The studies of stock transaction data, i.e., both the regularly-spaced high frequency data and the irregularly-spaced ultra-high frequency data, have been among the frontiers of modern financial data analysis. One of those data sets is the Trade and Quote (TAQ) data from the New York Stock Exchange (NYSE), which is a collection of all stock transaction information (e.g., the transaction date, time, prices and volumes, etc.) for every trading day. The analysis of the intraday transaction data still remains highly challenging today, especially on the statistical modeling aspects.

In this research, two new statistical modeling frameworks, namely, the Multi-Logit Mixture Autoregressive (MLMAR) models and the multivariate Mixture Transition Distribution (MMTD) models, are proposed respectively to handle above two types of financial data. The models are the univariate and multivariate generation of the MTD-type time series models.

The MLMAR time series model is a univariate time series model for the regularly-spaced intraday stock prices, which includes the exogenous information, such as the transaction volumes, the trading frequencies or any other market information, into the modeling framework. The MMTD model is a modeling framework for marked point processes in general, and ultra-high frequency transaction data in particular.

In both modeling frameworks, we solve a series of problems, which include the model specification, parameter estimation, prediction methodology and their applications to the stock transaction data. To show the capacity and advantage of the new models over the existing models, we also compare the new models with those benchmark models and show the new models' advantages in terms of either describing the underlying data generating process or prediction performance. For each class of time series model, potential extensions and related modeling issues are also discussed thereafter.

In reality many time series are non-linear and non-gaussian. They show the characters like flat stretches, bursts of activity and outliers. (Bivariate) Mixture Transition Distribution model are introduced to study these time series data. EM algorithm is used for point estimations of parameters. However as is known, for many mixture models, the likelihoods couldn't be maximized since they will go to infinity. Number of mixtures should be prefixed in this way but in many realities it is unknown.

In our research, Bayesian methods are used to solve these problems. When the posterior is got, EM algorithm is used to maximize the posterior. Under some conditions these estimates are proved to be consistent. The second method is using MCMC to sample from the posterior and now number of mixtures itself can be treated as a random variable. Two methods for MCMC sampling are used. The first is Birth-Death process: if a birth happens, a new mixture component is added; if a death happens, an existed mixture will disappear. The second is Dirichlet process mixtures where we choose Dirichlet process priors for the parameters. When using MCMC, not only point estimations but also interval estimations can be constructed. For all these methods we do simulations to compare Bayesian methods with non-Bayesian methods and to show the excellence of Bayesian methods.

We propose a model for the analysis of non-stationary point processes with almost periodic rate of occurrence. The model deals with the arrivals of events which are unequally spaced and show a pattern of periodicity or almost periodicity, such as the rate of financial transactions or customer/phone calls arrivals. The concept of almost periodicity is described and the purely periodic process is just a special case of the almost periodic process. We consider a non-homogeneous Poisson process and model its rate of occurrence as the sum of sinusoidal functions plus a base line. Given the number of sinusoidal functions which is denoted as K, a set of simple and consistent estimates of frequencies, phases and amplitudes which form the sinusoidal functions are constructed mainly by the Bartlett periodogram. The estimates are shown to be asymptotically normally distributed. Computational issues are discussed and it is shown that the frequency estimates have to be resolved with order o(T^-1) to guarantee the asymptotic unbiasedness and consistency of the estimates of phases and amplitudes, where T is the length of the observation period. The prediction of the next occurrence is also discussed. The proposed model is a finite approximation of the almost periodic process in terms of a finite value of K. In practice, the value of K is usually unknown, and we suggest to use the model selection criteria to determine K. Two criteria AIC and BIC are reviewed and discussed in the frame work of our model. Simulation and real data examples are used to illustrate the theoretical results and the utility of the model.

We propose analyzing our data with a model that exhibits errors-in-variables (EIV) in auxiliary information and which has an autoregressive covariance structure using a Bayesian methodology. The incorporation of these components into a model is often necessary and realistic in the study of many statistical problems. However, such an analysis usually mandates many simplifying and restrictive assumptions, especially when using a traditional probabilistic approach. Much research has been accumulated in this area. We will show that by taking a Bayesian approach, we can effectively deal with the complexity of these types of models. Such an approach cannot be found in the literature.

In fitting statistical models for the analysis of growth data, many curves and/or models have been proposed. We have collected an exhaustive list of the most important and frequently used growth curves, some of which are used in our model analysis. A motivating example is presented to show the applicability of our general approach. In addition, auxiliary covariates, both qualitative and quantitative, can be added into our model as an extension. These EIV growth curves with auxiliary covariates provide a very general framework for practical application.

We give several illustrative examples demonstrating how a Bayesian approach using MCMC (Metropolis Hastings in Gibbs sampler) techniques and goodness of fit statistics for model selections can be utilized in our analysis. Highest Density Regions (HDR's) are also used to facilitate Bayesian estimation and inference in these examples. Multivariate growth curve models are presented and detailed to study the relationship of the variables in models.

In the final chapter, we present growth curve models with auxiliary variables containing uncertain data distributions (i.e. mixtures of standard components, such as normal distributions) using Dirichlet Process Priors (DPP, which are composed of discrete and continuous components). We show that DPPs are appropriate in determining the number of components and in estimating the parameters simultaneously, and are especially useful and advantageous in the aforementioned multimodal scenario with respect to the goodness of fit of the model.