sur la moyenne empirique. Nous proposons egale- ment plusieurs exemples d'application comme les propri et es de grandes d eviations pr ecises pour le test du rapport de vraisemblance de Neyman-Pearson, pour la somme des carr es, pour l'estimateur de Yule-Walker du param etre d'un processus gaussien autor egressif stable, et nalement pour la fonction de r epartition spectrale empirique.

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Sharp large deviations for gaussian quadratic forms with applications

We develop order T−1 asymptotic expansions for the quasi-maximum likelihood estimator (QMLE) and a two-step approximate QMLE in the GARCH(l,l) model. We calculate the approximate mean and skewness and, hence, the Edgeworth-B distribution function. We suggest several methods of bias reduction based on these approximations.

An Asymptotic Expansion in the GARCH(l, 1) Model

Introducing a conditional mixing property, Gotze and Hipp’s theory is generalized to a continuous-time conditional ∈-Markov process satisfying this property. The Malliavin calculus for jump processes applies to random-coefficient stochastic differential equations with jumps with the aid of the support theorem to verify the non-degeneracy condition, i.e., a conditional type Cramer condition.

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Partial mixing and Edgeworth expansion

A survey of the first-order asymptotic theory for time series analysis higher order asymptotic theory for gussian arma processes validity of Edgeworth expansions in time series analysis higher order asymptotic sufficiency, asymptotic ancillarity in time series analysis higher order investigations for testing theory in time series analysis higher order asymptotic theory for multivariate time series some practical examples.

正信谷口

Papers

Higher order asymptotic theory for time series analysis

正信 谷口

Papers

Higher order asymptotic theory for time series analysis

正信谷口