The primary aim of this thesis is the elucidation of covariate effects on the dependence structure of random variables in bivariate or multivariate models. We develop a unified approach via a conditional copula model in which the copula is parametric and its parameter varies as the covariate. We propose a nonparametric procedure based on local likelihood to estimate the functional relationship between the copula parameter and the covariate, derive the asymptotic properties of the proposed estimator and outline the construction of pointwise confidence intervals. We also contribute a novel conditional copula selection method based on cross-validated prediction errors and a generalized likelihood ratio-type test to determine if the copula parameter varies significantly. We derive the asymptotic null distribution of the formal test. Using subsets of the Matched Multiple Birth and Framingham Heart Study datasets, we demonstrate the performance of these procedures via analyses of gestational age-specific twin birth weights and the impact of change in body mass index on the dependence between two consequent pulse pressures taken from the same subject.

The use of the local likelihood method (Tibshirani and Hastie, 1987; Loader, 1996) in the presence of interval-censored or aggregated data leads to a natural consideration of an EM-type strategy, or rather a local EM algorithm. In the thesis, we consider local EM to analyze the point process data that are either interval-censored or aggregated into regional counts. We specifically formulate local EM algorithms for density, intensity and risk estimation and implement the algorithms using a piecewise constant function. We demonstrate that the use of the piecewise constant function at the E-step explicitly results in an iteration that involves an expectation, maximization and smoothing step, or an EMS algorithm considered in Silverman, Jones, Wilson and Nychka (1990). Consequently, we reveal a previously unknown connection between local EM and the EMS algorithm. From a theoretical perspective, local EM and the EMS algorithm complement each other. Although the statistical methodology literature often characterizes EMS methods as ad hoc, local likelihood suggests otherwise as the EMS algorithm arises naturally from a local likelihood consideration in the context of point processes. Moreover, the EMS algorithm not only serves as a convenient implementation of the local EM algorithm but also provides a set of theoretical tools to better understand the role of local EM. In particular, we present results that reinforce the suggestion that the pair of local EM and penalized likelihood are analogous to that of EM and likelihood. Applications include the analysis of bivariate interval-censored data as well as disease mapping for a rare disease, lupus, in the Greater Toronto Area.

The use of the local likelihood method (Tibshirani and Hastie, 1987; Loader, 1996) in the presence of interval-censored or aggregated data leads to a natural consideration of an EM-type strategy, or rather a local EM algorithm. In the thesis, we consider local EM to analyze the point process data that are either interval-censored or aggregated into regional counts. We specifically formulate local EM algorithms for density, intensity and risk estimation and implement the algorithms using a piecewise constant function. We demonstrate that the use of the piecewise constant function at the E-step explicitly results in an iteration that involves an expectation, maximization and smoothing step, or an EMS algorithm considered in Silverman, Jones, Wilson and Nychka (1990). Consequently, we reveal a previously unknown connection between local EM and the EMS algorithm. From a theoretical perspective, local EM and the EMS algorithm complement each other. Although the statistical methodology literature often characterizes EMS methods as ad hoc, local likelihood suggests otherwise as the EMS algorithm arises naturally from a local likelihood consideration in the context of point processes. Moreover, the EMS algorithm not only serves as a convenient implementation of the local EM algorithm but also provides a set of theoretical tools to better understand the role of local EM. In particular, we present results that reinforce the suggestion that the pair of local EM and penalized likelihood are analogous to that of EM and likelihood. Applications include the analysis of bivariate interval-censored data as well as disease mapping for a rare disease, lupus, in the Greater Toronto Area.

The primary aim of this thesis is the elucidation of covariate effects on the dependence structure of random variables in bivariate or multivariate models. We develop a unified approach via a conditional copula model in which the copula is parametric and its parameter varies as the covariate. We propose a nonparametric procedure based on local likelihood to estimate the functional relationship between the copula parameter and the covariate, derive the asymptotic properties of the proposed estimator and outline the construction of pointwise confidence intervals. We also contribute a novel conditional copula selection method based on cross-validated prediction errors and a generalized likelihood ratio-type test to determine if the copula parameter varies significantly. We derive the asymptotic null distribution of the formal test. Using subsets of the Matched Multiple Birth and Framingham Heart Study datasets, we demonstrate the performance of these procedures via analyses of gestational age-specific twin birth weights and the impact of change in body mass index on the dependence between two consequent pulse pressures taken from the same subject.

Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.

Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

Dans cette thèse, nous nous intéressons à développer des outils mathématiques, afin de traiter une gamme des signaux biologiques et économiques. En premier lieu, nous proposons la transformée Dynalet, considérée comme une alternative, pour des signaux de relaxation sans symétrie interne, à la transformée de Fourier et à la transformée ondelette. L'applicabilité de cette nouvelle approximation est illustrée sur des données réelles. Ensuite, nous corrigeons la ligne de base des signaux biologiques spectrométriques, à l'aide d'une régression expectile pénalisée, qui, sur les applications proposées, est plus performante qu'une régression quantile. Puis, afin d'éliminer le bruit blanc, nous adaptons aux signaux spectrométriques une nouvelle approche combinant ondelette, seuillage doux et composants PLS. Pour terminer, comme les signaux peuvent être considérés comme des données fonctionnelles, d'une part, nous développons une vraisemblance locale fonctionnelle dont le but est d'effectuer une classification supervisée des courbes, et, d'autre part, nous estimons l'opérateur de régression pour une réponse scalaire positive non nulle, par minimisation de l'erreur quadratique moyenne relative. De plus, les lois asymptotiques de notre estimateur sont établies et son efficacité est illustrée sur des données simulées et sur des données spectroscopiques et économiques. / In this thesis, we focus on developing mathematical tools to treat a range of biological and economic signals. First, we propose the Dynalet transform for non-symmetrical biological relaxation signals. This transform is considered as an alternative to the Fourier transform and the wavelet transform. The applicability of the new approximation approach is illustrated on real data. Then, for spectrometric biological signals, we correct the baseline using a penalized expectile regression. Thus, the proposed applications show that our proposed regression is more efficient than the quantile regression. Then to remove random noise, we adapt to spectrometric data a new denoising method that combine wavelets, soft thresholding rule and PLS components. Finally, note that the biological signals may be often regarded as functional data. On one hand, we develop a functional local likelihood aiming to perform a supervised classification of curves. On the other hand, we estimate the regression operator with positive responses, by minimizing the mean squared relative error. Moreover, The asymptotic distributions of our estimator are established and their efficiency is illustrated on a simulation study and on a spectroscopic and economic data set.