Stochastic Models Involving Second Order Lévy Motions

This thesis is based on five papers (A-E) treating estimation methods
for unbounded densities, random fields generated by Lévy processes,
behavior of Lévy processes at level crossings, and a Markov random
field mixtures of multivariate Gaussian fields.

In Paper A we propose an estimator of the location parameter for a density

that is unbounded at the mode.

The estimator maximizes a modified likelihood in which the singular

term in the full likelihood is left out, whenever the parameter value

approaches a neighborhood of the singularity location.

The consistency and super-efficiency of this maximum leave-one-out

likelihood estimator is shown through a direct argument.

In Paper B we prove that the generalized Laplace distribution and

the normal inverse Gaussian distribution are the only subclasses of

the generalized hyperbolic distribution that are closed under

convolution.

In Paper C we propose a non-Gaussian Matérn random field models,

generated through stochastic partial differential equations,

with the class of generalized Hyperbolic

processes as noise forcings.

A maximum likelihood estimation technique based on the Monte Carlo

Expectation Maximization algorithm is presented, and it is

shown how to preform predictions at unobserved

locations.

In Paper D a novel class of models is introduced, denoted latent
Gaussian random filed mixture models, which combines the Markov random
field mixture model with the latent Gaussian random field models.

The latent model, which is observed under a measurement noise, is
defined as a mixture of several, possible multivariate, Gaussian
random fields. Selection of which of the fields is observed at each
location is modeled using a discrete Markov random field. Efficient
estimation methods for the parameter of the models is developed using
a stochastic gradient algorithm.

In Paper E studies the behaviour of level crossing of non-Gaussian
time series through a Slepian model. The approach is through
developing a Slepian model for underlying random noise that drives the
process which crosses the level. It is demonstrated how a moving
average time series driven by Laplace noise can be analyzed through
the Slepian noise approach. Methods for sampling the biased sampling
distribution of the noise are based on an Gibbs sampler.