Several truncated models for market weights are discussed. To summarize, the new truncated ordered stick breaking model introduced give restrictions on the ranks of the markets weights and show better fitting results for real data sets. / Thesis / Master of Science (MSc)

<p>In this thesis, we explore several stochastic models associated withthe two-parameter Poisson-Dirichlet distribution and population genetics.The impacts of mutation, selection and time onthe population evolutionary process will be studied by focusing on two aspects of the model:equilibrium and non-equilibrium. In the first chapter, we introduce relevant background on stochastic genetic models, andsummarize our main results and their motivations. In the second chapter, the two-parameter GEM distribution is constructedfrom a linear birth process with immigration. The derivationrelies on the limiting behavior of the age-ordered family frequencies. In the third chapter, to show the robustness of the sampling formula we derive the Laplace transform of the two-parameterPoisson-Dirichlet distribution from Pitman sampling formula. The correlationmeasure of the two-parameter point process is obtained in our proof. We also reverse this derivationby getting the sampling formula from the Laplace transform. Then,we establish a central limit theorem for the infinitely-many-neutral-alleles modelat a fixed time as the mutation rate goes to infinity.Lastly, we get the Laplace transform for the selectionmodel from its sampling formula. In the fourth chapter, we establisha central limit theorem for the homozygosity functions under overdominant selectionwith mutation approaching infinity. The selection intensity is given by a multiple of certain powerof the mutation rate. This result shows an asymptotic normality for the properly scaled homozygosities,resembling the neutral model without selection.This implies that the influence of selection can hardly be observed with large mutation. In the fifth chapter, the stochastic dynamics of the two-parameter extension of theinfinitely-many-neutral-alleles model is characterized by the derivation of its transition function,which is absolutely continuous with respect to the stationary distribution being the two-parameter Poisson-Dirichlet distribution.The transition density is obtained by the expansion of eigenfunctions.Combining this result with the correlation measure in Chapter 3, we obtain the probability generatingfunction of a random sampling from the two-parameter model at a fixed time. Finally, we obtain two results based on the quasi-invariance of the Gamma processwith respect to the multiplication transformation group.One is the quasi-invariance property of the two-parameter Poisson-Dirichletdistribution with respect to Markovian transformation group.The other one is the equivalence between the quasi-invarianceof the stationary distributions of aclass of branching processes and their reversibility.</p> / Doctor of Philosophy (PhD)

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.

The Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem.