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UBC Theses and Dissertations

On continuous-time generalized AR(1) processes : models, statistical inference, and applications to non-normal time series Zhu, Rong


This thesis develops the theory of continuous-time generalized AR(1) processes and presents their use for non-normal time series models. The theory is of dual interest in probability and statistics. From the probabilistic viewpoint, this study generalizes a type of Markov process which has a similar representation structure to the Ornstein-Uhlenbeck process (or continuous-time Gaussian AR(1) process). However, the stationary distributions can now have support on non-negative integers, or positive reals, or reals; the dependence structures are no longer restricted to be linear. From the statistical viewpoint, this study is dedicated to modelling unequally-spaced or equallyspaced non-normal time series with non-negative integer, or positive, or real-valued observations. The research on both the probabilistic and statistical sides contribute to a complete modelling procedure which consists of model construction, choice and diagnosis. The main contributions in this thesis include the following new concepts: self-generalized distributions, extended-thinning operators, generalized Ornstein-Uhlenbeck stochastic differential equations, continuous-time generalized AR(1) processes, generalized self-decomposability, generalized discrete self-decomposability, P-P plots and diagonal P-P plots. These concepts play crucial roles in the newly developed theory. We take a dynamic view to construct the continuous-time stochastic processes. Part II is devoted to the construction of the continuous-time generalized AR(1) process, which is obtained from the generalized Ornstein-Uhlenbeck stochastic differential equation, and the proposed stochastic integral. The resulting continuous-time generalized AR(1) process consists of a dependent term and an innovation term. The dependent term involves an extended-thinning stochastic operation which generalizes the commonly used operation of constant multiplier. Such a Markov process can have a simple interpretation in modelling non-normal time series. In addition, the family of continuoustime generalized AR(1) processes is surprisingly rich. Both stationary and non-stationary situations of the process are considered. In Part III, we answer the question of what kind of stationary distributions are obtained from the family of continuous-time generalized AR(1) processes, as well as the converse question of whether a specific distribution can be the stationary distribution of a continuous-time generalized AR(1) process. This leads to the characterization of distributions according to the extendedthinning operations. The characterization results are meaningful in statistical modelling, because under steady state, the marginal distributions of a Markov process are the same as the stationary distribution. They will guide us to choose appropriate processes to model a non-normal time series. The probabilistic study also shows that the autocorrelation function is of exponential form in the time difference, like that of the Ornstein-Uhlenbeck or Ornstein-Uhlenbeck-type process. Part IV deals with statistical inference and modelling. We have studied parameter estimation for various situations such as equally-spaced time, unequally-spaced time, finite marginal mean, infinite marginal mean, and so on. The graphical tools, the P-P plot and diagonal P-P plot, are proposed for use in identifying the marginal distribution and serial dependence, and diagnosing the fitted model. Three data examples are given to illustrate the new modelling procedure, and the application capacity of this theory of continuous-time generalized AR(1) processes. These time series are non-negative integer or positive-valued, with equally-spaced or unequally-spaced time observations.

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