Polynomial distribution functions on bounded closed intervals
Chirikhin, Andrey (2007) Polynomial distribution functions on bounded closed intervals. PhD thesis, University of Warwick.
WRAP_THESIS_Chirikhin_2007.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Official URL: http://webcat.warwick.ac.uk/record=b2242665~S15
The thesis explores several topics, related to polynomial distribution functions
and their densities on [0,1]M, including polynomial copula functions and their
densities. The contribution of this work can be subdivided into two areas.
- Studying the characterization of the extreme sets of polynomial densities
and copulas, which is possible due to the Choquet theorem.
- Development of statistical methods that utilize the fact that the density
is polynomial (which may or may not be an extreme density).
With regard to the characterization of the extreme sets, we first establish
that in all dimensions the density of an extreme distribution function is an extreme
density. As a consequence, characterizing extreme distribution functions
is equivalent to characterizing extreme densities, which is easier analytically.
We provide the full constructive characterization of the Choquet-extreme polynomial
densities in the univariate case, prove several necessary and sufficient
conditions for the extremality of densities in arbitrary dimension, provide necessary
conditions for extreme polynomial copulas, and prove characterizing
duality relationships for polynomial copulas. We also introduce a special case
of reflexive polynomial copulas.
Most of the statistical methods we consider are restricted to the univariate
case. We explore ways to construct univariate densities by mixing the extreme
ones, propose non-parametric and ML estimators of polynomial densities. We
introduce a new procedure to calibrate the mixing distribution and propose
an extension of the standard method of moments to pinned density moment
matching. As an application of the multivariate polynomial copulas, we introduce
polynomial coupling and explore its application to convolution of coupled
The introduction is followed by a summary of the contributions of this thesis
and the sections, dedicated first to the univariate case, then to the general
multivariate case, and then to polynomial copula densities. Each section first
presents the main results, followed by the literature review.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Distribution (Probability theory), Polynomials, Choquet theory, Density functionals|
|Official Date:||December 2007|
|Institution:||University of Warwick|
|Theses Department:||Department of Statistics|
|Supervisor(s)/Advisor:||Kendall, W. S. ; Jacka, Saul D.|
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