Likelihoods are widely used in the estimation of parameters, but the computation of their maximum can be quite complicated, and sometimes impossible in the presence of nuisance parameters. In such cases we seek a modified likelihood that will allow us to make inference on the parameters of interest with as little loss of information as possible. In the present work we will investigate some of these modified likelihoods, namely marginal likelihoods, conditional likelihoods, profile likelihoods, partial likelihoods, empirical likelihoods, and functions that exhibit properties similar to the ones of the likelihoods, more precisely the quasi-likelihoods. We will point out some cases where these modified likelihoods give rise to satisfactory estimations, and some other cases where they fail to give satisfactory estimations.

This thesis is concerned with the distributional problems related to quadratic forms in normal variables. Some functions of quadratic forms such as linear combinations and ratios will be considered. / Our main focus will be on the investigation of ratios of quadratic forms. The moments, the density functions and the distribution functions will be derived analytically and presented in closed forms. Some cases where independence or commutativity hold will be studied extensively. / Many new results are obtained and new derivations of known results are also proposed. Some particular cases as well as their applications are investigated. Some numerical examples are presented. / A fairly comprehensive bibliography on the subject is also included.

This dissertation principally addresses nonparametric maximal likelihood (NPML) estimation of the cumulative distribution function (CDF) given multivariate (arbitrarily) censored data (herein abbreviated MCD). / The CDF nonparametric maximal likelihood estimate (NPMLE) given MCD has support on the union of all maximal intersections of the data. The CDF NPMLE can be computed numerically using the clique matrix of the intersection graph of the data; these NPMLEs can be nonunique in both a representational and a mixture sense (see Peto 1973, Turnbull 1976, Gentleman & Vandal 2001 and Gentleman & Vandal 2002). / The fundamental methodology used in this dissertation consists in applying graph theory to the intersection graph of censored data and discrete mathematics to its linear algebraic representation. An optimal algorithm to determine the maximal intersections of MCD is proposed. A full discussion of measures of NPMLE mixture nonuniqueness and their computational implementations for the measures is provided. The iterative convex minorant (ICM) algorithm to obtain the NPMLE is extended to the case of MCD. The nonparametric likelihood maximization given MCD is simplified via the use of a reduction tree. The EM/X Algorithm is introduced to compute the NPMLE for large MCD set. Bounds on self-consistent estimates of the CDF (a class to which the CDF NPMLE belongs) given MCD are used to assess the degree of consistency of the CDF NPMLE. Constrained estimation and likelihood intervals computation given univariate censored data are discussed. The empirical likelihood method is also applied to construct CDF likelihood sets for MCD. An unbiased and consistent estimate is proposed for MCD with fixed censoring times.

Strategic planning is not only necessary in today's global economy where markets are becoming more susceptible to international competition, it is vital. The foresight of market reactions can lead to a competitive advantage. Market share losses can be minimized (and market share gains maximized) with the knowledge obtained from primary marketing research involving a stated preference study to examine consumer behaviour. Before launching a new product or providing a service, discrete choice analysis can empower strategic planners, managers and marketers with a tool which aids in optimizing products and services for a potential market with the end of maximizing sales and services. / Discrete choice analysis is a tool to understand human choice behaviour. It is employed for statistical inference on a model of choice behaviour from data obtained by sampling from a population of decision makers. This thesis gives an overview of the basic concepts of conjoint analysis which addresses discrete choice analysis for strategic product and service planning. The statistical model specification, the multinomial logit, is derived assuming that decision makers follow a choice rule called utility maximization, where these random utilities are Gumbel distributed. The model is applied to a stated preference study in which environmentally friendly vehicles are presented as possible vehicle choices.

An overview of the statistical procedures used in the analysis of electrocardiogram traces is presented in this thesis. / The purpose of these procedures is twofold: (i) they may suggest underlying mechanisms that influence heart rate (ii) they may be used as a means of classifying one or more patients into disease categories, by using objective criteria rather than the subjective approaches prevalent in current practice. / In an attempt to apply the methods discussed in this thesis, a selected group of patients was analyzed using spectral analysis. / Lack of information and of control of the patients' activities while they were being monitored precluded the possibility of obtaining definitive results.

We examine an 1880 theorem of Laguerre concerning polynomials with all real roots and a 1968 inequality of Samuelson for the maximum and minimum deviation from the mean, and establish their equivalence and present several proofs. We also study related inequalities attributed to (1) J. M. C. Scott (1936); (2) Brunk (1959); (3) Boyd (1971) & Hawkins (1971). / Also examined is a 1918 inequality of Szokefalvi-Nagy and some 1935 extensions of Popoviciu concerning the standard deviation and range of a set of real numbers and equivalent inequalities for the internally Studentized range due to K. R. Nair in 1947/1948 and G. W. Thomson in 1955, as well as related bounds on the standard deviation attributed to (1) Guterman (1962); (2) Margaritescu-Voda (1983); (3) Bhatia-Davis (1999). / Extensions and applications in statistics and matrix theory are provided, as well as biographical information and an extensive bibliography.

This is an expository thesis covering the area of matrix-variate statistical distributions. A survey of functions of one matrix argument and the allied one matrix-variate statistical distributions is carried out first. Then functions of several matrix arguments and generalized statistical distributions are surveyed. Recent contributions to hypergeometric functions of several matrix arguments are also examined. These results are collected, compiled, classified and presented in various chapters. Some minor simplifications and alternate derivations are also considered at various places.

During the course of follow-up studies designed to assess the time to an event of interest, longitudinal variables are usually monitored. These longitudinal variables are often prognostic and, as such, are a potential rich source of information regarding the true time-to-event of censored subjects. In this thesis I propose methodology that uses information from markers, these longitudinal prognostic variables, to enhance inference on (i) parametric time-to-event models, (ii) semi-parametric models, in particular, Cox's (1972) model and (iii) log rank tests. Using simulations I determine if and by how much markers can enhance inference on these models and tests, as a function of a marker's prognostic ability. These simulations are based on a novel method of random data generation whose properties I also examine.

In this thesis, we are concerned with the asymptotic properties of kernel estimates with variable bandwidth in mixed censorship-truncation models. A new approach has been developed, and the asymptotic normality of the hazard rate estimator $ lambda sbsp{n}{(2)}(x),$ which has not been established even in complete data cases, is established. By the same method, we derive the asymptotic normality of other hazard estimates, such as $ lambda sbsp{n}{(1)}(x), lambda sbsp{n}{(3)}(x),$ and $ lambda sbsp{n}{(5)}(x),$ as well as the density estimator $ f sb{n}(x).$ The second part of this thesis is devoted to the strong consistency and rates of convergence of the estimates mentioned above; these results improve and extend the previous results in Schafer (1985), Mielniczuk (1986), and Uzunogullari and Wang (1992). Lastly, we consider the consistency of the estimates $ beta sb{n}$ and $ Lambda sb0$ in the general relative risk regression (GRRR) models. We also consider the identifiability of $ beta sb0$ in GRRR models, and prove that most GRRR models possess the same properties as in the Cox regression model, and therefore should be used more often in practice.

In criminal investigations involving glass evidence, refractive index (RI) is the property of glass most commonly used by forensic examiners to determine the association between control samples of glass obtained at the crime scene, and samples of glass found on a suspect. Previous studies have shown that an intrinsic variability of RI exists within a pane of float glass. In this thesis, we attempt to determine whether this variability is spatially determined or random in nature, the conclusion of which plays an important role in the statistical interpretation of glass evidence. We take a Bayesian approach in fitting a spatial model to our data, and utilize the WinBUGS software to perform Gibbs sampling. To test for spatial variability, we propose two test quantities, and employ Bayesian Monte Carlo significance tests to test our data, as well as nine other specifically formulated data-sets.