# Department of Mathematics and Statistics Graduate Theses

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Item Generative Adversarial Networks Based on a General Parameterized Family of Generator Loss FunctionsVeiner, Justin; Mathematics and Statistics; Alajaji, Fady; Gharesifard, BahmanThis thesis introduces a unifying parameterized generator loss function for generative adversarial networks (GANs). We establish an equilibrium theorem for our resulting GAN system under a canonical discriminator in terms of the so-called Jensen-$f$-divergence, a natural generalization of the Jensen-Shannon divergence to the $f$-divergence. We also show that our result recovers as special cases several GANs from the literature, including the original GAN, least square GAN (LSGAN), $\alpha$-GAN and others. Finally, we systematically conduct experiments on three image datasets for different manifestations of our GAN system to illustrate their performance and stability.Item F-tests for Frequency Modulated Time Series Using Multitaper MethodsOtt, Ben; Mathematics and Statistics; Takahara, Glen; Burr, WesleyWe propose three new semiparametric multitaper tests for the detection of modulated line components where the modulation is assumed to be created by a polynomial of degree P. The first test, F_4, is a modification of the F-test found in [3] and uses a set of weights to obtain a better in-band concentration for a wider range of tapers. The second test, F_4', builds on the aforementioned test by fixing a problem that arises when choosing to evaluate the F-test with an even number of tapers. We derive the distributions for these two tests along with an examination of the final test based on an aggregation of F_4' tests, where the aggregation is over different multitaper orders. We derive an upper bound for the distribution of the Aggregate F-test statistic, and via simulation, we approximate the F-test's confidence level needed to obtain a pre-specified Type 1 error of the Aggregated test. These new F-tests are compared to the F-test in [3], and [30] via simulation comparing the probability of detecting a frequency-modulated signal in different types of noise and modulation widths. We also examine the functionality of these new test statistics through a study of the SoHO GOLF instrument data.Item Reinforcement and Preferential Attachment Models via Pólya UrnsSingh, Somya; Mathematics and Statistics; Alajaji, Fady; Gharesifard, BahmanIn this thesis, we devise two different types of discrete-time stochastic models using modified Pólya urn schemes. The first set of models concerns interacting contagion networks constructed using two-colour (red and black) finite memory Pólya urns in which reinforcing balls are removed M time steps after being added (where M is the “memory” of the urn). The urns interact in the sense that the probability of drawing a red ball (which represents an infectious state or an opinion) for a given urn, not only depends on the ratio of red balls in that urn but also on the ratio of red balls in other urns in a network representing the interconnections, hence accounting for the effect of spatial contagion. The finite memory reinforcement provides a diminishing effect of past draws which represents curing of an infection in an epidemic spread model, or lessening influence of a popular opinion in a social network. We examine the stochastic properties of the underlying Markov draw process and construct a class of dynamical systems to approximate the asymptotic marginal distributions. We also design a consensus achieving connected network of agents via two-color finite memory Pólya urns. The interaction between urns is time-varying and is represented via “super-urns” which combine for each node its own urn with its neighbouring urns. We obtain the consensus value in terms of the network’s reinforcement parameters and memory. In the second part of this thesis, we introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Unlike the Barabási-Albert model, the color-coded vertices in conjunction with the time-varying reinforcing parameter in our model allows for the vertices added (born) later in the process to potentially attain a high degree in a way that is not captured by the former. We study the degree count of the vertices in the graphs generated via our model by analyzing the draw vectors of the underlying stochastic process. Furthermore, we compare our model with the Barabási-Albert model via simulationsItem Advances in Inference for Minimization and Other Covariate-Adaptive Randomization MethodsZixuan, Zhao; Mathemetics and Statistics; Song Yanglei; Jiang WenyuCovariate-adaptive randomization (CAR) is widely used in randomized controlled experiments and can balance treatment allocation over important prognostic covariates. Despite its popularity, data collected under CAR designs are correlated, which makes subsequent statistical inference complicated. In this thesis, we address several issues concerning inference under CAR designs. Firstly, valid inference with data collected under covariate-adaptive randomization in many cases requires the knowledge of the limiting covariance matrix of within-stratum imbalances. While the limit is explicitly known for most CAR methods, this is not the case for Pocock and Simon’s minimization method, under which the existence of the limit is only recently established. This limit can be estimated by Monte Carlo methods if the distribution of stratification factors is known. However, this assumption may not hold in practice, resulting in invalid estimation methods. In the first work, we replace the usually unknown distribution with an estimator, such as the empirical distribution, in the Monte Carlo approach and establish consistency of the resulting covariance estimator. As an application, we consider adjustments to existing robust tests for treatment effects with survival data by the proposed covariances estimator in simulation studies. The result shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the minimization method. Secondly, we consider the inference of average treatment effect, a critical estimand in practice, under CARs. Several valid inferential procedures have been proposed with different working models for covariates adjustment. We focus on the approach that uses advanced machine learning methods to reduce estimators’ asymptotic variance. In the literature, a critical condition on the in-sample prediction error of these fitting methods is required to control a potential bias. We show, via a numerical example, that the condition fails to hold, and the estimation could be biased without sample splitting and cross-fitting. Further, we propose a cross-fitting procedure that fulfills this condition, and thus achieves asymptotically valid estimation. We corroborate the theory with several simulation studies.Item Higher Order Moments and Free Cumulants of Complex Wigner MatricesMunoz George, Daniel; Mathematics and Statistics; Mingo, James AThe main object of this thesis is a random matrix model known as the Wigner model. We investigate the higher order moments and free cumulants of a complex Wigner matrix. It is well known that the first order moments and free cumulants of this model are described by the Catalan numbers and the semicircle law, this constitutes part of the work done by Wigner in the 1950s. Later on, in 2020, Male, Mingo, Péché and Speicher answered this question for the second order case, these are known as the fluctuation moments. In this thesis, we provide a formula for the third order moments in terms of the set of partitioned permutations. Furthermore, we also present a simple expression for the third order free cumulants. In Chapter 6 we talk about the progress that has been made for any higher order case. We prove that the higher order moments and free cumulants exist as long as we ask for suitable conditions.