97 research outputs found
Asymptotic posterior normality of the generalized extreme value distribution
The univariate generalized extreme value (GEV) distribution is the most
commonly used tool for analysing the properties of rare events. The ever
greater utilization of Bayesian methods for extreme value analysis warrants
detailed theoretical investigation, which has thus far been underdeveloped.
Even the most basic asymptotic results are difficult to obtain because the GEV
fails to satisfy standard regularity conditions. Here, we prove that the
posterior distribution of the GEV parameter vector, given an independent and
identically distributed sequence of observations, converges to a normal
distribution centred at the true parameter. The proof necessitates analysing
integrals of the GEV likelihood function over the entire parameter space, which
requires considerable care because the support of the GEV density depends on
the parameters in complicated ways
On the smallest eigenvalues of covariance matrices of multivariate spatial processes
International audienceThere has been a growing interest in providing models for multivariate spatial processes. A majority of these models specify a parametric matrix covariance function. Based on observations, the parameters are estimated by maximum likelihood or variants thereof. While the asymptotic properties of maximum likelihood estimators for univariate spatial processes have been analyzed in detail, maximum likelihood estimators for multivariate spatial processes have not received their deserved attention yet. In this article we consider the classical increasing-domain asymptotic setting restricting the minimum distance between the locations. Then, one of the main components to be studied from a theoretical point of view is the asymptotic positive definiteness of the underlying covariance matrix. Based on very weak assumptions on the matrix covariance function we show that the smallest eigenvalue of the covariance matrix is asymptotically bounded away from zero. Several practical implications are discussed as well
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