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Gaussian Bridges: Modeling and Inference
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and five papers, dealing with the modeling of Gaussian bridges and membranes and inference for the α-Brownian bridge.

In Paper I we study continuous Gaussian processes conditioned that certain functionals of their sample paths vanish. We deduce anticipative and non-anticipative representations for them. Generalizations to Gaussian random variables with values in separable Banach spaces are discussed. In Paper II we present a unified approach to the construction of generalized Gaussian random fields. Then we show how to extract different Gaussian processes, such as fractional Brownian motion, Gaussian bridges and their generalizations, and Gaussian membranes from them.

In Paper III we study a simple decision problem on the scaling parameter in α-Brownian bridges. We generalize the Karhunen-Loève theorem and obtain the distribution of the involved likelihood ratio based on Karhunen-Loève expansions and Smirnov's formula. The presented approach is applied to a simple decision problem for Ornstein-Uhlenbeck processes as well. In Paper IV we calculate the bias of the maximum likelihood estimator for the scaling parameter and propose a bias-corrected estimator. We compare it with the maximum likelihood estimator and two alternative Bayesian estimators in a simulation study. In Paper V we solve an optimal stopping problem for the α-Brownian bridge. In particular, the limiting behavior as α tends to zero is discussed.

Place, publisher, year, edition, pages
Uppsala: Uppsala University, Department of Mathematics , 2014. , 32 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 86
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:uu:diva-232544ISBN: 978-91-506-2420-5 (print)OAI: oai:DiVA.org:uu-232544DiVA: diva2:749343
Public defence
2014-11-07, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2014-10-17 Created: 2014-09-19 Last updated: 2014-10-30Bibliographically approved
List of papers
1. Conditioning of Gaussian processes and a zero area Brownian bridge
Open this publication in new window or tab >>Conditioning of Gaussian processes and a zero area Brownian bridge
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on [0,1] pinned down in 0 at time 1 and conditioned to have vanishing area spanned by the sample paths. Finally, the generalization to arbitrary separable Banach spaces is studied.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-232543 (URN)
Available from: 2014-09-20 Created: 2014-09-19 Last updated: 2014-10-30
2. Gaussian processes, bridges and membranes extracted from selfsimilar random fields
Open this publication in new window or tab >>Gaussian processes, bridges and membranes extracted from selfsimilar random fields
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the class of selfsimilar Gaussian generalized random fields introduced by Dobrushin in 1979. These fields are indexed by Schwartz functions on Rd and parametrized by a self-similarity index and the degree of stationarity of their increments. We show that such Gaussian fields arise in explicit form by letting Gaussian white noise, or Gaussian random balls white noise, drive a shift and scale shot-noise mechanism on Rd, covering both isotropic and anisotropic situations. In some cases these fields allow indexing with a wider class of signed measures, and by using families of signed measures parametrized by the points in euclidean space we are able to extract pointwise defined Gaussian processes, such as fractional Brownian motion on Rd. Developing this method further, we construct Gaussian bridges and Gaussian membranes on a finite domain, which vanish on the boundary of the domain.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-232545 (URN)
Available from: 2014-09-23 Created: 2014-09-19 Last updated: 2015-04-23Bibliographically approved
3. Inference for α-Brownian bridge based on Karhunen-Loève expansions
Open this publication in new window or tab >>Inference for α-Brownian bridge based on Karhunen-Loève expansions
(English)Article in journal (Refereed) Submitted
Abstract [en]

We study a simple decision problem for the scaling parameter in the α-Brownian bridge X(α) on the interval [0,1]: given two values α0, α1 ≥ 0 with α0 + α1 ≥ 1 and some time 0 ≤ T ≤ 1 we want to test H0: α = α0 vs. H1: α = α1 based on an observation of X(α) until time T. The likelihood ratio can be written as a functional of a quadratic form ψ(X(α)) of X(α). In order to calculate the distribution of ψ(X(α)) under the null hypothesis, we generalize the Karhunen-Loève Theorem to positive finite measures on [0,1] and compute the Karhunen-Loève expansion of X(α) under such a measure. Based on this expansion, the distribution of ψ(X(α)) follows by Smirnov's formula.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-232541 (URN)
Available from: 2014-09-19 Created: 2014-09-19 Last updated: 2014-10-30
4. Bias-correction of the maximum likelihood estimator for the α-Brownian bridge
Open this publication in new window or tab >>Bias-correction of the maximum likelihood estimator for the α-Brownian bridge
2014 (English)In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 93, 78-86 p.Article in journal (Refereed) Published
Abstract [en]

The bias of the maximum likelihood estimator of the parameter α in the α-Brownian bridge is derived. A bias-correction which improves the estimator substantially is proposed. The corrected estimator and Bayesian estimators are compared in a simulation study.

Keyword
α-Brownian bridge, Bias-correction, Estimation, Scaled Brownian bridge
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-229177 (URN)10.1016/j.spl.2014.06.020 (DOI)000341479500012 ()
Note

Title also written as: Bias-correction of the maximum likelihood estimator for the alpha-Brownian bridge

Available from: 2014-08-04 Created: 2014-08-04 Last updated: 2017-12-05Bibliographically approved
5. Optimal stopping of an α-Brownian bridge
Open this publication in new window or tab >>Optimal stopping of an α-Brownian bridge
(English)Article in journal (Refereed) Submitted
Abstract [en]

We study the problem of stopping an α-Brownian bridge as close as possible to its global maximum. This extends earlier results found for the Brownian bridge (the case α = 1). The exact behavior for α close to 0 is investigated.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:uu:diva-232542 (URN)
Available from: 2014-09-19 Created: 2014-09-19 Last updated: 2014-10-30

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