School of Mathematics and Statistics - Theses

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    Parisian Ruin with Random Delays for Spectrally Negative Lévy Processes
    Nguyen, Duy Phat ( 2023-12)
    This thesis is devoted to studying Parisian ruin problems for spectrally negative Levy processes. The thesis consists of six chapters. Chapter 1 is a general introduction to spectrally negative Levy processes. We give definitions, examples and general properties of such processes. We also review the basics of stochastic calculus. Our exposition in this chapter often presents sketches of arguments rather than rigorous proofs. To compensate for the lack of rigour, we included references to more specialised texts where the proofs of the cited/used results can be found. Chapter 2 is devoted to the scale functions of spectrally negative Levy processes. We give basic definitions, examples and general properties. We also discuss some applications of scale functions. Chapter 3 presents previously known results of Parisian ruin theory and some related topics. We give the definitions and well-known facts in these areas. Chapter 4 introduces a new interesting and natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative Levy process with trajectories of locally bounded variation. The novel feature of this extension is that the distributions of the lengths of the random implementation delays can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. Moreover, this extension allows for the possibility of an immediate ruin when the deficit hits a certain subset. In this setting, we derive a closed-form expression for the Parisian ruin probability and the joint Laplace transform of the Parisian ruin time and the deficit at that time of ruin. Chapter 5 is devoted to extending the results obtained in Chapter 4 to the case of spectrally negative Levy processes with trajectories of unbounded variation. Chapter 6 deals with the Parisian ruin time with arbitrary delays being independent of the deficit for the compound Poisson risk model. We show that the absolute distance between two Parisian ruin probabilities with different delays is bounded from above by the Levy-Prokhorov distance between the distributions of the two delays multiplied by a function of the initial reserve value. This means that the Parisian ruin probability with an arbitrary delay distribution can be approximated by the Parisian ruin probability with delay windows following a (finite) mixture of Erlang distributions, and the latter probability admits a closed-form expression.
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    Mathematical approaches to pattern formation in dermatology
    Gilmore, Stephen. (University of Melbourne, 2005)
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    Aspects of mixed longitudinal growth analysis
    Matta, Alonso Alejandro. (University of Melbourne, 2010)
    This thesis presents practical approaches to the analysis of mixed longitudinal growth data. Longitudinal studies of the human population are specifically designed to investigate changes over a limited age range in a characteristic which is measured repeatedly for each study participant. This type of data poses several methodological challenges. First, models for the analysis of longitudinal data must recognize the relationship between the observations taken from each study participant. The mixed nature of the data calls for the use of random effects and variance and correlation structures for the within group errors. Secondly, the models must be flexible enough so that they can be easily differentiated for the timing of the population growth spurts. And thirdly, longitudinal growth data of human subjects is more often than not affected by the missing data problem. In practice, the missing data mechanism needs to be understood and taken into consideration when fitting the models. These aspects of mixed longitudinal growth analysis are covered in detail in this thesis using a comprehensive data set of repeated measures of human height of hundreds of Melbourne school children ranging form the ages of 5 to 18 years.
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    Pointwise axiomatic spectral theory in Banach algebras
    Lubansky, Raymond Alan. (University of Melbourne, 2008)
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    On sample size determination for discrete data
    Gordon, Ian Robert. (University of Melbourne, 1993)
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    On sample size determination for discrete data
    Gordon, Ian Robert. (University of Melbourne, 1993)
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    Modelling high-dimensional spatial-temporal data: a focus on nonstationary and nonlinear phenomena for a future-focused landslide early-warning system
    Zheng, Hangfei ( 2023-11)
    Rainfall-induced landslides are seeing an increase as the by-products of climate change and present significant damage to the environment, society, and human lives. These landslides are nonstationary and nonlinear phenomena often recorded as high-dimensional vector time series manifesting spatiotemporal dependence. Modelling and forecasting landslides are difficult, with the challenges coming from the complexity of the underlying time series and the remote-sensing techniques used in obtaining these monitoring data. Also, these time series may be of irregular frequency and contain missing values. We tackle these challenges by developing statistical forecasting tools for a future-focused landslide early-warning system (LEWS). These forecasting tools include our developed statistical models characterising complex time series, dimension reduction for efficient dynamic data representation, and three complementary risk assessment prongs to turn the derived forecasts into early-warning predictions of slope failure. Our proposed models are based on a novel spatial dimension reduction technique called empirical dynamic quantiles (EDQ). The idea behind this technique is to use a small number of representative EDQ series from the observed time series to surmise the whole dataset. We then perform various statistical analyses based on these representative EDQ series which will be computationally feasible. The general form of our time series model combined two advanced econometric methods error-correction cointegration (ECC), vector autogregression (VAR) and a nonlinear function $\boldsymbol{c}(t)$ with the EDQ method named ECC-VAR-$\boldsymbol{c}(t)$-EDQ model. We use this model to deal with these high-dimensional, spatial-temporal dependent vector time series with nonstationary and nonlinear phenomena. For different purposes in practice, we provide two methods to estimate the nonlinear function $\boldsymbol{c}(t)$ and further improve the forecasting accuracy. One is the \emph{empirical function-based method} and the other is \emph{physical-based method}. Once the form of $\boldsymbol{c}(t)$ has been determined, we can use the generalised least square (GLS) to estimate the unknown parameters involved in this ECC-VAR-$\boldsymbol{c}(t)$-EDQ model after performing our developed nonlinear cointegration test. The above-mentioned model is in a situation where there are no missing values and for high-frequency data. To apply the general time series analysis for low-frequency data with some missing values, we develop a model that combines the stochastic differential equations (SDE) and Markov chain Monte Carlo (MCMC) approach with the EDQ technique named SDE-MCMC-EDQ model. The basic idea behind this model is that we can convert these low-frequency time series to high-frequency by introducing some additional data between every two consecutive observations implies the estimation of these unknown data in addition to the SDE model parameters, where both these imputed data and the parameters in SDE are treated as random variables. The reproducibility and robustness of all these developed models are assessed by the application of different real-world ground motion data or simulation studies. Results found well fitted these different slope data with the goodness of fit statistic ($0.94\leq R^2\leq0.99$) close to 1. In addition to the forecast values derived from our developed models, we use three risk assessments in parallel to predict where, when, and risk of failure for supporting a complete future-focused LEWS which more accessible to the public.
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    A class of smooth, possibly data-adaptive, nonparametric copula estimators and corresponding resampling techniques with applications
    Yi, Bingqing ( 2023-08)
    Copulas are mathematical tools for modeling the dependence between the components of a random vector. They are frequentely used in fields such as finance, economics, and risk management. Chapter 1 and 2 of this thesis provide a review of the main results in the study of copulas including their basic properties, estimation methods, the empirical copula processes and appropriate resampling schemes for the latter. Chapter 3 proposes a broad class of smooth, possibly data-adaptive nonparametric copula estimators that contains empirical Bernstein copulas introduced by Sancetta and Satchell (2004) (and thus the empirical beta copula proposed by Segers et al. (2017)). A specific subclass that performs uniformly better than the empirical beta copula in Monte Carlo experiments is identified. Furtheremore, conditions under which related sequential empirical copula processes converge weakly are provided. Chapter 4 proposes two resampling techniques for the class of estimators considered in Chapter 3. One technique builds up on the work of Kiriliouk et al. (2021) and can be used to bootstrap related empirical copula processes in the i.i.d. case. The other technique is a smooth extension of the dependent multiplier bootstrap proposed in Bucher and Kojadinovic (2016) and can be used to bootstrap related empirical copula processes in the sequential time series case. In addition, two classes of smooth estimators of the first-order partial derivatives of the copula are also theoretically and empirically studied. The last chapter discusses potential future research directions, such as applying the studied estimators and corresponding resampling techniques for change-point detection and inference in factor copula models.
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    Mathematical model and heuristics for relief transportation in the aftermath of sudden-onset natural disasters
    Candida Cortez, Pamela Michele ( 2023-08)
    Floods and landslides are the most common natural disasters in Brazil. These disasters can have a long-term impact on the affected communities and leave whole populations in need of support. We study the efficient transportation of relief goods from depots to distribution points in the aftermath of such disasters. Although the field of humanitarian logistics has received increased attention in Operations Research, recent reviews highlighted gaps related to unrealistic assumptions, such as static post-disaster travel times, demands, and supply availability. As predicting future information in humanitarian logistics is harder than in its commercial counterpart, more research that accounts for information updates and dynamic parameters is needed. We addressed these gaps by proposing a more realistic model in order to study practical aspects of relief distribution, hoping to contribute to decision making in the real world. As black-box solvers can only handle very small instances of our proposed mathematical model, we developed heuristics that work within a fix-and-optimise framework, allowing for information updates in a rolling planning horizon. The proposed heuristics could adequately solve real-sized instances, enabling a comparison between deterministic and dynamic data approaches. Fairness is another key characteristic in humanitarian logistics which sets it apart from its commercial counterpart. To ensure fairness, we integrated a priority index in our constructive heuristic that prioritises underserved nodes over densely populated ones. In the improvement heuristics, fairness is considered through inequity penalties. Results indicate that fairness can be enforced without significantly increasing transportation costs.