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Stochastic Models Involving Second Order Lévy Motions

Wallin, Jonas LU (2014)
Abstract
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Rue, Håvard, Norwegian University of Science and Technology (NTNU), Norway
organization
publishing date
type
Thesis
publication status
published
subject
pages
162 pages
defense location
Lecture hall MH:A, Centre for Mathematical Sciences, Sölvegatan 18, Lund University Faculty of Engineering
defense date
2014-02-28 13:15
ISBN
978-91-7473-843-8
978-91-7473-842-1 (print)
project
MERGE
language
English
LU publication?
yes
id
7dcad679-7c5a-4cfb-b1f7-d3665eb9c439 (old id 4284726)
date added to LUP
2014-02-04 07:59:36
date last changed
2016-09-19 08:45:16
@phdthesis{7dcad679-7c5a-4cfb-b1f7-d3665eb9c439,
  abstract     = {This thesis is based on five papers (A-E) treating estimation methods<br/><br>
for unbounded densities, random fields generated by Lévy processes,<br/><br>
behavior of Lévy processes at level crossings, and a Markov random<br/><br>
field mixtures of multivariate Gaussian fields.<br/><br>
<br/><br>
In Paper A we propose an estimator of the location parameter for a density<br/><br>
<br/><br>
that is unbounded at the mode.<br/><br>
<br/><br>
The estimator maximizes a modified likelihood in which the singular<br/><br>
<br/><br>
term in the full likelihood is left out, whenever the parameter value<br/><br>
<br/><br>
approaches a neighborhood of the singularity location.<br/><br>
<br/><br>
The consistency and super-efficiency of this maximum leave-one-out<br/><br>
<br/><br>
likelihood estimator is shown through a direct argument.<br/><br>
<br/><br>
In Paper B we prove that the generalized Laplace distribution and<br/><br>
<br/><br>
the normal inverse Gaussian distribution are the only subclasses of<br/><br>
<br/><br>
the generalized hyperbolic distribution that are closed under<br/><br>
<br/><br>
convolution.<br/><br>
<br/><br>
In Paper C we propose a non-Gaussian Matérn random field models,<br/><br>
<br/><br>
generated through stochastic partial differential equations,<br/><br>
<br/><br>
with the class of generalized Hyperbolic<br/><br>
<br/><br>
processes as noise forcings.<br/><br>
<br/><br>
A maximum likelihood estimation technique based on the Monte Carlo<br/><br>
<br/><br>
Expectation Maximization algorithm is presented, and it is<br/><br>
<br/><br>
shown how to preform predictions at unobserved<br/><br>
<br/><br>
locations.<br/><br>
<br/><br>
In Paper D a novel class of models is introduced, denoted latent<br/><br>
Gaussian random filed mixture models, which combines the Markov random<br/><br>
field mixture model with the latent Gaussian random field models.<br/><br>
<br/><br>
The latent model, which is observed under a measurement noise, is<br/><br>
defined as a mixture of several, possible multivariate, Gaussian<br/><br>
random fields. Selection of which of the fields is observed at each<br/><br>
location is modeled using a discrete Markov random field. Efficient<br/><br>
estimation methods for the parameter of the models is developed using<br/><br>
a stochastic gradient algorithm.<br/><br>
<br/><br>
In Paper E studies the behaviour of level crossing of non-Gaussian<br/><br>
time series through a Slepian model. The approach is through<br/><br>
developing a Slepian model for underlying random noise that drives the<br/><br>
process which crosses the level. It is demonstrated how a moving<br/><br>
average time series driven by Laplace noise can be analyzed through<br/><br>
the Slepian noise approach. Methods for sampling the biased sampling<br/><br>
distribution of the noise are based on an Gibbs sampler.},
  author       = {Wallin, Jonas},
  isbn         = {978-91-7473-843-8},
  language     = {eng},
  pages        = {162},
  school       = {Lund University},
  title        = {Stochastic Models Involving Second Order Lévy Motions},
  year         = {2014},
}