Nonparametric estimation of the generating function of the intensity function process of a doubly stochastic Poisson process

Most statistical properties of a point process are determined by their intensity functions. Therefore, it is crucial to model and estimate intensity functions realistically and accurately. There are many approaches proposed in the literature by Cox, Diggle, Helmers, Kagan, Karr, Knopoff, Lewis, Marron, Ogata, Vere-Jones, Zitikis et al. which will be detailed described later.; Applications of point processes are numerous. They include the understanding of geophysical (earthquakes), economic (stock market), social (crime rate), and traffic (traffic accidents) phenomena. A new approach to modeling intensity function processes has been developed and tested using both real and synthetic data. A nonparametric estimator of the intensity function process when the point process is modeled by a Doubly Stochastic Poisson Process (DSPP) while the intensity process is composed of a homogeneous Poisson with constant rate $mH$ plus a nonhomogeneous part with rate which is a convolution of a non-negative generating function g and a homogeneous Poisson. This research effort focuses on the use of higher-order Fourier transform techniques to predict, and make inferences concerning, the occurrence of stock trades. Most of the prediction results based on existing technologies were suspect because they either make unrealistic assumptions, or ignore a critical aspect of the problem such as the historical nature of the data or the stochastic portion of the model. To rectify this situation, the following tasks were performed: (a) The development of a methodology for modeling intensity function processes as applied to understanding the dynamics and characteristics of stock data. (b) The extrapolation in time of a company's stock trading records considered as a stochastic point process, where estimates of occurrence time for predictions were considered. And (c) discussion of the application of theoretical results, obtained under the stock market paradigm, to other application areas (e.g. earthquake sequence data).; The completion of this project effectively provides an analyst with a collection of models capable of making inferences and predictions relative to his/her particular application.