In this dissertation, we apply Bayesian and Empirical Bayes methods for reliability growth models based on the power law process. We also apply Bayes methods for the study of microarrays, in particular, in the selection of differentially expressed genes. The power law process has been used extensively in reliability growth models. Chapter 1 reviews some basic concepts in reliability growth models. Chapter 2 shows classical inferences on the power law process. We also assess the goodness of fit of a power law process for a reliability growth model. In chapter 3 we develop Bayesian procedures for the power law process with failure truncated data, using non-informative priors for the scale and location parameters. In addition to obtaining the posterior density of parameters of the power law process, prediction inferences for the expected number of failures in some time interval and the probability of future failure times are also discussed. The prediction results for the software reliability model are illustrated. We compare our result with the result of Bar-Lev,S.K. et al. Also, posterior densities of several parametric functions are given. Chapter 4 provides Empirical Bayes for the power law process with natural conjugate priors and nonparametric priors. For the natural conjugate priors, two-hyperparameter prior and a more generalized three-hyperparameter prior are used. In chapter 5, we review some basic statistical procedures that are involved in microarray analysis. We will also present and compare several transformation and normalization methods for probe level data. The objective of chapter 6 is to select differentially expressed genes from tens of thousands of genes. Both classical methods (fold change, T-test, Wilcoxon Rank-sum Test, SAM and local Z-score and Empirical Bayes methods (EBarrays and LIMMA) are applied to obtain the results. Outputs of a typical classical method and a typical Empirical Bayes Method are discussed in detail.

In this thesis we investigate thinning of the renewal process. After multinomial thinning from a renewal process A, we obtain the k thinned processes, A_i , i =1,¡K, k. Based on some characterizations of the Poisson process as a renewal process, we give another characterizations of the Poisson process from some relations of expectation, variance, covariance, residual life of the k thinned processes. Secondly, we consider that at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the i-th new atom assigned to the process D_i , i =1,¡K, k. We also have characterizations of the Poisson process from some relations of expectation, variance of the process D_i , i =1,¡K, k.

This dissertation includes two topics. For the first topic about the housing price jump risk, we use EM gradient algorithms to estimate parameters of the jump diffusion model and test whether the US monthly housing price have jump risk during 1986 to 2006. Then, in order to obtain a viable pricing framework of mortgage insurance contracts, this paper uses the jump diffusion processes of Merton (1976) to model the dynamic process of housing price. Using this model, we investigate the impact of price jump risk on the valuation of mortgage insurance premium from jump intensity, abnormal volatility of jump size and normal volatility. Empirical results indicate that the abnormal volatility of jump size has the most significant impact on the mortgage insurance premium. For the second topic about the catastrophe risk, we investigate that, for catastrophic events, the assumption that catastrophe claims occur in terms of the Poisson process seems inadequate as it has constant intensity. We propose Markov Modulated Poisson process to model the arrival process for catastrophic events. Under this process, the underlying state is governed by a homogenous Markov chain, and it is the generalization of Cummins and Geman (1993, 1995), Chang, Chang, and Yu (1996), Geman and Yor (1997) and Vaugirard (2003a, 2003b). We apply Markov jump diffusion model to derive pricing formulas for catastrophe insurance products, included catastrophe futures call option, catastrophe PCS call spread and catastrophe bond. We use the data of PCS index and the annual number of hurricane events during 1950 to 2004 to test the quality of the fitting under the Markov Modulated Poisson process and the Poisson process. We reach the conclusion that the Markov Modulated Poisson process is fitter than the Poisson process and Weiner process in modeling the arrival rate of hurricane events when pricing three insurance products. Hence, if different status of climate environment has significant different arrival intensity in real economy, using jump diffusion model to evaluate CAT insurance products could cause significant mispricing.

Recurrent data are widely encountered in many applications. This thesis work focuses on how the recurrent hospital admissions relate to the air pollutants. In particular, we consider the data for two major cities in Saskatchewan. The study period ranges from January 1, 2005 to December 30, 2011 and involves 20,284 patients aged 40 years and older. The hospital admission data is from the Canadian Institute for Health Information (CIHI). The air pollutants data is from the National Air Pollution Surveillance Program (NAPS) from Environment Canada. The data set has been approved by the Biomedical Research Ethics Board, University of Saskatchewan. The gaseous pollutants included in this study are carbon monoxide (CO), nitrogen dioxide (NO2), sulfur dioxide (SO2), ozone (O3), as well as particulate matter PM2:5 (tiny particles in the air that are 2:5 microns in width). In the data analysis, we applied three different existing models to all respiratory diseases and asthma, respectively. The three models are the Poisson process model (also called Andersen-Gill model), the Poisson process model with the number of previous events as a covariate and the Poisson process model with shared gamma distributed frailties (random effects). For all respiratory diseases, the Poisson process model with random effects provides the best t in comparison to the other two models. The model output suggests that the increased risk of hospital readmission is significantly associated with increased CO and O3. For asthma, the Poisson process model provides the best t in comparison to the other two models. We found that only CO and O3 have significant effects on recurrent hospital admissions due to asthma. We concluded this thesis with the discussion on the current and potential future work.

<p>We explore the posterior inference available for Bayesian spatial point process models. In the literature, discussion of such models is usually focused on model fitting and rejecting complete spatial randomness, with model diagnostics and posterior inference often left as an afterthought. Posterior predictive point patterns are shown to be useful in performing model diagnostics and model selection, as well as providing a wide array of posterior model summaries. We prescribe Bayesian residuals and methods for cross-validation and model selection for Poisson processes, log-Gaussian Cox processes, Gibbs processes, and cluster processes. These novel approaches are demonstrated using existing datasets and simulation studies.</p> / Dissertation

This thesis covers non-homogeneous Poisson processes along with estimation of the intensity (rate) function and some selected simulation methods. In Chapter 1 the main properties of a non-homogeneous Poisson process are summarized. The main focus of Chapter 2 is the general maximum likelihood estimation procedure adjusted to a non-homogeneous Poisson process, together with some recommen- dations about calculation of the initial estimates of the intensity function param- eters. In Chapter 3 some general simulation methods as well as the methods designed specially for log linear and log quadratic rate functions are discussed. Chapter 4 contains the application of the described estimation and simulation methods on real data from non-life insurance. Furthermore, the considered sim- ulation methods are compared with respect to their time efficiency and accuracy of the simulations. 1

The thesis is concerned with capacity and coverage enhancement of OFDMA heterogeneous cellular systems with a specific focus on fractional frequency reuse (FFR), femtocells and amplify-and-forward (AF) relay systems. The main aim of the thesis is to develop new mathematical analysis for the spectral efficiency and outage probability of multi-cells multi-tier systems in diverse traffic, interference and fading scenarios. In the first part of the thesis, a new unified mathematical framework for performance analysis of FFR and soft frequency reuse (SFR) schemes is developed. This leads to new exact expressions of FFR and SFR area spectral efficiency in downlink and uplink scenarios which account for a mixture of frequency reuse factors in a homogeneous cellular system. The mathematical framework is extended to include modelling and performance analysis of FFR systems with elastic data traffic. Further analysis is carried out in relation to the performance of FFR and/or SFR schemes, in terms of energy efficiency and base station cooperation. The new proposed analytical framework can lead to a better understanding and computationally efficient performance analysis of next generation heterogeneous cellular systems. Next generation cellular systems are characterized by an increase in the spatial node density to improve the spectral efficiency and coverage, especially for users at home and at the cell edges. In this regard, relays and femtocells play a major role. Therefore, relays and femocells are the focus of the second part of the thesis. Firstly, we present a new and unified spectral efficiency analysis in dual-hop fixed-gain AF relay systems over generalised interferences models. The generalised interference models are either based on the Nakagami-m fading with arbitrary distance or on spatial Poisson Point Process in case of randomly deployed heterogeneous interferers. The models have been considered separately in the open literature due to the complexity of the mathematical analysis. Secondly, the outage probability is utilised to deduce the femtocell exclusion region for FFR system and a new static resource allocation scheme is proposed for femtocells which improve the capacity. The work presented in the thesis has resulted in the publication of seven scientific papers in prestigious IEEE journals and conferences.

Le but de cette thèse est de développer la théorie de Freidlin-Wentzell pour des modèles des épidémies, afin de prédire le temps mis par les perturbations aléatoires pour éteindre une situation endémique "stable". Tout d'abord nous proposons une nouvelle démonstration plus courte par rapport à celle établit récemment (sous une hypothèse un peu différente, mais satisfaite dans tous les exemples de modèles de maladie infectieuses que nous avons à l'esprit) par Kratz et Pardoux (2017) sur le principe de grandes déviations pour les modèles des épidémies. Ensuite nous établissons un principe de grandes déviations pour des EDS poissoniennes réfléchies au bord d'un ouvert suffisamment régulier. Nous établissons aussi un résultat concernant la zone du bord la plus probable par laquelle le processus solution de l'EDS de Poisson va sortir du domaine d'attraction d'un équilibre stable de sa loi des grands nombres limite. Nous terminons cette thèse par la présentation des méthodes "non standard aux différences finis", appropriés pour approcher numériquement les solutions de nos EDOs ainsi que par la résolution d'un problème de contrôle optimal qui permet d'avoir une bonne approximation du temps d'extinction d'un processus d'infection. / In this thesis, we develop the Freidlin-Wentzell theory for the "natural'' Poissonian random perturbations of the above ODE in Epidemic Dynamics (and similarly for models in Ecology or Population Dynamics), in order to predict the time taken by random perturbations to extinguish a "stable" endemic situation. We start by a shorter proof of a recent result of Kratz and Pardoux (under a somewhat different hypothesis which is satisfied in all the cases we have examined so far), which establishes the large deviations principle for epidemic models. Next, we establish the large deviations principle for reflected Poisonian SDE at the boundary of a sufficiently regular open set. Then, we establish the result for the most likely boundary area by which the process will exit the domain of attraction of a stable equilibrium of an ODE. We conclude this thesis with the presentation of the "non - standard finite difference" methods, suitable to approach numerically the solutions of our ODEs as well as the resolution of an optimal control problem which allows to have a good approximation of the time of extinction of an endemic situation.

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<p>This dissertation focuses on the simulation efficiency of the Markov process for two scenarios: Stochastic differential equations(SDEs) and simulated weather data. </p> <p><br></p> <p>For SDEs, we propose a novel Gibbs sampling algorithm that allows sampling from a particular class of SDEs without any discretization error and shows the proposed algorithm improves the sampling efficiency by orders of magnitude against the existing popular algorithms.  </p> <p><br></p> <p>In the weather data simulation study, we investigate how representative the simulated data are for three popular stochastic weather generators. Our results suggest the need for more than a single realization when generating weather data to obtain suitable representations of climate. </p>