Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Statistics and Actuarial Sciences

Supervisor

Provost, Serge B.

Abstract

This dissertation initially features distributional results related to copulas. Four distinct copula density estimation methodologies, including Bernstein’s polynomial approximation, are proposed and criteria for the selection of their tuning parameters are provided. These four approaches were found to produce similar density estimates, which validates their suitability. Moreover, the copula associated with a Wiener process and its running maximum is determined, and an illustrative numerical example is presented. The principal properties of Spearman’s, Kendall’s, Blomqvist’ and Hoeffding’s measures of association as well as their representations in terms of copulas are also discussed. Then, a novel method that is based on an arctangent transformation is introduced for classifying the tail behaviour of continuous probability laws. The resulting categories prove consistent with those obtained by applying available criteria. As well, approximations to the distributions of quadratic forms in gamma, inverse Gaussian, binomial and Poisson random variables, which hinge on the determination of their moments via a symbolic approach, are proposed and several applications are pointed out. Additionally, an accurate density approximation that relies of an extension of the generalized gamma distribution is introduced and the case of quadratic forms in Hermitian matrices in complex Gaussian vectors is also addressed. Finally, a methodology involving the use of Fritsch-Carlson monotonic interpolating splines and the Kulback-Leibler measure of divergence is proposed for quantifying the proportion of information that is contained in collections of distributional moments.

Summary for Lay Audience

In the first part of this dissertation, the focus is on copulas which are mathematical tools utilized to describe the relationships between random variables. Four distinct methods for estimating the distribution of copulas are proposed, with the objective consisting of capturing the patterns and characteristics of sets of observations on two variables. These methods were found to produce similar results, which confirms their effectiveness.The study also explores the copula associated with a specific probabilistic model called the Wiener process and its running maximum. An illustrative example is provided. The thesis further examines four measures of association and explains how they can be represented as copulas.

A novel methodology is introduced for classifying the behavior of the tails (extreme values) of continuous probability distributions. This method relies on an arctangent transformation and proves to be consistent with existing criteria.

The dissertation also presents approximations to the distributions of quadratic forms in various types of random variables. These approximations depend on the determination of their moments which are obtained by making use of a symbolic computational approach. The statistical applications of such quadratic forms are discussed. Furthermore, an accurate approximation involving the density of a generalized gamma distribution is introduced. The case of Hermitian quadratic forms in complex Gaussian vectors is addressed as well.

Finally, a methodology is proposed for quantifying the amount of information contained in sets of distributional moments. This approach relies on the use of a non-decreasing curve and a certain measure of divergence.

Overall, this research contributes to an improved understanding of copulas and other distributional concepts. The findings and methods presented in this thesis have the potential to enhance our ability to analyze complex data sets and make informed statistical modeling decisions.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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