Finding the distribution of a statistic is always an important problem that we face in statistical inference. Methods that are usually used for solving this problem are change of variable technique, distribution function technique and moment generating function technique. Among these methods change of variable technique is the most commonly used one. This method is simple when the statistic is a one-to-one transformation of the sample observations and if it is many-to-one, then one needs to compute the jacobian for each partition of the range for which the transformation is one-to-one. In addition, if we want to find the distribution of a statistic involving n random variables using the change of variable technique, we have to define (n-1) auxiliary variables. Unless these (n-1) auxiliary variables are carefully chosen, calculation of jacobian as well as finding the range of integration to obtain the marginal distribution of the statistic of interest become complicated. [See [3]]Au, Chi and Tam, Judy [1] proposed an alternative method of finding the distribution of a statistic by using Dirac generalized function. In this study we considera number of problems involving different probability distributions that are not quiet easy to solve by change of variable technique. We will illustrate the method by solving problems which include finding the distributions of sums, products, differences and ratios of random variables. The main purpose of the thesis is to show that using Dirac generalized function one can solve these problems with more ease. This alternative approach would be more suitable for students with limited mathematical background. / Department of Mathematical Sciences

This dissertation is divided into two parts. The first part addresses the following problem: Suppose 𝑣 is a finitely additive probability measure defined on the power set 𝒜 of the integer Z so that each singleton set gets measure zero. Let X be a product space Π/β∈B * Zᵦ where each Zₐ is a copy of the integers. Let 𝒜ᴮ be the algebra of subsets of X generated by the subproducts Π/β∈B * Cᵦ where for all but finitely many β, Cᵦ = Zᵦ. Let 𝑣_B denote the product measure on 𝒜ᴮ which has each factor measure a copy of 𝑣. A subset E of X is said to be 𝑣_B -measurable iff [sic] there is only one finitely additive probability on the algebra generated by 𝒜ᴮ ∪ [E] which extends 𝑣_B. The set E ⊆ X is said to be universally product measurable (u.p.m.) iff [sic] for each finitely additive probability measure μ on 𝒜 which gives each singleton measure zero,E is μ_B -measurable. Two theorems are proved along with generalizations. The second part of this dissertation gives a proof of the following theorem and some generalizations: There are 2ᶜ nonisomorphic subalgebras of the power set algebra of the integers (where c = power of the continuum).

The present thesis is devoted to the construction of a probability measure which counts the pairs of closed commuting subgroups in infinite groups. This measure turns out to be an extension of what was known in the finite case as subgroup commutativity degree and opens a new approach of study for the class of near abelian groups, recently introduced in [24, 27]. The extremal case of probability one characterises the topologically quasihamiltonian groups, studied originally by K. Iwasawa [30, 31] in the abstract case and then by F. K¨ummich [35, 36, 37], C. Scheiderer [45, 46], P. Diaconis [11] and S. Strunkov [48] in the topological case. Our probability measure turns out to be a useful tool in describing the distance of a profinite group from being topologically quasihamiltonian. We have been inspired by an idea of H. Heyer in the present context of investigation and in fact we generalise some of his techniques, in order to construct a probability measure on the space of closed subgroups of a profinite group. This has been possible because the space of closed subgroups of a profinite group may be approximated by finite spaces and the consequence is that our probability measure may be approximated by finite probability measures. While we have a satisfactory description for profinite groups and compact groups, the case of locally compact groups remains open in its generality.

In this thesis we study two problems concerning probability. The first is stochastic control problem, which essentially amounts to find an optimal probability in order to optimize some reward function of probability. The second is to approximate the solution of the Boltzmann equation. Thanks to conservation of mass, the solution can be regarded as a family of probability indexed by time. In the first part, we prove a dynamic programming principle for stochastic optimal control problem with expectation constraint by measurable selection approach. Since state constraint, drawdown constraint, target constraint, quantile hedging and floor constraint can all be reformulated into expectation constraint, we apply our results to prove the corresponding dynamic programming principles for these five classes of stochastic control problems in a continuous but non-Markovian setting. In order to solve the Boltzmann equation numerically, in the second part, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cutoff and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step, we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cutoff approximation method, our method results in higher order of accuracy.

We explore the connection between quantum error correction and quantum cryptography through the notion of conjugate (or complementary) channels. This connection is at the level of subspaces and operator subsystems; if we use a more general form of subsystem, the link between the two topics breaks down. We explore both the subspace and subsystem settings. Error correction arises as a means of addressing the issue of the introduction of noise to a message being sent from one party to another. Noise also plays a role in quantum measurement theory: If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Such a setup amounts to the introduction of noise to the measurement process, yet has the advantage of preserving the measurement statistics. Preprocessing by a quantum channel leads to the partial order "cleaner than" on quantum probability measures. Other meaningful partial orders on quantum probability measures exist, and we shall investigate that of cleanness as well as that of absolute continuity. Lastly, we investigate partial orders on vectors corresponding to quantum states; such partial orders, namely majorization and trumping, have been linked to entanglement theory. We characterize trumping first by means of yet another partial order, power majorization, which gives rise to a family of examples. We then characterize trumping through the complete monotonicity of certain Dirichlet polynomials corresponding to the states in question. This not only generalizes a recent characterization of trumping, but the use of such mathematical objects simpli es the derivation of the result. / The Natural Sciences and Engineering Research Council of Canada (NSERC)

La thèse est divisée en deux parties portant sur deux aspects relativement différents des approches bayésiennes non-paramétriques. Dans la première partie, nous nous intéressons aux propriétés fréquentistes (asymptotiques) de lois a posteriori pour des paramètres appartenant à l'ensemble des suites réelles de carré sommable. Dans la deuxième partie, nous nous intéressons à des approches non-paramétriques modélisant des données d'espèces et leur diversité en fonction de certaines variables explicatives, à partir de modèles qui utilisent des mesures de probabilité aléatoires. / This thesis is divided in two parts on rather different aspects of Bayesian statistics. In the first part, we deal with frequentist (asymptotic) properties of posterior distributions for parameters which belong to the space of real square sommable sequences. In the second part, we deal with nonparametric approaches modelling species data and the diversity of these data with respect to covariates. To that purpose, we use models based on random probability measures.

The aim of this thesis was to investigate the use of effect sizes to report the results of statistical credit rating models in a more practical way. Rating systems in the form of statistical probability models like logistic regression models are used to forecast the behaviour of clients and guide business in rating clients as “high” or “low” risk borrowers. Therefore, model results were reported in terms of statistical significance as well as business language (practical significance), which business experts can understand and interpret. In this thesis, statistical results were expressed as effect sizes like Cohen‟s d that puts the results into standardised and measurable units, which can be reported practically. These effect sizes indicated strength of correlations between variables, contribution of variables to the odds of defaulting, the overall goodness-of-fit of the models and the models‟ discriminating ability between high and low risk customers. / Statistics / M. Sc. (Statistics)

The aim of this thesis was to investigate the use of effect sizes to report the results of statistical credit rating models in a more practical way. Rating systems in the form of statistical probability models like logistic regression models are used to forecast the behaviour of clients and guide business in rating clients as “high” or “low” risk borrowers. Therefore, model results were reported in terms of statistical significance as well as business language (practical significance), which business experts can understand and interpret. In this thesis, statistical results were expressed as effect sizes like Cohen‟s d that puts the results into standardised and measurable units, which can be reported practically. These effect sizes indicated strength of correlations between variables, contribution of variables to the odds of defaulting, the overall goodness-of-fit of the models and the models‟ discriminating ability between high and low risk customers. / Statistics / M. Sc. (Statistics)

Cette thèse comporte deux parties indépendantes. Dans la première partie, nous nous intéressons à l’analyse asymptotique de quelques statistiques sur les arbres binaires de recherche (ABR). Dans la deuxième partie, nous nous intéressons à l’étude du problème du collectionneur de coupons impatient. Dans la première partie, en suivant le modèle introduit par Aguech, Lasmar et Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], on définit la profondeur pondérée d’un nœud dans un arbre binaire enraciné étiqueté comme la somme de toutes les clés sur le chemin qui relie ce nœud à la racine. Nous analysons alors dans ABR, les profondeurs pondérées des nœuds avec des clés données, le dernier nœud inséré, les nœuds ordonnés selon le processus de recherche en profondeur, la profondeur pondérée des trajets, l’indice de Wiener pondéré et les profondeurs pondérées des nœuds avec au plus un enfant. Dans la deuxième partie, nous étudions la forme asymptotique de la courbe de la complétion de la collection conditionnée à T_n≤ (1+Λ), Λ>0, où T_n≃n ln⁡n désigne le temps nécessaire pour compléter la collection. Puis, en tant qu’application, nous étudions les automates déterministes et accessibles et nous fournissons une nouvelle dérivation d’une formule due à Korsunov [Kor78, Kor86] / This Phd thesis is divided into two independent parts. In the first part, we provide an asymptotic analysis of some statistics on the binary search tree. In the second part, we study the coupon collector problem with a constraint. In the first part, following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyze the following statistics : the weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search procees, the weighted path length, the weighted Wiener index and the weighted depths of nodes with at most one child in a random binary search tree. In the second part, we study the asymptotic shape of the completion curve of the collection conditioned to T_n≤ (1+Λ), Λ>0, where T_n≃n ln⁡n is the time needed to complete accessible automata, we provide a new derivation of a formula due to Korsunov [Kor78, Kor86]

Indiana University-Purdue University Indianapolis (IUPUI) / Epidemiologic and clinical studies routinely collect longitudinal measures of multiple outcomes. These longitudinal outcomes can be used to establish the temporal order of relevant biological processes and their association with the onset of clinical symptoms. In the first part of this thesis, we proposed to use bivariate change point models for two longitudinal outcomes with a focus on estimating the correlation between the two change points. We adopted a Bayesian approach for parameter estimation and inference. In the second part, we considered the situation when time-to-event outcome is also collected along with multiple longitudinal biomarkers measured until the occurrence of the event or censoring. Joint models for longitudinal and time-to-event data can be used to estimate the association between the characteristics of the longitudinal measures over time and survival time. We developed a maximum-likelihood method to joint model multiple longitudinal biomarkers and a time-to-event outcome. In addition, we focused on predicting conditional survival probabilities and evaluating the predictive accuracy of multiple longitudinal biomarkers in the joint modeling framework. We assessed the performance of the proposed methods in simulation studies and applied the new methods to data sets from two cohort studies. / National Institutes of Health (NIH) Grants R01 AG019181, R24 MH080827, P30 AG10133, R01 AG09956.