# Stereology and Spatio-Temporal Models

PhD Dissertations | September 2020

Using the so-called Cavalieri estimator, the volume of a bounded 3-dimensional object can be unbiasedly estimated from area measurements on parallel section profiles. Mathematically, this corresponds to the problem of numerical integration on $\mathbb{R}$ when the integrand is known at a the points of a stationary and equidistant point process. Previously it has been shown that the variance of the Cavalieri estimator, which is simply a Riemann sum approximation, exhibits a slower decrease rate when the sampling nodes are not equidistant.

The first part of the dissertation concerns an alternative numerical estimation approach suitable for stationary and non-equidistant sampling. By using the information on the increments of the sampling points, we suggest using Newton-Cotes quadrature rules to approximate the integrand by a polynomial of a certain degree. It turns out that the variance inflation of the Cavalieri estimator under non-equidistant sampling can be avoided by this estimation approach. We investigate its variance properties under different sampling models, and we also suggest estimates of its variance based on sampling in a bounded interval in $\mathbb{R}$.

The second part of the dissertation mainly concerns spatial Lévy-driven moving average fields, which are integrals of a deterministic kernel function with respect to a Lévy basis. Throughout the dissertation we assume that the underlying Lévy measure is convolution equivalent, and we investigate the extremal behavior of the fields relative to that of the Lévy measure.

Firstly, we consider a space-time model, where the field is thought of as having a space and time component. Under reasonable assumptions on the Lévy measure and the integration kernel, we show that the tail of certain functionals (applied to the field) asymptotically equals the tail of the Lévy measure. As a complementary result we obtain that the field is continuous in the space component and càdlàg in time.

Secondly, we generalize classical extreme value results in $\mathbb{R}$ to higher-dimensional Euclidean space. In particular, assuming mixing conditions for a stationary random field, we show that an extremal types theorem holds for its normalized maximum on a large class of expanding sets. Furthermore, using similar proof techniques, we show that the normalized supremum of the Lévy-driven field converges to the Gumbel distribution as the index sets increase.

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