ESTIMATION OF SPATIAL GAUSSIAN DEPENDENCIES WITH A FOCUS ON EXTREMES

We provide theoretical advancements in the estimation of dependence structures corresponding to Gaussian processes. The first concerns the H"usler-Reiss distribution, which is known to be the limit of the maxima of properly linearly-scaled Gaussian distributions. In any application with finite sample sizes, we know that the true distribution of maxima of normal samples must not be exactly H"usler-Reiss, thus we seek to find what bias is introduced by assuming that our distribution is sufficiently large to substitute this asymptotic distribution into a maximum likelihood estimation procedure. We ultimately find a Central Limit Theorem-like result which states that we may control any bias to be finite. Building off of this result, we seek to extend this to a spatial setting regarding Brown-Resnick processes, which represent the limit distribution of linearly scaled Gaussian processes (as opposed to finite-dimensional Gaussian distributions). Finally, we derive a constrained fiducial distribution which allows for easy implementation of a Monte Carlo Markov Chain (MCMC) algorithm to estimate the parametric covariance of a Gaussian process. We implement this algorithm, which works for a very large family of parametric Gaussian models. As a proof of concept, we show well-performing results for the MA1 and Mat\'ern models.