In survival analysis, a competing risk model is a statistical method used to analyze time-to-event data in situations where multiple events of interest may occur and compete for occurrence. The...Show moreIn survival analysis, a competing risk model is a statistical method used to analyze time-to-event data in situations where multiple events of interest may occur and compete for occurrence. The events are considered ‘competing’ because the occurrence of one event prevents the occurrence of other events. Traditional survival analysis focuses on a single event of interest, such as death due to a particular cause. However, in real-world scenarios, there can be multiple events that individuals in a study population might experience. These events can have different causes. For example, in a study involving cancer patients, the events of interest could be death from cancer, death from other causes, and disease recurrence. In survival analysis, a cure model is a statistical model used when a proportion of the study population is considered ‘cured’, meaning they will never experience the event of interest. This concept is particularly important when studying diseases with a good prognosis. A notable example is paediatric oncology, where patients may be considered cured if they experience long event-free survival. Despite the growing recognition of the significance of considering cured fractions in statistical analysis, there remains limited research on the theoretical aspects of combining competing risks and cure models. The integration of these two approaches has not been extensively studied until now. This research aims to fill the existing gap in the field by focusing on the concept of identifiability. First, a general model that involves two competing events and cause-specific cure for both events is considered. The main objective is to identify the model parameters, particularly the dependence relationship between the two cure status indicators. A logistic model to estimate cure probabilities and a semi-parametric Cox model to assess cause-specific hazards (or subdistribution hazards) are employed. The results demonstrated that, under appropriate assumptions, certain parameters can be effectively identified. However, it is also revealed that the model becomes unidentifiable without these specific assumptions. It is further shown that the models previously proposed in the literature can be seen as special cases of this general model. The thesis presents a novel estimation procedure for the general model, utilizing the EM (ExpectationMaximization) algorithm. The flexibility of this procedure allows it to be applied to special cases of the model. Two simulation studies were conducted to investigate the performance of the estimation procedure and to study the practical identifiability properties of the model for cure and competing risks. The results showed good performance for most parameters of the model. In conclusion, this thesis provides valuable insights into the practical identifiability of parameters 2 through both theoretical and simulation-based analyses. This research significantly contributes to a better understanding of competing risks and cure models. The understanding of these statistical methods enables more accurate analysis of patient outcomes and treatment effects in diverse clinical and non-clinical contexts. Ultimately, this research positively impacts the field, facilitating better decision-making and improving overall outcomes for patients and individuals in various settings.Show less
In this thesis we study restricted digit sets for random L¨uroth expansions, a type of number expansion for numbers in the unit interval generalising the L¨uroth expansions introduced in [L¨ur83]....Show moreIn this thesis we study restricted digit sets for random L¨uroth expansions, a type of number expansion for numbers in the unit interval generalising the L¨uroth expansions introduced in [L¨ur83]. By studying a random transformation that generates these expansions and a corresponding iterated function system we find a general formula describing the Hausdorff dimensions of all of these restricted digit sets. We then study a family of two-dimensional fractal sets induced by the skew product representation of this random transformation. In particular, we find conditions under which the box-counting dimension of such a fractal equals the corresponding affinity dimension and use this to find upper and lower bounds.Show less
Bioluminescence Tomography is a growing field in optical imaging. Governed by the diffusion approximation of the Radiative Transfer Equation, Bioluminescence Tomography corresponds to the inverse...Show moreBioluminescence Tomography is a growing field in optical imaging. Governed by the diffusion approximation of the Radiative Transfer Equation, Bioluminescence Tomography corresponds to the inverse problem of reconstructing the source function q(x) from measurements ?(x). This inverse problem is ill-posed in the sense of Hadamard. The discrete form of the forward problem is given by A?1q = ?. Although it seems like obtaining q from ? is straightforward, through numerical experiments it is shown that additive Gaussian noise has a significant effect on reconstruction. Even for the simplest source function tested, i.e. q(x) = x 2 , regular inversion only suffices in the absence of noise. Both the Truncated Singular Value Decomposition solution and Tikhonov regularization provide a more adequate reconstruction of the source function in the case of additive noise. This is also seen for a source function made up of two Gaussian peaks. Although both methods are able to distinguish the two peaks in a similar fashion, Truncated Singular Value Decomposition may result in misinterpretation of extra peaks due to its forced sinusoidal form. Finally, in the case of a 2D image source function reconstruction we note that the regular inversion performs better than expected but that overall Tikhonov regularization provides the best reconstruction of the source function. Nevertheless, Tikhonov regularization is not perfect and may result in oversmoothing. Further research can be done to improve the reconstruction of source function q(x) from noisy data.Show less
In a 2007 paper, Charles, Lauter and Goren studied how one might use Ramanujan graphs to create cryptographic hash functions. One of the most well-known such graphs is the isogeny graph, whose...Show moreIn a 2007 paper, Charles, Lauter and Goren studied how one might use Ramanujan graphs to create cryptographic hash functions. One of the most well-known such graphs is the isogeny graph, whose vertices are indexed by the isomorphism classes of supersingular elliptic curves in a characteristic p. In this thesis, we study the spectra of these graphs. To start, we give two algorithms to compute these graphs for small p, and present data obtained from implementing these algorithms. This data provides some statistical evidence for several properties of the spectra of these graphs. We discuss the proof of two of these properties. Firstly, the fact that these supersingular isogeny graphs are Ramanujan, i.e. that they have large spectral gap. For this, we discuss the relation with Hecke operators and the Eichler–Shimura relation. Secondly, the distribution of the eigenvalues as p tends to infinity. This we prove via the relation between the graph and the Brandt matrices for the quaternion algebra ramified at p and ?. We sketch a proof of the Eichler–Selberg trace formula, and use this to conclude the proof.Show less
E-variables are a novel tool for constructing hypothesis tests that retain Type-I error guarantees when the sampling plan is not determined in advance, i.e. under optional stopping and optional...Show moreE-variables are a novel tool for constructing hypothesis tests that retain Type-I error guarantees when the sampling plan is not determined in advance, i.e. under optional stopping and optional continuation. We construct E-variables for null hypotheses that are univariate exponential families and point alternative hypotheses by calculating the Reverse Information Projection, abbreviated to RIPr, of the alternative on the set of mixtures over the null. We focus on RIPr’s that are simple; this means that they coincide with a single element of the null hypothesis rather than a mixture of such elements. We find that there is no unique simple way to determine the RIPr for the whole class of exponential families. We give conditions under which the RIPr is simple (and then also easy to calculate), and conditions under which it is not (and then it is hard to calculate), and we give several examples of each case. For the case that an E-variable for a specific exponential family null is given, we establish E-variables for other exponential families by 1-to-1 transformations of random variables. We approximate a more complex RIPr (i.e. a mixture of exponential distributions) when the sample space consists of two outcomes of the exponential distribution in a specific setting by programming in RShow less
