{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/37090"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/37090","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"On the Distribution of Quadratic Expressions in Various Types of Random Vectors","abstract":"Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful.","abstract_html":"Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful.","abstract_has_math":false,"creators":["Mohsenipour, Ali Akbar"],"institution":"The University of Western Ontario","degree_name":"Ph D","degree_level":null,"degree_discipline":"Statistics and Actuarial Sciences","degree_department":null,"school":null,"contributors":[],"advisors":["Serge B. Provost"],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-11-28","date_published":"2012-11-28","updated_at":"2026-07-27T21:56:01Z","subjects":["Hermitian quadratic forms","singular Gaussian vectors","elliptically contoured random vectors","uniform and beta random variables","exponential and gamma random variables","density approximation."],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/37090","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Serge B. 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In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful."]},{"key":"dc:title","label":"Title","values":["On the Distribution of Quadratic Expressions in Various Types of Random Vectors"]}]}],"canonical_facts":{"dc:contributor.advisor":["Serge B. Provost"],"dc:creator":["Mohsenipour, Ali Akbar"],"dc:date.accessioned":["2025-07-10T21:29:56Z"],"dc:date.available":["2025-07-10T21:29:56Z"],"dc:date.issued":["2012-11-28"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.abstract":["Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14721/37090"],"dc:language.iso":["en_ca"],"dc:publisher":["The University of Western Ontario"],"dc:subject":["Hermitian quadratic forms","singular Gaussian vectors","elliptically contoured random vectors","uniform and beta random variables","exponential and gamma random variables","density approximation."],"dc:title":["On the Distribution of Quadratic Expressions in Various Types of Random Vectors"],"dc:type":["thesis"],"thesis:degree_discipline":["Statistics and Actuarial Sciences"],"thesis:degree_name":["Ph D"]},"updated_at":"2026-07-27T21:56:01Z"}