The other topic that will be covered in this dissertation will be alternating trees and tiered trees. We will define a new generalization of alternating trees, which we will call tiered trees. We will also define a generalized weight system on these tiered trees. We will prove some enumerative results about tiered trees that demonstrate how they can be viewed as being obtained by applying certain procedures to certain types of alternating trees. We also provide a bijection between the set of permutations in S_{n} and the set of weight 0 alternating trees with n+1 vertices. We use this bijection to define a new statistic of permutations called the weight of a permutation, and use this weight to define a new q-Eulerian polynomial.

The first two chapters of this thesis deal with the study of the long-time behavior of parallel lattice Kinetic Monte Carlo (PL-KMC) algorithms for interacting particle systems. We introduce the relative entropy rate (RER) as a measure of long-time loss of information and illustrate that it is a computable* a posteriori* quantity. The RER can act as an information criterion (IC), discriminating between different parameter choices for the schemes and allowing comparisons at equilibrium. We make explicit how the RER scales with the time-step and the size of the system and that it captures details about the connectivity of the original process.

Another feature of long-time behavior is *time-reversibility*, which some physical systems naturally exhibit. Unfortunately, due to the domain and time-discretization, PL-KMC cannot conserve this property. To quantify the loss of reversibility, we introduce the entropy production rate (EPR) as an IC for comparisons between different schemes. We show that the EPR shares a lot of the properties of the RER and can be estimated efficiently from data.

The last chapter discusses uncertainty quantification for model bias. By connecting a recently derived goal-oriented divergence and concentration bounds, we define new divergences that provide *computable* bounds for model bias. The new bounds scale appropriately with data and become progressively more accurate depending on available information about the models and the quantities of interest. We discuss how the bounds allow us to bypass computationally expensive Monte Carlo sampling or specialized methods, e.g., Multilevel Monte Carlo.

Extreme value theory is concerned with probabilistic and statistical questions re- lated to unusual behavior or rare events. The subject has a rich mathematical theory and also a long tradition of applications in a variety of areas. We are interested in its application in risk management, with a focus on estimating and forcasting the Value-at-Risk of financial time series data. Extremal data are inherently scarce, thus making inference challenging. In order to obtain good estimates for risk measures, we develop a two-stage approach: (1) fitting the GARCH-type models at the first stage to describe the volatility clustering and other stylized facts of financial time series; (2) using the extreme value theory based models to fit to the tails of the residuals. Additionally, the performance measures provide information in terms of the comparison of the two-stage semi-parametric approach with the parametric methodologies, through robust backtesting.

Copula is a particular branch of probability theory, with which, given sufficient data, we can separate the marginal behavior of individual risks and their dependence structure from a multivariate random variable. Linear correlation is widely used to model dependence but has limitations as a measure of association and thus we opt to use copulas to analyze the dependence structure and build models for our different problems arising in risk management. For this part of the dissertation, we take a look at different copula families, highlight for some when they are most appropriate to use for a particular application, discuss some of their drawbacks as diverse scenarios occur in different risk management models, and explore the possibility of developing the copula modeling to reflect the complicated dependence structure of portfolios.

]]>Second, we describe some interesting properties of zeta functions of algebraic curves. Generally L-functions vanish only to the order required by their root number. However, we demonstrate that for a certain class of quaternion extensions of F_p(t), the zeta function vanishes at a higher order than the root number demands, indicating some other phenomenon at work.

]]>One of the common sampling methodology used in this context is respondent-driven sampling (RDS). Under RDS, the network connecting members of the target population is used to uncover the hidden members. Specialized techniques are then used to make inference from the data collected in this fashion. Our first objective is to correct traditional RDS prevalence estimators and their associated uncertainty estimators for misclassification of the outcome variable.

RDS also has the unusual characteristic that the participants are driving the sampling process by recruiting members into the survey. Since the researchers forfeit their control over the sampling process, the estimators are therefore susceptible to a great extent to participants' behavioral induced biases. Our second objective is therefore to provide a mathematical parametrization for a behavior referred to as differential recruitment and subsequently adjust the inference for potential induced bias.

Finally, a common issue encountered in the application motivating this thesis, that is, HIV prevalence estimation, is the derivation of a national prevalence estimate. Data are often collected at different study sites within a given country. Public health officials however commonly report national prevalence. Therefore, our last objective consists of using Bayesian hierarchical models to derive a national prevalence estimator from regional data.

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