Point processes and integer-valued time series

Chen, Zezhun (2023) Point processes and integer-valued time series. PhD thesis, London School of Economics and Political Science.

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Abstract

This thesis explores two primary themes across five scientific papers: Integer-value time series and their relationship with classical point processes. The first part of the thesis focuses on the development and application of Integer-valued autoregressive (INAR) models, extending from univariate to multivariate cases, with applications in financial and insurance count data. In Paper A, we introduce a new family of binomial-mixed Poisson INAR model of order one, INAR(1), by incorporating a mixed Poisson component to the innovation of the classical Poisson INAR(1). This allows for the capture of overdispersion and serial correlation evident in financial count data. Furthermore, we explore its distributional properties, estimation procedure and asymptotic properties and apply the model to iceberg count data from financial system. In Paper B, we extending beyond univariate case, introducing a novel family of multivariate mixed Poisson-Generalized Inverse Gaussian INAR(1), MMPGIG-INAR(1), regression models for modelling multivariate count time series. This family of models can accommodate a wide range of dispersion and cross-sectional correlation structures due to the flexibility in the parameter setting of the Generalized Inverse Gaussian. We then illustrate different members of the MMPGIG-INAR(1) through applying the model to Local Government Property Insurance Fund data from the state of Wisconsin. In Paper C, we develop novel Expectation-Maximization estimation algorithm for maximum likelihood estimation of bivariate mixed Poisson INAR(1) model. This method is readily extensible to the multivariate case. We examine three different mixing densities, univariate gamma, bivariate Lognormal and bivariate copula and demonstrate the algorithm through fitting the same used in Paper B. The second part of the thesis shifts focus to integer-valued approximation of classical point processes and applications of point process on Covid data modelling. In Paper D, we represent the Cox process and the dynamic contagion process, which is a Hawkes process whose immigration part is a Cox process, as limit of timeseries based point processes, namely integer-valued moving average model (INMA) and Integer-valued Autoregressive Moving Average model (INARMA). This would potentially facilitate the statistical inference of classical point processes. In Paper E, we propose a new type of univariate and bivariate Integer-valued autoregressive model of order one, INAR(1), to approximate univariate and bivariate linear birth and death process with constant rates. Due to the simplicity of Markov structure of INAR model, we demonstrate through simulation study that the parameters of linear birth and death process can be estimated through Quasi-likelihood function of INAR model.

Item Type:	Thesis (PhD)
Additional Information:	© 2023 Zezhun Chen
Library of Congress subject classification:	Q Science > QA Mathematics
Sets:	Departments > Statistics
Supervisor:	Dassios, Angelos and Tzougas, George
URI:	http://etheses.lse.ac.uk/id/eprint/4552

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