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Inference for α-Brownian bridge based on Karhunen-Loève expansions
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
(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: urn:nbn:se:uu:diva-232541OAI: oai:DiVA.org:uu-232541DiVA: diva2:748603
Available from: 2014-09-19 Created: 2014-09-19 Last updated: 2014-10-30
In thesis
1. Gaussian Bridges: Modeling and Inference
Open this publication in new window or tab >>Gaussian Bridges: Modeling and Inference
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:nbn:se:uu:diva-232544 (URN)978-91-506-2420-5 (ISBN)
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

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Görgens, Maik

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