Extensions of Markov Modulated Poisson Processes and Their Applications to Deep Earthquakes

The focus of this thesis is on the Markov modulated Poisson process (MMPP) and its extensions, aiming to propose appropriate statistical models for the occurrence patterns of main New Zealand deep earthquakes. Such an attempt might be beyond the scope of the MMPP and its extensions, however we hope its main patterns can be characterized by current models proposed in three parts of the thesis. The first part of the thesis is concerned with introductions and preliminaries of discrete time hidden Markov models (HMMs) and MMPP. The exibility in model formulation and openness in model framework of HMMs are reviewed in this part, suggesting also possible extensions of MMPP. The second part of the thesis is mainly about several extensions of MMPP. One extension of MMPP is by associating each occurrence of MMPP with a mark. Such an extension is potentially useful for spatial-temporal modelling or other point processes with marks. A special case of this type of extension is by allowing the multiple observations of MMPP synchronized together under the same Markov chain. This extension opens the possibility of modelling multiple point process observations with weak dependence. The third extension is motivated by the attempt to describe small scale temporal clustering existing in the deep earthquakes via treating the recognized aftershocks as marks which itself forms a finite point process. The rest of the second part focuses on some information theoretical aspects of MMPPs such as the entropy rate of the underlying Markov chain and observed point process respectively and their mutual information rate. A conjecture on the possible links between mutual information rate of MMPP and the Fisher information of the estimated parameters is suggested. The second part on extensions of MMPP is featured by the derivation of the likelihood and complete likelihood, parameter estimation via EM algorithm, state smoothing estimation and model evaluation through systematic applications of resca