We investigate conditions for the ergodicity of threshold autoregressive time series by embedding the time series in a general state Markov chain and apply a FosterLyapunov drift condition to demonstrate ergodicity of the Markov chain. We are particularly interested in demonstrating V uniform ergodicity where the test function V () is a function of a norm on the statespace. In this dissertation we provide conditions under which the general state space chain may be approximated by a simpler system, whether deterministic or stochastic, and provide conditions on the simpler system which imply V uniform ergodicity of the general state space Markov chain and thus the threshold autoregressive time series embedded in it. We also examine conditions under which the general state space chain may be classified as transient. Finally, in some cases we provide conditions under which central limit theorems will exist for the V uniformly ergodic general state space chain.

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We investigate conditions for the ergodicity of threshold autoregressive time series by embedding the time series in a general state Markov chain and apply a FosterLyapunov drift condition to demonstrate ergodicity of the Markov chain. We are particularly interested in demonstrating V uniform ergodicity where the test function V () is a function of a norm on the statespace. In this dissertation we provide conditions under which the general state space chain may be approximated by a simpler system, whether deterministic or stochastic, and provide conditions on the simpler system which imply V uniform ergodicity of the general state space Markov chain and thus the threshold autoregressive time series embedded in it. We also examine conditions under which the general state space chain may be classified as transient. Finally, in some cases we provide conditions under which central limit theorems will exist for the V uniformly ergodic general state space chain.

The problem of estimating latent or unobserved states of a dynamical system from observed data is studied in this thesis. Approximate filtering methods for discrete time series for a class of nonlinear systems are considered, which, in turn, require sampling from a partially specified discrete distribution. A new algorithm is proposed to sample from partially specified discrete distribution, where the specification is in terms of the first few moments of the distribution. This algorithm generates deterministic sigma points and corresponding probability weights, which match exactly a specified mean vector, a specified covariance matrix, the average of specified marginal skewness and the average of specified marginal kurtosis. Both the deterministic particles and the probability weights are given in closed form and no numerical optimization is required. This algorithm is then used in approximate Bayesian filtering for generation of particles and the associated probability weights which propagate higher order moment information about latent states. This method is extended to generate random sigma points (or particles) and corresponding probability weights that match the same moments. The algorithm is also shown to be useful in scenario generation for financial optimization. For a variety of important distributions, the proposed moment-matching algorithm for generating particles is shown to lead to approximation which is very close to maximum entropy approximation. In a separate, but related contribution to the field of nonlinear state estimation, a closed-form linear minimum variance filter is derived for the systems with stochastic parameter uncertainties. The expressions for eigenvalues of the perturbed filter are derived for comparison with eigenvalues of the unperturbed Kalman filter. Moment-matching approximation is proposed for the nonlinear systems with multiplicative stochastic noise.

<p>In this paper, we study the industrial production index of Sweden from Jan, 2000 to latest Feb, 2010. We find out there is a structural break at time point Dec, 2007, when the global financial crisis burst out first in U.S then spread to Europe. To model the industrial production index, one of the business cycle indicators which may behave nonlinear feature suggests utilizing a smooth transition autoregressive (STAR) model. Following the procedures given by Teräsvirta (1994), we carry out the linearity test against the STAR model, determine the delay parameter and choose between the LSTAR model and the ESTAR model. The results from the estimated model suggest the STAR model is better performing than the linear autoregressive model.</p>

In this paper, we study the industrial production index of Sweden from Jan, 2000 to latest Feb, 2010. We find out there is a structural break at time point Dec, 2007, when the global financial crisis burst out first in U.S then spread to Europe. To model the industrial production index, one of the business cycle indicators which may behave nonlinear feature suggests utilizing a smooth transition autoregressive (STAR) model. Following the procedures given by Teräsvirta (1994), we carry out the linearity test against the STAR model, determine the delay parameter and choose between the LSTAR model and the ESTAR model. The results from the estimated model suggest the STAR model is better performing than the linear autoregressive model.

abstract: The primary objective in time series analysis is forecasting. Raw data often exhibits nonstationary behavior: trends, seasonal cycles, and heteroskedasticity. After data is transformed to a weakly stationary process, autoregressive moving average (ARMA) models may capture the remaining temporal dynamics to improve forecasting. Estimation of ARMA can be performed through regressing current values on previous realizations and proxy innovations. The classic paradigm fails when dynamics are nonlinear; in this case, parametric, regime-switching specifications model changes in level, ARMA dynamics, and volatility, using a finite number of latent states. If the states can be identified using past endogenous or exogenous information, a threshold autoregressive (TAR) or logistic smooth transition autoregressive (LSTAR) model may simplify complex nonlinear associations to conditional weakly stationary processes. For ARMA, TAR, and STAR, order parameters quantify the extent past information is associated with the future. Unfortunately, even if model orders are known a priori, the possibility of over-fitting can lead to sub-optimal forecasting performance. By intentionally overestimating these orders, a linear representation of the full model is exploited and Bayesian regularization can be used to achieve sparsity. Global-local shrinkage priors for AR, MA, and exogenous coefficients are adopted to pull posterior means toward 0 without over-shrinking relevant effects. This dissertation introduces, evaluates, and compares Bayesian techniques that automatically perform model selection and coefficient estimation of ARMA, TAR, and STAR models. Multiple Monte Carlo experiments illustrate the accuracy of these methods in finding the "true" data generating process. Practical applications demonstrate their efficacy in forecasting. / Dissertation/Thesis / Doctoral Dissertation Statistics 2018

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The results of a statistical study of hard x-ray solar flares are presented in this dissertation. Two methods of analysis were used, the Diffusion Entropy (DE) method coupled with an analysis of the data distributions and the Rescaled Range (R/S) Method, sometimes referred to as "Hurst's method". Chapter one provides an introduction to hard x-ray flares within the context of the solar environment and a summary of the statistical paradigms solar astronomers currently work under. Chapter two presents the theory behind the DE and R/S methods. Chapter three presents the results of the two analysis methodologies: most notably important evidence of the conflicting results of the R/S and DE methods, evidence of a Levy statistical signature for the underlying dynamics of the hard x-ray flaring process and a possible separate memory signature for the waiting times. In addition, the stationary and nonstationary characteristics of the waiting times and peak intensities, are revealed. Chapter four provides a concise summary and discussion of the results.

The thesis concentrates on property of linearity in time series models, its definitions and possibilities of testing. Presented tests focus mainly on the time domain; these are based on various statistical methods such as regression, neural networks and random fields. Their implementation in R software is described. Advantages and disadvantages for tests, which are implemented in more than one package, are discussed. Second topic of the thesis is additivity in nonlinear models. The definition is introduced as well as tests developed for testing its presence. Several test (both linearity and additivity) have been implemented in R for purposes of simulations. The last chapter deals with application of tests to real data. 1