In many applications there are dynamic changes in the dependency structure between multivariate time series. Two examples include neuroscience and finance. The second and third chapters focus on neuroscience and introduce a data-driven technique for partitioning a time course into distinct temporal intervals with different multivariate functional connectivity patterns between a set of brain regions of interest (ROIs). The technique, called Dynamic Connectivity Regression (DCR), detects temporal change points in functional connectivity and estimates a graph, or set of relationships between ROIs, for data in the temporal partition that falls between pairs of change points. Hence, DCR allows for estimation of both the time of change in connectivity and the connectivity graph for each partition, without requiring prior knowledge of the nature of the experimental design. Permutation and bootstrapping methods are used to perform inference on the change points. In the second chapter of this work, we focus on multi-subject data while in the third chapter, we concentrate on single-subject data and extend the DCR methodology in two ways: (i) we alter the algorithm to make it more accurate for individual subject data with a small number of observations and (ii) we perform inference on the edges or connections between brain regions in order to reduce the number of false positives in the graphs. We also discuss a Likelihood Ratio test to compare precision matrices (inverse covariance matrices) across subjects as well as a test across subjects on the single edges or partial correlations in the graph. In the final chapter of this work, we turn to a finance setting. We use the same DCR technique to detect changes in dependency structure in multivariate financial time series for situations where both the placement and number of change points is unknown. In this setting, DCR finds the dependence change points and estimates an undirected graph representing the relationship between time series within each interval created by pairs of adjacent change points. A shortcoming of the proposed DCR methodology is the presence of an excessive number of false positive edges in the undirected graphs, especially when the data deviates from normality. Here we address this shortcoming by proposing a procedure for performing inference on the edges, or partial dependencies between time series, that effectively removes false positive edges. We also discuss two robust estimation procedures based on ranks and the tlasso (Finegold and Drton, 2011) technique, which we contrast with the glasso technique used by DCR.

The vector autoregressive (VAR) model has been widely used for describing the dynamic behavior of multivariate time series. However, fitting standard VAR models to large dimensional time series is challenging primarily due to the large number of parameters involved. In this thesis, we propose two strategies for fitting large dimensional VAR models. The first strategy involves reducing the number of non-zero entries in the autoregressive (AR) coefficient matrices and the second is a method to reduce the effective dimension of the white noise covariance matrix. We propose a 2-stage approach for fitting large dimensional VAR models where many of the AR coefficients are zero. The first stage provides initial selection of non-zero AR coefficients by taking advantage of the properties of partial spectral coherence (PSC) in conjunction with BIC. The second stage, based on $t$-ratios and BIC, further refines the spurious non-zero AR coefficients post first stage. Our simulation study suggests that the 2-stage approach outperforms Lasso-type methods in discovering sparsity patterns in AR coefficient matrices of VAR models. The performance of our 2-stage approach is also illustrated with three real data examples. Our second strategy for reducing the complexity of a large dimensional VAR model is based on a reduced-rank estimator for the white noise covariance matrix. We first derive the reduced-rank covariance estimator under the setting of independent observations and give the analytical form of its maximum likelihood estimate. Then we describe how to integrate the proposed reduced-rank estimator into the fitting of large dimensional VAR models, where we consider two scenarios that require different model fitting procedures. In the VAR modeling context, our reduced-rank covariance estimator not only provides interpretable descriptions of the dependence structure of VAR processes but also leads to improvement in model-fitting and forecasting over unrestricted covariance estimators. Two real data examples are presented to illustrate these fitting procedures.

This thesis focuses on developing nonlinear time series models and establishing relevant theory with a view towards applications in which the responses are integer valued. The discreteness of the observations, which is not appropriate with classical time series models, requires novel modeling strategies. The majority of the existing models for time series of counts assume that the observations follow a Poisson distribution conditional on an accompanying intensity process that drives the serial dynamics of the model. According to whether the evolution of the intensity process depends on the observations or solely on an external process, the models are classified into parameter-driven and observation-driven. Compared to the former one, an observation-driven model often allows for easier and more straightforward estimation of the model parameters. On the other hand, the stability properties of the process, such as the existence and uniqueness of a stationary and ergodic solution that are required for deriving asymptotic theory of the parameter estimates, can be quite complicated to establish, as compared to parameter-driven models. In this thesis, we first propose a broad class of observation-driven models that is based upon a one-parameter exponential family of distributions and incorporates nonlinear dynamics. The establishment of stability properties of these processes, which is at the heart of this thesis, is addressed by employing theory from iterated random functions and coupling techniques. Using this theory, we are also able to obtain the asymptotic behavior of maximum likelihood estimates of the parameters. Extensions of the base model in several directions are considered. Inspired by the idea of a self-excited threshold ARMA process, a threshold Poisson autoregression is proposed. It introduces a two-regime structure in the intensity process and essentially allows for modeling negatively correlated observations. E-chain, a non-standard Markov chain technique and Lyapunov's method are utilized to show the stationarity and a law of large numbers for this process. In addition, the model has been adapted to incorporate covariates, an important problem of practical and primary interest. The base model is also extended to consider the case of multivariate time series of counts. Given a suitable definition of a multivariate Poisson distribution, a multivariate Poisson autoregression process is described and its properties studied. Several simulation studies are presented to illustrate the inference theory. The proposed models are also applied to several real data sets, including the number of transactions of the Ericsson stock, the return times of Goldman Sachs Group stock prices, the number of road crashes in Schiphol, the frequencies of occurrences of gold particles, the incidences of polio in the US and the number of presentations of asthma in an Australian hospital. An array of graphical and quantitative diagnostic tools, which is specifically designed for the evaluation of goodness of fit for time series of counts models, is described and illustrated with these data sets.

In this article we investigate and develop the practical model assessment and selection methods for Bayesian models, when we anticipate that a promising approach should be objective enough to accept, easy enough to understand, general enough to apply, simple enough to compute and coherent enough to interpret. We mainly restrict attention to the Kullback-Leibler divergence, a widely applied model evaluation measurement to quantify the similarity between the proposed candidate model and the underlying true model, where the true model is only referred to a probability distribution as the best projection onto the statistical modeling space once we try to understand the real but unknown dynamics/mechanism of interest. In addition to review and discussion on the advantages and disadvantages of the historically and currently prevailing practical model selection methods in literature, a series of convenient and useful tools, each designed and applied for different purposes, are proposed to asymptotically unbiasedly assess how the candidate Bayesian models are favored in terms of predicting a future independent observation. What's more, we also explore the connection of the Kullback-Leibler based information criterion to the Bayes factors, another most popular Bayesian model comparison approaches, after seeing the motivation through the developments of the Bayes factor variants. In general, we expect to provide a useful guidance for researchers who are interested in conducting Bayesian data analysis.

This thesis deals with a general class of transformation models that contains many important semiparametric regression models as special cases. It develops a self-induced smoothing method for estimating the regression coefficients of these models, resulting in simultaneous point and variance estimations. The self-induced smoothing does not require bandwidth selection, yet provides the right amount of smoothness so that the estimator is asymptotically normal with mean zero (unbiased) and variance-covariance matrix consistently estimated by the usual sandwich-type estimator. An iterative algorithm is given for the variance estimation and shown to numerically converge to a consistent limiting variance estimator. The self-induced smoothing method is also applied to selecting the non-zero regression coefficients for the monotone transformation models. The resulting regularized estimator is shown to be root-n-consistent and achieve desirable sparsity and asymptotic normality under certain regularity conditions. The smoothing technique is used to estimate the monotone transformation function as well. The smoothed rank-based estimate of the transformation function is uniformly consistent and converges weakly to a Gaussian process which is the same as the limiting process for that without smoothing. An explicit covariance function estimate is obtained by using the smoothing technique, and shown to be consistent. The estimation of the transformation function reduces the multiple hypotheses testing problems for the monotone transformation models to those for linear models. A new hypotheses testing procedure is proposed in this thesis for linear models and shown to be more powerful than some widely-used testing methods when there is a strong collinearity in data. It is proved that the new testing procedure controls the family-wise error rate.

The model within the present study incorporated Bayesian Multidimensional Scaling and Markov Chain Monte Carlo methods to represent individual preferences and threshold parameters as they relate to the influence of survey items popularity and their interrelationships. The model was used to interpret two independent data samples of ordinal consumer preference data related to purchasing behavior. The objective of the procedure was to provide an understanding and visual depiction of consumers' likelihood of having a strong affinity toward one of the survey choices, and how other survey choices relate to it. The study also aimed to derive the joint spatial representation of the subjects and products represented by the dissimilarity preference data matrix within a reduced dimensionality. This depiction would aim to enable interpretation of the preference structure underlying the data and potential demand for each product. Model simulations were created both from sampling the normal distribution, as well as incorporating Lambda values from the two data sets and were analyzed separately. Posterior checks were used to determine dimensionality, which were also confirmed within the simulation procedures. The statistical properties generated from the simulated data confirmed that the true parameter values (loadings, utilities, and latititudes) were recovered. The model effectiveness was contrasted and evaluated both within real data samples and a simulated data set. The two data sets analyzed were confirmed to have differences in their underlying preference structures that resulted in differences in the optimal dimensionality in which the data should be represented. The Biases and MSEs of the lambdas and alphas provide further understanding of the data composition and Analysis of variance (ANOVA) confirmed the differences in MSEs related to changes in dimensions were statistically significant.

Item response theory (IRT) models are a conventional tool for analyzing both small scale and large scale educational data sets, and they are also used for the development of high-stakes tests such as the Scholastic Aptitude Test (SAT) and the Graduate Record Exam (GRE). When estimating these models it is imperative that the data set includes many more examinees than items, which is a similar requirement in regression modeling where many more observations than variables are needed. If this requirement has not been met the analysis will yield meaningless results. Recently, penalized estimation methods have been developed to analyze data sets that may include more variables than observations. The main focus of this study was to apply LASSO and ridge regression penalization techniques to IRT models in order to better estimate model parameters. The results of our simulations showed that this new estimation procedure called penalized joint maximum likelihood estimation provided meaningful estimates when IRT data sets included more items than examinees when traditional Bayesian estimation and marginal maximum likelihood methods were not appropriate. However, when the IRT datasets contained more examinees than items Bayesian estimation clearly outperformed both penalized joint maximum likelihood estimation and marginal maximum likelihood.

This thesis consists of two parts. The first part uses Gaussian Copula and Student's t Copula as the main tools to model the credit risk in securitizations and re-securitizations. The second part proposes a statistical procedure to identify changepoints in Cox model of survival data. The recent 2007-2009 financial crisis has been regarded as the worst financial crisis since the Great Depression by leading economists. The securitization sector took a lot of blame for the crisis because of the connection of the securitized products created from mortgages to the collapse of the housing market. The first part of this thesis explores the relationship between securitized mortgage products and the 2007-2009 financial crisis using the Copula method as the main tool. We show in this part how loss distributions of securitizations and re-securitizations can be derived or calculated in a new model. Simulations are conducted to examine the effectiveness of the model. As an application, the model is also used to examine whether and where the ratings of securitized products could be flawed. On the other hand, the lag effect and saturation effect problems are common and important problems in survival data analysis. They belong to a general class of problems where the treatment effect takes occasional jumps instead of staying constant throughout time. Therefore, they are essentially the changepoint problems in statistics. The second part of this thesis focuses on extending Lai and Xing's recent work in changepoint modeling, which was developed under a time series and Bayesian setup, to the lag effect problems in survival data. A general changepoint approach for Cox model is developed. Simulations and real data analyses are conducted to illustrate the effectiveness of the procedure and how it should be implemented and interpreted.

In this thesis, we consider systems of interacting diffusion processes which we call Generalized Volatility-Stabilized processes, as they extend the Volatility-Stabilized Market models introduced in Fernholz and Karatzas (2005). First, we show how to construct a weak solution of the underlying system of stochastic differential equations. In particular, we express the solution in terms of time-changed squared-Bessel processes and argue that this solution is unique in distribution. In addition, we also discuss sufficient conditions under which this solution does not explode in finite time, and provide sufficient conditions for pathwise uniqueness and for existence of a strong solution. Secondly, we discuss the significance of these processes in the context of Stochastic Portfolio Theory. We describe specific market models which assume that the dynamics of the stocks' capitalizations is the same as that of the Generalized Volatility-Stabilized processes, and we argue that strong relative arbitrage opportunities may exist in these markets, specifically, we provide multiple examples of portfolios that outperform the market portfolio. Moreover, we examine the properties of market weights as well as the diversity weighted portfolio in these models. Thirdly, we provide some asymptotic results for these processes which allows us to describe different properties of the corresponding market models based on these processes.

This thesis focuses on the statistical modeling of the dynamics of limit order books in electronic equity markets. The statistical properties of events affecting a limit order book -market orders, limit orders and cancellations- reveal strong evidence of clustering in time, cross-correlation across event types and dependence of the order flow on the bid-ask spread. Further investigation reveals the presence of a self-exciting property - that a large number of events in a given time period tends to imply a higher probability of observing a large number of events in the following time period. We show that these properties may be adequately represented by a multivariate self-exciting point process with multiple regimes that reflect changes in the bid-ask spread. We propose a tractable parametrization of the model and perform a Maximum Likelihood Estimation of the model using high-frequency data from the Trades and Quotes database for US stocks. We show that the model may be used to obtain predictions of order flow and that its predictive performance beats the Poisson model as well as Moving Average and Auto Regressive time series models.