Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Master of Science

Program

Statistics and Actuarial Sciences

Supervisor

Provost, Serge B.

Abstract

This thesis comprises various results that rely on the moments of a distribution or the sample moments associated with a set of observations. Since a sample of size n is uniquely specified by its first n moments, it is pertinent to make use of sample moments for modeling, classification or inference purposes. Three density mixtures are approximated by adjusting in various ways an initial density approximation referred to a base density by means certain moment-based functions, and the accuracy of the resulting density approximants are compared. A similar study is carried out in the context of density estimation. Moreover, it is explained that methodologies that are based on moments are, in fact, ideally suited to model massive data sets. Various types of quasi-Monte Carlo deterministic samples are then compared to randomly generated samples with respect to their distributional representativeness. As well, a novel methodology depending on an arctangent transformation is introduced for classifying the tail behaviour of probability laws. Finally, certain approximations to the distributions of quadratic forms in gamma, inverse Gaussian, binomial and Poisson random variables, which rely on a symbolic expansion of their moments, are proposed.

Summary for Lay Audience

Various statistical results of interest that are based on the moments of a distribution are presented in this thesis. In fact, the moments associated with a sample of observations contain all the distributional information therein available. Several types of adjustments and samples are investigated in order to determine which ones will provide the most accurate representations of a given distribution. As well, a simple new criterion is proposed to categorize the tail behaviour of probability laws. Finally, an efficient approach is proposed for approximating the distribution of quadratic forms in several types of random variables, which are utilized in connection with contingency tables and generalized linear models.

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