M.Sc. Mathematics and Statistics
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Item Open Access On the Extendibility of a D(4)-Pair of Pell Numbers(Brock University) Emanuel, David; Department of MathematicsA Diophantine m-tuple with property D(ℓ) is a set of m integers such that the product of any two integers plus ℓ results in a perfect square. This thesis establishes that a particular family of D(4) pairs of Pell numbers can be extended to a D(4) triple by exactly one Pell number. A similar result has been found for the Diophantine triples of Fibonacci numbers, a discussion of which is included in the first chapter of this thesis. This chapter finishes with a statement of the main result of my thesis, and the subsequent chapters discuss several topics in number theory which were used to prove the main result in chapter 5. Specifically, results about continued fractions, Pell-type equations, and linear forms in logarithms were used. These topics are the subjects of chapters 2, 3 and 4, which contain some history and discussions of the important results. The conclusion of this thesis discusses some possible generalizations.Item Open Access Distributed Supervised Statistical Learning(Brock University) khalili Mahmoudabadi, Amir; Department of MathematicsWe live in the era of big data, nowadays, many companies face data of massive size that, in most cases, cannot be stored and processed on a single computer. Often such data has to be distributed over multiple computers which then makes the storage, pre-processing, and data analysis possible in practice. In the age of big data, distributed learning has gained popularity as a method to manage enormous datasets. In this thesis, we focus on distributed supervised statistical learning where sparse linear regression analysis is performed in a distributed framework. These methods are frequently applied in a variety of disciplines tackling large scale datasets analysis, including engineering, economics, and finance. In distributed learning, one key question is, for example, how to efficiently aggregate multiple estimators that are obtained based on data subsets stored on multiple computers. We investigate recent studies on distributed statistical inferences. There have been many efforts to propose efficient ways of aggregating local estimates, most popular methods are discussed in this thesis. Recently, an important question about the number of machines to deploy is addressed for several estimation methods, notable answers to the question are reviewed in this literature. We have considered a specific class of Liu-type shrinkage estimation methods for distributed statistical inference. We also conduct a Monte Carlo simulation study to assess performance of the Liu-type shrinkage estimation methods in a distributed framework.Item Open Access Convergence Analysis of Heterogeneous Decision-making Populations Under the Coordinating Best-response and Imitation Update Rules(Brock University) Hasheminejad, Nazanin Jr; Department of MathematicsThis thesis emphasis is on coordination games. In a coordination game, selecting the same strategy or decision as the opponent is mutually beneficial for both parties. We studied the problem of equilibrium convergence in such games in both discrete and continuous (time) cases. In the first Chapter, we provide a brief introduction to the field of game theory. We discuss different categories of agents based on their levels of rationality and decision-making strategies, along with a variety of games. Additionally, we address important issues and challenges within this field. The second Chapter of this work is dedicated to a heterogeneous mixed population of imitators and best-responders. In this model, agents’ update rules are assumed to be discrete functions of time. Imitators refer to agents who simply replicate the strategy of another agent with the highest payoff, while best-responders pick the strategies that maximise their individual outcomes. Suggesting the concept of ’sections’--a consecutive sequence of agents with similar strategies– helped us in establishing convergence to an equilibrium state. This convergence was demonstrated under any arbitrary asynchronous activation sequence within a linear network. The proof was then extended to networks with ring, starike, and sparse-tree structures. However, the question of equilibrium convergence for other network structures remains an open challenge. In the third Chapter, we examined a large well-mixed population of imitators within a coordination context. Our analysis is grounded in the assumption that imitation here is driven by dissatisfaction. Equivalently, agents with lower payoffs are more dissatisfied and have more tendency to change and imitate higher earners within the population. The analysis reveals the presence of three fixed points, of which two are stable and one is a saddle point. The stable manifold of the unstable fixed point is also calculated. Additionally, It is demonstrated that starting from any initial state, the population eventually converges towards one of these introduced fixed points.Item Open Access Convergence Analysis of Heterogeneous Decision-making Populations Under the Coordinating Best-response and Imitation Update Rules(Brock University) Hasheminejad, Nazanin Jr; Department of MathematicsThis thesis emphasis is on coordination games. In a coordination game, selecting the same strategy or decision as the opponent is mutually beneficial for both parties. We studied the problem of equilibrium convergence in such games in both discrete and continuous (time) cases. In the first Chapter, we provide a brief introduction to the field of game theory. We discuss different categories of agents based on their levels of rationality and decision-making strategies, along with a variety of games. Additionally, we address important issues and challenges within this field. The second Chapter of this work is dedicated to a heterogeneous mixed population of imitators and best-responders. In this model, agents’ update rules are assumed to be discrete functions of time. Imitators refer to agents who simply replicate the strategy of another agent with the highest payoff, while best-responders pick the strategies that maximise their individual outcomes. Suggesting the concept of ’sections’--a consecutive sequence of agents with similar strategies– helped us in establishing convergence to an equilibrium state. This convergence was demonstrated under any arbitrary asynchronous activation sequence within a linear network. The proof was then extended to networks with ring, starike, and sparse-tree structures. However, the question of equilibrium convergence for other network structures remains an open challenge. In the third Chapter, we examined a large well-mixed population of imitators within a coordination context. Our analysis is grounded in the assumption that imitation here is driven by dissatisfaction. Equivalently, agents with lower payoffs are more dissatisfied and have more tendency to change and imitate higher earners within the population. The analysis reveals the presence of three fixed points, of which two are stable and one is a saddle point. The stable manifold of the unstable fixed point is also calculated. Additionally, It is demonstrated that starting from any initial state, the population eventually converges towards one of these introduced fixed points.Item Open Access AdaBoost And Its Variants: Boosting Methods For Classification With Small Sample Size And Brain Activity In Schizophrenia(Brock University) Perry, Brittany; Department of MathematicsAdaBoost is an ensemble method that can be used to boost the performance of machine learning algorithms by combining several weak learners to create a single strong learner. The most popular weak learner is a decision stump (low depth decision tree). One limitation of AdaBoost is its effectiveness when working with small sample sizes. This work explores variants to the AdaBoost algorithm such as Real AdaBoost, Logit Boost, and Gentle AdaBoost. These variants all follow a gradient boosting procedure like AdaBoost, with modifications to the weak learners and weights used. We are specifically interested in the accuracy of these boosting algorithms when used with small sample sizes. As an application, we study the link between functional network connectivity (as measured by EEG recordings) and Schizophrenia by testing whether the proposed methods can classify a participant as Schizophrenic or healthy control based on quantities measured from their EEG recording.Item Open Access Fitting AdaBoost Models From Imbalanced Data with Applications in College Basketball(Brock University) Romaniuk, Raymond; Department of MathematicsData imbalance is an important consideration when working with real world data. Over/undersampling approaches allow us to gather more insight from the limited data we have on the minority class; however, there are many proposed methods. The goal of our study is to identify the optimal approach for over/undersampling to use with Adaptive Boosting (AdaBoost). Based on a simulation study, we’ve found that combining AdaBoost with various sampling techniques provides an increased weighted accuracy across classes for progressively larger data imbalances. The three Synthetic Minority Oversampling Technique’s (SMOTE) and Jittering with Over/Undersampling (JOUS) performed the best, with the JOUS approach being the most accurate for all levels of data imbalance in the simulation study. We then applied the most effective over/undersampling methods to predict upsets (games where the lower seeded team wins) in the March Madness College Basketball Tournament.Item Open Access Enhancing Lexical Sentiment Analysis using LASSO Style Regularization(Brock University) Blanchard, Jeremy; Department of MathematicsIn the current information age where expressing one’s opinions online requires but a few button presses, there is great interest in analyzing and predicting such emotional expression. Sentiment analysis is described as the study of how to quantify and predict such emotional expression by applying various analytical methods. This realm of study can broadly be separated into two domains: those which quantify sentiment using sets of features determined by humans, and approaches that utilize machine learning. An issue with the later approaches being that the features which describe sentiment within text are challenging to interpret. By combining VADER which is short for Valence Aware Dictionary for sEntiment Reasoning; a lexicon model with machine learning tools (simulated annealing) and k-fold cross validation we can improve the performance of VADER within and across context. To validate this modified VADER algorithm we contribute to the literature of sentiment analysis by sharing a dataset sourced from Steam; an online video game platform. The benefits of using Steam for training purposes is that it contains several unique properties from both social media and online web retailers such as Amazon. The results obtained from applying this modified VADER algorithm indicate that parameters need to be re-trained for each dataset/context. Furthermore that using statistical learning tools to estimate these parameters improves the performance of VADER within and across context. As an addendum we provide a general overview of the current state of sentiment analysis and apply BERT a Transformer-based neural network model to the collected Steam dataset. These results were then compared to both base VADER and modified VADER.Item Open Access CEO Overconfidence and the Probability of Bankruptcy(Brock University) Amin, Ruhul; Faculty of Business ProgramsThis thesis examines the relation between CEO overconfidence and the probability of bankruptcy. In addition to the main research question, we develop two additional hypotheses. We evaluate the potential link or channel between CEO overconfidence and the probability of bankruptcy. In the relationship between CEO overconfidence and the probability of bankruptcy, we seek for any interaction effects of CEO dominance. It is not uncommon for CEOs to be overconfident about their firms' prospects. In our sample, we use data from the year 2000 to 2019 for US companies. We proxy the bankruptcy probability using Altman’s Z Score. We use a stock option-driven measure of overconfidence, and this measure assumes that non-overconfident CEO will exercise their stock options if it is in the money, while overconfident CEOs will hold stock options beyond a rational threshold. We construct both continuous and indicator-based measures of overconfidence to test the hypotheses. The empirical findings reveal that CEO overconfidence increases the probability of bankruptcy. We do not find any evidence in favor of overinvestment which we consider as a channel through which overconfidence leads to increased bankruptcy risk. We also find that dominant and overconfident CEOs are suited for innovative firms, implying that giving an overconfident CEO a dominant position can minimize a firm's probability of bankruptcy. The implications of this study are that firms should be cautious in hiring overconfident CEO and they should take measures to reduce the negative effects of CEO overconfidence like the probability of bankruptcy. One way to reduce the probability of bankruptcy in innovative firms is to appoint overconfident CEO into a dominant position.Item Open Access Implications of Non-Operating Room Anesthesia Policy for Operating Room Efficiency(Brock University) Liang, Yihang; Faculty of Business ProgramsThis thesis focuses on examining the use of Non-Operating Room Anesthesia (NORA) policy in Operating Room (OR) scheduling. A NORA policy involves a practice whereby the administration of anesthesia stage is performed outside the OR. The goal of the thesis is to determine whether NORA policy can improve OR efficiency measured by the performance of total costs, which consists of a weighted sum of patient waiting time, OR overtime and idle time. A simulation optimization method is adopted to find near-optimal schedules for elective surgeries in an outpatient setting. The results of a traditional OR scheduling model, where all stages of the surgery are performed in the OR, will be compared to the results of a NORA OR model where the initial anesthesia stage is performed outside of the OR. Two cases are considered for the NORA model given the decrease on mean durations: (1) a model with the same number of surgery appointments and shorter session length and (2) a models with the same session length and more surgery appointments. . The impact of a NORA policy on OR performance is further analyzed by considering scenarios that capture Surgery duration variability and mean surgery durations which are two traits for surgeries that have been shown to impact OR performance. This thesis aims to investigate how a NORA policy performs when standard deviations and mean surgery durations change. The results show that NORA policy can improve OR efficiency in all settings.Item Open Access A Study on Immersion and Emotions’ Influence on Impulse Buying in Virtual Environments(Brock University) Selcuk, Cem; Faculty of Business ProgramsImpulse buying has always been an interesting phenomenon that is observed in our daily lives. Statistics have shown that impulse purchases make up almost 40% of all purchases made online. Many studies have examined impulse buying, and they have found that emotions accompany impulsive behaviors naturally. With the recent development in virtual reality (VR) technology, this phenomenon is observable in online virtual environments. Retailers can create immersive virtual shops where the customer can walk among the aisles of a virtual store and make purchases. This study examines whether the effects of emotions on impulse buying vary across different immersion levels (2D vs. VR) and gender. To test our hypotheses, we collected data from the 2D and VR setting using experiments. The results provide evidence that gender plays a significant role in the three-way relationship between positive/negative emotions, immersion, and impulse buying. The unique setting of our research extends the literature on impulse buying, marketing, and virtual reality. The results offer valuable insights to marketers and retailers who want to develop virtual shops and influence impulse buying in these virtual shops.Item Open Access Examining The Influence of Social Augmented Reality Apps on Customer Relationships: The Mediating Role of Shared Social Experience(Brock University) Nguyen, Oanh; Faculty of Business ProgramsThe development of augmented reality (AR) has provided firms with increasing opportunities to improve customer experiences, especially in a shared context where customers are encouraged to communicate with others. This study investigates the effectiveness of social AR in building relationships among customers through a shared social experience, one which includes shared sense of place, social interaction, and social identity. Data was collected from 378 active users of a social AR application and was analyzed using the partial least squares structural equation modelling (PLS-SEM) and Hayes’ PROCESS Macro. Results from this study show that shared sense of place, social interaction, and social identity mediate the influence of social AR past usage on customer-to-customer relationships, which consequently enhance customers’ continuance intention to use the social AR application. Additionally, the results of the moderated mediation analysis reveal that the indirect effect of social AR past usage on continuance intention is positively moderated by extraversion, such that at higher level of extraversion the mediated relationship becomes stronger. These findings offer important contributions to the AR marketing literature and add valuable insights for practitioners to advance the use of AR technology.Item Open Access The Effect of Perceived Deception on Consumer Repurchase Intention(Brock University) Wang, Xinyue; Faculty of Business ProgramsOnline commerce changes the way products are displayed. Bounded by less chance to present information of the product, e-retailers always face misunderstandings on the consumer side, and consequently, unfavourable consumer behaviour. This makes online retailing prone to perceived deceptive practice. Past research has mainly integrated perceived deception into existing consumer behavior theories. In the same vein, this research further examines the factors moderating the relationship between perceived deception and repurchase intention. Specifically, we tested how product type (hedonic versus utilitarian), consumer regulatory focus (promotion versus prevention), and their interaction can help mitigate perceived deception's negative effect on consumer repurchase intention. This research expands the literature on perceived deception. With the prior work establishing the negative effect of perceived deception on consumer purchase behaviour, this research further investigates the factors that may attenuate the unfavourable outcome. It also helps marketers increase repurchase rates by emphasizing the hedonic attribute and instigating promotion intention to help mitigate the negative effects of perceived deception.Item Open Access How Virtual Reality Leads to Positive Responses to Persuasion Attempts: The Implications of VR Brand Placement(Brock University) Rabbani Movarekh, Ahmadreza; Faculty of Business ProgramsFacebook has started to test advertising in virtual reality, yet consumers’ responses toward this phenomenon have been neglected in the virtual reality and consumer behaviour literature. Most of the previous research has focused on VR as the primary tool for representing the service or product and not a medium for advertising purposes. Therefore, brand placement in virtual environments, as one of the most common persuasive advertising efforts by brands, is the focus of this study. More specifically, this research analyzes the effect of brand placement context (VR, 360 or 2D) and placement congruity on consumers’ persuasion knowledge and their responses towards brands, using the cognitive load theory and persuasion knowledge model to predict and explain the effect. The research model was tested using PLS-SEM and the PROCESS macro with a sample of 209 participants. The results confirmed that participants who experienced a higher sense of telepresence and interactivity (VR condition) were more likely to report lower persuasion knowledge and better brand evaluations and behavioural intentions. It was also found that compared to the 360 condition, in VR and 2D environments, participants were more likely to recall the brand embedded into the environment. Placement congruity was found to moderate the underlying mechanism through which interactivity and telepresence affect persuasion knowledge. These findings provide helpful insights to marketers and brand managers, who think of VR as an advertising tool, on how the technology factor impacts consumers’ responses to their persuasion attempts, such as brand placements.Item Open Access The Influence of Perceived Value on Exploratory Behaviour Towards Future Patronage Intention in M-Commerce: An S-O-R Approach.(Brock University) Pouyan, Mohammad Mahdi; Faculty of Business ProgramsThe exploratory behaviour issue has received considerable attention in both online and brick-and-mortar consumer behaviour literature so far. However, regarding the widely prevalent use of mobile commerce in daily life, surprisingly, mobile exploratory behaviour has seldom been investigated. It is unclear to what extent mobile commerce characteristics can facilitate explorative behaviour. Thus, this study aims to fill the gap in the extant literature by examining the positive relationship between the perceived value, namely, functional, emotional and social and exploration (diversive and specific), which in turn, directly impacts future patronage intention. Due to the pivotal role of flow state in computer-mediated and online behaviour in the extant literature, the current study set out to examine the mediation role of flow between the relationship of perceived values and divisive vs specific exploration. This thesis begins with a brief overview of the recent history of noted research elements and proposes the conceptual model based on the stimulate-organism-response (S-O-R) model. It then discussed the hypotheses development. The remaining part of the paper proceeds with details on the data collection process and the methodological approach adopted to test these relationships.Item Open Access Firm Performance and CEO Compensation: CEO Pay Slice vs Pay-Performance Sensitivity(Brock University) Hasan, S M Muyeed; Faculty of Business ProgramsI study the relationship between CEO incentive compensation and firm performance in the presence of CEO dominance to examine how incentive compensation improves firm performance by reducing agency conflicts between shareholders and managers. I estimate pay-performance sensitivity (PPS) as a measure of CEO incentive compensation and the CEO pay slice (CPS) as a measure of CEO dominance. Controlling for standard control variables, I conduct multiple OLS regressions and find that at the high level of CPS, PPS improves firm performance, but at the low level of CPS, impact of PPS diminishes. This shows that determining stand-alone associations of PPS or CPS to firm value—a popular practice in the literature—might not be adequate because of an unexplored interaction effect between executive incentive and executive dominance. To address the potential endogeneity issues, I conduct robustness check by employing instrumental variables with a two-stage least square (2SLS) estimation procedure. As an additional robustness check, I account for the year effect and confirm that the results still stand to the same level of significance.Item Open Access When Moneyball Meets the Beautiful Game: A Predictive Analytics Approach to Exploring Key Drivers for Soccer Player Valuation(Brock University) Li, Yisheng; Faculty of Business ProgramsTo measure the market value of a professional soccer (i.e., association football) player is of great interest to soccer clubs. Several gaps emerge from the existing soccer transfer market research. Economics literature only tests the underlying hypotheses between a player’s market value or wage and a few economic factors. Finance literature provides very theoretical pricing frameworks. Sports science literature uncovers numerous pertinent attributes and skills but gives limited insights into valuation practice. The overarching research question of this work is: what are the key drivers of player valuation in the soccer transfer market? To lay the theoretical foundations of player valuation, this work synthesizes the literature in market efficiency and equilibrium conditions, pricing theories and risk premium, and sports science. Predictive analytics is the primary methodology in conjunction with open-source data and exploratory analysis. Several machine learning algorithms are evaluated based on the trade-offs between predictive accuracy and model interpretability. XGBoost, the best model for player valuation, yields the lowest RMSE and the highest adjusted R2. SHAP values identify the most important features in the best model both at a collective level and at an individual level. This work shows a handful of fundamental economic and risk factors have more substantial effect on player valuation than a large number of sports science factors. Within sports science factors, general physiological and psychological attributes appear to be more important than soccer-specific skills. Theoretically, this work proposes a conceptual framework for soccer player valuation that unifies sports business research and sports science research. Empirically, the predictive analytics methodology deepens our understanding of the value drivers of soccer players. Practically, this work enhances transparency and interpretability in the valuation process and could be extended into a player recommender framework for talent scouting. In summary, this work has demonstrated that the application of analytics can improve decision-making efficiency in player acquisition and profitability of soccer clubs.Item Open Access Risk-Taking and CEO Compensation: CEO Pay Slice Versus Pay-Volatility Sensitivity(Brock University) Ye, Fan; Faculty of Business ProgramsThis paper aims to analyze the impacts of compensation incentives and CEO power on firm’s risk-taking by using stock return volatility (Srisk) and earnings volatility (Erisk) as the proxies of firm’s risk-taking level, and by using pay-volatility sensitivity (PVS) and CEO-pay slice (CPS) as the proxies of compensation incentives and CEO power, respectively. By applying ordinary least square (OLS) regression and two-stage least square (2SLS) regression on obtained data, this paper provides strong empirical evidence that PVS and CPS have negative impact on earnings volatility and stock return volatility. In addition, the negative impact of PVS on managerial risk-taking is greater for CEOs with lower CPS than that for CEOs with higher CPS. That is, EBC discourages CEOs from taking more risks, and more powerful CEOs are less risk-averse than less powerful CEOs when granted EBC.Item Open Access Some New Estimator in Linear Mixed Models with Measurement error(Brock University) Yavarizadeh, Bahareh; Department of MathematicsLinear mixed models (LMMs) are an important tool for the analysis of a broad range of structures including longitudinal data, repeated measures data (including cross-over studies), growth and dose-response curve data, clustered (or nested) data, multivariate data, and correlated data. In many practical situations, the observation of variables is subject to measurement errors, and ignoring these in data analysis can lead to inconsistent parameter estimation and invalid statistical inference. Therefore, it is necessary to extend LMMs by taking the effect of measurement errors into account. Multicollinearity and fixed-effect variables with measurement errors are two well-known problems in the analysis of linear regression models. Although there exists a large amount of research on these two problems, there is by now no single technique superior to all other techniques for the analysis of regression models when these problems are present. In this thesis, we propose two new estimators using Nakamura's approach in LMM with measurement errors to overcome multicollinearity. We consider that prior information is available on fixed and random effects. The first estimator is the new mixed ridge estimator (NMRE) and the second estimator is the weighted mixed ridge estimator (WMRE). We investigate the asymptotic properties of these proposed estimators and compare the performance of them over the other estimators using the mean square error matrix (MSEM) criterion. Finally, a data example and a Monte Carlo simulation are also provided to show the theoretical results.Item Open Access Distress Effects in Stock Returns(Brock University) Arnott, Spencer; Faculty of Business ProgramsThis thesis addresses a fundamental topic in financial economics: the effects of distress risk in the cross section of equities returns. Initial results show that both raw and risk-adjusted excess returns are rising in distress risk, and the remainder of this thesis examines the general robustness of the distress premium. Accordingly, the additional excess returns to stocks having heightened levels of financial distress are contingent upon the stock price being low. These findings are then extended to demonstrate that these same stocks are also microcap firms, thus attributing the anomalous behaviour of distressed stocks to a common factor with many other market anomalies. The economic implication is that arbitrage profits are likely to be limited due to the high transaction costs alongside the limited investment capacity with associated low-priced, microcap stocks.Item Open Access Linear Forms in Logarithms and Fibonacci Numbers(Brock University) Earp-Lynch, Benjamin; Department of MathematicsThe main work included in these pages is from a paper co-written by myself and my brother, Simon Earp-Lynch, under the supervision of Omar Kihel, pertaining to Diophantine triples of Fibonacci numbers. To go along with this will be introductory material not included in said paper which establishes the mathematical concepts therein and offers some historical perspective and motivation. The initial aim of the paper was to explore the possibility of a generalization of the main result in [2] on D(4)-Diophantine triples of Fibonacci numbers. The paper managed to extend the ideas in [2] to results for D(9)-Diophantine triples and D(64)-Diophantine triples. A generalization of Lemma 1 of [1] was also found, a lemma on Diophantine triples and Pellian equations which is key in establishing the main result in [2]. This paper includes this result and its proof, which involves a correction of the proof of Lemma 1 of [1]. This result may prove useful in the extension of the results in the paper, and potentially others as well. I will begin by introducing Diophantine equations, leading to Diophantine triples, followed by a section on the necessary preliminaries on Fibonacci num- bers, which concludes with the statements of our main results. Following this, I establish the primary machinery used in the proof of the main result, linear forms in logarithms. I then move to the generalization of the aforementioned Lemma 1 of [1], before finally commencing the proof of the main results.