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Yim Tsz-ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 47-49). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Local Polynomial Modeling --- p.4 / Chapter 2.1 --- Local Polynomial Fitting --- p.4 / Chapter 2.1.1 --- Methodology --- p.4 / Chapter 2.1.2 --- The kernel K --- p.6 / Chapter 2.1.3 --- The bandwidth h --- p.7 / Chapter 2.1.4 --- The order p --- p.10 / Chapter 2.2 --- Estimation of Conditional Density --- p.11 / Chapter 3 --- Bandwidth Selection --- p.14 / Chapter 3.1 --- Rule of Thumb --- p.14 / Chapter 3.2 --- Bootstrap Bandwidth Selection --- p.15 / Chapter 3.3 --- A Cross-Validation Method --- p.16 / Chapter 4 --- A Theoretical Justification --- p.18 / Chapter 4.1 --- Proof of (4.1) --- p.19 / Chapter 4.2 --- Proof of (4.2) --- p.22 / Chapter 5 --- Simulation Studies --- p.25 / Chapter 6 --- Real Data Applications --- p.38 / Chapter 6.1 --- Case Study With Canadian Lynx Data.............................. --- p.38 / Chapter 6.2 --- Case Study With U.S. Twelve-Month Treasury Bill Data.......... --- p.41 / Chapter 7 --- Conclusions --- p.45 / Bibliography --- p.47

The purpose of this thesis is to test whether the probability of falsely rejecting a true null hypothesis of a model intercept being equal to zero is consistent with the chosen significance level when modelling the variance of the error term using GARCH (1,1), TGARCH (1,1) or IGARCH (1,1) models. We test this by estimating “Jensen’s alpha” to evaluate alpha trading, using a Monte Carlo simulation based on historical data from the Standard & Poor’s 500 Index and stocks in the Dow Jones Industrial Average Index. We evaluate over simulated daily data ranging over periods of 3 months, 6 months, and 1 year. Our results indicate that the GARCH and IGARCH consistently reject a true null hypothesis less often than the selected 1%, 5%, or 10%, whereas the TGARCH consistently rejects a true null more often than the chosen significance level. Thus, there is a risk of incorrect inferences when using these GARCH-type models.

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As of now, not many pharmaceutical companies are producing new categories of antibiotics to fight bacterial infections. Therefore, bacteria are building up a resistance to the medications commonly used. Often, antibiotic resistance begins within a hospital. To combat resistance, researchers completed several studies using cycling of the medications that are already in place, but they found either no improvement or the resistance increased with this type of setting. In addition, although preventative infection control measures have been shown to decrease antibiotic resistance for some antibiotics, the level of antibiotic resistance found in hospitals is still extremely high. This motivates the main goal of this thesis: to quantify how much the overall resistance can be lowered by simply adding one new drug to the regimen. The process of adding a new antibiotic can be quantified using mathematical models that show the flow of patients colonized with various types of bacteria into, out of, and within the hospital. Deterministic models can be used to model the spread of resistant bacteria in hospitals with a relatively large number of beds. However, not all hospitals are large enough to accurately determine the effects using a deterministic model; thus, we must use stochastic models, where mathematical formulations include probability in ways that describe intrinsic random fluctuations, typical of infection processes at smaller scales. In examining the addition of a new antibiotic within a hospital, we consider different administration protocols, either assuming that physicians are equally likely to prescribe the new antibiotic as they are to prescribe existing antibiotics or that physicians prescribe the new antibiotic to only a targeted population of patients. We will examine the variation in the expected level of overall resistance in a hospital depending on the administration procedure as well as the whether the hospital is large (deterministic model) or small (stochastic model). We will conclude with initial results for fitting these models to simulated data using common inverse problem methodology.

Experiments to learn about the effect of light, sex, and diet on the activity of flies generate great quantities of data that is necessary to analyze. Since different researches and students participate in the analysis of those experiments, it is convenient to have a methodology to analyze the experimental data using software so that the data can be analyzed in a uniform way. Being a double major in mathematics and biology, I am interested in:Deciding which statistical procedure to use to analyze the data so that the research questions of the researchers in biology are answered.To recommend how to implement those procedures using software in an efficient way.To write a prototype for the interpretation of the results.Those are the objectives of this work. In the thesis, we first applied two-way ANOVA to analyze the effect of two selected factors, sex (female and male) and diet (liver and non-liver), on the fly activity under dark condition and under light condition, respectively. Next, we employed the repeated measures to capture how fly activity changes over time (day in this case) and to relate the changes to the selected factors, sex and diet, also under dark condition and under light condition, respectively. Finally, we did a little research on the analysis of circadian rhythms and compared the results with that obtained from honey bee activity experiments carried out before.

We introduce a novel stochastic Petri net formalism where discrete and continuous phase-type firing delays can appear in the same model. By capturing deterministic and generally random behavior in discrete or continuous time, as appropriate, the formalism affords higher modeling fidelity and efficiencies to use in practice. We formally specify the underlying stochastic process as a general state space Markov chain and show that it is regenerative, thus amenable to renewal theory techniques to obtain steady-state solutions. We present two steady-state analysis methods depending on the class of problem: one using exact numerical techniques, the other using simulation. Although regenerative structures that ease steady-state analysis exist in general, a noteworthy problem class arises when discrete-time transitions are synchronized. In this case, the underlying process is semi-regenerative and we can employ Markov renewal theory to formulate exact and efficient numerical solutions for the stationary distribution. We propose a solution method that shows promise in terms of time and space efficiency. Also noteworthy are the computational tradeoffs when analyzing the "embedded" versus the "subordinate" Markov chains that are hidden within the original process. In the absence of simplifying assumptions, we propose an efficient regenerative simulation method that identifies hidden regenerative structures within continuous state spaces. The new formalism and solution methods are demonstrated with two applications.

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