This thesis is a contribution to the covering times problems for random walks on graphs. By considering uniform random walks on finite connected graphs, the covering time is defined as the time (number of steps) taken by the random walk to visit every vertex. The motivating problem of this thesis is to find bounds for the expected covering times. We provide explicit bounds that are uniformly valid over all starting points and over large classes of graphs. In some cases the asymptotic distribution of the suitably normalized covering time is given as well.

Markov Chain Monte Carlo (MCMC) methods are employed to sample from a given distribution of interest, pi, where either pi does not exist in closed form, or, if it does, there exist no efficient methods to simulate an independent sample from it. MCMC methods create an ergodic Markov chain {Xn, n = 1, 2...} with stationary distribution pi such that as n tends to infinity, the distribution of Xn approaches pi. A wealth of diagnostic tools for convergence assessment of MCMC methods have been proposed, yet none has proved to be completely dependable and easy to implement. This thesis will review the literature on MCMC algorithms and diagnostic tools, and it will present a new convergence assessment method based on the principle of detailed balance. Moreover, the proposed diagnostic tool will be implemented as a stopping criterion for the optimization algorithm known as simulated annealing, which finds the global maximum or minimum of a function through an iterative improvement approach.

Sample size determination is one of the most important statistical issues in the early stages of any investigation that anticipates statistical analyses. / In this thesis, we examine Bayesian sample size determination methodology for interval estimation. Four major epidemiological study designs, cohort, case-control, cross-sectional and matched pair are the focus. We study three Bayesian sample size criteria: the average length criterion (ALC), the average coverage criterion ( ACC) and the worst outcome criterion (WOC ) as well as various extensions of these criteria. In addition, a simple cost function is included as part of our sample size calculations for cohort and case-controls studies. We also examine the important design issue of the choice of the optimal ratio of controls per case in case-control settings or non-exposed to exposed in cohort settings. / The main difficulties with Bayesian sample size calculation problems are often at the computational level. Thus, this thesis is concerned, to a considerable extent, with presenting sample size methods that are computationally efficient.

In brain mapping, the regions of the brain that are 'activated' by a task or external stimulus are detected by thresholding an image of test statistics. Often the experiment is repeated on several different subjects or for several different stimuli on the same subject, and the researcher is interested in the common points in the brain where 'activation' occurs in all test statistic images. The conjunction is thus defined as those points in the brain that show 'activation' in all images. We are interested in which parts of the conjunction are noise, and which show true activation in all test statistic images. We would expect truly activated regions to be larger than usual, so our test statistic is based on the volume of clusters (connected components) of the conjunction. Our main result is an approximate P-value for this in the case of the conjunction of two Gaussian or chi2 test statistic images. The results are applied to a functional magnetic resonance experiment in pain perception.

It is well known that when subjects with a disease are identified through a cross-sectional survey and then followed forward in time until either failure or censoring, their estimated survival function of the true survival function from onset are biased. This bias, which is caused by the sampling of prevalent rather than incident cases, is termed length bias if the onset time of the disease forms a stationary Poisson process. While authors have proposed different approaches to the analysis of length-biased survival data, there remain a number of issues that have not been fully addressed. The most, important of these is perhaps that of how to include covariates into length-biased lifetime data analysis of the natural history of diseases, that are initiated by cross-sectional sampling of a population. One aspect of that problem, which appears to have been neglected in the literature, concerns the effect of length-bias on the sampling distribution of the covariates. If the covariates have an effect on the survival time, then their marginal distribution in a length-biased sample is also subject to a bias and is informative about the parameters of interest. As is conventional in most regression analyses one conditions on the observed covariate values. By conditioning on the observed covariates in the situation described above, however, one effectively ignores the information contained in the distribution of the covariates in the sample. We present the appropriate likelihood approach that takes into account this information and we establish the consistency and asymptotic normality of the resulting estimators. It is shown that by ignoring the information contained in the sampling distribution of the covariates, one can still obtain, asymptotically, the same point estimates as with the joint likelihood. However, these conditional estimates are less efficient. Our results are illustrated using data on survival with dementia; collected as part of the Canadian Study of Health an Aging.

We implement a general procedure for incomplete data estimation problems proposed by Gu & Li (1998) and Gu & Kong (1998). The procedure can be used to find the Maximum Likelihood Estimate (MLE) or solve estimating equations in problems such as estimations with censored or truncated regression model, nonlinear structural measurement error model and random effects model. The procedure is based on the general principle of stochastic approximation (Robbins & Monroe 1951) and Markov Chain Monte Carlo method (Metropolis et al. 1953, Hastings 1970). A new stopping criterion is proposed and simulation studies indicate that the algorithm converges consistently to the MLE for the mixed effects logistic regression model.

Natural Exponential Families (NEF) belonging to the Grand-Babel class have variance functions (VF) of the form V$ sb{F}$(m) = P(m)$ Delta(m)$ + Q(m)$ sqrt{ Delta(m)},$ with P,Q and $ Delta$ polynomials with deg P $ geq$ 3, deg Q $ geq$ 2 and deg $ Delta$ $ geq$ 2. Although the members of this class have not as yet all been enumerated, several useful sub-classes have been fully described, namely the Morris class, with at most quadratic polynomial VFs: the Mora class, with cubic polynomial Vfs; the Babel class, with deg P = 0, deg Q $ leq$ 1 and deg $ Delta$ $ leq$ 2; and the Seshadri class, with deg P = degQ = deg$ Delta$ = 1. In order to motivate a uniform presentation of each member of these classes in compendium form, the basic properties of NEFs are surveyed, with special insistence on extension models such as convolution families, exponential dispersion models and affinities of NEFs. The Grand-Babel NEFs are presented with both a canonical parametrization which emphasizes the link to their basis measure and a more familiar or utilitarian parametrization. Expressions for the variance function, the cumulant transform, the mean-domain mapping, the density (when available), the Legendre transform and some asymptotics are given for each NEF, thus providing links to the theories of Likelihood and Quasi-likelihood, Generalized Linear Models, Saddlepoint approximation, Large deviations, Distributions and Asymptotic approximation. The notion of Canonical Caste Member (CCM), an easily identifiable representative of the equivalence class of all affinities of a NEF, is introduced; correspondingly, a table of variance functions for the CCMs of the currently classified Grand-Babel NEFs is provided.

In a screening survey for a particular disease, suppose that results from one or more imperfect diagnostic tests are available. Using a Bayesian approach, methods for simultaneous inferences about the population prevalence of the disease and the characteristic parameters of the diagnostic tests are developed. Marginal posterior densities of all parameters are estimated using the Gibbs sampler, a Markov Chain Monte Carlo technique. The methods are applied to data from Kampuchean refugees who arrived in Montreal, Canada, during the years 1982-1983, to estimate the prevalence of Strongyloides infection in this population, as well as the test properties of two diagnostic tests for Strongyloides, stool examinations and serological testing.

We examine the problem of finding the exact distributions of linear functions of k independent generalized gamma variables, X 1, X2,..., Xk. Special cases of generalized gamma distributions include the exponential, gamma and Weibull distributions. A linear function of such variables is often a quantity of interest in the analysis of survival data, reliability of certain systems and stochastic processes and hence we present this problem in the context of life testing. The exact distributions of these linear functions are needed to compute survival functions, hazard functions and other important functions in practical problems. Stacy (1962) obtained some exact results involving generalized gamma variables and Huzurbazar and Huzurbazar (1999) used saddlepoint approximations where the input variables are gamma or Weibull. We examine this problem where the k independent real scalar random variables, X1, X 2,..., Xk, are of gamma type with general parameters. For this case, various exact distributions are obtained and it is shown that most of these representations are easily computable. These exact results are compared with the usual saddlepoint approximations. We also examine numerically inverting the Laplace transform in this context, showing that it is one of the most efficient and accurate ways of estimating the exact distribution for certain cases. Results of this thesis are being published and presented in co-authorship with A. M. Mathai in Koulis and Mathai (2000).

Although the single-path changepoint problem has been extensively treated in the statistical literature, the multi-path changepoint problem has been largely ignored. / In the multi-path changepoint setting it is often of interest to assess the impact of covariates on the changepoint itself as well as on the parameters before and after the changepoint. This thesis is concerned with including covariates in the changepoint distribution, a topic never before addressed in the literature. The model we introduce, based on the hazard of change, enjoys features which allow one to establish asymptotic results needed for estimation and testing. Indeed, we establish consistency of the maximum likelihood estimators of the parameters of our model. / As the proposed model is a mixture model, two of the difficulties associated with such models are addressed. They are identifiability, and positive definiteness of the information matrix. It is shown that under suitable conditions the set of zeros of the determinant of the information matrix is a nowhere dense set, thus partially compensating for the impossibility of directly establishing positive definiteness. / A limited simulation, using simulated annealing, is carried out to assess how the estimation procedure works in practice. In the example presented, the estimators appear to follow an approximately normal distribution even for moderate sample sizes. The maximum likelihood estimators appear to approximate their parameter counterparts well.