Stochastic Models Involving Second Order Lévy Motions
(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:
https://lup.lub.lu.se/record/4284726
- author
- Wallin, Jonas LU
- supervisor
- opponent
-
- Professor Rue, Håvard, Norwegian University of Science and Technology (NTNU), Norway
- organization
- publishing date
- 2014
- 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 13:15:00
- ISBN
- 978-91-7473-843-8
- 978-91-7473-842-1 (print)
- language
- English
- LU publication?
- yes
- id
- 7dcad679-7c5a-4cfb-b1f7-d3665eb9c439 (old id 4284726)
- date added to LUP
- 2016-04-04 13:27:06
- date last changed
- 2019-03-12 11:47:47
@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}},
school = {{Lund University}},
title = {{Stochastic Models Involving Second Order Lévy Motions}},
url = {{https://lup.lub.lu.se/search/files/6123183/4284740.pdf}},
year = {{2014}},
}