Fisher information and maximum-likelihood estimation of covariance parameters in Gaussian stochastic processes

The inverse of the Fisher information matrix is commonly used as an approximation for the covariance matrix of maximum-likelihood estimators. We show via three examples that for the covariance parameters of Gaussian stochastic processes under infill asymptotics, the covariance matrix of the limiting...

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Bibliographic Details
Published in:Canadian journal of statistics 1998-03, Vol.26 (1), p.127-137
Main Authors: Abt, Markus, Welch, William J.
Format: Article
Language:English
Subjects:
Approximation
Consistency
Consistent estimators
Correlations
Covariance
Covariance matrices
equivalent measures
Estimators
Fisher information
Gaussian random measures
Identifiability
infill asymptotics
Maximum likelihood estimation
orthogonal measures
Stochastic processes
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Summary:The inverse of the Fisher information matrix is commonly used as an approximation for the covariance matrix of maximum-likelihood estimators. We show via three examples that for the covariance parameters of Gaussian stochastic processes under infill asymptotics, the covariance matrix of the limiting distribution of their maximum-likelihood estimators equals the limit of the inverse information matrix. This is either proven analytically or justified by simulation. Furthermore, the limiting behaviour of the trace of the inverse information matrix indicates equivalence or orthogonality of the underlying Gaussian measures. Even in the case of singularity, the estimator of the process variance is seen to be unbiased, and also its variability is approximated accurately from the information matrix. /// L'inverse de la matrice d'information de Fisher est communément utilisée comme approximation de la matrice de covariance des estimateurs du maximum de vraisemblance. Nous démontrons, à l'aide de trois exemples, que pour les paramètres de covariance de processus stochastiques Gaussiens sous des asymptotiques "infill", la matrice de covariance de la distribution limite de leurs estimateurs du maximum de vraisemblance égale la limite de la matrice d'information inverse. Cela sera prouvé soit de façon analytique, soit justifié par simulation. De plus, le comportement limite de la trace de la matrice d'information inverse indique l'équivalence ou l'orthogonalité des mesures gaussiennes sous-jacentes. Même dans le cas de la singularité, l'estimateur de la variance du processus est non-biaisée et sa variabilité est approximée précisément à partir de la matrice d'information.
ISSN:0319-5724
1708-945X
DOI:10.2307/3315678