Department of Mathematics and Statistics Graduate Theseshttp://hdl.handle.net/1974/7582017-06-29T10:44:32Z2017-06-29T10:44:32ZNon-commutative Independences for Pairs of FacesGu, Yinzhenghttp://hdl.handle.net/1974/158982017-06-14T07:14:35ZNon-commutative Independences for Pairs of Faces
Gu, Yinzheng
In non-commutative probability theories, many different notions of independence arise in various contexts. According to the early classification work, there are only five such notions that are universal/natural in the sense of Speicher and Muraki, with free independence playing a prominent role.
This thesis follows Voiculescu's recent generalization of free independence to bi-free independence to allow the simultanous study of left and right non-commutative random variables. We consider similar generalizations of other types of independence in the literature and show that many of the important concepts such as cumulants, convolutions, transforms, limit theorems, and infinite divisibility have counterparts in the new setting. While many of the results in this thesis were to be expected, some peculiarities and difficulties do occur in this left-right framework as things become much more complicated.
Information and Estimation Theoretic Approaches to Data PrivacyAsoodeh, Shahabhttp://hdl.handle.net/1974/158722017-05-29T15:12:04ZInformation and Estimation Theoretic Approaches to Data Privacy
Asoodeh, Shahab
Warner [145] in 1960s proposed a simple mechanism, now referred to as the randomized response model, as a remedy for what he termed “evasive answer bias” in survey sampling. The randomized response setting is as follows: $n$ people participate in a survey and a statistician asks each individual a sensitive yes-no question and seeks to find the ratio of "yes" responses. For privacy purposes, individuals are given a biased coin that comes up heads with probability $a\in(0,\frac{1}{2})$. Each individual flips the coin in private. If it comes up heads, they lie and if it comes up tails, they tell the truth. Warner derived a maximum likelihood unbiased estimator for the true ratio of "yes" based on the reported responses. Thus the parameter of interest is estimated accurately while preserving the privacy of each user and avoiding survey answer bias.
In this thesis, we generalize Warner's randomized response model in several directions:
(i) we assume that the response of each individual consists of private and non-private data and the goal is to generate a response which carries as much "information" about the non-private data as possible while limiting the "information leakage" about the private data, (ii) we propose mathematically well founded metrics to quantify the tradeoff between how much the response leaks about the private data and how much information it conveys about the non-private data, (iii) we make no assumptions on the alphabets of the private and non-private data, and (iv) we design optimal response mechanisms which achieve the fundamental tradeoffs.
Unlike the large body of recent research on privacy which studied the problem of reducing disclosure risk, in this thesis we formulate and study the tradeoff between utility (e.g., statistical efficiency) and privacy (e.g., information leakage). Our approach (which is two-fold: information-theoretic and estimation-theoretic) and results shed light on the fundamental limits of the utility-privacy tradeoff.
Statistical Methods For Biomarker Threshold Models in Clinical TrialsGavanji, Parisahttp://hdl.handle.net/1974/153452017-03-30T14:26:56ZStatistical Methods For Biomarker Threshold Models in Clinical Trials
Gavanji, Parisa
In clinical trials, the main objective is to investigate the treatment effects on patients. However, many molecularly targeted drugs or treatments tend to benefit a subset of patients more, identified by a certain biomarker. The cut-point value defining patient subsets is often unknown. For this situation, the ordinary likelihood ratio test cannot be applied for testing treatment-biomarker interaction because of the model irregularities.
We develop a residual bootstrap method to approximate the distribution of a proposed test statistic to test for treatment-biomarker interaction in survival data. Simulation studies show that the residual bootstrap test works well. The proposed method is applied to BIG 1-98 randomized clinical trial of breast cancer with Ki-67 as biomarker to consider the treatment effects on patients in two subsets. We also extend the residual bootstrap method to clustered survival data with an application to data from the I-SPY 1 clinical trial with the estrogen receptor total score as a biomarker.
Another research topic of the thesis is deriving the asymptotic distribution of a penalized likelihood ratio test statistic for testing biomarker effect and treatment-biomarker interaction in binary data. The model can be viewed as a mixture of logistic regression models with unknown cut-point for which the regularity conditions of ordinary likelihood methods are not satisfied. We first approximate the indicator function defining biomarker subgroups by a smooth continuous function. To overcome irregularities, we develop a penalized likelihood method, introducing a new idea of using random penalty term. Proposing a new set of regularity conditions helps us to study the properties and limiting distributions of the maximum penalized likelihood estimates of the parameters. We further prove that the penalized likelihood ratio test statistic has an asymptotic $\chi^{2}_{3}$ distribution under the null hypothesis. Extensive simulation studies show that the proposed test procedure works well for hypothesis testing. The proposed method is applied to a clinical trial of prostate cancer with the serum pro-static acid phosphatase (AP) as a biomarker.
An impulsive differential equation model for Marek's diseaseRozins, Carlyhttp://hdl.handle.net/1974/149442017-03-30T14:26:45Z2016-09-22T00:00:00ZAn impulsive differential equation model for Marek's disease
Rozins, Carly
Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are referred to as impulsive differential equations, systems of differential equations coupled with discrete mappings in state space. In this thesis we employ impulsive differential equations to model disease transmission within an industrial livestock barn.
In particular we focus on the poultry industry and a viral disease of poultry called Marek's disease. This system lends itself well to impulsive differential equations. Entire cohorts of poultry are introduced and removed from a barn concurrently. Additionally, Marek's disease is transmitted indirectly and the viral particles can survive outside the host for weeks. Therefore, depopulating, cleaning, and restocking of the barn are integral factors in modelling disease transmission and can be completely captured by the impulsive component of the model.
Our model allows us to investigate how modern broiler farm practices can make disease elimination difficult or impossible to achieve. It also enables us to investigate factors that may contribute to virulence evolution.
Our model suggests that by decrease the cohort duration or by decreasing the flock density, Marek's disease can be eliminated from a barn with no increase in cleaning effort. Unfortunately our model also suggests that these practices will lead to disease evolution towards greater virulence. Additionally, our model suggests that if intensive cleaning between cohorts does not rid the barn of disease, it may drive evolution and cause the disease to become more virulent.
Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-09-22 10:30:23.342
2016-09-22T00:00:00Z